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.