tag:blogger.com,1999:blog-9986730716319985572018-03-05T08:28:23.435-08:00Thoughts on Gifted EducationA father's perspective raising a girl highly gifted in areas traditionally dominated by menPablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.comBlogger35125tag:blogger.com,1999:blog-998673071631998557.post-57052788848006129692010-07-26T11:31:00.000-07:002010-08-23T15:48:47.313-07:00Last Post on BloggerI have managed to move my archive of Blogger posts to my WordPress self-hosted setup. You should check out my new site. Coinciding with the start of Paulina's homeschooling, I have launched a treasure trove of resources. My site includes everything from my posts, to research in gifted educations, my problem sets for Paulina, and even a forum for parents and K12 students to collaborate on math and physics problems.<br /><br />So, please, visit and enjoy. All the posts from this site have been moved to the new site, which is fully searchable.<br /><br /><a href="http://perezhortinelafamily.us/">http://PerezHortinelaFamily.us</a><br /><br />Enjoy,<br /><br />PabloPablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-53594691899452631932010-07-13T19:23:00.000-07:002010-07-13T19:31:29.009-07:00Moving the BlogI have decided to continue hosting this blog on my personal server. I have more control over the formatting and other matters. I also am able to host files, which makes it easy for me to share the problem sets I write for Paulina. Visit my new site from now on:<div><br /></div><div><a href="http://PerezHortinelaFamily.us">http://PerezHortinelaFamily.us</a></div><div><br /></div><div>Thanks for reading. Enjoy.</div>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-63300724479673036222010-07-12T23:38:00.001-07:002010-07-16T22:25:52.481-07:00Exercising the Young MindI have posted another problem set in my website. The first one on exponents was successful. I got a few questions from Paulina, but she was able to do most of the problems. At least, I was successful in getting her to sit down on her own to think hard for a while.<div><br /></div><div>I made this one a little harder. I wanted Paulina to think carefully about:</div><div><ol><li>the determinants of dimension</li><li>how measurements change when enlarging along one, two, or three dimensions</li><li>the relationship between volume and area</li></ol><div>The last item can get quite complicated. In fact, there is a whole area of research into isoperimetrical inequalities. However, young kids can answer simple questions about the surface area of a solid when the dimensions are changed. For example, how does the surface area of a unit cube change when we double its dimensions?</div></div><div><br /></div><div>This problem set is #2 in <a href="http://perezhortinelafamily.us/MathProblemSets/Main_Frame.htm">http://perezhortinelafamily.us/MathProblemSets/Main_Frame.htm</a>. I recommend using Lego blocks with this problem set.</div>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-78449965770008993412010-07-11T17:06:00.000-07:002010-07-11T22:00:34.750-07:00Starting a Problem Set and Resource RepositoryI have decided to write problem sets to supplement my daughter's regular curriculum. My aim is enough difficulty to keep her interested and to avoid frustration. Given the limitations of Blogger, I am hosting these problems sets on my personal server. Since I don't generate RSS feeds from my website, I will announce it here when I post new material. I will collect the following:<div><ol><li>Problems sets I write for my daughter</li><li>Useful math and science websites. These are resources that offer courses, tools, and other useful stuff</li><li>Other educational resources</li></ol><div>Check back here often.</div></div>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-34291781501325072362010-06-29T23:41:00.001-07:002010-06-29T23:58:00.370-07:00Difficulty vs. Acceleration in Gifted Mathematics Education<span xmlns=""><p>My idea of proper mathematical education may be different from that of most other people, or I may be biased because of my experience getting a Ph.D. in mathematics. Whatever the issue may be, I have an uneasy feeling that most gifted math program focus too much on acceleration and too little on difficulty and exploration. I am not against acceleration. In fact, educating a mathematically talented daughter who hates repetition and "boring" stuff, I am acutely aware of the importance of cruising through the basics as fast as possible to get to the "fun" and "interesting." Unfortunately, I have yet to find an elementary school program that uses difficulty and exploration systematically. I find this disturbing because it may be impossible to get truly good at math without solving increasingly hard problems and learning to explore by asking what-if questions. The key question here is what parents can do to foster the problem solving talent of their children? I don't have a wholly satisfactory answer. All I can do is describe the problem and offer suggestions. I am not an education expert, but I am smart and educated enough to realize that conventional methods don't work. This blog post is as much about my observations and ideas as it is about asking for suggestions from those who have gone down this path before me. I am passionate about and always happy to discuss this subject. Hit me by email at <a href="mailto:Pablo_A_Perez_Fernandez@yahoo.com">Pablo_A_Perez_Fernandez@yahoo.com</a> if you have an answer to my question.<br /></p><p>What do I mean by difficulty vs. acceleration? The easiest way to describe it is by example:<br /></p><ul><li><strong>The Mirman School for the Gifted – </strong>When first looking for the right school for our daughter, my family went through the application process at the Mirman School for the Gifted. It looked like the ideal place for Paulina until we got to the last step. While she was escorted to a placement test, I waited in the library along with a group of other eager and anxious parents. Mirman's head of school took the opportunity to spend an hour or so answering questions from the group. Fifteen minutes into the Q&A, I asked how Mirman taught math to kids. I was told that kids work at their own pace. If they are ahead of their peers, they could move to more advanced "rooms." I remarked that this was great and then asked what else was done besides letting kids progress faster than their peers. I got a blank stare. I asked if the school used custom curriculum or specialized books. I was told that they used standard mathematics books but allowed kids to work faster than normal and are encouraged to participate in competitions. I did not ask any more questions.<br /></li><li><div><strong>EPGY and CTY's Distance Education Courses –</strong> Both programs use Stanford's adaptive software for K through pre-algebra. My daughter started in K and now is in the middle of 6<sup>th</sup> grade. I find EPGY to be carefully thought out, rigorous, and complete relative to the California's DEO standards. EPGY does a good job teaching concepts like variables, equations, and the representation of English statements as mathematical equations. However, I would not characterize EPGY as challenging. Paulina so far has cruised through the program, and I know that she is working below her problem solving potential. I am disturbed by this.</div><p>I investigated this issue a few months ago. I discovered that EPGY provides software for non-gifted school programs, and it is identical to EPGY's gifted track. The only difference between the gifted and non-gifted tracks is a teacher-controlled flag that triggers acceleration. There is no switch for difficulty or depth.<br /></p></li><li><strong>My Yale University Experience –</strong> Let's fast-forward to my freshman year at Yale University. There were three tracks for freshman physics and math. Each subject offered easy, traditional, and advanced paths. The easy classes were also known as physics and math for poets. They introduced the basic ideas without the "torture" of really difficult problems. The traditional tracks resembled a traditional university course. The advanced courses were much more difficult. They covered much more material and from a far more theoretical perspective. They also required killer problem sets that few students could finish on their own and on time. Bear in mind that the students taking these advanced classes were some of the best in the world. A number of my classmates participated and won honors at the International mathematical Olympiad and other top-level competitions. The biggest difference between the advanced courses and the others was the combination of acceleration and problem solving difficulty.<br /></li></ul><p>I hope it is clear by now what I mean by the difference between difficulty and acceleration. Gifted kids need both. It is not okay to facilitate one but not the other. So, what can we do to help? This question is difficult to answer. Here are some suggestions:<br /></p><ol><li><strong>Use Problems to Teach Material – </strong>There is no sense in explaining things through endless lectures. This has the potential to bore the most enthusiastic students. One learns math best by doing math. Some of the best math classes I ever took asked me to discover math my solving problems.<strong><br /> </strong></li><li><strong>Challenge the Student – </strong>Arithmetic drills are rarely challenging for gifted students. This suggests that increasingly difficult problems are necessary to push these kids toward their potential.<strong><br /> </strong></li><li><strong>Ask Open-Ended or Broad Questions –</strong> This is the only way to truly challenge a smart kid. Give them an open problem. See how far their minds can go. See what questions they come up along the way.<strong><br /> </strong></li></ol><p>I hope this post gives you a few ideas and helps you ask the right questions when thinking about your kid's education.</p></span>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com2tag:blogger.com,1999:blog-998673071631998557.post-84057696137049710292010-06-07T15:34:00.000-07:002010-06-07T20:31:21.604-07:00The Outcome of Radical Acceleration: The Stanley Kids from the 1980s<span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size: medium;"><span class="Apple-style-span" style="font-family:arial;">I have been looking for evidence for or against radical acceleration, and it occurred to me to search for articles about the 1980s group of precocious kids under the guidance of the late Stanley Julian at Johns Hopkins University. The bottom line is that 70% of the participants seem to have benefited from the program. </span></span></span><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; "><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size: medium;"><span class="Apple-style-span" style="font-family:arial;">More importantly, fewer than 10% of participants reported negative impact from acceleration.</span></span></span></span><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; "><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size: medium;"><span class="Apple-style-span" style="font-family:arial;"><br /></span></span></span></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; "><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size: medium;"><span class="Apple-style-span" style="font-family:arial;">Here is the link to a web-based version of a June 1997 article on Stanley's kids from the 1980s. Enjoy.</span></span></span></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; "><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size: medium;"><span class="Apple-style-span" style="font-family:arial;"><br /></span></span></span></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; "><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size: medium;"><span class="Apple-style-span" style="font-family:arial;">Take a look a the picture at the top of the article. Amazing... Not a single girl in the group.</span></span></span></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; "><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size: medium;"><span class="Apple-style-span" style="font-family:arial;"><br /></span></span></span></span></div><div><span class="Apple-style-span" style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; "><a href="http://www.jhu.edu/~jhumag/0697web/whiz.html"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size: medium;"><span class="Apple-style-span" style="font-family:arial;">http://www.jhu.edu/~jhumag/0697web/whiz.html</span></span></span></a></span></div><div><span class="Apple-style-span" style="color:#000099;"><span class="Apple-style-span" style="font-family:arial;"><br /></span></span></div><div><span class="Apple-style-span" style="font-family:'Times New Roman';color:#000099;"><br /></span></div>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com1tag:blogger.com,1999:blog-998673071631998557.post-60830957868992714922010-06-06T01:07:00.000-07:002010-06-06T20:27:17.471-07:00We Saw the Light When Our Daughter Looked into a Black Hole!The title of this post probably makes little sense now, but it will be clear when you are done reading. The end of the school year is upon us, and partially home schooling Paulina has been an exhilarating experience. It has been a good test bed for our theories, to hone our techniques, and to correct our mistakes before committing full time to our roles as primary teachers. My wife and I speak often about how this year went and how we will educate our daughter the next. I guess it is only natural because in just two weeks we will transition out of the traditional academic setting. Hence, this probably is a good time to reflect on our progress and to plan for the future.<div><br /></div><div><span class="Apple-tab-span" style="white-space:pre"> </span><b>June 2009</b><div><ul><li>Graduates from K at a regular school</li><li>37% done with 2rd grade English language arts</li><li>74% done with 3th grade math</li><li>Lots of children's television and little reading</li><li>Lots of "I am bored"</li><li>Expresses lots of interest in black holes and other space phenomena</li><li>Math is boring</li></ul><div><span class="Apple-tab-span" style="white-space:pre"><b> </b></span><b>May 2010</b></div></div></div><div><ul><li>Graduates from 2nd grade from a top 5% school in California (SAS, distinguished school, API scores above 900) after skipping first grade</li><li>65% done with 5th grade English language arts</li><li>Starts 6th grade math now to follow with pre-algebra in September.</li><li>Lots of reading and virtually no television other than Friday movies and science and history programs</li><li>Lots of arts and science experiments</li><li>Asks an astronomer how it is possible for black holes to eject matter through its poles when nothing can escape the event horizon</li><li>Lots of "I like interesting math and competitions like the Math Kangaroo"</li></ul><div>In other words, despite our limited after-school schedule, Paulina skipped from K to 2nd grade, excelled at her classes, completed more that two full years of elementary school math and English language arts. We went from just acquiring information on Black Holes to infer that it is impossible for matter to shoot out of a black hole despite the fact she had been taught this is the case.</div></div><div><br /></div><div>As we asses Paulina's progress over the past year, we become more and more convinced that home schooling is the right setting for her. Our thinking became crystallized this Friday while attending Andre Ghez's lecture on super massive black holes at the Museum of Natural History of Los Angeles. At the end of the lecture, Paulina approached Dr. Ghez eager to ask a question. She pushed her way through the swarm of adult, science groupies and patiently waited her turn. She asked how matter can shoot out of the poles of black holes because she had learned that nothing can escape the event horizon. The scientist smiled and told her that hers was very good question. Dr. Ghez then proceeded to explain how the jets form, and Paulina was happy. We left the museum. We did not speak about black holes again that night. However, I admitted to my wife when we got home that Paulina's question never occurred to me. I just had never wondered. Paulina's question was original, carefully thought out, and proof positive to me as a parent and teacher that Paulina needs to be challenged beyond what is possible in a traditional classroom.</div><div><br /></div><div>The past year has been wonderfully rewarding for our family, but we don't know what the future holds. We only know that the path we are taking is best for our daughter today. We have learned over the past year that deep parental involvement and support is a key to instilling the joy of learning. As a result, convincing parents to be actively involved in teaching their children has become a mission of mine. We facilitated and encouraged Paulina to explore her interests. The effort paid off. We saw the light when our daughter looked into a black hole.</div>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-52265472831183568722010-05-21T13:12:00.000-07:002010-05-26T05:56:41.603-07:00Teaching Social Ethics<span class="Apple-style-span" style=" ;font-family:'Times New Roman';font-size:medium;"><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; background- color:transparent;"><span id="internal-source-marker_0.6679040230810642" style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Last week, Paulina worked on a civil rights project. Her topic was Ruby Bridges and the integration of the public school system. We went on YouTube so she could watch Ruby Bridges, The Movie. The following dialogue ensued ten minutes after she finished watching.</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><hr /><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Pablo</span></span></span></p><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /></span></span></span></p><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">What did you think about the Ruby Bridges movie? Did you like it?</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Paulina</span></span></span></p><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">I don’t know. Why do you ask me so many questions?</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Pablo</span></span></span></p><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Well, I really care about what you think. I like talking to you, and you were glued to the computer watching the movie. Seems to me that you liked it.</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Paulina</span></span></span></p><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">It made me mad. I don’t like the way people treated her. It wasn't fair.</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Pablo</span></span></span></p><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">What wasn’t fair? Why wasn’t it fair? What made you mad?</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Paulina</span></span></span></p><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">It is unfair that some people could not get a good education because it is very hard to get ahead without a good education.</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><hr /><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Paulina left the room and came back ten minutes later. Our ensuing exchange is below. Bear in mind while reading that Paulina is an interracial child. She is 50% Spanish and 50% Filipino, with 50% of her Spanish background from Puertorican descendants of Spanish settlers. So, she is fairly light skinned but darker than Anglo-Saxon kids.</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><hr /><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /></span></span><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Paulina</span></span></span></p><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Papa, is my skin dark or light? </span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /></span></span><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /></span></span></span></p><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Pablo</span></span></span></p><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Your skin is perfect. You look like papa and mama. Your skin is beautiful. What are you worried about?</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Paulina</span></span></span></p><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Well, would I have had problems in Ruby’s time?</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><p style="text-align: center; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">Pablo</span></span></span></p><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">I don’t think you would have had problems because of the color of your skin. My dad looked a lot like I do, and he went to college in Ole Miss in the 1950s. He was never asked to ride in the colored section. However, the fact that you mom and I come from different racial groups may have cause some problems. Back then, it was not common to see mixed couples, but we don’t have to worry about it because things are different today. Society still discriminates, but we are all a lot more tolerant today than we were 50 years ago.</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><hr /><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">This series of conversations took me by surprise. Race has never been an issue in my household. Traci and I are a mixed couple, and we have lived since we met in big, multi-cultural cities like Los Angeles, New York, and London. We have many friends from all over the world with whom we interact on a regular basis. Paulina knows other interracial kids. So, I was a bit surprised by her preoccupation.</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">I think I handled the situation well. First, I did not appear flustered or worried. Second, I brought up something people could have used to discriminate against her. Third, we talked about how discrimination is used as a tool to gain power. I explained that some people will use anything you can imagine to single out a group of people and be unfair to them. Finally, I explained to her why it is important to speak out against discrimination.</span></span></span><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="font-size:medium;"><span class="Apple-style-span" style="color:#330033;">I am certain that I will have more conversations like this one soon. I am glad this one went well, and I hope I handle the next one similarly.</span></span></span></div></span>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-19768459054581141662010-05-16T15:30:00.000-07:002010-05-16T21:03:55.809-07:00Emotions and Teaching: Mastery Goals vs. Performance Goals-Based Teaching<span class="Apple-style-span" style=" ;font-family:'Times New Roman';font-size:medium;"><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; background- color:transparent;"><p id="internal-source-marker_0.7984604723751545" style="text-indent: 36pt; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">What is the best way to teach highly gifted children? Some people advocate home schooling. Others posit that regular schools combined with grade and/or subject-specific acceleration is sufficient. However, I firmly believe that arguing for one approach over any other clearly misses what is almost certainly the single biggest determinant of long-term success: emotions. It is wrong in my opinion to argue that a one-size-fits-all approach is the best for gifted children. Yes, I believe that my daughter will flourish at home. However, what works for my child may not for others regardless of intellectual capacity. I just finished reading </span></span></span><span style="font-style: italic; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">Science Education for Gifted Learners</span></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">. The chapter titled </span></span></span><span style="font-style: italic; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">The Emotional Lives of Fledgling Geniuses</span></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;"> tackles the issue of matching educational approach and emotional personality. The key thesis is that the choice of educational approach should be dictated largely by the emotional characteristics of the student. My wife and I have chosen to home school Pauline next year. After objectively reading </span></span></span><span style="font-style: italic; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">The Emotional Lives of Fledgling Geniuses</span></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">, I feel comfortable with our decision to home school because it best matches our daughter’s personality, emotions, and approach to learning.</span></span></span></p><p id="internal-source-marker_0.7984604723751545" style="text-indent: 36pt; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;"><span class="Apple-style-span" style="color: rgb(0, 0, 0); white-space: normal; font-family:'Times New Roman';"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;"><br /></span></span></span></span></span></span></span></p><p id="internal-source-marker_0.7984604723751545" style="text-indent: 36pt; margin-top: 0pt; margin-bottom: 0pt; "><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;"><span class="Apple-style-span" style="color: rgb(0, 0, 0); white-space: normal; font-family:'Times New Roman';"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">There are many ways to categorize teaching styles. </span></span></span><span style="font-style: italic; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">The Emotional Lives of Fledgling Geniuses</span></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;"> argues that one may view teaching as split into two camps:</span></span></span></span></span></span></span></p><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span></span></span><ul><li style="list-style-type: disc; font-style: normal; text-decoration: none; vertical-align: baseline; font-family:Arial;font-size:11pt;color:transparent;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">performance goals-based</span></span></span></li><li style="list-style-type: disc; font-style: normal; text-decoration: none; vertical-align: baseline; font-family:Arial;font-size:11pt;color:transparent;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">mastery goals-based</span></span></span></li></ul><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">Real-world teaching may mix the two approaches, but it is instructive to think about the implications of these two and how they relate to emotions. Performance goals-based teaching focuses on the tangible and measurable like grades and test scores. Mastery goals-based puts the emphasis on learning and understanding, brushing aside grades as unnecessary and possibly outdated. Mastering arithmetic or learning enough to be able to understand a research paper or solve an opened problem are examples of mastery goals-based learning. Some kids flourish under performance goals-based teaching because they are very competitive and/or because they need a structured environment. Other kids prefer abstract, long-term goals and to study what they care about. Finally, there are kids who enjoy both types of teaching. Hence, it is important to understand your children and try to structure the teaching style around their personality. I am not arguing here for one philosophy over the other. I believe that both are important, but a curriculum could be structured with a bias towards the philosophy that benefits your children the most. This is the key message of this blog post. Get to know the emotional personality of your child and then structure his or her learning environment to optimize the learning potential.</span></span></span><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;"><br /><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"></span><br /></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;"><span class="Apple-tab-span" style="white-space:pre"> </span>Let’s use my daughter as an example. Paulina does well in exams. I did too when I was a kid, but she is one of those people who seems to do well in tests without even trying. She is extremely competitive, and she has started attending contests. For instance, she participated in the Math Kangaroo this year and came out very excited, asking to do it again next year. She always wants to get the highest score in every test she takes and practices incessantly whenever she has a performance. On the other hand, she already has long-term goals. A good example is her passion for black holes. I don’t remember how this started, but she became fascinated with black holes when she was five. She would ask me to read her everything we could find on black holes. She now reads by herself everything she can find on the subject, and she has been speeding through the math curriculum as fast as possible since I explained that it is a key to understanding black holes. In the process, she has discovered probability, graph theory, and other subjects that interest her, but her goal remains to learn math because it will allow her to understand black holes and other astronomical phenomena.</span></span></span></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; background- color:transparent;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;"><br /></span></span></span></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; background- color:transparent;"><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;"></span></span></span><span style="font-style: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap; font-family:Arial;"><span class="Apple-style-span" style="color:#330033;"><span class="Apple-style-span" style="font-size:small;">The point of the above example is that my daughter needs both performance goals and mastery goals-based teaching. In a way, I think the former appeals to her competitive nature and the latter to her interest in particular subjects and her search for depth of knowledge. The exact reason is irrelevant to me as a father and teacher. What I must do is keep in mind is her need for both types of teaching and how to use them appropriately. I have met highly gifted kids who are happy in performance goals-based environments. I have met others with personalities to thrive in a mastery goals-based setting. Finally, some like my daughter prefer a mixed environment. The thing to remember is to understand your child well enough to foster the right teaching environment.</span></span></span></div></span>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-48879689950202013772010-05-01T10:36:00.000-07:002010-07-05T00:25:53.060-07:00On the Importance of Making Math Fun<div>“Dad, I hate math! I hate boring math. Once I know something, why do I have to do a ton of homework on it?” This is the beginning of my conversation with Paulina a few days ago when I asked her to do her school homework. This was a worrisome warning sign in my opinion. If I continue subjecting my daughter to traditional classroom instruction, I will kill her interest in math, and squander any chance she may have of developing her considerable talents.</div><div><br /></div><div>It is sometimes hard to understand what gifted children have to endure in a regular classroom. Let us do a thought experiment together to try to see the problem. Consider the following scenario. You sit through a one hour lecture on arithmetic. You then spend a whole afternoon doing repetitious drills on problems that are clearly too easy and don’t teach you anything. Now, repeat this every week for an entire school year. Then, do this year after year until graduating from high school. Sounds fun. Doesn’t it? Take a minute or two to imagine how you would feel. Now, do you agree with me?</div><div><br /></div><div>It am convinced that the best way to kill a child’s interest in math is to teach him or her in the traditional way. I advocate a different approach based on the concept behind math circles because they are particularly well suited for gifted kids. Yes, arithmetic is important because it is core knowledge. However, gifted kids can go through it very quickly and benefit most from creative problems sets introducing advanced material. The rest of this article describes the material covered at UCLA’s junior math circle over the past four weeks. I hope you agree with me when you are done that the math-circle approach is far more educational and fun than the way our kids are been tortured today.</div><div><br /></div><div>What do you think of when you read the following topics?</div><div><ul><li>Graphs (not to be confused with the X-and-Y variety)</li><li>Trees</li><li>Degree of a vertex</li><li>Isomorphic graphs</li><li>Circuits in graphs</li><li>Euler circuits</li><li>Planar graphs</li><li>Restating problems involving maps using graphs</li><li>The Four Color Problem</li></ul></div><div>You may have never heard or know what any of the above terms mean. You may know a few, but, unless you have studied theoretical computer science or discrete mathematics, you probably don’t know much about them. Would you be surprised to learn that this is what my daughter and a group of other like-minded kids have been learning in the UCLA math circle? All of these topics are generally considered advanced based on the grade level when they are typically taught. However, they can be introduced early on because the terminology is intuitive and simple. This does not mean that graph theoretic problems are easy. In fact, some of these seemingly simple problems are at the cutting edge of research, and this is the beauty of graph theory. It can introduce young minds to cutting-edge research and concepts without spending countless years getting up to speed. This can make math fun and interesting, and I believe this approach should be adopted simultaneously with the teaching of the so-called fundamental concepts.</div><div><br /></div><div>I am implementing a learning program for Paulina based on the ideas I discuss here. I am done with traditional classrooms. It is time to let her mind fly where it wants to go.</div>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-84592192165654970582010-04-10T22:37:00.001-07:002010-04-10T22:45:29.097-07:00A 21 Setup to Teach Probabilistic Decision Making<div><span style="font-size:100%;">I want to expand on our discussion of games of chance as teaching tools. I picked the classic casino game of 21 because it is relatively simple to learn. I invented a simple variation and tested it successfully with my daughter. Hence, I assume this should work with other kids as well. The rest of this article explains the setup.</span></div><span style="font-size:100%;"><br /></span><div><span style="font-size:100%;"><b>Rationale for My Version of the Game of 21</b></span></div><div><span style="font-size:100%;">This is a modified version of 21. It removes some of the complexities that make the game inappropriate for teaching young kids about probabilistic decision making. Students say "hit"to be dealt additional cards until the probability of going bust (i.e. getting more than 21) is unacceptable. At this point, they say "stay." The point of this game is not to teach betting but to make the computation of probabilities a bit more fun than traditional classroom teaching.</span></div><span style="font-size:100%;"><br /></span><div><span style="font-size:100%;">Playing this game should teach:</span></div><span style="font-size:100%;"><br /></span><ol><li><span style="font-size:100%;">Computing probabilities on a discrete space.</span></li><li><span style="font-size:100%;">Computing conditional probabilities when there the probability space changes.</span></li><li><span style="font-size:100%;">Determining if a bet is unfavorable, roughly fair, or favorable.</span></li><li><span style="font-size:100%;">Making decisions based on probabilities.</span></li></ol><span style="font-size:100%;"><br /></span><div><span style="font-size:100%;">As you can probably surmise from the above, the goals of my game are a bit ambitious. However, this game could give you a start to help your little one learn tools that could prove very valuable later in academia and life.</span></div><span style="font-size:100%;"><span style="font-size:100%;"><br /></span><b>Step 1</b><br /></span><span style="font-size:100%;">Get a big piece of construction paper -- the kind used for school projects and presentations. Arrange a deck of cards along columns on the construction. The leftmost column should have the twos. The next column over should hold the threes and so on. Instead of using a deck of cards, you could simply draw the cards on the construction paper. You should have 13 columns total. I will call this the "board" throughout the rest of this article.</span><br /><br /><div><span style="font-size:100%;">The purpose of the board is to keep track of the cards that have been played. To this end, use a spare deck of cards facing down to cover all the locations on the board corresponding to cards that have either being played or are "in play." This makes it easy to visually determine how many and which cards remain on the deck.<br /></span><span style="font-size:100%;"><br /></span><div><span style="font-size:100%;"><b>Step 2</b></span></div><div><span style="font-size:100%;">Explain the rules of this version of 21:</span></div><span style="font-size:100%;"><br /></span><ol><li><span style="font-size:100%;">At the beginning of a hand, each player is given two cards.</span></li><li><span style="font-size:100%;">Each player gets only one card at a time after the initial two. Additional cards are only handed out when it is the player's turn and the player asks to be hit.</span></li><li><span style="font-size:100%;">The game continues until there are no cards left in the deck or there are not enough cards to start a new hand.</span></li><li><span style="font-size:100%;">There is a common pile of chocolate chips for betting. All players draw chips from the same pile.</span></li><li><span style="font-size:100%;">The winner of the game is the one who ends with the biggest pile of chocolate chips.</span></li><li><span style="font-size:100%;">A player should be dealt as many cards as desired until either he or she goes bust or decides to "stay."</span></li></ol><br /><div><span style="font-size:100%;"><b>Step 3</b></span></div><span style="font-size:100%;">Explain how to play a hand.</span></div><b><br /></b><div><ol><li><span style="font-size:100%;">Start by getting two cards.</span></li><li><span style="font-size:100%;">Continue saying "hit me" (i.e. asking for another card) until you think you will go bust.</span></li><li><span style="font-size:100%;">Say "stay" when you are done with your hand.</span></li><li><span style="font-size:100%;">Choose to hit or stay based on the probability of going bust by picking one more card. <span style="font-size:100%;">This step is crucial. One of the key reasons for the this game is to teach how to compute conditional probabilities (i.e. probability under variable change).</span></span></li><li><span style="font-size:100%;">If your total exceeds 21, you are out of this hand.</span></li></ol><br /><div><span style="font-size:100%;"><b>Step 4</b></span></div><div><span style="font-size:100%;">Explain the rules for betting and scoring in this game.</span></div><span style="font-size:100%;"><br /></span><ol><li><span style="font-size:100%;">Points are earned or lost in each hand.</span></li><li><span style="font-size:100%;">A bet is from 0 to 4 chips.</span></li><li><span style="font-size:100%;">The person who comes closest but not over 21 wins the hand and takes all the chips bet in the round.</span></li><li><span style="font-size:100%;">If two or more players tie, they split the pile. If the pile does not split evenly, remove the smallest number of chips so it splits evenly. Add the removed chips to the next hand's pile.</span></li></ol><span style="font-size:100%;"><br /></span><span style="font-size:100%;"><b>Step 5</b></span></div><span style="font-size:100%;">Teach some betting guidelines.</span><br /><br /><ol><li><span style="font-size:100%;">Bet 0 chips if there is a high probability of losing the hand.</span></li><li><span style="font-size:100%;">Bet 1 or 2 chips if there is low probability of winning the hand.</span></li><li><span style="font-size:100%;">Bet 2 chips if the probability of winning is high. </span></li><li><span style="font-size:100%;">Bet 4 chips if you are very sure to win.</span></li></ol><br /><b><span style="font-size:100%;">A Few Additional Details</span></b><span style="font-size:100%;"><br />The game could be played with different degrees of sophistication. For instance, students could be asked to base the size of their bets on the probability of winning the hand. However, this requires computational skills beyond the skills of most seven year olds. A better way to place bets is based on intuition.</span><br /><span style="font-size:100%;"><br /></span><div><span style="font-size:100%;">This game probably would be a lot of fun to play with a group of kids. I explained the setup to my daughter's teacher, and she thought it would be fun to play in the classroom since they are learning the basics of probability.</span></div><span style="font-size:100%;"><br /></span><div><span style="font-size:100%;">Have fun,</span></div><div><span style="font-size:100%;"><br />Pablo</span></div>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-1315544562554801772010-04-07T18:57:00.000-07:002010-04-08T23:11:06.490-07:00Vegas at HomeI just finished three initiation reports -- I follow stocks for a living -- and needed a diversion. So, I decided to teach Paulina how to play cards. It occurred to me earlier today that card games are a great vehicle to teach basic probability concepts. Conditional probability, in particular, arises naturally. We tried it, and Paulina had a lot of fun.<div><br /></div><div>Here is what we did. I first taught her the various suits and the cards in each suit. That took very little time. I then asked her the following questions:</div><div><ol><li>Before dealing any cards, what is the chance of dealing a heart?</li><li>Assuming that a heart is the only card that has been dealt, what is the probability of dealing another heart?</li><li>Assuming that a heart is the only card that has been dealt, what is the probability of dealing a spade?</li><li>Assuming that a heart has been dealt, what is a better bet for the next card? A heart or a diamond?</li></ol><div>Shuffle the deck. Deal two cards for you and two for your child. Forget about the dealer's cards. This just complicates the setup. Now is when things get educational. Let the child be the first player. Ask the kid "hit" or "stay." He or she is going to look at you funny. Explain that you say "hit" if you want more cards and '"stay" if you are done. Remind the child that an "A" works as a 1 or an 11. Remind the child going over 21 gets you busted. Tell the kid to look at all the cards that have been dealt and ask for the probability of going bust. Chances are the kid won't have a clue. In fact, Paulina had no idea where to start. However, she got it quickly once I showed her the thought process. If your child has been paying attention, he or she will get it too. Here is an example. Let's say that you dealt:</div><div><ul><li>a 10 of hearts and a 5 of clubs for your child</li><li>a 2 of diamonds and 5 of diamonds for you</li><li>a 10 of hearts and a card facing down</li></ul><div>You can compute the probability of your child going bust as follows:</div><div><ul><li>To bust, you must deal a card with a value of 7 or higher.</li><li>Given what has already being dealt, the following cards will get you busted: 4 sevens, 4 eights, 4 nines, 4 tens, 4 jacks, 4 queens, and 4 kings</li></ul><div>This is a total of 28 cards out of 48 cards that have yet to be dealt. This means that the probability of going bust is 28 out of 48 or 7/12. Your child should know that this is more than 1 out of every two cards. This is not a lot over 1/2, but one is more likely than not to go bust at this point by taking one more card.</div><div><br /></div><div>You could continue playing the game. As you work through the deck, you can ask your child to remember what has been dealt and to decide on every play whether or not it is a good idea to take another card.</div><div><br /></div><div>Remember that this is not Vegas. You can count cards here. Lay out in front of you the cards that have been used so your kid can tell what is left in the deck. Don't be too serious. That's not the point. Have fun. Raise the stakes by using chocolate chips to bet. Let me repeat this. HAVE FUN. EAT CHOCOLATE. COMPUTE. THINK. If you make this too serious, you will fail.</div><div><br /></div><div>There are a million questions you could ask. For example, you could ask the probability of getting a 21 when dealt the first two cards from a deck. However, you should just play and ask the kid to make educated guesses about whether or not to bet. Once again. Raise the stakes by using chocolate. Trust me. Chocolate works when teaching math.</div><div><br /></div><div>Have fun,</div><div><br /></div><div>Pablo</div></div></div></div>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-75793996465196769582010-03-25T15:26:00.001-07:002010-03-26T07:05:43.757-07:00Teaching the Concept of Conditional ProbabilityI still remember it like it was yesterday. Paulina was three. We picked two cards each at random from the top of a deck. The winner of the hand was the one with the highest total. We would repeat until we had gone through the entire deck. The winner of the game was the one who had won the most hands. The thing is that Paulina could not handle losing. She did not understand that this game involved zero skill. I explained it countless times, but it did not matter. Her competitive spirit got in the way. We stopped playing games of chance after a short while because Paulina could not deal with randomness. I felt at the time that we would never be able to play games of chance. Fortunately, I was wrong. She now enjoys thinking about probability and gets the idea of computing them to take good bets. Given her new-found fondness for probability, I decided to try teaching her conditional probability.<br /><br /><span style="font-weight: bold;">Prerequisites:</span><br /><ol><li style="font-weight: bold;">understand fractions<span style="font-weight: normal;"></span><br /></li><li><span style="font-weight: bold;">understand the concept of outcomes - </span>The possibilities for the given problem. An outcome is something that may happen. It does not necessarily mean that it has happened.<br /></li><li><span style="font-weight: bold;">understand the concept of sets and events - </span>Once needs to understand what sets are to tackle events. An event is just a set of outcomes. An event is used to narrow down the set of possible outcomes in a given probabilistic universe.<br /></li><li><span style="font-weight: bold;">understand how to compute probabilities in finite spaces - </span>Know how to compute probabilities by counting. We only need the very simplest computational skills at this point.<br /></li></ol><span style="font-weight: bold;">Step 1 - Drawing Probability Trees<br /></span>Teach your child how to draw spaces of probability outcomes using trees. I have found that they are simpler for young children to understand than simply listing out all possible outcomes.<br /><br /><span style="font-weight: bold;">Example 1: </span>Assume that a family has two children. What are all the possible outcomes of pairs of children?<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_f1F9pEcIKmI/S6wcQ-ekUrI/AAAAAAAACEY/jg9rCysI0Sg/s1600/Pic1.JPG"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 257px; height: 361px;" src="http://2.bp.blogspot.com/_f1F9pEcIKmI/S6wcQ-ekUrI/AAAAAAAACEY/jg9rCysI0Sg/s400/Pic1.JPG" alt="" id="BLOGGER_PHOTO_ID_5452764326813520562" border="0" /></a>Moving from the top to the bottom, we can now read out the four outcomes: {BB, BG, GB, GG}.<br /><br /><span style="font-weight: bold;">Example 2:</span> Assume that a family has three children. What are all the possible sets of children?<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_f1F9pEcIKmI/S6wf6PJAvrI/AAAAAAAACEg/3mV212ps54s/s1600/Pic2.JPG"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 333px; height: 400px;" src="http://3.bp.blogspot.com/_f1F9pEcIKmI/S6wf6PJAvrI/AAAAAAAACEg/3mV212ps54s/s400/Pic2.JPG" alt="" id="BLOGGER_PHOTO_ID_5452768334196031154" border="0" /></a><br />Once again, by reading from the top to the bottom of the tree, we can list all the outcomes {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}.<br /><br />It is very important at this point to make sure that your child understands how to compute the number of outcomes without actually listing them out.<br /><br /><span style="font-weight: bold;">Example 3:</span> Introduce an example of choosing randomly <span style="font-weight: bold; font-style: italic;">without replacement</span>. This means that once an object is picked, it cannot be picked again. I would suggest using colored M&Ms here. Use two green and two red M&Ms. Explain that you want to build an outcome tree as follows. Pick one M&M at random. Leave it out of the box. Pick another M&M at random. Now, draw the tree of outcomes.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_f1F9pEcIKmI/S6wmG9MqCTI/AAAAAAAACEo/oUthR_JiaoA/s1600/Pic3.JPG"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 217px; height: 400px;" src="http://4.bp.blogspot.com/_f1F9pEcIKmI/S6wmG9MqCTI/AAAAAAAACEo/oUthR_JiaoA/s400/Pic3.JPG" alt="" id="BLOGGER_PHOTO_ID_5452775149787547954" border="0" /></a>The key point when drawing this tree is to realize that once an M&M has been picked, it remains out of the set. Hence, there are only three choices when picking the second M&M. This means that the full set of outcomes is {GG, GR, GR, GG, GR, GR, RG, RG, RR, RG, RG, RR}.<br /><br /><span style="font-weight: bold;">Step 2 - Teaching the Difference Between Normal and Conditional Probability</span><br />Most people never really learn the concept of conditional probability because their teaches did not understand it either. Students are usually taught the formula P(A|B) = P(A and B) / P(B). The issue is that this makes little sense to most people. This is sad because math is about concepts, not symbolic manipulation. The only difference between regular probability and conditional probability is the set over which one computes.<br /><br />Compute the probability of getting a green followed by a red in the last example. Looking at the tree corresponding to the last example, this event is equal to {RGG, RGG, RGG, RGG}. This is the case because there are four paths in the tree with RGG outcomes. Hence, the probability of one green followed by a red is 4/12 or 1/3.<br /><br />Compute the probability of getting a green on the second pick given that the first pick was a red. This conditional probability is computed by counting outcomes in the sub-tree with Rs in the first pick. This is because we are told to assume that the first pick was an R.<br /><br />This can be computed by looking at the correct portion of the tree, which is circled in red below.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_f1F9pEcIKmI/S6w0sVTMs8I/AAAAAAAACE4/5lgTREp_edo/s1600/Pic4.JPG"><img style="display: block; margin: 0px auto 10px; text-align: center; cursor: pointer; width: 400px; height: 364px;" src="http://4.bp.blogspot.com/_f1F9pEcIKmI/S6w0sVTMs8I/AAAAAAAACE4/5lgTREp_edo/s400/Pic4.JPG" alt="" id="BLOGGER_PHOTO_ID_5452791185075385282" border="0" /></a><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_f1F9pEcIKmI/S6wu0ShrHYI/AAAAAAAACEw/HrMeblxjGng/s1600/Pic4.JPG"><br /></a><br />The possible second picks when the first one is an R are {G, G, R, G, G, R}. Hence, the conditional probability is 4/6 or 2/3.<br /><br /><span style="font-weight: bold;">Step 3 - Repeat Many Times</span><br />The trick to teach conditional probability to a kid is to give lots of concrete examples.<br /><br />I hope this post helps you in some way.Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com3tag:blogger.com,1999:blog-998673071631998557.post-17202376368166946352010-03-12T20:09:00.001-08:002010-03-24T20:30:38.990-07:00When Homeschooling is the ONLY OptionTraditional education was good while it lasted. My wife and I knew this day was coming, and we held out as long as possible. However, we have come to the realization that home schooling will arrive in our household much earlier than we expected. We now find ourselves planning for next year. Fortunately, we have been learning and preparing for this moment for the past two years. Paulina's school has been better than we expected when this school year started, but the academic environment is simply not challenging. Paulina skipped from K to 2nd grade, but she caught up with her classmates rather quickly and is now growing bored and tired of the long weekly homework assignments that teach her little. She is approaching 6th math and language arts and is on track to start pre-algebra in September. We have little choice but to home school her. She may never fit in a normal school, but her mother and I are fortunate to have the flexibility to be deeply involved in her education.<div><br /></div><div>We suspected this day was coming, but we thought we could postpone it for a few years. Paulina's homeroom teacher offered to have her take third grade math this year, but Paulina is finishing 5th grade now. Her school is a typical K through 5. It makes no difference if she takes second, third or fifth grade math. She is done it before. Either one would be torturous repetition. As a result of the above considerations, we chose to keep her in second grade with her homeroom for all her classes. We thought she could attend third or fourth grade next year, but the gap between her and her classmates is widening. It becoming particularly wide in math. However, it has become patently clear that we will have this problem until she goes out to college. She will never fit in a traditional, primary education classroom.</div><div><br /></div><div>We have chosen to home school Paulina next year. I am lucky enough to work from home and only travel two weeks per quarter to visit clients. I handle math and science, and EPGY allows my wife to supervise Paulina when I am away. My wife is highly educated, with advanced degrees in the arts and business, which rounds up what I can contribute to my daughter's education. We have no idea what the future holds, but home schooling looks like the only option to us now. Paulina has had six months to think about it. After countless conversations about how her days would be, she has decided that she would much prefer studying at home than at school. This has been a family decision, and we are ready to take the plunge.</div>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-17835246452282966502010-03-11T22:54:00.000-08:002010-03-13T10:16:30.949-08:00Teaching Multiplication of Fractions VisuallyThis is a very short post outlining how to teach the multiplication of fractions using a geometrical interpretation. I have tested this with kids across a wide range of the ability spectrum, and it has always helped. I hope it works for you too.<br /><br />While fractions are easy for most kids, multiplication of fractions can sometimes be a little tricky. What I mean is that kids learn how to multiply fractions easily without really learning why it works. I wanted to make sure my daughter understood the concept, so I resorted to a geometrical interpretation.<div><br /></div><div>Here are the basic prerequisites for this approach:</div><div><ol><li>Understand what a fraction is</li><li>Understand how to compute areas. Basically, that A = L x W</li></ol><div>Here are a few examples of how to interpret multiplication of fraction geometrically.</div><div><br /></div><div><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_f1F9pEcIKmI/S5r3wD0y9lI/AAAAAAAACEA/QIN5VEgMNBs/s1600-h/Frac+1.JPG"><img src="http://3.bp.blogspot.com/_f1F9pEcIKmI/S5r3wD0y9lI/AAAAAAAACEA/QIN5VEgMNBs/s400/Frac+1.JPG" alt="" id="BLOGGER_PHOTO_ID_5447939104290567762" style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 247px; height: 208px;" border="0" /></a></div><div>This diagram shows a "unit" square. The length of the shaded rectangle is 1/2 and the width 1/2. Simple visual inspection shows that we have divided the unit square into 4 equal pieces. Hence, the product 1/4.<br /><br />Here is another example. Multiply 2/3 and 3/5.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_f1F9pEcIKmI/S5r8g3wG-CI/AAAAAAAACEI/YGbztLUcaCU/s1600-h/Frac+2.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 319px;" src="http://2.bp.blogspot.com/_f1F9pEcIKmI/S5r8g3wG-CI/AAAAAAAACEI/YGbztLUcaCU/s400/Frac+2.JPG" alt="" id="BLOGGER_PHOTO_ID_5447944340909783074" border="0" /></a><br />The length is 2/3. The width is 3/5. The product is 6/15. You kid should figure this out by counting.<br /><br />Finally, let me show how to illustrate multiplication of fractions involving improper fractions.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_f1F9pEcIKmI/S5vV4diMLmI/AAAAAAAACEQ/uIG-f-NekNY/s1600-h/Frac+3.JPG"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 400px; height: 188px;" src="http://3.bp.blogspot.com/_f1F9pEcIKmI/S5vV4diMLmI/AAAAAAAACEQ/uIG-f-NekNY/s400/Frac+3.JPG" alt="" id="BLOGGER_PHOTO_ID_5448183340212170338" border="0" /></a>Hopefully, it is clear by now that this geometrical interpretation works equally well for heterogeneous fractions.<br /><br />The point of the geometrical interpretation is that it can be used to teach multiplication right after understanding the concept of a fraction. As always, the key to learning things well is to use lots of examples and to spend enough time thinking about the concepts.<br /></div></div>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-59053543964743783822010-02-27T16:00:00.000-08:002010-03-01T05:46:16.969-08:00Teaching Fractional Arithmetic VisuallyI was recently asked to teach fractions to a group of kids in my daughter's second grade class. I think most kids are quick to learn the concept of a fraction, and as a consequence, they get bored with the endless drills of coloring exercises. Many stop paying attention, which leads to problems later on. Trying to figure out how to do things differently, I recalled an afternoon two years ago when I taught my daughter how to add and subtract fractions. I figured she that if she understood the concepts of a whole and a part of a whole, she would be able to tackle basic fractional arithmetic. I used visual representations, and she quickly learned how to add and subtract of homogeneous and heterogeneous fractions. Given my prior success, I decided to try this in my daughter's classroom.<br /><br /><span style="font-weight: bold;">Step 1:</span><br />Explain that fractions are the part of a whole that you are talking about. Give a ton of examples. This is a critical step before moving on. Kids must understand what a fraction is and how to read and write them using the standard notation.<br /><br /><span style="font-weight: bold;">Step 2:</span> Review how to represent fractions as pictures and vice versa. For example,<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_f1F9pEcIKmI/S4m4rMWrwzI/AAAAAAAACCo/DVudwEk--es/s1600-h/Frac+1.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 238px; height: 147px;" src="http://3.bp.blogspot.com/_f1F9pEcIKmI/S4m4rMWrwzI/AAAAAAAACCo/DVudwEk--es/s400/Frac+1.bmp" alt="" id="BLOGGER_PHOTO_ID_5443084676844208946" border="0" /></a><br />Use a few more examples. Teach the kid how to draw squares divided into thirds, fifths, tenths, and a few other common denominators.<br /><br /><span style="font-weight: bold;">Step 3:</span> Explain what it means to add fractions. Say that it means adding parts of a whole. Explain that if we add parts of a whole, it is easier to add pieces of the same size. Give the example of adding 1/2 and 1/4. 1/2 and 1/4 have different sizes. So, how do you tell how much of the whole you have when you put together 1/2 and 1/4? The best thing to do here is to use four blocks of the same size. Manipulate two of the four blocks as 1/2 of a whole. Give the kid enough time to realize that 1/4 is 1/2 of 1/2.<br /><br /><span style="font-weight: bold;">Step 4:</span> Once the child understands that cutting a square into pieces of the same size is the key to adding fractions (i.e. making the fractions homogeneous), proceed with a few exercises such as the following. Draw the two fractions we used before.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_f1F9pEcIKmI/S4m4rMWrwzI/AAAAAAAACCo/DVudwEk--es/s1600-h/Frac+1.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 238px; height: 147px;" src="http://3.bp.blogspot.com/_f1F9pEcIKmI/S4m4rMWrwzI/AAAAAAAACCo/DVudwEk--es/s400/Frac+1.bmp" alt="" id="BLOGGER_PHOTO_ID_5443084676844208946" border="0" /></a><br />Ask your child to figure out how to divide both fractions into pieces of the same size. Tell him that he is not allowed to erase lines already drawn. Tell him that he is allowed to draw new lines, but that the goal is both fractions to divided into pieces of the same size. Clearly, the answer here is to subdivide the 1/2 horizontally as follows:<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_f1F9pEcIKmI/S4rPZE2J1lI/AAAAAAAACCw/E8oTkJttuZs/s1600-h/Frac+2.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 269px; height: 166px;" src="http://2.bp.blogspot.com/_f1F9pEcIKmI/S4rPZE2J1lI/AAAAAAAACCw/E8oTkJttuZs/s400/Frac+2.bmp" alt="" id="BLOGGER_PHOTO_ID_5443391129335223890" border="0" /></a><br />Now (this is critical), make sure the child understands that how much is shaded in the square on the left does not change simply because we drew another line. It should be clear by now that 1/2 and 2/4 are the same fraction. Make sure this is clearly understood before proceeding.<br /><br /><span style="font-weight: bold;">Step 5: </span>It is now time to learn how to convert heterogeneous to homogeneous fractions. I would suggest easy cases first, followed by slightly more complicated ones. Let's start with the following two fractions. Always draw one fraction using vertical lines and the other using horizontal lines.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_f1F9pEcIKmI/S4rbdTTQqrI/AAAAAAAACC4/45kBg7Y4MoU/s1600-h/Frac+3.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 217px; height: 147px;" src="http://2.bp.blogspot.com/_f1F9pEcIKmI/S4rbdTTQqrI/AAAAAAAACC4/45kBg7Y4MoU/s400/Frac+3.bmp" alt="" id="BLOGGER_PHOTO_ID_5443404396074412722" border="0" /></a><br />The answer should be<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_f1F9pEcIKmI/S4rgjU08SXI/AAAAAAAACDI/qm1u7_aozyA/s1600-h/Frac+4.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 223px; height: 153px;" src="http://2.bp.blogspot.com/_f1F9pEcIKmI/S4rgjU08SXI/AAAAAAAACDI/qm1u7_aozyA/s400/Frac+4.bmp" alt="" id="BLOGGER_PHOTO_ID_5443409997121472882" border="0" /></a><br />By drawing vertical lines in one fraction and horizontal in the other, it becomes clear how to draw new lines to divide both pictures into pieces of the same size.<br /><br />Let's try one more example.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_f1F9pEcIKmI/S4rh1Y3Su7I/AAAAAAAACDQ/8McZYVEOFW4/s1600-h/Frac+5.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 221px; height: 150px;" src="http://4.bp.blogspot.com/_f1F9pEcIKmI/S4rh1Y3Su7I/AAAAAAAACDQ/8McZYVEOFW4/s400/Frac+5.bmp" alt="" id="BLOGGER_PHOTO_ID_5443411406954347442" border="0" /></a><br />The answer now is<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_f1F9pEcIKmI/S4rig_MX73I/AAAAAAAACDY/ZjEP6dwKTOw/s1600-h/Frac+6.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 221px; height: 150px;" src="http://3.bp.blogspot.com/_f1F9pEcIKmI/S4rig_MX73I/AAAAAAAACDY/ZjEP6dwKTOw/s400/Frac+6.bmp" alt="" id="BLOGGER_PHOTO_ID_5443412155977690994" border="0" /></a><br />Make sure your child understands that 3/5 is equal to 6/10. Likewise, make sure it is understood that 1/2 = 5/10.<br /><br /><span style="font-weight: bold;">Step 6: </span>It is now time to introduce the pictorial representation of improper fractions. Ask your kid to draw the following fractions: 3/2, 5/4, 4/2, 5/2, etc. The answers follow:<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_f1F9pEcIKmI/S4rjzrZWNzI/AAAAAAAACDg/_mcdzTm8QRU/s1600-h/Frac+7.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 194px; height: 131px;" src="http://1.bp.blogspot.com/_f1F9pEcIKmI/S4rjzrZWNzI/AAAAAAAACDg/_mcdzTm8QRU/s400/Frac+7.bmp" alt="" id="BLOGGER_PHOTO_ID_5443413576592537394" border="0" /></a><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_f1F9pEcIKmI/S4rn5VLj8hI/AAAAAAAACDo/lUGc7tLHlVA/s1600-h/Frac+8.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 211px; height: 142px;" src="http://1.bp.blogspot.com/_f1F9pEcIKmI/S4rn5VLj8hI/AAAAAAAACDo/lUGc7tLHlVA/s400/Frac+8.bmp" alt="" id="BLOGGER_PHOTO_ID_5443418071754863122" border="0" /></a><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_f1F9pEcIKmI/S4roUSKA_rI/AAAAAAAACDw/Xt6xN3aPLo4/s1600-h/Frac+9.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 221px; height: 148px;" src="http://2.bp.blogspot.com/_f1F9pEcIKmI/S4roUSKA_rI/AAAAAAAACDw/Xt6xN3aPLo4/s400/Frac+9.bmp" alt="" id="BLOGGER_PHOTO_ID_5443418534799539890" border="0" /></a><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_f1F9pEcIKmI/S4ro1b0DN6I/AAAAAAAACD4/hqajKeiBSRU/s1600-h/Frac+10.bmp"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer; width: 335px; height: 158px;" src="http://1.bp.blogspot.com/_f1F9pEcIKmI/S4ro1b0DN6I/AAAAAAAACD4/hqajKeiBSRU/s400/Frac+10.bmp" alt="" id="BLOGGER_PHOTO_ID_5443419104327448482" border="0" /></a>Do as many examples as necessary until the child is proficient at drawing improper fractions.<br /><br /><span style="font-weight: bold;">Step 7:</span> It is now time to bring it all together to add and subtract fractions visually. Tell the child to draw each fraction in a problem. Ask the child to complete the problem visually. Finally, ask the child to convert the drawing representing the answer to a written fraction.<br /><br />As you probably realize, you can teach reduction to lowest terms visually as well.<br /><br />I hope this blog entry helps you introduce fractions faster and earlier than is typically done in schools. It took only 40 minutes to teach a group of second graders in my daughter's school how to add and subtract homogeneous and heterogeneous fractions. All they knew before I taught them was what a fraction is and how to write them down in standard notation.<br /><img src="file:///C:/DOCUME%7E1/Pablo/LOCALS%7E1/Temp/moz-screenshot-2.png" alt="" />Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-18221250381532106742010-02-01T10:54:00.001-08:002010-02-01T11:21:06.441-08:00A Sample Problem Set for Second GradeMost of my posts are about issues surrounding gifted education. Some of them are about dealing with the complications of raising a girl gifted in areas typically dominated by men. I figure it was time to give a sample problem set.<br /><br />I wrote the problem set attached to the bottom of this post for my daughter's second grade class. I help the teacher once a week by splitting the classroom into three groups. I write problems sets to challenge the "gifted" group. I work with the "average" kids to make sure they are proficient on the topics mandated by the State of California -- essentially preparing them for the CST exams by reinforcing what is taught in class. Finally, I tutor the bottom third of the class to help it understand the basic concepts of arithmetic. There are clear cognitive differences between the three groups, and this makes my job quite difficult -- what works for one third of the class does not work for the other two.<br /><br />The point of the including the problem set below is to give an example of how to teach the basic ideas of proof construction to kids in early elementary school. Notice the structure of the problem set. I try to emulate the way college math books are structured:<br /><ol><li>Definitions</li><li>Example and computational exercises</li><li>Proof construction as a vehicle to learning math and deepen understanding</li></ol>I am guided by three principles. First, computation is important. Second, learning how to derive new ideas from definitions and first principles is central to the philosophy of mathematics. Finally, there is no substitute for learning by discovering, and proof construction is the door to the wonderful world of mathematical discovery. I firmly believe even 6 and 7 year olds should be taught how to construct logical arguments.<br /><br /><span style="font-size:130%;"><br /><span style="font-weight: bold;">Sample Problem Set for Second Grade Statistics and Set Theory</span></span><br /><span style="font-weight: bold;">Prerequisites:</span> Addition, subtraction, an informal understanding of the concept of a set, and Venn diagrams. We do not assume any knowledge of multiplication, division, or fractions. We assume students do not know anything about negative numbers.<br /><br /><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Definition: </b><span style="font-weight: normal;">The </span><b>range</b><span style="font-weight: normal;"> of a set of numbers is the distance between the biggest and the smallest.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"> </span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 1: </b><span style="font-weight: normal;">Find the range of the set of even numbers between 0 and 100. Assume that 0 and 100 are in the set.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"> </span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 2: </b><span style="font-weight: normal;">Define a set of numbers using the following four properties:</span></span></p><ol style="font-family:Times New Roman;"><li><span style="font-size:100%;"><span style="font-weight: normal;"> </span><span style="font-weight: normal;">Every number in the set is bigger than or equal to 30 and smaller than or equal to 60.</span></span></li><li><span style="font-size:100%;"><span style="font-weight: normal;">You can get every number in this set by counting by tens.</span></span></li><li><span style="font-size:100%;"><span style="font-weight: normal;">You can get every number in this set by counting by fives.</span></span></li><li><span style="font-size:100%;"><span style="font-weight: normal;">You can get every number in this set by counting by twenties.</span></span></li></ol><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"> </span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;">Find the set of numbers defined by the above properties. Compute the range of the set.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"> </span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 3: </b><span style="font-weight: normal;">Which one of the following two sets has the bigger range?</span></span></p><ul style="font-family:Times New Roman;"><li><span style="font-size:100%;"><span style="font-weight: normal;"></span><b>Set 1: </b><span style="font-weight: normal;">The set of even numbers between 10 and 20, including 10 and 20.</span></span></li><li><span style="font-size:100%;"><b>Set 2:</b><span style="font-weight: normal;"> The set of odd numbers between 10 and 22.</span></span></li></ul><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 4:</b><span style="font-weight: normal;"> What is the range of the set of numbers equal to their doubles?</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"> </span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 5: </b><span style="font-weight: normal;">What is the set of numbers not equal to themselves:? What is the range of this set?</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"><br /></span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Definition: </b><span style="font-weight: normal;">The </span><b>mode</b><span style="font-weight: normal;"> of a set is defined as the element that appears most often. A set may have no mode, one mode, or more than one mode. We say that a set is </span><b>bi-modal</b><span style="font-weight: normal;"> if it has exactly two modes.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"> </span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 6:</b><span style="font-weight: normal;"> Find the mode of {100, 99, 50, 3, 2, 1, 60 ,1 ,85}</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"><br /></span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 7: </b><span style="font-weight: normal;">Find the mode of {Pablo, Paulina, Alex, Kolane, Pablo, Paulina, 2, 3, 10, 1}</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"><br /></span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 8: </b><span style="font-weight: normal;">Find the mode of {2, 3, 4, 5, 6, 7, 8}</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"><br /></span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Definition: </b><span style="font-weight: normal;">The </span><b>median</b><span style="font-weight: normal;"> of a set is the number for which half the elements in the set are smaller and half bigger than the mode. Sometimes, the mode is part of the set. Sometimes, the mode is not.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"> </span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 9: </b><span style="font-weight: normal;">Find the median of {1,2,3}.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"><br /></span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Example 1:</b><span style="font-weight: normal;"> The median of the set in the exercise 9 was a member of the set. However, the median is often not a member of the set. For example, 3 is median of the {2,4}. There are only two numbers in the set {2,4}. We pick the number right in between these two. That number is 3. Half the elements in {2,4} are smaller than 3 and half bigger than 3.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"> </span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 10:</b><span style="font-weight: normal;"> Find the median of {100, 1, 80, 52, 48, 10, 12, 15}</span></span></p><span style="font-size:100%;"><span style="font-weight: normal;"> </span></span><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 11: </b><span style="font-weight: normal;">If you throw away the smallest and the largest numbers in a set, does the median change?</span></span></p><span style="font-size:100%;"><span style="font-weight: normal;"> </span></span><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 12: </b><span style="font-weight: normal;">Assume the following things:</span></span></p><ol style="font-family:Times New Roman;"><li><span style="font-size:100%;"><span style="font-weight: normal;">There are twenty students in the classroom.</span></span></li><li><span style="font-size:100%;"><span style="font-weight: normal;">Every student likes either apples, pears, or both.</span></span></li><li><span style="font-size:100%;"><span style="font-weight: normal;">15 students like apples.</span></span></li><li><span style="font-size:100%;"><span style="font-weight: normal;">10 students like pears.</span></span></li></ol><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;">How many students like both apples and pears? Prove your answer using a Venn diagram.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"><br /></span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 13: </b><span style="font-weight: normal;">Assume the following three things:</span></span></p><ol style="font-family:Times New Roman;"><li><span style="font-size:100%;"><span style="font-weight: normal;"> </span><span style="font-weight: normal;">There are twenty students in the classroom.</span></span></li><li><span style="font-size:100%;"><span style="font-weight: normal;">A student likes only one type of fruit.</span></span></li><li><span style="font-size:100%;"><span style="font-weight: normal;">15 students like apples.</span></span></li><li><span style="font-size:100%;"><span style="font-weight: normal;">10 students like pears.</span></span><span style="font-size:100%;"><span style="font-weight: normal;"><br /></span></span></li></ol><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;">How many students like neither apples nor pears? Prove your answer using a Venn diagram.<br /></span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><span style="font-weight: normal;"><br /></span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Definition: </b><span style="font-weight: normal;">The </span><b>mean </b><span style="font-weight: normal;">of a set of numbers is one that repeated as many times as there are elements in the set gives you the same number as adding all the numbers in the set.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Example 2: </b><span style="font-weight: normal;">Given the set {2,3,4}, computer the mean.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Solutions: </b><span style="font-weight: normal;">First, we add all the numbers in the set. 2+3+4 = 9. We have to find a number that added to itself three times equals 9. That number is 3, because 3+3+3 = 9. Hence, 3 is the mean of {2,3,4}.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Example 3: </b><span style="font-weight: normal;">Compute the mean of {1,2,3,4,5,6,7}.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Solution: </b><span style="font-weight: normal;">1+2+3+4+5+6+7=28. What number can I add to itself 7 times to get 28? You can find out by trial and error, but the answer is 4 because 4+4+4+4+4+4+4=28. Hence, the mean of the set is 4.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Example 4:</b><span style="font-weight: normal;"> Show that the mean of {2,3,4,5,6,7,8}=35 is bigger than or equal to 2 without computing the sum of the numbers?</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Solutions: </b><span style="font-weight: normal;">If the mean were smaller than 2, the first choice would be 1. This would mean that adding 1 to itself seven times would equal the sum of the elements in the set {2,3,4,5,6,7,8}. However, 1 is smaller than every element in the set. Hence, adding seven ones cannot equal the sum of the elements in the set. This implies that the mean must be larger than or equal to 2, the smallest element in the set.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Example 5:</b><span style="font-weight: normal;"> Show that the mean of {2,3,4,5,6,7,8}=35 is smaller than or equal to 8 without computing the sum of the numbers?</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Solutions: </b><span style="font-weight: normal;">If the mean were bigger than 8, the first choice would be 9. This would mean that adding 9 to itself seven times would equal the sum of the elements in the set {2,3,4,5,6,7,8}. However, 9 is larger than every element in the set. Hence, adding seven 9s must be larger than then sum of the elements in the set. This means that 9 must be smaller than or equal to 8, the largest element in the set.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 14:</b><span style="font-weight: normal;"> Can the mean of a set be smaller than the smallest number in the set?</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Hint: </b><span style="font-weight: normal;">Look at the prior two examples. This problem is solved the same way. Assume that the mean is smaller than the smallest number in the set. Compare it to every number in the set. What do you see? </span></span></p><span style="font-weight: bold;"></span><span style="font-size:100%;"><b><br /></b></span><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 15: </b><span style="font-weight: normal;">Can the mean of a set be bigger than the largest number in the set?</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Hint: </b><span style="font-weight: normal;">Assume that the mean is bigger than the biggest number in the set. Compare it to every number in the set. What do you see?</span></span></p><span style="font-size:100%;"><b> </b></span><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 16: </b><span style="font-weight: normal;">Show that the mean of any three numbers is both</span></span></p><ul style="font-family:Times New Roman;"><li><span style="font-size:100%;"><b> </b><span style="font-weight: normal;">larger than or equal to smallest element of the set</span></span></li><li><span style="font-size:100%;"><span style="font-weight: normal;">smaller than or equal to the largest element of the set</span></span></li></ul><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Hint: </b><span style="font-weight: normal;">Use what you did in exercises 14 and 15.</span></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b> </b></span></p><p style="margin: 0pt;font-family:Times New Roman;"><span style="font-size:100%;"><b>Exercise 17:</b><span style="font-weight: normal;"> Give an example of a set for which the mean and the median are NOT equal.</span></span></p>Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com2tag:blogger.com,1999:blog-998673071631998557.post-47837977410459063992010-01-11T21:19:00.001-08:002010-01-12T06:30:29.579-08:00Dealing with the Impact of Societal Biases on Mathematically/Scientifically Gifted GirlsI have a daughter highly gifted in areas typically dominated by men, and I constantly worry about the impact that societal biases could have on her passions and career choice. Fortunately, I have not encountered any problems so far, but I am concerned that this period of bliss may not last. Is there anything I could do to help her explore and develop her talents while preparing her to deal successfully with society's stereotypical biases?<br /><br />Many parents are well aware that little girls are far more verbal than boys, but I believe that behavior, attitudes, and learning styles are far more diverging. It is instructive to think about how girls socialize and compare it with how boys do it. Girls can sit quietly for long stretches of time focused on tasks that would drive insane a boy of any age. I was largely unaware of this until I observed my daughter playing with other kids. A few girls-only birthday parties drove the stereotype even deeper into my mind. Girls and boys really do come from different planets. While boys are happiest running around, pushing each other, and generally competing physically to establish some sort of pecking order, girls can sit for hours talking and playing pretend. My seven year-old daughter sums it up better than anybody else:<br /><blockquote>Girls are smarter than boys because boys are coo-coo and just like to run around and cannot pay attention in school.<br /></blockquote><br />My key observation here is that societal expectations and intrinsic, gender differences must have deep implications for learning styles. Yet, we teach boys and girls exactly the same way and do little to encourage kids to explore all possible carriers. For instance, very few boys go into nursing despite a clear, current and projected shortage of front-line, medical professionals. An small number of girls choose carriers in math, science, and engineering. Studies suggests that this phenomenon can be partially explained by the impact of society's expectations for each gender. However, it is also a fact, in my opinion, that boys and girls learn differently, and this leads to girls becoming bored with scientific in high school. Child rearing also has an impact, but it generally does not come into play until after college. It follows that societal pressures, stereotypes, and differences in learning styles may be the most important explanatory variables. There could be biological reasons as well, but I tend to discount this explanation and believe that while there could some relationship, it is probably much weaker than generally believed. Hence, we are back to my original question. Is there anything I could do to help my daughter explore and develop her talents while preparing her to deal effectively with society's stereotypical biases?<br /><br />I have had pre-conceptions about gender-specific aptitudes for as long as I can remember, but I always have tried to keep an open mind. Unfortunately, as is the case with race, we are bombarded daily with stereotypical messages. Everybody is a bigot to some degree, but most of our biases are unconscious. However, this is irrelevant. The point is that we have them, and they affect the way we raise our children. This is a particularly tough problem for parents of girls gifted in areas typically dominated by men. I have never thought that men are smarter than women. For instance, my mother is, by far, one of the smartest people I have ever known. However, I always have had a feeling that men tend to be better than women at math. The reality is that so few women go into math and even fewer make it into the upper echelons of the profession. What worries me now is that I have a daughter who is extremely talented in math and will undoubtedly face some of the same stereotypes that I and the rest of society have always forced onto women. What can I do to help her deal with this issue? I do not know if my daughter will become a scientist, mathematician, or something altogether different. Yes, iIt would make me happy if she chose such a career because I value the pursuit of knowledge, and she has innate abilities. However, Paulina also loves to dance, make up stories, and create works of art. Hence, she could end up doing pretty much anything -- other than singing, since it is pretty clear she will not make a good living performing. However, I cannot let societal stereotypes and misconceptions influence her choice of profession.<br /><br />My daughter has exhibited abilities in multiple areas, but it does appear that mathematics is her biggest strength,and I have observed that people tend to specialize in subjects that leverage their skills. Hence, it is a reasonable guess that my daughter could choose a career in math, science, or engineering. Hence, I wonder how I can help her gain sufficient confidence to fight stereotypes. I also want her to avoid the "dumb-down" tendencies common in many smart teenage girls who avoid being intellectual hoping to be accepted by boys. I know of no easy answer to my question.<br /><br />My wife and I have thought about sending our daughter to a private, all-girls school. My daughter is friends with many boys, and she participates in lots of extracurricular activities where she gets to interact with the opposite sex. Hence, we are not very concerned with the socialization problems typically associated with single-sex schools. Another approach we are employing is to introduce her to as many female role models as we possibly can. We think this may help convince our daughter that she can do whatever she chooses to. Finally, we divide who works on what with Paulina. My wife handles writing and the arts. I handle math, science, and computers. We do it this way simply because those are the areas where each of us excels. However, it may turn out to be a lucky choice after all. Doing math and science with my daughter may persuade her that it is acceptable for women to pursue careers in math or science. Beyond this, we are not sure what we could or should do.Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-67885406816371715832009-12-16T10:22:00.000-08:002009-12-16T17:22:22.399-08:00On the Necessity of Ability-based Grouping in Mixed-Ability ClassroomsI have always believed that ability-based grouping is beneficial, but I never thought that failing to do so could be tremendously detrimental for students. As readers of this blog know, I have been helping my daughter's second grade by teaching math once a week to a small group of gifted kids. A few weeks ago, the homeroom teacher and I decided to try the supplementary material with the entire class. It is our opinion that the experiment was a complete failure despite every effort to choose problems carefully. I believe the reasons are the same reason that cause large, mixed ability classrooms to fail the majority of students.<br /><br />The normal bell curve, or some variation thereof, has a funny way of manifesting itself despite loud proclamations by skeptics that ability-based grouping hurts education in general. My observations so far this year in my daughter's classroom have matched academic theory so closely as to be shocking! There are a few very smart kids, a few disengaged due to boredom, many of average ability, and a few clearly requiring special attention. Math is best learned by doing it. Hence, I always design my lessons as sets of carefully chosen problems. The idea behind my approach is to explain a few key concepts at the beginning of each lesson and then let the students learn the material by discovering the math while working through the problems. Unfortunately, this approach -- which is optimal for teaching mathematics -- fails miserably in a mixed-ability setting because the distribution of skill sets quickly interferes with everybody's learning process. The failure arises because learning speed and comprehension varies widely. Some kids cruise through the materials, while others work on it at the expected pace. Finally, a few struggle to the point where it becomes clear they do not even understand one-digit addition. My method calls for students to work independently, raising their hands when they get stuck. Unfortunately, many get stuck simultaneously in completely different places of the problem set. This puts such high, simultaneous demand on the teacher's attention, that no kid really benefits much from the session. Is my teaching method to blame, or is the organization of the classroom the real problem? I believe the latter is.<br /><br />Today, my daughter's home room teacher and I decided to cluster kids according to ability starting in January. We will split the 24 students into three groups: one requiring additional help, one capable of average achievement, and one whose members are "gifted" learners. I write "gifted" in quotation marks, but my daughter is the only one in the class who has been for IQ or subject-specific aptitude. All we know is that the "gifted" group seems to learn faster, finish problems earlier, and work farther into the problems than the rest of the class. The key observation here is that we are grouping after assessing the kids. We did spend a ton of money getting everyone tested. I will report back in a few months on how this new experiment turns out. I am optimistic because things worked out very well earlier this trimester when I only taught the "gifted" group. Hence, I expect few problems when we come back from the holidays because I also have experience and have been successful with other levels of achievement.<br /><br />One thing bothers me more than anything else. Why don't public schools group based on ability? There is no incremental cost over a mixed-ability setting. Take my daughter's school as a example. There are six second grade classrooms. This means that each could hold 16.7% of the second grade student body. Why is it so hard to dedicate 1 classroom for students with 130+ IQ and one for pupils requiring additional attention? Wouldn't these two populations perform better in classrooms designed to meet their intellectual needs. Bear in mind that this is a problem affecting not only gifted children but also those with modest intellect. I am not advocating a room for highly gifted kids (i.e. 145+). I am merely asking that we group together students of similar intellectual capabilities. Why is it so hard to realize that kids requiring additional help should be taught using appropriate methods and curriculum? The same goes for gifted kids. One argument against my idea is that it costs a lot of money to test for aptitude or IQ. Recall what I wrote above? We are grouping according to ability in my daughter's class without spending a small fortune on testing. The problem with the cost argument is that students are much better behaved and require less supervision then they are properly matched to the curriculum and the teaching methods (i.e. larger classrooms) and they are properly challenged. Hence, re-organizing schools around ability-based grouping could save money. In fact, small schools could be merged into larger campuses with five or six classrooms per grade, eliminating redundancies and providing a population big enough to exhibit clear distribution of abilities along a significant portion of the spectrum. It makes little sense -- other than political -- to ignore mountains of sound academic research as well as the clear evidence in our children's own classrooms.Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com1tag:blogger.com,1999:blog-998673071631998557.post-42673060667385969582009-11-11T21:39:00.000-08:002009-12-05T16:16:24.232-08:00Don't Think Your School Is Good Enough, Get InvolvedMany parents believe their public schools are not good enough. They complaint incessantly about the deteriorating state of public eduction. The rants are warranted in many cases, but it is my experience that few parents do anything about it. By this I mean that very few donate money and time. My daughter's public elementary is one of the best in California. It ranks consistently in the top 5%. It has earned the designations of school of advanced studies and distinguished school, but it still is too slow to meet my daughter' s intellectual needs. There are many potential solutions to this problem, but most involve after-school, enrichment activities. Other potential solutions center on advocacy, but this tends to be confrontational and ineffective. As a result, homeschooling is growing in popularity. While I believe that homeschooling is ideal for some children, others like my daughter are extremely social and crave close, daily interaction with other children. This leaves parents like myself with two options: fight the system or make it work for you. I have opted for the second option, by getting involved in my daughter's classroom, and the results have been surprisingly positive.<br /><br />LAUSD is crumbling due to California's budget crisis. Teaching assistants have been eliminated in many grades, and class sizes have increased markedly. Teachers are more overworked than ever and even have to clean their own classrooms because LAUSD has cut janitorial staff. I offered my daughter's homeroom teacher to teach the weekly computer class. She accepted gladly because my technical background is extensive, and kids like working with me. Computer lab worked out well. I taught the kids about graphs, data analysis, logic puzzles, simulations, and other fun topics. Things well so well that the teacher asked me to pull out a group of gifted learners for a weekly math class. The experience has been extremely rewarding. Over the past three weeks, my group has learned binary arithmetic and how it relates and compares to decimal arithmetic, various topics involving the platonic solids -- including Euler's formula --, and the concept of measure in 1D, 2D, and 3D. I am planning to introduce them to graphs (i.e. the discrete mathematical kind) and their relationship to the wire frames of the platonic solids. This will lead into a discussion of how to represent various problems in terms of graphs. I am excited because the kids have been enjoying our sessions and are always ready to work on problems and discover things on their own.<br /><br />The moral of this little story is that you can always make a direct difference in your child's education by getting involved. Most parents complain, but they rarely get invest enough energy to make a difference. Getting involved means giving as much money and time as reasonable. Failing to do so is perpetuating the very problems you complain about. I am not certain what the next few years hold, but this one is turning out much better than expected.Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-76690882030054448922009-10-12T09:02:00.000-07:002009-11-02T17:05:25.225-08:00The Power of Math CirclesI recently discovered the wonderful world of math circles, and the activity has quickly become a favorite of my daughter. I approached the UCLA math circle with the idea of having Paulina do interesting math with other kids equally interested in the subject. I was also looking for a chance to step back from the role of parent/teacher and let her experience collaborative math without having my looking over her shoulder. I got the schedule wrong for the first meeting, but she managed to spend 30 mins working on cool introductory problems. I wanted Paulina to experience the joy of solving problems and working with other people, but I promised myself that I would not force her to return if she did not want to. To my surprise, Paulina said that the circle was fun and that she wanted to keep going. She now has attended five sessions, and she even fell asleep holding her book of problems the night of the second session. It appears that the math circle may provide Paulina with a source of intellectual challenge, excitement, as well as a place to enjoy math with other like-minded kids.<br /><br />Why do I like the math circle? The answer is that the it approaches the subject very differently from the boring, tradition-bound, mind-numbing torture imposed on kids all across the US. The last few sessions have centered around binary numbers (i.e. base 2). This is a very simple concept for middle and high school kids, but few six, seven, and eight year olds have ever imagined that one could do arithmetic using anything but the decimals. The circle's leader -- a UCLA professor with kids in the circle -- started the afternoon by asking if anybody knew a way to represented numbers in any way other than using decimal notation. A few kids offered interesting, albeit impractical solutions, and one of them said that you could just use as many sticks as needed, counting each once. The instructor then suggested Roman numerals. The kids worked on writing Roman numbers and ended the discussion by figuring out how to correct the following equation by moving just one stick:<br /><br />II + III = VI.<br /><br />Clearly, the point of Roman numerals was to teach that numbers can be represented in different ways. As such, binary numbers were introduced, and this is where the teaching got clever. Instead of using powers of two, the instructor wrote the following sequence on the blackboard: 1, 2, 4, 8, 16, 32, 64... Every kid understood that you get the next number by doubling the preceding one. The instructor then asked the kids to imagine that each number represents a weight and to figure out how to balance an object using only those weights, assuming that each weight could be used only once. One way to think about it is to say that you are trying to balance a given weight with counterweights of 1, 2, 4, 8, ... It actually helps to draw a scale and ask the student to draw the counter weights required to balance both sides.<br /><br />The beautiful thing about the balance / weight metaphor for binary is that it allows very young kids to understand binary without resorting to exponents and other more advanced material, and binary is, in my opinion, a great way to teach the basic principles of arithmetic.<br /><br />Binary is just one of the topics taught at the UCLA Math Circle. Challenging word problems, principles of algebra, and other "advanced" material are introduced weekly to kids as young as six. The sad part of this is that most of the participants only get to enjoy fun math once a week. If it was up to me, math class would be abolished, and math circles would become the norm.Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-64178093271568064432009-09-13T14:53:00.001-07:002009-09-13T14:53:46.706-07:00Using Adaptive Programs to Teach MathIt is my position that math education must be as individualized as possible. However, it is impossible to do this in classrooms with 15 to 30 kids. This is where adaptive computer software fits in. My daughter has been using Stanford University's Education Program for Gifted Youth (EPGY) since December 1, 2008. Things went well with math , so we enrolled her in the English track this past May. The results have been outstanding, and I believe many other children could benefit from the system. In fact, I am convinced that a slightly slower pace would make EPGY suitable for kids at many different levels of cognitive development.<br /><br />EPGY's math program -- at least the highly gifted version -- is probably a bit too fast for normal kids. It aims to cover K through 6th at a recommended rate of 2 grade levels per 2.5 quarters (approximately 7 to 8 months or nearly a full school year). My daughter has moved much faster because numbers and logic are one of her strengths. There are three elementary school courses:<br /><ul><li>K through 2nd grade</li><li>3rd / 4th grade</li><li>5th / 6th grade</li></ul>However, EPGY continues through advanced undergraduate mathematics. This could allow an accelerated learner to go through the bulk of an undergraduate math degree before entering college.<br /><br />Acceleration, adaptability, and early introduction of "advanced" concepts are key features of EPGY. The strong emphasis on acceleration is supported by decades of research showing that gifted children learn faster and make deeper abstract connections than the population within two standard deviations from the mean IQ. Adopting an accelerated curriculum need not lead to early college admission, but it provides a way out of monotonous concepts such as arithmetic so more abstract topics can be tackled early and in great depth. In practice, acceleration works well with gifted children they require fewer drills than normal children. This opens access to increasingly sophisticated material earlier than would be possible otherwise.<br /><br />Adaptability is single, biggest reason why I love EPGY and why systems like it should be become part of the mainstream education system. The online platform tracks progress across six different strands:<br /><ul><li>Number Sense: Integers</li><li>Number Sense: Decimals and Fractions </li><li>Geometry</li><li>Logic and Reasoning</li><li>Measurement</li><li>Data/Statistics/Probability</li></ul>A student could be at different levels in each strand. However, all strands must "graduate"to the next grade simultaneously. For instance, if a student completes the statistics strand one month before the others, the system increases the number of non-statistical questions until all strands make it to the next grade. The system adapts the difficulty and number of questions in a given topic according to the student's progress. Hence, every single student moves at a different speed and through a unique set of problems. The system's goal is to ensure comprehension and proficiency in as little time as possible. I find that the system works best when the student works with an adult to review difficult material. Additional explanations, exercises, and discussion can reinforce the learning process so the student moves as quickly as possible through the material. The most beautiful part of the EPGY system is that it requires zero homework. Because teaching happens through short lectures and the material is learned via exercises, there is no need for homework. Learning happens by doing. EPGY offers virutal classrooms as well as access to and feedback/guidance from very capable instructors. If EPGY were integrated into traditional classrooms, very little homework would be required. This would free up valuable time for reading, playing, and just relaxing.<br /><br />I love EPGY's early introduction of "advanced" concepts. Ideas such as variables, equations, positive/negative numbers, proper survey design methodologies (i.e. avoiding leading questions, etc.), and statistical concepts surface as early as first and second grade. Early introduction could eliminate the shock suffered by many middle school students when confronted by these topics. Proper teaching techniques allow young children to understand what these things mean and how to use them. By the time second grade ends, variables and simple linear equations are second nature to EPGy students. I don't believe that the concept of variables is any more abstract that multiplication itself, but I have very little data to support my hypothesis (i.e. my daughter is my entire population). However, I believe that many children could handle at least some of the concepts if taught using appropriate techniques. Regardless of one's position on the early introduction of advanced concepts, some should be presented as early as possible. Some students will not understand what is going on, but many are likely to benefit greatly.<br /><br />It turns out that I am not the only one who thinks that adaptive software has a place in "traditional" classrooms. EPGY conducted studies in California (click <a href="http://epgy.stanford.edu/research/EPGYvsCSTReport_June05.pdf">here </a>for PDF of study) to determine the effectiveness of EPGY's variables as a predictor of performance on CST (i.e. California Standards Test). Clearly, the purpose of the study was to determine if there was a statistically significant correlation between performance in EPGY and CST. However, it was very instructive to see the impact the program had on Title I students. The bottom line is that the overall population sample benefited greatly. Furthermore, because EPGY maps into California standards, there would few, if any, legal repercussions if a school adopted EPGY. Finally, EPGY offers a school-wide option to use EPGY, as well as grants and financial aid for students with modest resources. This all means that there is little reason to avoid using computerized, adaptive systems, and EPGY is an excellent option.<br /><br />I would advice parents of gifted children to look into EPGY (click <a href="http://epgy.stanford.edu/">here</a> for the program's website). If money is an issue, apply for financial aid. Some homeschooling charters like the Sky Mountain Academy give you as much as $3,000 toward curriculum materials, and EPGY is one of the approved curriculum providers.<br /><br />As always, I hope you find this useful.Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com2tag:blogger.com,1999:blog-998673071631998557.post-56715414686302177172009-09-12T23:17:00.000-07:002009-09-13T14:54:40.462-07:00Having the Talk at Age 6!We just had THE TALK with Paulina. She is only six, and this definitely happened with her earlier than it did with me. I remember getting a great book from my parents when I was eight. I read it intently. I asked a lot of questions, but I was two years older than her when it happened to me. Paulina has been asking questions about babies for a long time. I think she was about three when she asked the first serious questions. We were very factual in describing the various systems of the human body. She especially loved the cardio-pulmonary system and, as she termed it, the "baby system." The questions got more and more probing until we simply could not avoid answering them precisely. Hence, we got an age-appropriate book and had the talk.<br /><br />"The talk" went better than expected. There were a few of the giggles triggered by talking about sex with children and teenagers. There was very little interest on the intercourse section -- which will almost certainly change in a few years. Paulina was fascinated by the fact that the sperm and egg carry the DNA that defines a baby. She has known about DNA for quite some time and that half of the material comes from each parent, but it was a revelation for her that the sperm and the egg are the vessels.<br /><br />I always wondered when we would have the talk with Paulina. It happened sooner than I expected, but things rarely happen when you want them to. It went better than I imagined it would, but I think things may get a bit more interesting when other kids in her class start talking about sex. In any case, she is the youngest second grader in her school, and it makes sense to help her prepare to handle the situation by teaching her facts and helping her understand what they all mean.<br /><br />In case somebody is looking for a book appropriate for young kids, we bought<br /><br /><span style="font-style: italic;">What's the Big Secret</span> by Laurie Krasny Brown and Marc Brown.Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0tag:blogger.com,1999:blog-998673071631998557.post-30679280835799024232009-09-11T16:44:00.001-07:002009-09-11T19:54:28.789-07:00Want to Skip Grades in LAUSD, Be Nice to Your Principal and Homeroom Teacher!Our local public school agreed to skip my daughter from K to 2nd grade. I did not push for 3rd grade because she needs a transition year for her writing to catch up to her radical acceleration in other subjects. I estimate that it would take her approximately five or six months to catch up to third grade spelling expectations. Given her perfectionist tendencies, skipping one more year now could prove counterproductive. On the other hand, I am convinced that by the end of this school year she will be ready to skip to fourth or even fifth grade. Skipping grades in one-year vs. multi-year increments gives my daughter enough time to adapt, and I get plenty of time to figure out how to navigate my district's impenetrable bureaucracy.<br /><br />I met with my daughter's homeroom teacher this afternoon to discuss grade the recent grade skipping decision. To my surprise, she was very well informed about gifted education and for many years has handled clusters of gifted kids, sprinkled with the rare highly and exceptionally gifted. We reviewed my daughter's test scores, academic record from EPGY, list of books read since last year, as well as her own assessment academic readiness. I was left speechless when she argued that my daughter could be skipped to third grade and that it could be arranged if I requested it. Say what?????? Did I hear the teacher argue for the radical acceleration of my child? Is this possible in the LAUSD? I explained to the teacher that I thought it best to allow one year for my daughter's writing to rise to third grade standards. We agreed that the best course of action would be to skip over third grade next year provided that the writing proficiency goal is accomplished.<br /><br />The surprises continued this afternoon after I got to my house. I received an email from the homeroom teacher following a meeting with the principal. She informed me that my daughter will be accelerated to third grade math. Logistically, this means that Paulina will leave her homeroom every day to take math in one of the third grade classrooms and then return to second grade for the remainder of the day. Moreover, my wife and I will be allowed to come to class to help proctor Paulina while she spends part of her English and math classes working on EPGY. In exchange, we have offered to help the teacher since budget cuts mean she has no teaching assistant this year.<br /><br />Here is a bullet point summary of what I learned today:<br /><ul><li>radical acceleration is possible in the LAUSD</li><li>homeroom teachers and school principals make the final decision to accelerate</li><li>it is possible to do single-subject acceleration simultaneously with grade skipping</li><li>this seems to work best when the teachers and principal are well-informed</li><li>offer to help when the school accommodates your child</li></ul>The basic fact is that my family got lucky. Our experience is uncommon. Our public school is one of the best in California. The principal is well-informed. The school has a high concentration of high-achieving students, and many of the teachers have some training in gifted education.<br /><br />I am having a bit of trouble coming up with a prescription for success. I did some things right. Good luck played a big role. However, I also believe that "Chance favors the prepared mind." This implies that you can best advocate for your child by being ready:<br /><ul><li>Talk to parents of current students to find out how the school has handled acceleration and grade skipping in previous years<br /></li><li>Learn the rules and regulations governing grade skipping and acceleration in your district</li><li>Document your child's talents by collecting IQ test scores, grades from prior courses, scores from standardized exams, transcripts from gifted and talented programs (i.e. Stanford's Education Program for Gifted Youth, John Hopkins' Center for Talented Youth, etc.), evaluations from former teachers, etc.</li><li>Enroll the help of gifted education advocates. You may want to contact the <a href="http://www.davidsongifted.org/youngscholars/">Davidson Institute</a>. The Davidson Institute's Davidson Young Scholars offers guidance, free consulting services, and may help you communicate with local school officials.</li><li>Read as much research as possible on the benefits of grade skipping, acceleration, ability-based grouping, etc. Become an expert. Knowledge is the most powerful tool at your disposal.</li><li>Become a relentless advocate for your child's rights.<br /></li></ul>There is no magic bullet here. Prepare yourself. Look out for opportunities and seize them when they present themselves.<br /><br /><br />Hope this helps,<br /><br />PabloPablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com4tag:blogger.com,1999:blog-998673071631998557.post-18627077344499285202009-09-07T09:34:00.000-07:002009-09-10T20:04:00.161-07:00Skipping Grades in the LAUSD (i.e. Los Angeles)Is it possible to skip grades in the Los Angeles Unified School District (LAUSD)? The answer is yes but only within very limited constraints. It is possible because my daughter was allowed to go straight from K to 2nd grade. I am fairly certain that she will have to skip grades a few more times over the next few years. Hence, I have set out to learn how LAUSD and the California Department of Education handle radical acceleration -- defined as three or more years. My findings have been discouraging so far. The LAUSD's Board of Education enforces minimum age limits for admission to various grade levels. I summarize my findings below and will update the information when I learn more.<br /><br />I have to admit that my experience so far has been much more pleasant than I expected. My daughter was recently allowed to skip first grade, but her school is one of the best in California. I believe that our school's willingness to accommodate my daughter is the result, among other things, of its administration's focus on academics and the large concentration of high-achieving students. A thorough review of my daughter's academic record, including her EPGY scores, convinced the principal and first grade teacher that skipping to second grade was optimal. Unfortunately, I fear that my daughter will only be able to skip one more year before violating LAUSD's minimum age requirements. I believe my daughter will be allowed to skip from second to fourth grade because the principal is open minded and LAUSD's rules allow one more year for somebody in my daughter's situation. Unfortunately, her school is a K through 5th grade elementary, and I fear I will not find a public middle school that will allow her to skip from 4th to 6th or 7th.<br /><br />Here is a summary of the pertinent age limitations for grade skipping within LAUSD:<br /><ul><li><span style="font-weight: bold;">Age of Admission to K - </span>For admission to kindergarten, during the first school month of the school year, the fifth birthday of the child must be on or before December 1 of that calendar year. (Education Code, Section 48000) For good cause, a child of proper age may be admitted to a class after the first school month.</li><li><span style="font-weight: bold;">Age of Admission into 1st Grade</span> - A student who is at least five years of age and who has been lawfully admitted to a kindergarten class in the Los Angeles Unified School District may be placed in the first grade, in accordance with regulations established by the Superintendent of Schools, when the administration determines the child is ready for first grade work. For admission to first grade during the first school month of the school year, the sixth birthday of the student must be on or before December 1 of that calendar year. Verification of age shall be required as provided in Board Rule 2001. For good cause, a student of proper age may be admitted to a class after the first school month.</li><li><span style="font-weight: bold;">Minimum Age to Enter Middle School - </span>The minimum age for students entering middle high school who have been accelerated because of superior mental ability is 10-9 years of age on September 1 of the school year.</li><li><span style="font-weight: bold;">Minimum 6th Grade Attendance Before Middle School</span> - Students transferring from regular Los Angeles Unified School District elementary classes to middle high school must have been enrolled in grade 6 for a minimum of one semester.</li><li><span style="font-weight: bold;">Exceptions to Rules for Entering Middle School </span>- If exceptions to this policy become necessary for the overage student, the elementary and the middle high school principals involved must confer prior to the transfer of the student. It is understood that the final decision relative to exceptions shall rest with the elementary school principal.</li><li><span style="font-weight: bold;">Minimum Age for Senior High School</span> - The minimum age for students entering senior high school who have been accelerated because of superior mental ability is 13-9<br /></li></ul>The single biggest implication of the above rules is that It makes no difference to LAUSD if your child is the smartest, most accomplished person in the planet. Your second grader already knows pre-algebra and reads seventh-grade books. Who cares? The poor child will be forced to study addition, subtraction, and how to write the simplest of sentences.<br /><br />The following list summarizes the required ages by December 2nd of each grade level:<br /><br /><span style="font-weight: bold;">Elementary</span><br />1st grade, 6<br />2nd grade, 7<br />3rd grade, 8<br />4th grade, 9<br />5th grade, 10<br />6th grade, 11<br /><br /><span style="font-weight: bold;">Middle School</span><br />7th grade, 12<br />8th grade, 13<br />9th grade, 14<br /><br /><span style="font-weight: bold;">High School</span><br />10th grade, 15<br />11th grade, 16<br />12th grade, 17<br /><br />LAUSD's minimum, allowed age for seventh grade is eleven, provided the birthday happens on or before December 2. This is because the student must be 10 years and 9 months old by September 1st (i.e. the start of the 7th grade school year). Under this rule, the smartest kid in the world could only accelerate one year ahead of the expected, seventh grader. As you can see from the age rules above, high school admission is regulated more or less the same way.<br /><br />It is very important to understand that the California Department of Education is powerless to enforce grade skipping. In fact, the California Department of Education takes the following position.<br /><br />-----------START OF CITATION ---------------------<br />A child who was not age-eligible for kindergarten (that is, the child turned five after December 2 in the school year) and who attended a California private school kindergarten for a year is viewed by the CDE as not legally enrolled in kindergarten, pursuant to <em>EC</em> Section 48000 requirements. Therefore, this child, upon enrollment in public school, is enrolled in kindergarten, assessed, and may (but is not required to) be immediately promoted to first grade if the child meets the following State Board of Education criteria, pursuant to Title 5, Section 200: <br /> <ul><li>The child is at least five years of age.</li><li> The child has attended a public school kindergarten for a long enough time to enable school personnel to evaluate the child's ability.</li><li> The child is in the upper 5 percent of the child's age group in terms of general mental ability.</li><li> The physical development and social maturity of the child are consistent with the child's advanced mental ability.</li><li>The parent or guardian has filed a written statement with the district that approves placement in first grade.</li></ul> <p>A statement, signed by the district and parent/guardian, is placed in the official school records for these five-year-olds who have been advanced to first grade (<em>EC</em> Section 48011). This action prevents a subsequent audit exception for first grade placement of an age-ineligible student.</p><p>Source: http://www.cde.ca.gov/ci/gs/em/kinderinfo.asp<br /></p><p>-----------END OF CITATION-------------------<br /></p>The above verbiage means that it does not matter if your child has already completed K. It is entirely at the discretion of your school's administration to let your kid into first grade. The State of California does not even recognize private K as a real K for legal purposes. The California Department of Education recommends alternatives when grade skipping is not approved. For instance, individualized instruction, multi-classroom settings (i.e. moving to a higher grade for a single subject), etc. However, my research into the subject shows these alternatives are seldom if ever implemented in LAUSD.<br /><br />-----------START OF CITATION----------------------------<br />Local districts have discretion regarding promotion and retention. According to California Code EC48070, "each school district and each county superintendent of schools shall adopt policies regarding pupil promotion and retention. A pupil shall be promoted or retained only as provided in the policies adopted pursuant to this article." <a href="http://www.leginfo.ca.gov/cgi-bin/displaycode?section=edc&group=48001-49000&file=48070-48070.5">EC48070 Promotion and Retention</a> EC48070 is fairly precise about retention policies, but it is glaringly vague on acceleration.<br /><br />Source: http://www.accelerationinstitute.org/Resources/Policy/By_State/Show_Policy.aspx?StateID=6<br /><br />-----------END OF CITATION--------------------------<br /><br />The age limits for grade acceleration can be found at<br />http://www.lausd.net/lausd/board/secretary/BoardRules/BoardRules3-08.pdf<br /><br />I hope this information helps you advocate for your child. Inform yourself. It is the best tool at your disposal.Pablo A. Perez-Fernandezhttp://www.blogger.com/profile/17664939441743111860noreply@blogger.com0