How do you predict that a group of students (9th Graders, Geometry, nearly all are comfortable with scaling) would respond to this prompt? Do you think they’ll disagree? Converge on one option? What reasons do you think they will bring to support their answers? Do you think that their responses will differ significantly from the responses that a group of teachers would give? If so, how?

Sheesh, that’s a lot of prompts. Let’s condense that:

1. What do you predict students will respond?
2. How do you predict that a group of teachers will respond?
3. How would you respond?
• Hao Ye

I would guess Feedback #1 would be pretty popular, because it’s specific about the error, and is the most encouraging for a fix. Of the three options, I think I prefer Feedback #2 the most, since it at least gives some sense that a different approach might be needed. I don’t like it very much though, because “prove” is a loaded term and isn’t being used in the mathematical sense here, and I think showing that 68 is wrong is a bit of a roundabout way of getting the student to think of the problem in the correct manner.

• I like feedback #2 best except for it being so direct about this answer being wrong. I like the way it asks for more work as a form of encouragement — “try this!” — instead of “I know you can do it” which to me seems like a setup for possible failure if they don’t really understand the proportionality this well.

I’d rather say “What does it mean for the poster to be the same shape as the original picture?” or something like that. Or if I want a less pushy hint, “How are we supposed to know the height of the poster?” or “How does the poster need to relate to the original picture?”

• I think that was a response to prompt #3. I feel like students will pick feedback#1 and teachers #2, and I’m curious to hear what happens in the actual classroom. Students (and maybe teachers) will like pointing out what’s wrong as something with more clarity to get people moving and thinking they need another approach.

It occurs to me that asking about a smaller photo with base 4cm might be my favorite way to give a hint here.

• Chester Draws

#1 is just saying it is wrong — and then encouragement to get it right — but no actual help getting there and misleading in its direction.. (You don’t say “you shouldn’t subtract 56 here”, you need to say “you shouldn’t subtract” or they will get the idea that you subtract — just not 56.)

#2 might help the student find out the answer was wrong, still doesn’t help them get it right, and it wastes time if they can’t find a way to show it is wrong. (A few cleverer, older students might appreciate being told not only that it is wrong, and that they should be able to work out why, but most people find the “it’s wrong, but I’m not going to tell you how” type of feedback to be immensely irritating.)

#3 is also worse than just saying “you did it wrong” because the use of “improve” suggests that they did something was right, albeit not quite right. That is exactly what you don’t want them to think because they will keep on trying additive methods on the basis that it only needs “improving”.

None of them help the student to the realisation that attacking these questions additively is totally wrong. I would just say “remember that all scale and conversion problems use multiplication or division”. The quicker they are put right the better, and advice that doesn’t steer them that way is just taking longer to get them to where they need to be.

Otherwise, how about suggesting that they double it, which is easy, then treble it, then quadruple it, etc? That might strike the note that multiplying holds the key, if saying it directly isn’t your thing..

• Paul Hartzer

I don’t like #2 because I feel like it could be taken as mockery. The issue at hand for me is that the particular method used is not correct; it’s possible to get the correct answer using an incorrect method, and it’s possible to get an incorrect answer using a correct method and making errors. So “prove it’s not 68” doesn’t get at the core problem. It does at least reinforce number sense, although I prefer those reinforcements to come with answers that are much more obviously incorrect (for instance, when a student finds the third side of a 3-4-? triangle to be 10).

I would probably use a variation of #1. I might draw (or have ready) a third version with a length of 4 cm and ask them what the height would be. Hopefully that would help them see that subtraction is not a useful strategy in this case.

How teachers in general would respond would depend on the type of teacher they are.

I don’t know how the described students would respond because, sadly, I have yet to work with groups of students at that skill level.

• Paul Hartzer

Also, short of knowing how to do the problem correctly or using a 4 cm x ? example, no, I can’t think of a way to prove it couldn’t be 68 cm high. In both of the cases, the height is less than the width, so it passes the most basic of sense checks. So, for me, #2 amounts to, “It’s wrong. If you knew how to do it correctly, you’d know that. So do it correctly.”

• Nk

It’s 54

• Jeremy Galvagni

First, it could be 68 with a little stretching and cropping. No one would notice a bit of missing sky.

Ask the student to scale the photo down to 4cm wide using the same method. From that you can have a useful conversation about the difference between multiplication and addition.