What’s the Point?

The other day, I refused my kids’ very reasonable request to skip a problem in math class. They argued, from the only point of view available to them, that of a fourth grader and a seventh grader, that they had already mastered the technique necessary to solve the problem. They said they’d simply need to use the same process they had used for the previous problem, only with different numbers.

I hesitated.

I knew they could likely solve the problem correctly and without any of my help, even though it was slightly different beyond the use of different numbers. We had also skipped problems before, or solved them only orally, discussing the process without writing it all down, when I felt we could move on without it causing issues later. But this time, I thought there were important things for them to learn yet. Things that I wanted them to keep practicing, repeating until they become automatic habits of their minds. I knew they handled calculations both on paper and in their heads with enough confidence at this point that skipping one problem wouldn’t be a big deal, but in my gut, I felt the lessons to be learned here were not about basic operations and mathematical processes.

Luckily for me, they accepted my eventual response that no, they should not skip the problem, because I would like them to spend the ten minutes it will take to complete the task. They probably realized that asking for and listening to a more detailed reasoning from me would likely take longer than working out the solution to the problem. So they set to work while I tried to figure out a less-than-ten-minute summary of everything I hoped they’d learn — not from this single problem, but the repeated practice of many similar problems.

In the problem to be skipped or not skipped, we had to divide 190 and 1/8 to four parts. Not four equal parts. According to the task, adding five and half to the first number had to equal the second number minus five and a half, the third number multiplied by five and a half, and the fourth one divided by five and a half.

Spoiler alert — in the rest of this story, I will work through the problem. If you enjoy math, you may want to grab a piece of paper and a pen, and solve the problem yourself before reading on. If you are of the opinion that math sucks, or that you just didn’t get the “math gene,” bear with me. I won’t tell you the answers right now, but the four numbers that add up to our lovely solution of 190 and 1/8 will be almost irrelevant here anyway. As it is the case so often, the journey toward that solution will be more important than the destination.

No. 1. Strong foundations are essential

Let’s call our unknown numbers a, b, c, and d.

If we know that

a + 5.5 = b – 5.5 = c × 5.5 = d ÷ 5.5; we also know the following:

b = a + 11 (from a + 5.5 = b – 5.5; adding 5.5 to both sides of the equation)

c = (a + 5.5) ÷ 5.5 (using the same logic as above); and

d = (a + 5.5) × 5.5 (again, using the same process).

If we add the number a, plus b, c, and d as expressed above, we end up with an addition that has no other unknowns but ‘a,’ equalling 190 and 1/8.

To arrive at this point, we have to understand some basic math. For example that subtracting x from a number then adding x to the result leaves us with our original number. We should be familiar with the workings of parentheses in math, multiplying and dividing by decimals (or mixed numerals), etc.

This is a classic example of mastering some skills that enable us to work through more complex processes — nothing too amazing here, though the concept is not always obvious, either.

I was talking to my son’s mentor one day when I was picking up my budding woodcarving artist from his lesson. Don said, “when I teach a workshop, I always take two carvings with me. They are both of a Black-capped Chickadee, same pattern, same tools used.” He said the difference between his first bird and the last is immediately obvious to anyone looking at the carvings. “But people don’t get that. They come to me wanting to learn, but then they expect their first work to look like my competition birds.” Why do people think that they can do anything complex without mastering the basics?

I’m not saying that math is the only way to learn this lesson, or any of the lessons listed here. Anyone who stood with a baseball bat waiting for (and missing) the ball, or tried to shoot baskets on the court, or painted, drew a picture, … or did anything at all that required practice and mastery of a sub-skill to achieve the original goal should know what I’m talking about. Without the foundational skills of holding the baseball bat or the basketball, hitting and throwing, having good control of the brush or the pencil, our attempt on the sports field or creating a masterpiece of art will be less than successful.

No. 2. Verbal comprehension is just as important

The problem presented here happens to be task no. 1,270 in a workbook for kids in middle school, covering all material but including no explanations (Matematika feladatok — Összefoglaló feladatgyűjtemény 10-14 éveseknek). It is 382 pages of nothing but problems, presented in a system so that each task builds on the previous ones. Here are a few more examples, in translation, of the 223 tasks from chapter 11, on problems that can be solved with first-degree equations or inequalities.

Anita and Eva went to the farmers’ market. Anita bought a watermelon that weighs 3 kilograms, and pears for 22 HUF (Hungarian Forints). With this purchase, she paid 7 HUF more than Eva, who purchased a 4-kilogram watermelon and sweet peppers for 5 HUF. How much did one kilogram of watermelon cost at the market and how much money did Anita have if she spent 25% of her money?
[For those of you planning a trip to Hungary, the copy of the book we are using was published in 2013, not that long ago, but it is the 17th unchanged edition, and produce prices have gone up considerably since the first edition.]

A tub can hold 950 liters of water. A faucet allows 500 liters of water to pour into the tub every hour, a different faucet allows 300 liters per hour. One of the two drains allows 200 liters of water to escape from the tub every hour, the second 500 liters in the same amount of time. How fast would the tub fill if we opened both faucets but failed to close either drain for the first half an hour, then realized our mistake but closed only the second drain?

Antal and Bela are buying tickets for the movies. Bela went to the end of the shorter line. Antal stood back, considered the two lines and realized that there were twice the number of people in one line compared to the other. After four people left from the first line and three joined the second, there were more people in the first one than in the second. How many people could have been waiting in line at each ticket counter when the boys arrived?

These problems relate to real-world issues, and it is impossible to solve any of them without understanding the meaning of words in the language they are written and without having an accurate grasp of grammar. Not the prescriptive system of rules presented to us in grammar class, but an intuitive understanding of the discrete combinatorial system that human grammar is. “Intuitive” here means we can skip conscious explanations, but it does not mean it is available to us without practice. Lots of practice.

There is some evidence that some languages make it easier to establish mathematical thinking, such as Chinese with its one-syllable number names. However, by this level of math problems, the difference cannot be blamed on our native language — it is all about what we had been doing with that language from preschool to middle school. Video games, with their sophisticated vocabulary of “Bang!” “Oh, no!” and “Congratulations! Level 5.” won’t do it. Solving math problems like this requires no calculations beyond simple addition, subtraction, multiplication, and division, but verbal understanding is a must. With a background in cognitive psychology and linguistics, I know only one sure way to improve language comprehension — kids must engage in lots of conversations with people of different ages, experiences, and backgrounds.

Of course, the issue here is more nuanced than simply setting the kids down to listen to grandpa, with his endless stories about “back in the day.” There are plenty of word problems in math with cultural references that may be unfamiliar to the kids in the classroom. When I was going through these problems as a child, I had a difficult time imagining a 1,000 liter tub. It is likely that today’s kids would buy (or rather, watch their parents buy) produce or movie tickets on their phone, so it might be the other two problems that seem far from reality to them.

As I was writing this story, I learned that the Florida Department of Education rejected 71% of submitted materials for K-5 and 20% of materials for grades 6-8 for reasons such as “incorporating prohibited topics or unsolicited strategies, including critical race theory.” Though at this time, I have not seen any specific examples of what was wrong with the books exactly, the issue is obviously more complex (and political) than my writing here indicates.

Whatever the “story” is that we use to describe the math problem, however, the kids should be able to understand the text well enough to turn it into an equation or inequality, especially if they have a teacher who can help them explain how life works or used to work — or maybe one that can come up with a creative framework to present the same math problem in a culturally more relevant way.

Some people worry about giving the kids “crutches”, helping them out in the beginning with examples they can touch, see, imagine. Others argue against word problems in general, saying we should just do away with them altogether, avoiding the (political) issue of “whose reality it is” completely. However, word problems, such as the examples given here, present an opportunity for exercising the power of moving from the concrete to the abstract in the first step of their solution — an important skill if we want to be able to generalize from our experiences, whether we gain them in school, on the baseball field, or anywhere else in life. With my kids, I have found that once specific examples were mastered, it was much easier to move towards the abstract than trying to handle everything in the abstract from the very first encounters with mathematics. It seems to me that it would be silly to leave out all word problems just because we want to avoid cultural references — the loss in (mathematical) thinking and understanding is much greater than the risk of “indoctrination.”

Of course, if we return to the problem we started working out here, we only have to worry about language comprehension, not cultural issues — the problem is described wholly within the world of numbers. To solve it, kids should have reached a level of abstraction in their mathematical thinking that makes real-world references unnecessary and even confusing. If they had not reached this level, I think the remedy lies in building strong foundations not only in math, but also language comprehension.

No. 3. Among the many paths to a solution, some are better than others

Let’s get back to solving our problem. So far, we have:

a + a + 11 + (a + 5.5) ÷ 5.5 + (a + 5.5) × 5.5 = 190 and 1/8

At this point, we can be ready to start on the busywork of solving the equation, and as long as we don’t mess up our calculations, we can arrive at the solution for ‘a.’ However, with 1,269 problems under their belt, the kids noticed that this equation is not especially “pretty.”

So we stopped.

After looking at me for clues but getting nowhere, they set to work at figuring out whether they could make their lives a bit easier. Of course, even as we were reading the problem in the very beginning, we didn’t expect “pretty” — when dealing with division by decimals and an addition that equals 190 and 1/8 (still a relatively innocent number, if we look back at it from later problems in the book), one is prepared for certain “ugliness.” But still, it’s worth looking…

Once they got rid of the parentheses, they realized that the only part they didn’t like was dividing the number ‘a’ with 5.5. We could simply multiply both sides of our equation by 5.5 and get rid of this division, but that would result in higher numbers, more decimals…

The division came from the way we expressed our number ‘c.’ What if we expressed all other numbers using ‘c’ instead? When they tried, this is what they got:

5.5c – 5.5 + 5.5c + 5.5 + c + 5.5 × 5.5c = 190 and 1/8.

Not bad. We can figure out the square of 5.5, then immediately simplify our equation to

42.25c = 190 and 1/8.

While there isn’t always a way to reach a solution in a math problem easier than by using the immediately obvious route, and calculations are often inevitable, it is always worth checking. When you have 1,270 chances to pause and anticipate issues before you blindly start on a memorized technique, “thinking” becomes second nature. Now that’s a lesson math problems can teach us much quicker, more directly, and with less pain than many other learning opportunities in life.

No. 4. Feedback is invaluable

Once we reach the 42.25c = 190 and 1/8 point in our process, we are only a simple division away from finding ‘c.’ While it would be worthwhile to discuss the benefits of completing this calculation on paper instead of using the calculator in middle school (and that’s what my kids do), that’s not what I would like to talk about here.

Dividing 190 and 1/8 by 42.25 and knowing that c = 4.5 allows us to figure out all the other numbers, if we just take a look at where we started. So we arrive at a = 19.25, b = 30.25, and d = 136.125. With all four numbers available to us, there is one more step to take — adding a, b, c, and d together to see whether we really end up with 190 and 1/8.

While not all math problems come with a built-in opportunity such as this one for immediate feedback, many at this level do, which is another reason why everyone should become proficient enough in math to work at this level with confidence. When we complete the addition, we don’t need a teacher to tell us whether we did okay. There is no need for anyone to sugarcoat or “sandwich” anything. No grades, no meaningless “good job” from someone who doesn’t know what it takes to complete the work. Whether we did everything right or not is there for us to see, plain and clear.

We might not like what we see. We might not understand where we made our mistake. But we know for sure what our work is worth, we know where we are at. And where we are is nothing but our starting point to become better — either by finding our mistake and learning from it, or by moving on to the next, more difficult task ahead. Was there a problem with our way of thinking, or did we mess up a calculation? A teacher, another adult, or a peer can help, and we can correct the issue and move on without any great loss. That is, unless we were bragging to our sibling about our math prowess previously, and now the evidence of our fallibility is obvious to everyone… Most importantly, to ourselves.

I think feedback is crucial to improvement, and getting used to looking for and receiving feedback is a great benefit of these math problems. If you always get everything correct, you cannot gain true confidence in yourself, because you won’t know what it is exactly you did right, and conversely, what you would need to pay close attention to in order not to make mistakes. If we complete enough (progressively more difficult) math problems, we can get used to failing. If we fail enough, we have a chance to learn what comes after failing… getting back up, trying again, eventually succeeding. Along the way, we can learn not to be scared of failures and to appreciate honest feedback. Instead of avoiding it, we will look for feedback in every area of life, whether it comes in the form of “correct” and “incorrect,” or the more sophisticated constructive criticism regarding something more complex.

More lessons

While working on this problem, my son was reminded of another important thing — in math, every step you take must follow the rules exactly. He tried to calculate 190 and 1/8 divided by 42.25 by dividing 190 and 1/8 by 42, then dividing the result by 1/4. Since he knew that dividing by 1/4 is the same as multiplying by 4, the error in his thinking became obvious to him quickly. He realized he must stick to the correct application of processes in order to arrive at a valid solution. Sometimes this lesson is more difficult to master, that’s another reason I’m glad our book gives us many opportunities for failure while working on the different steps of over three thousand tasks.

This lesson goes hand in hand with the requirement to pay attention to every little detail. What happens, for example, if our problem asks which four whole numbers satisfy the five criteria specified about a, b, c, d, and their sum? We could have simply concluded, without any calculations at all, that there are no such four numbers — there isn’t always a solution. (We cannot possibly add four whole numbers and end up with 190 and 1/8 as the sum.)

Of course, the list could go on about things to learn, habits of mind to develop, but I think my ten minutes is up, so I’ll just stop here. I’m not a teacher of math by profession, but I hope to connect with math teachers in the future to see what they think — what are the values they think kids can learn from math?

About this story

As a homeschooling parent, the state I live in allows me wonderful flexibility in what and how I teach my kids. While there are great homeschool curricula out there, I hardly ever use them, and so far, we have avoided textbooks as much as possible. This sometimes feels like an overwhelming responsibility, other times it is a real blessing — either way, it makes me continuously aware of why we do the things we do when we sit down (or stand up and go places) to learn.

When I was doing research for this story, I realized that there are many ways to teach math. The lessons to be learned could include memorization, obedience to authority, and following instructions blindly, without questioning anything along the way — much different from what I’m trying to do, and for this reason, I’m excited to hear from others about the benefits of math. If you have an opinion, you can reach me through Flywheel Parenting on Facebook.