Practice 1 has often been presented together with Practice 6 ("Attend to Precision") as one of the two "overarching" practices that work in conjunction with all the rest, which makes a certain amount of sense.
I also like the way that Grace Kelemanik and Amy Lucenta have presented it, with Practice 1 as a sort of "umbrella" practice that becomes a gateway to the others.
(Kelemanik & Lucenta, 2016)
In any case, I think most people agree that the other practices don't much come into play if you can't make sense of a problem to begin with, and persevering through it is a key part of eventually solving it, or even just learning something new from it, even if you don't ever arrive at an answer. In that sense, it's sort of the "gateway" practice.Practice 1 has two parts:
- Make sense of problems.
- Persevere in solving problems.
We'll start by talking this week about what it means to make sense of problems; next time, we'll discuss 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?"
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.
For example, here is a problem I used to give 9th graders at the beginning of the year. (You may be familiar with it as it is quite accessible even for upper elementary students.) 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.
The Broken Eggs
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.
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 = 6x_1 + 1 = 5x_2 + 1 = 4x_3 + 1 = 3x_4 + 1 = 2x_5 + 1..." etc. etc. etc.
Or they might start by drawing pictures, or by putting some coins into different size groups to see how many are left over. There are many, many valid ways a student might start to make sense of this problem, or entry points, as we often refer to ways of "getting into" a problem.
Here's another example:Tony had 18 Pokemon cards and gave 6 to his friend Carlos. Now how many does Tony have?
It is easy for an adult to look at such a problem and think there isn't much to make sense of here. 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, they must 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.
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.
Here's a high school example: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. That reminds me of inequalities, 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."
That said, that is not all we want them to be able to do, because it isn't possible to classify every math problem that could ever exist into a category for which there is a quick and straightforward strategy; students still need to be able to make sense of a problem first in order to recognize that "Oh, this is a _____ type of problem"; and clean, efficient strategies only make sense when you've first spent some time grappling and tinkering inefficiently.
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 fully 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 around anyway.
What NOT making sense looks like:There are 125 sheep and 5 dogs in a flock. How old is the shepherd?
How many eighth graders do you think 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 grabbed some numbers from the problem and started calculating.
This is why, when we teach students tricks or shortcuts (like "all together" means add, or "how many more" means subtract, or you can solve percents by writing "is over of equals number over 100") we are ultimately doing them a disservice. In those situations, we are not teaching students mathematics; we are teaching them how to avoid doing mathematics.
Tricks and mnemonics like this are not harmless. Yes, it's possible for students to use them once they've fully made sense of the underlying mathematics and understood it, because they then 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 they 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.
Next time: What does it mean to persevere in solving a problem?