I'm half finished writing this issue and I realize that I want to write an article on every question from every released test. Instead...
Every time we practice a problem, I want to write an article on the process, because it's worth an article. Instead of a 30 article issue, I'm going to summarize the process in this article. In our case, we're tackling the team level questions from state level competition. Me and my 10 year old buddy. We both pitch in - it's a team sport, after all - and nail each question. Why schools don't teach this framework is the latest in a long series of shortcomings in math education.
Practicing rigorous math at home is a team effort between parent and child. In competition, the team is comprised of 5 students, and they can divvy up ownership of the 5 steps below. All 5 members contribute to each step, but one child is assigned the role of gate keeper. With the parent-child team, I like to let the child do all the work and be the gate-keeper for all 5 steps.
Here is my example question for this article. I'm paraphrasing the question. On an actual test, the question would be worded in a much more confusing, vague style with more sentences and clauses.
The math coach passes out 10 felt tip markers to each team member. Bill and Joe will always receive the same number of felt tip markers. How many different ways can the coach distribute markers to the team?I like to jump right into solving the problem and get it wrong a few times because I missed subtleties, implications, and missing parts of the directions. Don't we all. The first and most important task is to get the question right. That's more than half the battle and this step is the hard one.
There is one complexity in the problem above and one missing element that is critical to the answer. Step 1 continues through step 2. Reading this question should take some time.
In the previous article, understanding the question means trying a few values of x to see how the equation behaves. In this problem, understanding the question means brainstorming different distributions of markers or laying out at a high level groups of distributions. If a diagram were applicable (think geometry), the diagram would be mandatory and might solve the problem with no additional work.
During this process, my teammate said 'OK, Bill and Joe each get zero and the other 10 pens go to the other three kids.' After he said this, I realized that I read the problem wrong. Where does it say that you can't give zero pens to one of the kids? Is zero a valid distribution? Yes it is. Kids who read questions quickly might miss or forget that Bill and Joe get the same number of pens. Kids who haven't screwed up numerous problems in the past might miss the trivial solution, and the trivial solution is about 40% of the total count of permutations. Sometimes the trivial solution is referred to as the corner solution. (What's the maximum value the graph y = 4x + 3 can take if y is less than or equal to 30? Draw the graph and the constraint and you'll see the corner.)
We have a rule in this house. "Always start with the easiest possible answer, and the easiest possible answer is always zero." This rule doesn't apply to this problem. The rule that applies to this problem is 'Don't ignore trivial solutions' and trivial solutions often involve zero.
After half our team re-read the question again, we determined that we both understood the question.
This step will take many issues to elaborate because these strategies are fundamental to success in all types of endeavors. Success in math does not mean that the child can win math competitions. It means that the child does well in all subjects that depend on strategy, including doctoral level history and literature, not to mention science. Math is the raw application of logic and problem solving that will be used elsewhere.
In this case, the question is the last one on the state competition test. The kids probably have 4 minutes or less to solve it. The strategy is to group permutations in buckets, solve the buckets, and aggregate the permutations. Divide and conquer. We keyed off of Bill and Joe, with buckets of 0-0, 1-1, ... , 5-5. Order least-to-greatest or greatest-to-least is imperative. This was my contribution because I felt bad about missing the trivial solution and wanted to pull my weight.How did we get to this strategy? Our overriding objective in math is to find the fastest easiest cheatiest way to the answer. Lots of counting is tedious and error prone. Grouping and order is required.
Success doesn't result from perfection. Success results from fixing mistakes. Mistakes can only be fixed if the kid is not flustered by their own incompetence and dispassionately continues the struggle.
This problem can only be solved by an ordered list. We started with 0-0-10-0-0, 0-0-9-1-0, 0-0-9-0-1, 0-0-8-2-0, etc. At one point, we skipped 0-0-6-4-0 and went right to 0-0-6-2-2. It wasn't intentional, but we found groupings of 6 and groupings of 3 - a huge time saver - and in the excitement of overloaded working memory lost our disciplined order. In my role as strategy enforcer I raised the flag. Getting out of numerical order almost guarantees permutations will be skipped.
Most problems have 2 or 3 viable strategies, one of which will yield the answer and 2 which require a do-over. With different categories of problems, this step might be called 'get it wrong and try again' or 'get nowhere and go with plan B.' A child who works with problems that are time consuming, frustrating and error prone, instead of the spoon feeding from school text books, is a big leap closer to life success.
On a math exam or a test like the SAT, this step can be worth 15% of the final score. It usually involves recalculating or calculating a different way. In our case, we were delaying a pizza dinner, and stopped as soon as we had our first version of the answer. The team member in charge of checking already had his coat and car keys. My older son, a veteran of math competition and the SAT, mentioned that no one ever finishes, let alone checks answers. Under these conditions, our easy-cheaty way of working is superior because easy-cheaty means less mistakes.
I don't remember our solution. The trivial distribution where Bill and Joe each get zero is 72 distributions and the total is around 170. This article provides a partial list of all the things the team learned doing the problem. That is why solutions are so pointless to education. If we had 172 and the actual answer was 184, would I diminish the achievements in learning by ending 20 minutes of effort with 'You got it wrong.'
To be more precise, the diameter of my child's brain grew 1/2 inch during this problem. He improved on at least 6 different sub skills and I didn't have to yell at him like I always do on certain elements of technique. Knowing the solution isn't going to add any additional benefits. If more parents took this approach, Stanford would have to put trailers in the parking lot to handle the overflow in math majors. Instead, they have 2 or 3 freshman sign up for the math department every year.
During 2nd and 3rd grade, while we were learning to make mistakes and check the work, I repeated "Wrong, try again" over and over again until neither of us cared. In this case, I said 'Good job. I think we either got the right answer or got close. Get in the car so we can go to dinner.'
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