I wonder what my 8th grade geometry kids think of this question.
I give them Version A on a strip of paper and ask them to work on it alone for 5 minutes. I tell them that I’m interested in learning if they understand the question as is, therefore I’m not answering any clarifying questions about it. After the 5 minutes, I put them into random groups, and they work on the problem for another 10 minutes.
Then I show them Versions B and C and ask for their preference and reason for each version.
Version A: 18 likes, 14 dislikes
Highlighted reasons for LIKING:
It gives every detail you need to know. It tells you directly all of the information. It also seems easiest to solve.
It isn’t as confusing as looking at fast-motion pictures of a circle and a square. Doing math is more exact than visual guessing. [Did he translate the animation to mean guessing?]
I am able to make my own diagram and I can try to solve an equation to find out the answer.
It is simple and not confusing. It allows me to think the way I want to and not be misled by a moving picture.
With the given information, you could construct the two shapes by working backwards.
There’s enough given information to make the problem interesting and hard.
I like this one the most because you can actually read the problem and refer back to it.
I think it can be solved using an equation, and be solved more easily than B and C.
It’s a more accurate way to find their areas and make them equal.
I like this version because I understand the problem.
Instead of a picture of it on paper, you have to visualize it in your head first.
You can get an exact answer. It is challenging.
Version B: 18 likes, 14 dislikes
Highlighted reasons for LIKING:
It would be cool to build the animation in GSP and solve it that way.
It’s a lot more simple. It provides an image and idea of what it looks like.
I can visually see when they are equal. It will be easier to see when they are equal instead of having to do a load of math.
It is visual.
I feel I have a higher chance of answering the question with a right answer.
You can easily see when the shapes have the same area.
Version C: 15 likes, 17 dislikes
Highlighted reasons for LIKING:
It seems easier because you can just count the candies and see if they’re equal.
Anyone can count how many candies there are, then subtract the extra space to get the correct area.
Some students like and dislike more than one version. My takeaway on their responses:
LIKES: (see above)
DISLIKES: Not understanding the question, or “I’m a visual learner, so I like Version B better.”
LIKES: It’s visual. It’s easy.
DISLIKES: Too fast and hard to follow. One student, “The movement is distracting and confusing. I feel like it’s too abrasive and violent. Math should be more elegant than this.”
LIKES: You just count the number of candies. It’s visual.
DISLIKES: Too fast to follow. It seems too easy. There’s space between the candies. One student, “You can’t get the exact answer… And the leftover space in one shape may be more than the leftover space in the other.”
I collect all their papers before telling them which version I like. I like Version A for its simplicity. I’m curious if the stated question is enough information for them to understand. This student’s reason nails it for me: “It allows me to think the way I want to and not be misled by a moving picture.”
We all have students who struggle with word problems. I don’t think this means we should give them fewer word problems. I think it means we should give them better word problems — ones that are written with just enough information and not embedded in contrived contexts that either confuse or insult the students. And for students who need help with the question, they get to hear an explanation from a classmate.
Version B is okay, but I don’t want to start with it because I feel I’d be wasting a perfectly good question in Version A! I’d reach for a piece of string to explain this question, if needed. Version C gives me a headache.
At least one person in each group understands the question, and they do their best to make sense of it in just the short 10 minutes that we have. They’re trying. And making mistakes.
Our whole-class discussion at this 8th grade level:
- That “arbitrary” point P is pretty darn close to the middle of A and B. You can roughly tell from using a piece of string. Or you can tell from arriving at y2/(4pi) = x2/16 (where x is distance AP forming the square’s perimeter, and y is distance PB forming the circle’s circumference) — the denominators are almost the same.
- Likewise, P cannot be at the center because pi doesn’t reconcile nicely in the equation.
- We can solve for x and y using some arbitrary distance AB, and we find y to be slightly shorter.
- We can ask a related question: A circle’s circumference and a square’s perimeter are equal, what is the area enclosed by each? Kids can certainly think about optimization and do a little bit of calculation outside of a formal calculus class.
In addition to asking the students which versions they like, I also pose Dan’s exact question to them: Which of these is a “real world” math problem? Or is none of them a real-world math problem?
Their answers vary as widely as those of math educators’. However, I find this correlation that doesn’t surprise me: kids who like math more do not care if the problem is real-world or not.
This [Version C] is the most “real-world” solely because of the fact that it involves a material object which in this case is the candy. However, the thing you’re solving for in this question is not very “real-world” at all. Personally, I don’t care at all if a problem is “real-world” or not; I just like to solve problems.
If a problem didn’t have to do with “real-world” I will still do it if I like it. It doesn’t really matter.
I don’t think any of these problems are “real world” math problems. I like how they make me think. But I don’t think I need them in the “real world.”
I wouldn’t care if it is a real-world problem because I was there to learn. I think all versions can be a real-world problem because it can be needed in some situations.
I feel like all of these problems are real world… But honestly it doesn’t matter at all to me. It doesn’t matter if it’s real world or not, it doesn’t affect me wanting to solve the problem.
So, one from the kids.