Issue #7

This issue explores the topic of math competition from the standpoint of general academic skills. Each article will break down the sub-skills of a state level question...
Issue #7
We kicked off At Home Schooling academic work at age 4. focusing on the core academic skills like reading the question, getting the wrong answer, being baffled and the like. In this issue, I'm going to show how these skills mature in the context of much more difficult problems. Spoiler alert! They don't mature at all. I took these skills from Poyla's seminal work on how to solve high school geometry proofs and Poyla took them from career mathematicians.
From the Editor
Issue #7 has three themes.

We're getting into competitive math. The first theme is roughly 'all competitive math all the time'. In this issue, I'm going to break down how we tackle these problems in the context of higher order problem solving techniques. Only one competitive math website mentions the skill set, and they only mention 4 of the skills. This issue covers foundational skills in anticipation of problem solving strategies.

In this year's Competitive Parent Magazine, I'm going to demonstrate foundational skills, applications to all subjects (not just math), and then delve into advanced solution strategies, not necessarily in any order. The second theme is problem solving strategies. The third theme is application to all subjects. Regardless of the topic, all themes show up in bits and pieces every time we do anything, math or otherwise.

A Time for Guessing

Sometimes the best way to do an problem is the easist. The easiest way is simply to guess and check, adjust and repeat until you have the right answer.

A Time for Guessing

There are three integer values of x that make the equation x3 + 6x2 + 11x + 6 = 0.

This is a difficult problem for any child in grade school. It's nearly impossible for a 10 year old. This class of problems shows up on all tests, including the SAT. Yet my kids have figured out a way to get these problems right and frustrate my ability to teach them math concepts. In this article, I'm going to describe the direct way of doing this problem (their way), the problem solving way (the right way), and the math way which I teach after they get the problem right quickly with no discernible effort (the penalty box way).

The right approach

If this problem shows up on the SAT, my lazy child will just try all 4 answer choices and circle the one that works. This is a real skill with a name and this is how it works.

  • I recognize the math concepts in the problem and go about solving the equation.
  • My son, meanwhile, notices that he doesn't recognize the math concepts and goes about plugging in random numbers until he gets one that works.
  • In the process of plugging in random numbers, sometimes the result is too high, sometimes too low, so he'll adjust accordingly. Sometimes he sees a pattern between 'too high' and 'too low' and adjusts with intent.
  • Just as I'm beginning my work, he announces the correct answer and makes me look stupid and slow.

The problem solving skill is called estimation, and the sub-skill is to recognize a problem that makes this technique the most expedient. Advanced concepts usually fit this category, and younger children have an advantage recognizing things they haven't learned versus an adult who has already learned the concept.

The second sub-skill is iterate and adjust. This takes some practice. A child will adjust the wrong way and make things worse and start to learn what adjusting is all about. In this case, starting with -1, 0, and 1 should point to an 'ah ha' moment because -1 works but the other's don't. A survey of the equation again (aka reread the question) shows why this is the case.

An alternate approach

The acceptable approach is to spend some time understanding the equation under the heading 'read the question'. To understand how this equation works, one plugs in a few numbers to build a mental graph. Since this is a timed test, there are 3 numbers that need to be plugged in; any other number is usually wrong. These numbers are -1, 0, and 1. The correct numbers are the easiest to work with - thus small or trivial ones.

I'm OK with 1,2 and 3, provide it's followed by -1, 0, and 1, but 4 is definitely the wrong approach and results in a lecture. The little one always says 4 to buy some thinking time while I lecture him. I always fall for it.

In every problem we've seen so far, the numbers we plug in are very close to the answers. I think the test makers intentionally put a time waster like this on the test to give the advantage to kids who stop, think, and go through the Big Five problem solving techniques. 'Estimation' and 'Try An Easier Problem' are somewhat related on these problems.

The penalty box

In the event that a child guesses correctly within seconds and ruins my plans for a good 20 minute math problem, I'll follow up the problem with a lecture in question form. They should know all of the answers, because they've heard them before, but they don't regurgitate terminology like '3rd order polynomial with complex roots' quickly no matter how many times they've heard it. Nonetheless, sitting through a lecture - in question form - is good practice for getting thrown into a math class beyond their comfort level.

I'll spare you the questions. Here are the answers.

  • This is a 3rd degree polynomial and therefore has 3 roots according to the fundamental theorem of algebra.
  • I'll draw a few examples. If it only crosses the x axis 2 times, this means that one root is complex
  • New York gives a test in high school with a famous name I forget that expects you to decompose the equation into it's roots, but we are not going to do that because it's tedious and boring.
  • In a 3rd degree polynomial, like a 2nd degree polynomial, once you get past the outside roots, y continues to grow in magnitude.
  • If I'm in a bad mood, we'll delve into the slope of the curve, a 2nd degree polynomial on it's side is not a function, or anything else I can think of until we get to the 20 minute mark.

There's a big difference between a kid sitting in class hearing '3rd degree polynomial with complex roots' for the first time and one who's heard it all before and finally gets to learn what it really means.

The answer

The answer to this question is the roots -1, -2, and -3. The purpose of this question is to test the child's ability to apply basic arithmetic to a complicated, unfamiliar topic and move on quickly to the next question.

I've outlined 3 approaches above that you can use as you're child's academic coach to impart learning skills. If you take the time to not teach math, your child will pick up a variety of really create cognitive and learning skills. The tree examples above involve cognitive sub-skills, resilience in the face of complexity, and becoming inured to tops that are age-inappropriate.

My todo list includes preparation for a future math competition. Based on past experience, we'll never get that far down on the priority list.

Final note - I 'borrowed' the picture above from I like this website. It's thought provoking.

Back to algebra

Until we get to algebra, returning to algebra is a consistent theme in our daily practice until we take the step from solving algebra problems to using algebra to solve them.

Back to algebra

Teaching algebra to a child who is not ready for algebra is like teaching arithmetic to a 2nd grader without letting the child memorize math facts. In the latter situation, the child builds number sense. In the former, the child is building algebra sense.

We started doing algebra problems about a year ago, and now can silently look at a problem that requires algebra and derive the equation not using algebra. All of the basic problem solving skills are required to get to this point, especially the ability to work diligently for an extended period in the face of frustration. The frustration in this case is me watching my child not use algebra. It's worth it.

Here is the problem for this article: Jack London wrote Moon Face in 68 days. If he would have written an extra page a day, it would have taken only 51 days. How many pages are in the book?

False start

A child steeped in problem solving might say 'n times 17 = 51'. Is this the correct answer? I don't know. I have to work it out, because I didn't have an academic coach with the forsight to not teach me algebra when algebra is so obviously needed.

I presume that the child split this problem into 2 sub problems on the way to 17n = 51 but can't untwist the jumble of thinking to explain how he did it.

Where genius is born

If you withhold prepackaged math concepts from your child like carry the one in double digit addition, long division, and in this case algebra, you're opening the door to advanced thinking. Once you provide pre-packaged solution methods, the child's brain will shut down while they follow the steps in order and reduce the complexity of advanced math to applying arithmetic steps.

The time is coming in their academic career when the child will be required to think. The whole purpose of an academic career is to get the child to the point of thinking. A child who applies formulas and mechanisms to solve problems year-after-year will get to the point where thinking is required and fall apart academically. It usually happens in pre-algebra, if you are lucky and there's time to rectify the problem, but may happen in high school where the only recourse left is a tutor or the school's psychologist.

Coaching genius

At some point the child will a) have the correct answer, b) have an incorrect answer, or c) be totally stuck.

What's the difference between coaching a genius and coaching a complete math moron? Nothing other than how many of these problems you've done. Every math moron is a genius waiting to happen. If you think you have a math genius, you're using problems that are too easy.

Working out the answer

When it's time to check the answer, I'll work out the answer verbally and with step-by-step algebraic equations. I ask a question for each step like 'How many pages is in the book if Jack wrote for 68 days?' 68p = n. Write the equation to describe writing an extra page a day for 51 days (not me, I'll wait 30 minutes for this) which is 51(p+1) = n. Now what do we do?

We do about 3 of these problems a week. At some point last year, I explained that n =n, and the fundamental theorem of algebra means that you can't solve 1 equation with 2 variables (does it?) so 68p = 51(p+1).

If it takes 10 minutes to answer this problem, and I don't see a step-by-step solution, it will take 20 minutes to do it properly with algebra. I'm inching toward the point where I can ask for the child to write down the step-by-step equations without my assistance. We're not there yet. I think my older one is there, because he's studying algebra in math class, but I'm out of the picture so I can't say for sure.

Getting to algebra

How long will it take for my little student to stop working out problems mentally and start using algebra? When these problems are no longer challenging, he'll start using algebra. It's as if his brain is trying to eek out perfect comprehension before substituting mental effort with abstract equations.

The bonus

A child with number sense knows he's wrong when he gets 15 x .3 = 45. A child with algebra sense needs no formulas. He will eventually need algebraic equations, but topics like profit and probability are just known without help. He'll have a dozen or so internal algorithms that he created himself. Occasionally I see one of these in action and I'm in awe.

It is impossible to use this pedagogy in a classroom setting. If I were to use this method in the average American 6th grade, parents would watch their children get most problems wrong between September and May, and these kids would probably fail a pre-algebra test on May 31. Then by June 15, they would ace anything on the SAT. I would be fired in October, so it's a mute point.

Solving all questions

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...

Solving all questions

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?
Step 1 - Read the question

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.

Step 2 - Understand the question

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.

Step 3 - Devise the strategy

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.

Step 4 - Get off track and adjust

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.

Check the work

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.

The solution

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.'

Not Really Math

The first question uses only two skills, but demonstrates them in a powerful way...

Not Really Math

How many of the ways that 27 cents can be made using quarters, dimes, nickels and pennies use and odd number of coins?

I rate this question about a 5 out of 10 because it has a few things going on at once and it's poorly worded so that the child has to read it a few times and think. This is warm up questions for a nervous middle school team, but a great question for 5th grader. On a 40 minute, 8 question test, this question should take no more than a few minutes. We spent about 15 minutes on it.

The skill

There are two ways to do this problem. You can spend 15 minutes on the question and 1 minute getting the answer right, or one minute on the question and 15 minutes getting it wrong. We don't use solutions. If the child does the problem correctly, the answer will be obvious to parent and child or easily provable. If it's not, I consider the question wrong. The 'right way' is the payoff, the solutions never show more than a single number, and I'm interested in the learning process and not the final number.

When I'm not helping as the missing team member, I usually look at the answer and declare it wrong. My trainee does it again, usually getting a better answer or learns to verbalize his defense.

The right way

There is only one correct way to do this problem. That would be the easiest cheatiest way or organize the work so that the least amount of effort is expended finding the qualifying combination of coins. Solving this problem is like coming up with 'alphabetizing' the qualifying combinations.

The actual result

We started brainstorming combinations. This technique is a way to understand the problem and works in other cases. In this case, it led to an unverifiable mess of combinations.

The only correct answer - the answer that a 5th grader is unlikely to come up with on the first try, is to start with quarters (one), them move to dimes (2 and then 1, in that order), and on to nickles. Any other approach is too much work for me to grade.

The problem seems deceptively easy if you've never seen one of these before (which is the whole point). I asked 'How do you know you have the correct answer' and with a bit of staring, we found some missing combinations. The faster this problem goes, the worse the result.