Monday, July 26, 2010

Last Post on Blogger

I 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.

So, please, visit and enjoy. All the posts from this site have been moved to the new site, which is fully searchable.



Tuesday, July 13, 2010

Moving the Blog

I 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:

Thanks for reading. Enjoy.

Monday, July 12, 2010

Exercising the Young Mind

I 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.

I made this one a little harder. I wanted Paulina to think carefully about:
  1. the determinants of dimension
  2. how measurements change when enlarging along one, two, or three dimensions
  3. the relationship between volume and area
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?

This problem set is #2 in I recommend using Lego blocks with this problem set.

Sunday, July 11, 2010

Starting a Problem Set and Resource Repository

I 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:
  1. Problems sets I write for my daughter
  2. Useful math and science websites. These are resources that offer courses, tools, and other useful stuff
  3. Other educational resources
Check back here often.

Tuesday, June 29, 2010

Difficulty vs. Acceleration in Gifted Mathematics Education

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 if you have an answer to my question.

What do I mean by difficulty vs. acceleration? The easiest way to describe it is by example:

  • The Mirman School for the Gifted – 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.
  • EPGY and CTY's Distance Education Courses – Both programs use Stanford's adaptive software for K through pre-algebra. My daughter started in K and now is in the middle of 6th 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.

    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.

  • My Yale University Experience – 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.

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:

  1. Use Problems to Teach Material – 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.
  2. Challenge the Student – Arithmetic drills are rarely challenging for gifted students. This suggests that increasingly difficult problems are necessary to push these kids toward their potential.
  3. Ask Open-Ended or Broad Questions – 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.

I hope this post gives you a few ideas and helps you ask the right questions when thinking about your kid's education.

Monday, June 7, 2010

The Outcome of Radical Acceleration: The Stanley Kids from the 1980s

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. More importantly, fewer than 10% of participants reported negative impact from acceleration.

Here is the link to a web-based version of a June 1997 article on Stanley's kids from the 1980s. Enjoy.

Take a look a the picture at the top of the article. Amazing... Not a single girl in the group.

Sunday, June 6, 2010

We 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.

June 2009
  • Graduates from K at a regular school
  • 37% done with 2rd grade English language arts
  • 74% done with 3th grade math
  • Lots of children's television and little reading
  • Lots of "I am bored"
  • Expresses lots of interest in black holes and other space phenomena
  • Math is boring
May 2010
  • Graduates from 2nd grade from a top 5% school in California (SAS, distinguished school, API scores above 900) after skipping first grade
  • 65% done with 5th grade English language arts
  • Starts 6th grade math now to follow with pre-algebra in September.
  • Lots of reading and virtually no television other than Friday movies and science and history programs
  • Lots of arts and science experiments
  • Asks an astronomer how it is possible for black holes to eject matter through its poles when nothing can escape the event horizon
  • Lots of "I like interesting math and competitions like the Math Kangaroo"
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.

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.

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.

Friday, May 21, 2010

Teaching Social Ethics

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.


What did you think about the Ruby Bridges movie? Did you like it?


I don’t know. Why do you ask me so many questions?


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.


It made me mad. I don’t like the way people treated her. It wasn't fair.


What wasn’t fair? Why wasn’t it fair? What made you mad?


It is unfair that some people could not get a good education because it is very hard to get ahead without a good education.

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.


Papa, is my skin dark or light?


Your skin is perfect. You look like papa and mama. Your skin is beautiful. What are you worried about?


Well, would I have had problems in Ruby’s time?


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.

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.

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.

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.

Sunday, May 16, 2010

Emotions and Teaching: Mastery Goals vs. Performance Goals-Based Teaching

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 Science Education for Gifted Learners. The chapter titled The Emotional Lives of Fledgling Geniuses 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 The Emotional Lives of Fledgling Geniuses, I feel comfortable with our decision to home school because it best matches our daughter’s personality, emotions, and approach to learning.

There are many ways to categorize teaching styles. The Emotional Lives of Fledgling Geniuses argues that one may view teaching as split into two camps:

  • performance goals-based
  • mastery goals-based
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.

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.

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.

Saturday, May 1, 2010

On the Importance of Making Math Fun

“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.

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?

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.

What do you think of when you read the following topics?
  • Graphs (not to be confused with the X-and-Y variety)
  • Trees
  • Degree of a vertex
  • Isomorphic graphs
  • Circuits in graphs
  • Euler circuits
  • Planar graphs
  • Restating problems involving maps using graphs
  • The Four Color Problem
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.

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.

Saturday, April 10, 2010

A 21 Setup to Teach Probabilistic Decision Making

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.

Rationale for My Version of the Game of 21
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.

Playing this game should teach:

  1. Computing probabilities on a discrete space.
  2. Computing conditional probabilities when there the probability space changes.
  3. Determining if a bet is unfavorable, roughly fair, or favorable.
  4. Making decisions based on probabilities.

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.

Step 1
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.

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.

Step 2
Explain the rules of this version of 21:

  1. At the beginning of a hand, each player is given two cards.
  2. 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.
  3. The game continues until there are no cards left in the deck or there are not enough cards to start a new hand.
  4. There is a common pile of chocolate chips for betting. All players draw chips from the same pile.
  5. The winner of the game is the one who ends with the biggest pile of chocolate chips.
  6. A player should be dealt as many cards as desired until either he or she goes bust or decides to "stay."

Step 3
Explain how to play a hand.

  1. Start by getting two cards.
  2. Continue saying "hit me" (i.e. asking for another card) until you think you will go bust.
  3. Say "stay" when you are done with your hand.
  4. Choose to hit or stay based on the probability of going bust by picking one more card. 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).
  5. If your total exceeds 21, you are out of this hand.

Step 4
Explain the rules for betting and scoring in this game.

  1. Points are earned or lost in each hand.
  2. A bet is from 0 to 4 chips.
  3. The person who comes closest but not over 21 wins the hand and takes all the chips bet in the round.
  4. 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.

Step 5
Teach some betting guidelines.

  1. Bet 0 chips if there is a high probability of losing the hand.
  2. Bet 1 or 2 chips if there is low probability of winning the hand.
  3. Bet 2 chips if the probability of winning is high.
  4. Bet 4 chips if you are very sure to win.

A Few Additional Details
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.

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.

Have fun,


Wednesday, April 7, 2010

Vegas at Home

I 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.

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:
  1. Before dealing any cards, what is the chance of dealing a heart?
  2. Assuming that a heart is the only card that has been dealt, what is the probability of dealing another heart?
  3. Assuming that a heart is the only card that has been dealt, what is the probability of dealing a spade?
  4. Assuming that a heart has been dealt, what is a better bet for the next card? A heart or a diamond?
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:
  • a 10 of hearts and a 5 of clubs for your child
  • a 2 of diamonds and 5 of diamonds for you
  • a 10 of hearts and a card facing down
You can compute the probability of your child going bust as follows:
  • To bust, you must deal a card with a value of 7 or higher.
  • 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
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.

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.

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.

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.

Have fun,


Thursday, March 25, 2010

Teaching the Concept of Conditional Probability

I 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.

  1. understand fractions
  2. understand the concept of outcomes - The possibilities for the given problem. An outcome is something that may happen. It does not necessarily mean that it has happened.
  3. understand the concept of sets and events - 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.
  4. understand how to compute probabilities in finite spaces - Know how to compute probabilities by counting. We only need the very simplest computational skills at this point.
Step 1 - Drawing Probability Trees
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.

Example 1: Assume that a family has two children. What are all the possible outcomes of pairs of children?

Moving from the top to the bottom, we can now read out the four outcomes: {BB, BG, GB, GG}.

Example 2: Assume that a family has three children. What are all the possible sets of children?

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}.

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.

Example 3: Introduce an example of choosing randomly without replacement. 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.

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}.

Step 2 - Teaching the Difference Between Normal and Conditional Probability
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.

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.

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.

This can be computed by looking at the correct portion of the tree, which is circled in red below.

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.

Step 3 - Repeat Many Times
The trick to teach conditional probability to a kid is to give lots of concrete examples.

I hope this post helps you in some way.

Friday, March 12, 2010

When Homeschooling is the ONLY Option

Traditional 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.

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.

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.

Thursday, March 11, 2010

Teaching Multiplication of Fractions Visually

This 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.

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.

Here are the basic prerequisites for this approach:
  1. Understand what a fraction is
  2. Understand how to compute areas. Basically, that A = L x W
Here are a few examples of how to interpret multiplication of fraction geometrically.

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.

Here is another example. Multiply 2/3 and 3/5.

The length is 2/3. The width is 3/5. The product is 6/15. You kid should figure this out by counting.

Finally, let me show how to illustrate multiplication of fractions involving improper fractions.

Hopefully, it is clear by now that this geometrical interpretation works equally well for heterogeneous fractions.

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.

Saturday, February 27, 2010

Teaching Fractional Arithmetic Visually

I 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.

Step 1:
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.

Step 2: Review how to represent fractions as pictures and vice versa. For example,

Use a few more examples. Teach the kid how to draw squares divided into thirds, fifths, tenths, and a few other common denominators.

Step 3: 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.

Step 4: 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.

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:

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.

Step 5: 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.

The answer should be

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.

Let's try one more example.

The answer now is

Make sure your child understands that 3/5 is equal to 6/10. Likewise, make sure it is understood that 1/2 = 5/10.

Step 6: 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:

Do as many examples as necessary until the child is proficient at drawing improper fractions.

Step 7: 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.

As you probably realize, you can teach reduction to lowest terms visually as well.

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.

Monday, February 1, 2010

A Sample Problem Set for Second Grade

Most 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.

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.

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:
  1. Definitions
  2. Example and computational exercises
  3. Proof construction as a vehicle to learning math and deepen understanding
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.

Sample Problem Set for Second Grade Statistics and Set Theory

Prerequisites: 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.

Definition: The range of a set of numbers is the distance between the biggest and the smallest.

Exercise 1: Find the range of the set of even numbers between 0 and 100. Assume that 0 and 100 are in the set.

Exercise 2: Define a set of numbers using the following four properties:

  1. Every number in the set is bigger than or equal to 30 and smaller than or equal to 60.
  2. You can get every number in this set by counting by tens.
  3. You can get every number in this set by counting by fives.
  4. You can get every number in this set by counting by twenties.

Find the set of numbers defined by the above properties. Compute the range of the set.

Exercise 3: Which one of the following two sets has the bigger range?

  • Set 1: The set of even numbers between 10 and 20, including 10 and 20.
  • Set 2: The set of odd numbers between 10 and 22.

Exercise 4: What is the range of the set of numbers equal to their doubles?

Exercise 5: What is the set of numbers not equal to themselves:? What is the range of this set?

Definition: The mode 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 bi-modal if it has exactly two modes.

Exercise 6: Find the mode of {100, 99, 50, 3, 2, 1, 60 ,1 ,85}

Exercise 7: Find the mode of {Pablo, Paulina, Alex, Kolane, Pablo, Paulina, 2, 3, 10, 1}

Exercise 8: Find the mode of {2, 3, 4, 5, 6, 7, 8}

Definition: The median 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.

Exercise 9: Find the median of {1,2,3}.

Example 1: 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.

Exercise 10: Find the median of {100, 1, 80, 52, 48, 10, 12, 15}

Exercise 11: If you throw away the smallest and the largest numbers in a set, does the median change?

Exercise 12: Assume the following things:

  1. There are twenty students in the classroom.
  2. Every student likes either apples, pears, or both.
  3. 15 students like apples.
  4. 10 students like pears.

How many students like both apples and pears? Prove your answer using a Venn diagram.

Exercise 13: Assume the following three things:

  1. There are twenty students in the classroom.
  2. A student likes only one type of fruit.
  3. 15 students like apples.
  4. 10 students like pears.

How many students like neither apples nor pears? Prove your answer using a Venn diagram.

Definition: The mean 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.

Example 2: Given the set {2,3,4}, computer the mean.

Solutions: 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}.

Example 3: Compute the mean of {1,2,3,4,5,6,7}.

Solution: 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.

Example 4: 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?

Solutions: 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.

Example 5: 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?

Solutions: 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.

Exercise 14: Can the mean of a set be smaller than the smallest number in the set?

Hint: 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?

Exercise 15: Can the mean of a set be bigger than the largest number in the set?

Hint: 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?

Exercise 16: Show that the mean of any three numbers is both

  • larger than or equal to smallest element of the set
  • smaller than or equal to the largest element of the set

Hint: Use what you did in exercises 14 and 15.

Exercise 17: Give an example of a set for which the mean and the median are NOT equal.