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.

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Pablo

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

Paulina

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

Pablo

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.

Paulina

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

Pablo

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

Paulina

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

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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.

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Paulina

Papa, is my skin dark or light?

Pablo

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

Paulina

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

Pablo

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.

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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.

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.

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.

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,

Pablo

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,

Pablo

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.

Prerequisites:

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.

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.

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.

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.

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.

Dealing with the Impact of Societal Biases on Mathematically/Scientifically Gifted Girls

I 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?

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:

Girls are smarter than boys because boys are coo-coo and just like to run around and cannot pay attention in school.

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?

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.

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.

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.

On the Necessity of Ability-based Grouping in Mixed-Ability Classrooms

I 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.

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.

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.

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.

Don't Think Your School Is Good Enough, Get Involved

Many 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.

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.

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.

The Power of Math Circles

I 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.

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:

II + III = VI.

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.

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.

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.