Tuesday, January 14, 2014

Making Sense of Solving Word Problems

At around 5th-grade, there is a transition from learning to read to reading to learn. Before the transition, students are taught explicitly how to read. After the transition, students use their reading skills to read and learn new things. Students who don't have the reading skills they need when this transition occurs struggle and often fall behind.

There is a similar transition in math at around 10th-grade. After Algebra II, we start using algebra to model and solve problems not just to learn math, but to learn other things. Unfortunately, only about 10% of us make this transition successfully. I talk to a lot of adults about their math experiences, and many highly intelligent people who got A's and B's in math in school report being absolutely traumatized by word problems in high school. I also talk to a lot of college professors, and many lament that they can't rely on students being able to do math to learn even in technical fields such as pre-med or computer science.

There are two basic approaches for teaching students how to solve word problems using algebra. In the first approach, the student literally memorizes recipes for solving a handful of classic word problem types, such as coin or train problems. (I can feel some of you tensing up just at the mention of train problems.) The hope is that, at some point in the future, students will generalize from those recipes and figure out how to solve word problems on their own.

In the second approach, the student is immersed in a single context and asked to solve a set of problems within that context. Instead of giving the student a set of recipes to use, the student is expected to use his or her deep understanding of the problem context to solve the problems. A premium is placed on using multiple representations, so the student will work with tables, graphs, and equations. Again, the hope is that, at some point in the future, students will generalize what they learned from this problem context and figure out how to solve word problems in other contexts on their own.

I prefer the educational philosophy behind the second approach, but neither approach has been successful and both suffer from the same fatal flaw: students are left to generalize on their own. Starting from a deep understanding of a single problem context is an effective way to help students generalize if the learning experiences and tasks follow a logical progression, but what we always get instead is a shotgun approach where the curriculum designer throws everything against the wall and it is up to the teacher to see what sticks and then build on it. What's worse, generalization is suppose to happen at some future and undetermined date, so no one can assess if the curriculum or instruction are working or not.

I feel like it is the job of the curriculum designer to provide opportunities for sense-making within the curriculum itself, and it is the job of the teacher to monitor the progress of each student's sense-making and adjust the curriculum and instruction as necessary.

When I examined my own strategy for solving word problems, I realized that I use the same basic strategy I use when solving word problems by guess-and-check.

Kirsten bought 36 cans of soda on sale for 20¢ each. She sold some of them to Luigi for 50¢ each, she sold some of them to Melania for 40¢ each, and she drank the rest. Melania bought 8 more cans than Luigi. If she made a profit of $5.90, how much soda did Kirsten drink?

To solve this problem using guess-and-check, I would start by guessing the number of sodas that Luigi or Melania bought.

Let's guess that Luigi bought 5 cans of soda.

If that's the case, then Luigi paid Kirsten $2.50 and Melania bought 13 cans of soda.

If that's the case, then Melania paid Kirsten $5.20 and Kirsten drank 18 cans of soda.

If that's the case, then Luigi and Melania paid Kirsten $7.70.

Hmm… Kirsten made a profit of $5.90, how would I figure out her profit here? Oh! I need to know how much she paid for the soda, which is $7.20. So Kirsten's profit is $0.50.

My guess is incorrect, so I need to guess again.

By starting with a guess, I am able to work forward through the problem. Each time I figure out a value, I scan through the remaining unknowns to see if any of them are easy for me to figure out next. As a curriculum designer, you can't know for sure if this step is intuitive for students or not until you test it out. I have found that every 6th- and 7th-grader that I've worked with has been capable of starting with a guess and then determining if that guess is correct or not. Students have years of experience solving this type of problem (and generally love doing it), so I am leveraging a fairly robust mental model here. However, if I ran into a student who struggled with this step, then I would have to figure out how to start at a more basic level.

I should also point out that I had to do some problem solving in order to figure out how to calculate the profit. To get to the profit, I had to find how much Kirsten paid for the soda first, which is not obvious. The guess-and-check strategy works if the student understands the problem context and is willing to do some simple problem solving. We shouldn't expect students to solve word problems in unfamiliar contexts.

Now that I've done one round of guess-and-check, I have a roadmap I can follow if I want to solve the same problem using algebra. Instead of guessing a random value for the number of sodas Luigi bought, I can guess the correct value and call it x.

Luigi bought x cans of soda.

Luigi paid Kirsten 50x and Melania bought x + 8 cans of soda.

Melania paid Kirsten 40(x + 8) and Kirsten drank 36 − x − (x + 8) cans of soda.

Luigi and Melania paid Kirsten 50x + 40(x + 8).

Kirsten made 50x + 40(x + 8) − 720 in profit.

Since x is the correct guess, then 50x + 40(x + 8) − 720 = 590.

Solving for x, we get x = 11.

So if Luigi bought 11 cans of soda, then Melania bought 19 and Kirsten drank 6.

This is the basic approach that I have used to teach students how to solve word problems using algebra. I have tested this curriculum once in a classroom setting with a heterogenous group of sixth-graders. It was my first time teaching this curriculum, but I was still able to get about 60% of the students to mastery, which means that they could solve any word problem that was limited to linear equations and understand what they were doing. The other 40% of the students could also solve any word problem, but they were less confident in their understanding and more prone to making errors. It was probably a combination of less automaticity with the algebra (more cognitive overhead), the tendency of 6th-graders to make a careless error somewhere in a multistep problem (which can affect confidence and cause second-guessing), and an incomplete understanding of one or more of the subtasks. I would have to refine and re-teach this unit to sort all of that out.

Before I wrap up, I want to point out a few things that trip students up. First, there is the question of which unknown to guess. There are six unknowns in the problem, and I can solve the problem by guessing any one of them. But I know that my life will be much easier if I guess the number of sodas Luigi or Melania bought than if I guessed the number of sodas Kirsten drank or (god forbid) how much Luigi and Melania paid Kirsten. At the beginning, I usually tell the students which unknown to guess, then ask them to choose later once they are comfortable with the guess-and-check process. I tell them that if they guess an unknown and run into complications, then they should just start over again. Over time, this will help them develop an intuitive sense of which unknown to use as their starting point. That's what I'm doing, but I'm doing it so quickly that it seems like I just know which unknown to pick.

Second, some students will want to start adjusting values within a guess instead of starting over with a fresh guess if a guess is wrong. For example, if they guessed that Luigi bought 5 cans of soda and figured out that the guess is wrong because Kirsten's profit is only $0.50, they may try increasing the number of sodas Melania bought without recognizing how that affects the number of sodas Luigi bought and the number of sodas Kirsten drank. They are focusing on one relationship at a time. You need to show them how each variable is linked to every other variable so it isn't possible to isolate a single pair and tweak them. If their guess that Luigi bought 5 cans of soda is wrong, then they need to start fresh with a new guess for Luigi. Seeing how everything is interconnected will help students understand how a system responds to change and why they need algebra to model it.

Once students are organizing their guesses in a table, they can look for patterns and guess more efficiently.

Luigi soda Luigi paid Melania soda Melania paid Kirsten soda Kirsten cost Profit
5 $2.50 13 $5.20 18 $7.20 $0.50
6 $3.00 14 $5.60 16 $7.20 $1.40
7 $3.50 15 $6.00 14 $7.20 $2.30

Here, they can see that each time the number of sodas that Luigi bought goes up by one, Kirsten's profits go up by 90¢, making it possible to determine the correct guess without any more guesswork. This is a nice intermediate step before going straight to guessing x and using algebra.

Third, I like to isolate the subtask of translating computations into algebraic expressions using order of operations. I call it solving word problems without computation.

Kirsten bought 36 cans of soda on sale for 20¢ each. She sold some of them to Luigi for 50¢ each, she sold some of them to Melania for 40¢ each, and she drank the rest. Melania bought 8 more cans than Luigi. If Luigi bought 5 cans of soda, how much profit did Kirsten make?

This problem does not require algebra or guess-and-check to solve. Basically, I am supplying what would have been the guess and removing the check. I ask students to solve the problem without doing any computations, which means writing the answer as a numerical expression.

Luigi soda = 5
Luigi paid = 5 × 50
Melania soda = 5 + 8
Melania paid = (5 + 8) × 40
Kirsten soda = 36 − 5 − (5 + 8) or 36 − (5 + 5 + 8)
Kirsten cost = 36 × 20
Kirsten profit = 5 × 50 + (5 + 8) × 40 − 36 × 20

Besides practicing their metacognition, students are also practicing writing expressions with order of operations and finding values for a series of unknowns.

Wrap Up

So what should you take away from all of this? Learning how to solve word problems using algebra is a serious pain point for both students and teachers. A 90% failure rate is abysmal. It is also a major pain point for anyone who thinks that we don't have enough people entering STEM careers. Most students who lack the skills to transition from learning to do math to doing math to learn opt-out of math completely at this point. There should be some serious urgency around fixing this problem.

Despite all of this, the two approaches we have for teaching students how to solve word problems using algebra are utterly ineffective, and instead of trying something new, we keep trying to refine them. It seems obvious that if we want students to come up with a general problem-solving strategy, then we should figure out the strategy we use and try teaching them that. But I don't see that happening anywhere. Instead, we continue to expect students to generalize and make sense of things on their own outside of the classroom and the curriculum.

By identifying my own problem-solving strategy and trying it with students, I've been able to help over half of my sixth-grade students understand how to solve word problems using algebra. Imagine what those sixth-graders could be doing by the time they are in high school taking Algebra II. I still need to refine the curriculum and my instruction, but that is a simple matter of closely observing students and helping them overcome stumbling blocks to understanding. It is a logical and straightforward process that, again, I don't see anyone else doing. Why is that? And why aren't effective innovations bubbling up to the surface?

Bonus Problem

Finally, a bonus problem in case anyone wants to see my problem-solving strategy applied to a second problem.

Humphrey leaves Chicago at 2:00 pm and drives west at a speed of 55 mph. Iris leaves Chicago at 6:00 pm and drives east at a speed of 65 mph. When Iris finally stops for gas, she and Humphrey are 580 miles apart. When did Iris stop for gas?

Start by guessing a time: Iris stops for gas at 8:00 pm.

If that is the case, then Iris has been driving for 2 hours and Humphrey has been driving for 6 hours.

If that is the case, then Iris has driven 130 miles and Humphrey has driven 330 miles.

Since Iris is driving east from Chicago and Humphrey is driving west from Chicago, then they must be 460 miles apart.

The guess was wrong (460 miles ≠ 580 miles), so guess again.

Using algebra, we can guess correctly on our first try: Iris stops for gas at x pm.

(There are issues that students will have to deal with when using time, but they should be able to handle it if the context is intuitive to them. The traditional approach is to guess hours instead of time, but either will work as long as the students are thinking about what they are doing and not just operating on cruise control.)

Iris has been driving for x − 6 hours and Humphrey has been driving for x − 2 hours.

Iris has driven 65(x − 6) miles and Humphrey has driven 55(x − 2) miles.

They are 65(x − 6) + 55(x − 2) miles apart, so 65(x − 6) + 55(x − 2) = 580.

x = 9

Iris stopped for gas at 9 pm.

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