Figuring out why you can't solve a math problem
Observing what type of "exercise problem category" you are solving helps you figure out what to do next
Learning math involves learning new techniques
Much of the math our kids do in online classes and the math we do with them at home involves learning new problem-solving methods.
Sure, we do practice problem sessions at home from time to time, but for the most part, “Math Time” usually means learning new techniques.
We discovered that the kids would occasionally stumble during the session, but it wasn't entirely clear to us why or when they had gotten confused and stopped being able to follow what was happening.
It felt like they were following the lesson, book, or video just fine, but then they suddenly stopped following the process and solving the problems.
This problem makes sense. This problem DOESN’T make sense.
To understand what had happened, we taught them to review the examples/problems they had been working on and determine when things stopped making sense.
Problem 1 - This makes sense.
Problem 2 - This makes sense.
Problem 3 - This makes sense.
Problem 4 - This DOESN’T make sense.
etc…
Okay - that gives us a clue.
Something about Problem 4 is different from the previous problems, and we/they should try to figure out what changed.
We would then work together to figure out what they hadn’t understood and/or forgotten about the technique.
That worked for the specific technique we were studying, but it didn’t seem to be helping them progress and figure things out independently.
The reason?
Each Math Technique they learned had a different “This DOESN’T make sense” moment
Each time they ran into an issue, they would have to go through the problems one by one to figure out what needed to be fixed.
Sometimes, the break from making sense to not making sense occurred at the start of the technique's practice problems.
Sometimes, it was near the end of the practice problems.
Sometimes, it happened at the start and the end.
Sometimes, it happened every few problems.
And sometimes, it didn’t happen at all.
As you can imagine, and we found this very often, this can be frustrating because each time they started a new problem, they didn’t quite know if they would hit a wall.
So we took a step back, squinted our eyes a bit, and tried to think through what the example problems were trying to do to see if we could figure out a pattern.
In retrospect, this should have been blindingly obvious, and you probably already know it, but we didn’t think about it, so I’m writing it here. =)
10,000 Foot View of Exercise Problems
When teaching a new problem-solving technique, you do the following:
Introduce it (definition, how it extends previous work, and what types of problems it solves)
Show a few examples
Give students some exercise problems to work on
So far, so good.
The question we had to consider was: If we were writing the lessons, what types of exercise problems would we give students?
We concluded that we would give the students problems that answered the following questions:
Can they do the basic, most straightforward technique application?
Can they do a more challenging technique application?
Can they recognize when the technique isn’t going to work?
Can they recognize when they can apply the new technique they learned?
Can they combine the new technique with prior techniques they’ve learned?
Can they derive it using tools they already know (an example here is being able to derive the quadratic formula by using the completing the squares technique with ax^2 + bx + c = 0)
You probably already knew that, but it surprised us because it was relatively straightforward and logical.
Using these questions, let us examine the exercise problems through a new lens.
This meant we could categorize the questions by what the questions were trying to do.
This meant we could help our kids look at the bigger picture of where they had lost understanding.
Problem Solving Technique Guide Posts
When our kid ran into a “doesn’t make sense” problem, we could have them zoom out to figure out what category of question they were struggling with.
It was no longer, “I can’t understand why this problem doesn’t make sense,” “I’m not sure how to set up the problem,” or “I’m not sure why I can’t get the right answer,” or something like that.
Now, it was a “how can I tell that the technique can be applied here” or “I’m not sure how to derive it” or “I don’t understand how to do the basic calculation”.
Example
A recent example, from the AoPS Pre-Calculus class from a section of Trigonometry and Angle Addition Formulas.
My kid went through the video and book and then was going through the homework problems.
The question: What is sin(15°)?
They could have done this with the calculator, but the goal was to use the technique they had learned in the class (angle addition formulas).
This stumped them.
There wasn’t an angle addition [sin(x+y), cos(x+y), tan(x+y)], so what was going on?
After much thinking, they realized this was a “Can they recognize when they can apply the new technique they learned?”
Given that they are in the chapter of angle addition formulas, is there a way to rewrite sin(15°) as an addition of two angles?
Eventually, they realized that they could write sin(15°) as sin( 45°+ (-30°) ).
They could then use the technique they learned in the chapter to figure out the value without a calculator.
Observing where they were (in what chapter) and what the question was trying to get them to do (recognize when to apply the technique) helped them figure out what to do next.
If you are lost, figure out where you are
There’s a big difference between “I don’t know what to do” and “I don’t know what to do, but what I think they want me to do is [x].”
Helping your kid who loves math figure out those clues (first with your help and then later by themselves) will make a huge difference.
Give it a try and let me know what you and your kid who loves math think!
That’s all for today :) For more Kids Who Love Math treats, check out our archives.
Stay Mathy!
All the best,
Sebastian Gutierrez