Using Adaptive Programs to Teach Math

It 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.

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:

• K through 2nd grade
• 3rd / 4th grade
• 5th / 6th grade

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.

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.

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:

• Number Sense: Integers
• Number Sense: Decimals and Fractions
• Geometry
• Logic and Reasoning
• Measurement
• Data/Statistics/Probability

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.

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.

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 here 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.

I would advice parents of gifted children to look into EPGY (click here 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.

As always, I hope you find this useful.

Having 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.

"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.

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.

In case somebody is looking for a book appropriate for young kids, we bought

What's the Big Secret by Laurie Krasny Brown and Marc Brown.

Want 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.

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.

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.

Here is a bullet point summary of what I learned today:

• radical acceleration is possible in the LAUSD
• homeroom teachers and school principals make the final decision to accelerate
• it is possible to do single-subject acceleration simultaneously with grade skipping
• this seems to work best when the teachers and principal are well-informed
• offer to help when the school accommodates your child

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.

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:

• Talk to parents of current students to find out how the school has handled acceleration and grade skipping in previous years
  
• Learn the rules and regulations governing grade skipping and acceleration in your district
• 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.
• Enroll the help of gifted education advocates. You may want to contact the Davidson Institute. The Davidson Institute's Davidson Young Scholars offers guidance, free consulting services, and may help you communicate with local school officials.
• 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.
• Become a relentless advocate for your child's rights.
  

There is no magic bullet here. Prepare yourself. Look out for opportunities and seize them when they present themselves.


Hope this helps,

Pablo

Skipping 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.

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.

Here is a summary of the pertinent age limitations for grade skipping within LAUSD:

• Age of Admission to K - 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.
• Age of Admission into 1st Grade - 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.
• Minimum Age to Enter Middle School - 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.
• Minimum 6th Grade Attendance Before Middle School - 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.
• Exceptions to Rules for Entering Middle School - 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.
• Minimum Age for Senior High School - The minimum age for students entering senior high school who have been accelerated because of superior mental ability is 13-9
  

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.

The following list summarizes the required ages by December 2nd of each grade level:

Elementary
1st grade, 6
2nd grade, 7
3rd grade, 8
4th grade, 9
5th grade, 10
6th grade, 11

Middle School
7th grade, 12
8th grade, 13
9th grade, 14

High School
10th grade, 15
11th grade, 16
12th grade, 17

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.

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.

-----------START OF CITATION ---------------------
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 EC 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:

• The child is at least five years of age.
• The child has attended a public school kindergarten for a long enough time to enable school personnel to evaluate the child's ability.
• The child is in the upper 5 percent of the child's age group in terms of general mental ability.
• The physical development and social maturity of the child are consistent with the child's advanced mental ability.
• The parent or guardian has filed a written statement with the district that approves placement in first grade.

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 (EC Section 48011). This action prevents a subsequent audit exception for first grade placement of an age-ineligible student.

Source: http://www.cde.ca.gov/ci/gs/em/kinderinfo.asp

-----------END OF CITATION-------------------

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.

-----------START OF CITATION----------------------------
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." EC48070 Promotion and Retention EC48070 is fairly precise about retention policies, but it is glaringly vague on acceleration.

Source: http://www.accelerationinstitute.org/Resources/Policy/By_State/Show_Policy.aspx?StateID=6

-----------END OF CITATION--------------------------

The age limits for grade acceleration can be found at
http://www.lausd.net/lausd/board/secretary/BoardRules/BoardRules3-08.pdf

I hope this information helps you advocate for your child. Inform yourself. It is the best tool at your disposal.

Using Graph Theory With Young Kids

I recently spoke about the challenges of teaching math with a good friend from college who is now a computer science professor at Yale University. I have fond memories of our freshman year sitting at a pizza joint doing math on napkins. Our lives took completely different directions, although we both got advanced, math-related degrees. He stayed in academia, where he has been very successful, and I went to work in Wall Street. We kept in touch throughout the years, and we now find ourselves with young daughters at similar stages of development. Our conversation focused on how to to spark interest in math and develop abstraction using topics accessible, yet absent from the traditional, elementary school curriculum. He said he had been trying a bit of graph theory with his daughter. This sparked my interest. While I never liked discrete mathematics, I now realize that it is probably the most accessible area in the discipline. Graphs are simple to explain, and, most importantly, many opened problems in the subject could be explained to a smart six or seven year old.

Let me give you an example. Show your kid the following star. Ask your kid to figure out if it is possible to draw the star without lifting the pencil or going over the same line twice.


Obviously, it is possible to draw a five-point start in the manner described above. That's not the point. Look at the following star-shaped graph.


Have you ever heard about the traveling salesman problem? The problem aims to find the shortest tour of a group of cities without repetition. Clearly, this is equivalent to draw the star without lifting the pencil or going over the same line twice. In other words, you are introducing your kid to a major area of graph theoretic research by asking how to solve a cute, little drawing problem.

Here are two additional graphs to try. Determine if it is possible to draw the following two figures without lifting your pencil or going over the same path twice.


If a solution is found, ask if there is a shorter way to draw the figure. Depending on the maturity of the child, you may draw little circles at the corners of the diagram and then explain that this is really a graph that can be used to represent lots of real-world problems such as how to travel to a bunch of cities along given roads in the least amount of time. Whether or not you choose to tell your kid that he is working with graphs and what they mean is irrelevant. The key here is to get the kid to think about optimal ways to draw these shapes. At the end of the day, you are teaching your child to think about optimal algorithms. Who cares if you achieve this by drawing silly figures or proper graphs?