This is the first follow-up post to a previous one aimed at parents and other non-math teachers, The Real Power of Studying Mathematics, about the significance of the Standards for Mathematical practice.
I was talking to someone once about this practice (MP 1, "Make sense of problems and persevere in solving them") and she asked me, "Isn't that just, like, doing math?"
Well, yes and no.
Yes, in that if you can't make sense of a problem, you're probably not going to get very far toward solving it. And if you can't persevere, you probably won't get to a solution even if you can make sense of the problem. In some ways, MP 1 is a kind of "Gateway Practice"; if you struggle to make sense and persevere, it's difficult to get much of any math done at all.
No, in that MP 1 is more of a starting point, a 'necessary' condition for doing powerful mathematics, but it is far from sufficient. You made sense of a problem enough to get started? Great! You're hanging in there and persevering? Also great! But there is a lot more to be said regarding how students are able to make sense of problems, and a lot to be said about how effective and efficient our perseverance is, and more yet to be said about what we do once we have solved a problem. All of that is what we get from the other seven practices.
Let us break MP 1 down into its two constituent parts:
- 1) Make sense of problems.
2) Persevere in solving problems.
Today we'll focus on what it means to make sense of problems; in the next post we'll come back to what it means to persevere in solving them.
So...what does it mean to make sense of a problem?
Ask any math teacher; if s/he had a nickel for every time a kid read a problem and threw up their hand to proclaim "I don't get it!," s/he'd have a whole lot of nickels.
To make sense of a problem means to read or look at a problem, comprehend the information that's being communicated, and be able to ask yourself, "What mathematics is involved in solving this problem? How will I know when I have an answer, and what will the answer mean?"
Essentially, making sense of a math problem means finding a meaningful way into working on it. A student might make complete sense of a problem right away and know exactly how they are going to approach it, or they may only understand a small part of it at first and need to "tinker" their way around it for a while, making sense here and there as they go.
Example. Here is a problem I used to give 9th graders at the beginning of the year. It appeared in our current-at-the-time IMP Year 1 textbook, but it's a classic problem that's been around in one form or another since antiquity.
A farmer has packed up her eggs to take to market to sell, but on the way there, another farmer accidentally tips her cart over, and all her eggs are broken. The other farmer offers to pay for the eggs and asks her how many she had. She doesn't remember exactly, but remembers some things from when she tried packing them in different ways.
She knows that when she packed them two at a time, there was one egg left over. The same thing happened when she tried packing them in threes, fours, fives, and sixes. But when she tried packing them in sevens, there were no leftover eggs. Can you figure out how many eggs the farmer had?
I'm sure you can imagine some 9th graders reading this problem and immediately thrusting a hand up to declare, "Miss(ter), I need help! I don't get it!" These students are not currently making sense of the problem.
When kids say, "I don't get it," what they're saying is, "I don't see a way into this problem" and/or "I don't know what I'm supposed to figure out." Instead of bits of information that each pack a chunk of mathematical meaning, they're instead seeing something closer to "information soup"--a bunch of homogeneous facts they can't parse.
(At this point you might be asking questions like, "Why does that happen to some kids and not others?" or "What can teachers do when kids have that experience?" Which are both great questions! Alas, they're beyond the scope of this particular post, but perhaps we'll tackle them another day. Our focus today is mostly on understanding what MP 1 means, what it looks like when kids are doing it, and why it is an important skill to teach.)
If we could listen in on the brain of a student who IS making sense of this problem, we might hear something like the following:
"Okay, what I need to do here is figure out how many eggs the farmer had. And it tells me some things about how she packed them, so that must be how I'm going to figure it out. Let's see....If she had one left over when she packed them in two's, that means that she must have had an odd number. So 1, 3, 5, 7, 9, etc. If she had one left over when she packed them in three's, that means she couldn't have had three eggs, but she could have had four, because four is one group of three and one left over. She couldn't have five, because that would be two left over. And she couldn't have six because that would be none left over. But seven would be two groups of three with one left over. Okay, I see a pattern here...My answer has to be an odd number that's one more than a multiple of three. So I can write out all the odd numbers and then circle the ones that are one more than multiples of three. But then we also know that she had one left over when she packed them in groups of four..." etc. etc. etc.
"All right, this problem is about the number of eggs a farmer had. We also know the size of the groups she tried packing them in and the number of leftover eggs when she packed them in two's, three's, four's, all the way up to sevens. The part about the other farmer paying her doesn't matter. The main connection between the number of eggs the farmer started with and the leftovers when she packed them in different ways is that there was always one left over, except when she packed them in sevens. So that tells me that the answer has to be a multiple of 7. So it could be 7, 14, 21, 28, 35, 42, 49, etc. Let me write a bunch of those out. If there was always one leftover when she tried the other ways, that means that the number can't be a multiple of two, three, four, five, or six, so I'll cross those out..." etc. etc. etc.
"Let's see, I want to know how many eggs the farmer started with. We know a bunch of stuff about how she packed the eggs, so that might tell me something about the number. Okay, when she packed the eggs in pairs, there was one left over. That means the number could be written as 2x + 1. But I also know that there was one left over when she packed them in threes, so it also has to be able to be written as 3x + 1. Actually, it looks like that's true for all the different size groups up to six. So whatever the number is, I have to be able to write it as 2x + 1, and 3x + 1, and 4x + 1, etc. Except when I get to seven. She could divide the eggs into groups of seven evenly, so the number also has to look like 7x. But all the x's are going to be different numbers, I guess, so let me write it as 7x = 6x1 + 1 = 5x2 + 1 = 4x3 + 1 = 3x4 + 1 = 2x5 + 1..." etc. etc. etc.
Or they might start by drawing pictures. Or they might start by taking a bunch of coins and putting them in different size groups to see how many are left over. Honestly, there are tons of different ways a student might start to make sense of this problem, a ton of ways that they might "get into" the problem, or, as many math teachers are fond of saying, a ton of different "entry points." And all of them are completely valid, logical ways to get started with it.
Example. How about at the early elementary level? What does it mean to "make sense" of a problem like "Tony had 18 Pokemon cards and gave 6 to his friend Carlos. Now how many does Tony have?"
It is easy for an adult--or even, ahem, a secondary math teacher--to look at such a problem and think, "Well, what is there to make sense of, really? You have some and then you subtract the amount that's taken away." Ask any first grade teacher, though; at some point, for all children, this problem is non-trivial. Before they can even get to the point of subtracting 6 from 18, at the very least they need to make sense of (1) what it means to have 18 Pokemon cards, (2) what the mathematical significance of giving some away is, (3) what does that mean I should do with the 18 and the 6. (And who knows, I am not an elementary specialist, there may be even more pieces involved than that! Elementary math gurus please chime in plzthnx.)
A first grader who is making sense of this problem might act it out with a partner, or draw pictures, or use a number line. Even if they draw the wrong number at first, or do the wrong operation at first, or have to think for a bit about what to do with a number line, these are all strategies for "getting into" the problem; they are all valid entry points.
Example. It works the same for older students.
- The Soccer Team is selling charm bracelets to raise money to go to the State Tournament. They need to raise at least $500. Fancy bracelets cost $3, and simple bracelets cost $2. They have enough materials to make up to 180 bracelets. How many of each type of bracelet do you think they should they make? What are the options?
A student who is making sense of this problem might think, "Okay, they need to make at least $500. So the amount they need has to be greater than $500. Innnnnteresting, this is starting to sound like an inequality problem. So [amount needed] > 500. Oh, but it would be okay if they made exactly $500 too, so [amount needed] ≥ 500. Fancy bracelets cost $3 and simple ones cost $2, so they could make only 250 simple bracelets and then they'd make enough money. But even more than that would be okay. Or, they could sell....Let's see, $500 divided by 3 is 166.666 repeating, so they could also just sell 167 fancy bracelets. But they only have enough materials to make 180 bracelets, so they can't sell only simple bracelets. With inequality problems with more than one variable, sometimes graphs help me see better, so let me call the number of simple bracelets x and the number of fancy bracelets y..." etc. etc. etc.
(You may be thinking, "Some of these methods seem really inefficient. Shouldn't we teach kids the quickest, cleanest way to do problems, instead of slogging around the long way or just trying a bunch of things until they find something that works?"
Yes and no; efficiency is certainly a goal in mathematics, and ultimately we DO want kids to be able to look at certain type of problems and say "Ah, this is a _____ type of problem and that means that I can solve it quickly using _____ strategy." BUT
- a) that is not all we want them to be able to do, because there is no way we could ever pigeon hole every possible type of problem that exists into a category for which there is a quick and straightforward strategy;
b) they still need to be able to make sense of a problem in order to recognize that "Oh, this is a _____ type of problem"; and
c) clean, efficient strategies only make sense when you first approach them the long way 'round and spend a little time grappling and tinkering.
The road to "I don't get it" is paved with the absolute best intentions of teachers who said "Let me just show you the easy way to do this, memorize it," and whose students, because they didn't have the opportunity to really make sense what they were being told, sooner or later forgot the easy way or skipped a step or confused it with some other strategy for a different type of problem and ended up having to go the long way 'round anyway. Trust.)
And now, since we've come this far, what does it look like not to make sense of a problem?
- There are 125 sheep and 5 dogs in a flock. How old is the shepherd?
I dare you to guess how many eighth graders read this problem and got straight to work calculating an answer.
(Answer: Too many to not horrify you.)
These students didn't start by asking themselves, "What do I need to figure out? What information am I given? How are those pieces of information related?" They just pulled some numbers out and started calculating.
- "'In all,' 'all together,' or 'sum' means add; 'many more,' 'left,' or 'difference' means subtract."
At Madison Elementary, fourth grade students play either soccer or baseball. There are two fourth grade teachers. In Ms. Lassen's class, there are 30 students and 12 of them play baseball. In Mr. Ruiz's class, there are 28 students and 9 play baseball. The principal is ordering soccer shirts, so she wants to know how many students play soccer all together.
Hot dogs come in packages of 10 and Tania bought 7 packages. How many hot dogs does she have in all?
John had 14 marbles in his left pocket. He had 37 marbles in his right pocket. How many marbles did John have? (source: 13 Rules That Expire, NCTM 2014)
When students approach word problems by looking for key words to clue them in to what operation to use, they aren't making sense of mathematics; they're actively avoiding the mathematics via shortcuts. And guess what, the shortcut isn't even reliable. Looking for key words in word problems is not a math strategy, it's a strategy for avoiding the work necessary for getting good at MP 1.
- "A negative and a negative makes a positive."
Then what about...
Find the sum of -9 and -12.
Approaching calculation problems using memorized rules is fraught with peril. Again, giving students rules or shortcuts to memorize can seem like a kindness at the time, because the cognitive work of, say, extending one's entire body of knowledge of arithmetic to an entirely new system of numbers is long and subtle and complex and will almost certainly involve a good bit of struggle and frustration. But again, tips and tricks like this do students no favors in the long run. Yes, they will get some answers right along the way, but ultimately, relying on rules without really understanding what they mean, why they work, and the circumstances under which they do and do not apply only robs students of the opportunity to work on MP 1 and instead lets them do an end-run around the actual mathematics.
- "'Is' over 'of' equals percent over 100."
As in, "15 is what percent of 75?" 15 is near "is" and 75 is near "of," so kids knew to write 15/75 = x/100. Or, "What is 35% of 120?" 'What' (the unknown) is near "is" and 120 is near "of," so write x/120 = 35/100.
I heard this one from students my first year of teaching 9th grade Algebra Support (ie, students who had failed Algebra I at least once already). Except guess what, even if the clue words worked out in a particular problem and they were able to write a correct proportion, most students would still go on to do something like multiply the two sides of the equation, or divide out common factors from places that they can't be divided out of (because something something cross cancelling), or lord help them, invert and multiply the two ratios. (Yours is not to reason why, indeed.)
Say it with me: Someone somewhere with the absolute best of intentions thought, "Trying to really understand fractions and ratios and percents is going to be really hard for these kids, let me help them out by giving them an easy way to remember how to do it," or maybe even, "No one ever taught me a really good way to teach this, but here is a good shortcut that will help them pass the test." It happens! But the result was that instead of learning to really grapple with what the problem is asking and make mathematical sense of it, these kids learned a trick for getting an answer. A trick that a) doesn't even always work, b) most of them couldn't execute reliably anyway, and c) got them no closer to understanding ratios and percents than they were before.
Which is all to say, tricks and mnemonics like this are not harmless. Is it okay to use a memory trick or a shortcut once you've made sense of the underlying mathematics and fully understood it? Absolutely, because you have the understanding to recognize when such a shortcut does and does not apply or is or is not the most efficient way or when you need to adjust it a bit for some mathematical reason.
That is learning mathematics. Memorizing tricks for getting answers is not.
And this is Part 1 of why we need MP 1. Solving problems in a way that helps kids mature mathematically and develop skill and confidence around approaching both familiar and unfamiliar types of problems requires making sense, not memorizing shortcuts or procedures for every kind of problem imaginable.
What Can Parents Do?
(**Caveat**: It's always a little tricky to make suggestions to parents because a) not all parents have the luxury of being available regularly to support their kids with homework; b) even for those who do, or can shuffle things around to make the time, it often comes at great opportunity costs; c) parents have a limited amount of control over what and how much work kids are being assigned, how it's being presented at school, etc.; and d) when kids are struggling with school work, emotions between parents & kids can run high & parents trying to help can start to feel like torture for both parties.
So I don't want to come across as if I'm suggesting you're responsible for doing your kid's homework, or teaching them what they didn't learn in class for whatever reason, or even sitting patiently next to them night after night until they're done. But if you do have interactions with your kids around their math work, particularly if they're struggling with something, here are a few things to keep in mind that help support students' ability to focus on first making sense of a problem.)
- Emphasize sense- and meaning-making, not what rule to use or trying to determine what "type" of problem this or that one is. (There is a time and place for that kind of thinking but if a kid is still in the "I don't get it!" phase of solving a problem, they need to do the sense-making first and worry about efficiency and categorizing types of problems later.) A good question is, "Can you tell me in your own words what's going on in this problem? What's it about? What are you trying to find out?"
- Get comfortable with the idea of productive struggle, then frame working on challenging math problems that way for your student. Remind them that the point is to understand mathematics more deeply, which mostly comes from grappling with problems we're not necessarily sure what to do with first. Good mathematicians tinker with hard problems at first without getting too caught up in "how do I do this one."
- Forget about speed. Time pressure is the enemy of deep, careful sense-making. Remind your student that being "good" or "successful" at math is not about how fast they can solve a problem but about the effort they put into understanding it at deep level.
Next time: What does it mean to persevere in solving a problem?
(Thank you for reading this far! I consider these posts to be living works in progress, so please do feel free to share any thoughts, suggestions, or questions. I want to make them as accurate and useful as possible!)