Learning How to Struggle... Productively

It seems counter-intuitive to say that teachers want their students to struggle. That doesn't make any sense. It is the teacher's job to make sure their students learn the concepts, which should mean that the students aren't struggling anymore. The key piece of that is the "anymore" bit. The goal is, ideally, to get every student to the point that they no longer struggle with concepts that were once new and confusing to them. The only way to really make sure this is the outcome, however, is to let them struggle in the beginning.

The concepts are new to students when you first introduce them, it's only right that they don't understand it immediately. So you start with some basic, trivial examples to get them started. Now that they have an idea of the concept you are trying to teach them, you have a choice.

1. Keep giving them easy examples.
• They get more practice with the method
• They gain confidence because they always find the correct answer
2. Give them examples that aren't easy where the answer isn't immediately obvious.
• They get more practice with the method using examples that are more meaningful and closer to what they may have to do later on
• They can't always find the correct answer right away, leading them to put some real thought into the problem and apply the mathematical skills you've been teaching them
• They work together with peers to discuss problems
• They struggle now so that they don't struggle later

I've tutored a lot of students who come in before a test asking for help with homework. After talking through some of their homework problems and they've gotten the right answers, I always ask if they understand what we did to get the answers. Could you tell me why we did that? Would you be able to do this alone on a slightly different problem? etc. Almost always I get a "yes" answer back. "Yeah, it makes so much sense the way we did it here". That's good, right? The students seem to genuinely understand how to solve the problems presented to them.

A week later I seem the same students after the test and ask them how it went. "It didn't go too well, the teacher put stuff on there that we never learned". I hear this nearly every week from one student or another. If I didn't know better I would be seriously worried about our teaching staff, constantly testing on stuff they didn't even teach! Fortunately, I have more faith that that in our teachers, so I press for further information.

"What kind of stuff was on the test that you think they didn't teach you?" or "Can you recall a problem from the test that you weren't taught how to do?"

When presented with problems on a test that are "impossible" based on what they were taught, students are usually bitter. So much so that they can recite the problem word for word back to me days later, both the general set up and the exact given values. So, I always get some good material to work with when I ask them to recall a problem from the test. Every time, however, it ends up being about something they definitely learned that we went through together the week before and they assured me they understood how to do.

After talking through the problem, they usually begrudgingly agree that it is just the concept that they were using before but the problem was still somehow unfair.

"All of the examples we did in class and on the homework were easy and so much simpler than this. She shouldn't have put such a complicated problem on the exam." And there lies our real issue. Easy examples in class or on homework let the students believe that they understand a concept better than they really do because they are able to easily get the answer. Once presented with harder problems on a test, in a later class, etc. it's clear that they don't actually understand what's going on.

Let's look at a specific example. I was observing a class the other day and they had a homework due soon. One of the problems was

and everyone was confused. So, the professor decided to give a "small" hint to the class to help them along. She put the following problem on the board and asked them to do it.

 

They all started working, actual smiles on their faces because this was so easy and they knew exactly what to do. She had people yell out how they solved the easy problem, listing in order the steps they took.

1. Distribute
2. Combine like terms (with x in them)
3. Divide to get x alone

Then she told them do try the other problem again, and was met with complaints still. The students didn't understand that the two problems were nearly identical and, thus, didn't appreciate the rather large hint that she gave them. So, either they didn't understand the distributive property or combining like terms (though they all seemed to understand the easy example) or they didn't understand how the trig functions worked. It was different for different students. One divided immediately by x+8 to get both terms with x onto the same side, forgetting that that's not how it worked in the other problem. One student distributed the t from cos(t) and ended up with a cos(xt+8t), so their final answer involved an arccos function. One student canceled the cos functions. Not cos(θ) cancelling with cos(t), but dividing by "cos" and being left with xθ=t(x+8).

Regardless of where the misconception was (distributive property or trig functions) the students genuinely believed that they understood how to do any problems regarding those topics until this problem came along. I'm not trying to say anything negative about the professor here. I've been observing her for a while with this class and I think she is doing a great job. All of my notes on her are positive and I'm definitely going to steal some strategies from her for when I have my own classroom. That being said, these students aren't being challenged enough when they are learning the concepts. So, when they are learning how to do straight forward examples rather than learning how to recognize when the methods are necessary and how to carry them out in less straightforward problems.

Included here is a list compiled by the MIND Research Institute of 8 things that teachers can do to promote productive struggling for their students.

1. Call on students who may not have the right answer.
• Then guide students in the process of questioning their thinking.
• We should definitely stress the importance of questioning your own or someone else's thinking, especially when something doesn't make sense. This method though of calling out a student you know will give a wrong answer seems strange, however. First, it says a lot about your confidence in your students. Second, you risk embarrassing a student by forcing them to give answers you know will be incorrect. Perhaps we can call on students who volunteer, even if we think they will be correct and then get them to explain why they got the answer that they did.
2. Praise students for perseverance in problem solving, not for being correct.
• They'll be more motivated to face challenging problems.
• This applies to in the classroom during activities as well as on tests, I believe. It isn't enough to tell a student that at least trying something they aren't sure on and getting the wrong answer is better than not trying unless you know you'll get the right answer. Challenging themselves and making an attempt will appeal to them more if they know there is a possibility for partial credit on tests and homework for attempted answers. We can't grade all-or-nothing based only on if the final answer is correct.
3. Display work that shows creative problem solving, not the highest scores.
• Students will learn to value the struggle toward a learning goal.
• We really want to stress how important problem solving and critical thinking are over strictly knowing how to do a specific problem. Displaying creating problem solving examples is a great idea, especially if it is work done by the current students in the class. That way, they can see that you value their effort.
4. Provide non-routine problems that can't be solved with a memorized formula.
• This challenges students to make sense of the problem then figure out the math needed to solve it.
• Problems in real life aren't going to be cookie-cutter type issues where you have all the information needed and are basically told how to solve it. "Plug and chug" problems do nothing to help students hone their problem solving skills.
5. Give students informative feedback.
• Provide context to help students course correct toward the solution.
• If a student gets a wrong answer, it won't help at all to simply give them the right answer with no explanation. There is probably a misconception somewhere that they need help working through. This doesn't mean to write out the correct work either though. Leading questions to get students thinking about stuff they may have forgotten or neglected the first time through allows them to still struggle on their own with the problem and come up with the right way of solving it, rather than trying to memorize your work.
6. Don't give easier work to struggling students.
• This gives the message that some students can't handle challenging work.
• One of the biggest hurdles to get over is confidence. I can't begin to know how many times I had people come in for tutoring and tell me that they had no idea how to approach a problem, then I asked them to take a guess and they got it right or were at least on the right path. If students are struggling and you just give them easier problems that shows them that you aren't confident in their abilities either and that they shouldn't be challenging themselves.
7. Allow students time to ask questions and tinker with ideas.
• Slowing down the learning process optimizes retention.
• Obviously it is better to spend more time on any given subject (to a point). Allowing students to discover ideas on their own makes a huge difference in how well they actually understand the concepts moving forward. It isn't very practical, however. While ideal, there is just too much to cover in too little time to allow for every student to rediscover mathematical concepts from scratch.
8. Encourage having a growth mindset.
• Remind students that everyone has the ability to be mathematical problem solvers.
• A growth mindset means that you weren't born with abilities, you learn them. This means that someone who struggles initially isn't doomed to failure because they just "weren't born to be a mathematician". This is not only harmful to students to hear of their classmates, but also of themselves.
• "He was born good at math, that's why he's better than me. I'll never be that good no matter how hard I try"
• "I was born good at math which is why it has always come so easy to me. Now that it isn't so easy it must be because I wasn't born with this specific skill and I'll never be able to understand it like I did earlier concepts"

Overall, productive struggle is essential to students understanding of concepts. Teachers really need to be careful with how they talk to and treat students while they are struggling to ensure that they are doing so productively; not struggling too much, but also not too little.