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My six-year old daughter was given this maths problem for her homework:

Given a regular square grid of 4 × 4 dots, how many different triangles with one dot in the middle can you draw?

We were given no additional information other than that stated in the page.

Faced with this, my immediate conclusion was that there was no way of answering the question. Not enough information is given to create a clear problem statement.

For example it is not stated whether we are limited to equilateral triangles or can use different kinds of triangles. It is also not obvious as to what "different" means in this context: are mirror images and other transpositions of the same triangle on the grid "different" or not.

Because the sample used an equilateral triangle, we followed that and presumed that transpositions were allowed. Now we see the solution, that's not what was required.

It seems to me this sort of issue is going to cause confusion for children rather than foster good understanding of maths. Is that correct? Is it fair of me to complain to the teacher that it's a poorly set problem?


Note, here is a solutions page discussing what other groups did to approach the problem.

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    $\begingroup$ When I read the directions I didn't give abusing everything a second thought. Maybe that's just me, but if all they required is to have one and only one point in the triangle, reflections, translations, etc. are fair game. But maybe that's just because I haven't studied geometry in a very long time so I didn't even think these could be an issue when I read the question. $\endgroup$ – Ovi Nov 28 '16 at 21:27
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    $\begingroup$ The sample is not equilateral. $\endgroup$ – JTP - Apologise to Monica Nov 29 '16 at 4:11
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    $\begingroup$ I too assumed that translations and rotations were ok. It's also further frustrating with having the second question of "How do you know have found them all?". Even the answers don't give you an answer to this. Only proof by exhaustion seems to be the answer. If you'd found all 9 you could still spend hours searching to see if there was another triangle you'd missed. $\endgroup$ – icc97 Nov 29 '16 at 11:52
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    $\begingroup$ I agree that it's unclear what "different" means but "triangle" means "triangle". Why would you assume that "triangle" means "equilateral triangle"? That seems like responding to the question "How many pieces of fruit are in this box?" with "Do you mean all fruit or just apples?" $\endgroup$ – David Richerby Nov 29 '16 at 17:01
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    $\begingroup$ @DanielWagner The question asks about a 4x4 peg board. You are not free to pick any vertices you like. $\endgroup$ – Jessica B Nov 30 '16 at 7:18

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Given the solution they were looking for, yes this is a very poorly posed problem.

The usual convention for problems of this general type is, for example, that all 16 of the translates and rotations of the sample triangle count as different triangles. And any other shape of triangle counts as well.

A well-posed problem following that convention ought to actually indicate that somehow, but could be given a pass if that detail is left implicit and the reader can be expected to have some awareness of the convention.

However, the website is asking for congruence classes of triangles; it is not following the standard convention for such problems. Even a posteriori, absolutely nothing in the problem indicates that fact.

In isolation, this is wholly unreasonable. It might be reasonable as one of a series of problems where the nonstandard convention has already been established so that this one can be assumed to continue the pattern, but that does not seem to be the case here.

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    $\begingroup$ I recall this problem as a kid, and was lucky enough to be in a school where the teachers were focused enough to catch why it tripped me up and realize I was following the rules, not the sloppy assumptions. I had already found 16 triangles without trying, and was working on things that WEREN'T clones of the original when the teacher realized and announced the poor assumption to the class. Different SHAPED triangles. Sadly, too few schools are able to do this for various reasons. $\endgroup$ – Nanban Jim Nov 29 '16 at 20:00
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This is a bit of a can of worms. Let's unpack a little.

We were given no additional information other than that stated in the page.

This is your daughter's homework, not yours. Be careful with this distinction, because coloring a child's question with adult interpretations can lead to trouble (and does in this case).

it is not stated whether we are limited to equilateral triangles or can use different kinds of triangles

Your daughter likely makes no distinction between different classifications of triangles, and this isn't something that she would even consider, let alone trip up on, without your help. But for what it's worth, the absence of a restriction implies that there's no restriction - why imagine one?

(As an aside, the example triangle isn't equilateral. In fact, it's impossible to make an equilateral triangle on a pegboard like that.)

are mirror images and other transpositions of the same triangle on the grid "different" or not[?]

In this context, yes, they are different. Again, this isn't a point-of-clarification that your daughter is in a position to seek out. Children at that age don't sort objects / shapes by equivalence classes.

Imaging your daughter looking at a page with some squares drawn on it. Now imagine asking her to count the squares.

Did she answer "there's one square, transposed, rotated, and scaled all over the page"?

is going to cause confusion for children rather than foster good understanding of maths

Another aside, but I consider causing confusion to be the hallmark of a good question. There is opportunity here for your daughter to recognize and exploit symmetries, translations, etc, but it question doesn't rely on these faculties either.

Now we see the solution, that's not what was required.

If you (and your daughter) take an activity like this as a set of ideas to play with rather than a piece of answer getting to do, it's likely to be much more rewarding for her, and less frustrating for you both. Whatever formulation of the question you eventually tackled had value in its own right.

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    $\begingroup$ "take an activity like this as a set of ideas to play with rather than a piece of answer getting to do" Unfortunately, not all teachers think that way. True, at this age any homework is most likely completion-only or entirely ungraded, but that will change with time. That said, I absolutely think the most salient point is "your daughter should do her own homework and you should answer her questions by teaching her how to find the answer rather than trying to find it for her". $\endgroup$ – Tin Man Nov 28 '16 at 22:30
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    $\begingroup$ "Now give your daughter a cookie and a glass of milk. Does she spill the milk everywhere because she used too weak of an equivalence structure again?" $\endgroup$ – djechlin Nov 29 '16 at 0:46
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    $\begingroup$ @Walt the homework is graded and a "correct" answer expected, which was nine. FWIW I didn't guide her on this to begin with in any way. She confidently told me she understood so I let her get on with it. Then, as she read the question, her understanding melted in the face of the ambiguities and she came to me for help. $\endgroup$ – Bob Tway Nov 29 '16 at 9:03
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    $\begingroup$ A lot of bad assumptions about the parent and unneeded condescension in this answer. $\endgroup$ – Davor Nov 29 '16 at 13:14
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    $\begingroup$ A distinction can be made between critiquing a question and advising a parent on "what to do" in the face of a certain question. My answer was about the latter, @Hurkyl's was about the former. I upvoted Hurkyl's - it's absolutely correct. If bad assumptions were made, I hope they can be understood as being made in good faith. My aims were to A) discourage the parent from reasoning for the child, and B) discourage an adversarial relationship with problems / homework / education. The question raised alarm bells on both fronts - not addressing them would be a disservice to the daughter. $\endgroup$ – NiloCK Nov 29 '16 at 17:38
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I've come across some creative 10 year olds who produced solutions like enter image description here

So when there are questions set that seem closed you can allow it to be opened out and the child's exploration can begin, helping them to understand it's ok to think and not just follow convention.

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    $\begingroup$ Hahaha! Gold star for that student $\endgroup$ – Jessica B Nov 30 '16 at 22:23
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    $\begingroup$ Yes, this kind of thinking is great, and what we should be encouraging. $\endgroup$ – Dov Dec 10 '16 at 14:55
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I think the key issue is with this part of the question:

How many different triangles with one dot in the middle can you draw?

Which depends on the context of how the subject is being taught

I looked at the other UK Key Stage 1 level Maths questions focussing on triangles. Assuming this is an okay overview of the course structure, spatial transformations aren't dealt with at this level. This is meant to be the basic building blocks - i.e. what shapes are and how they are constructed.

The learning path following through the problem sets (only triangle shape ones) looks like:

  1. How can you make triangles shapes with straight lines of different lengths?
  2. What triangle shapes (constructed with straight lines of varying length) are similar to other triangle shapes?
  3. How many unique triangle shapes can you make with x limitations?

In the context of that specific path of learning - the question and solutions do make sense. The question you have posted is a lateral thinking problem about drawing triangle shapes. Which is following on from the previous questions. i.e. using different lengths of straight lines to make triangle shapes.

However, anyone with any understanding of spatial transformations will get different answers in the current way it's being asked. Apparently it's assumed no-one at KS1 level knows about spatial transformations. Is this a mistake/misleading? Depends... I can totally see a child getting confused if they have knowledge of spatial transformations.

A better wording might have been:

How many triangles that are not the same shape (read: different angles and line lengths) and one dot in the middle can you draw?

tl;dr it's all about context

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I think you're focusing too much on 'the answer'. I suspect the teacher was tired and busy, and quickly needed to come up with an exercise of a suitable level that was worth completing.

If I was marking this, I wouldn't be looking for the answer '9'. I'd be looking for a solution, and particularly what understanding has been demonstrated by the pupil in their solution.

As I see it, the question tests different aspects: mathematical creativity to create different shape triangles; understanding of Euclidean isometries to either list all translates/rotations or point out that they don't count; systematic thinking to find all answers; creation and communication of proof to explain why their answer is complete.

A six year old is not going to have perfectly mastered all of these, by any means. But their solution should demonstrate at least some understanding of one or more of these, and help to answer the question 'what level is this pupil working at?', which is the question that should be being applied for a six year old (not 'how many marks did they get?').

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  • $\begingroup$ "I suspect the teacher was tired and busy, and quickly needed to come up with an exercise of a suitable level that was worth completing." Unfortunately, this exercise likely came straight out of a book or teaching manual. I've seen people complain about this exact problem before. $\endgroup$ – Mast Dec 3 '16 at 13:58
  • $\begingroup$ @Mast That's the point! Teachers don't have time to carefully think through every single piece of work. Sometime they need to pull something off the internet. $\endgroup$ – Jessica B Dec 3 '16 at 23:05
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Think of the pedagogical goal of the problem as being to generate discussion about assumptions being made, just as you are doing. The goal is not to come up with a pre-determined correct number.

Einstein is purported to have said, “The formulation of the problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.”

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    $\begingroup$ I wish you were correct. Unfortunately I know that in this case that's not the point of the question. The answer they're looking for is 9 - I feel it is an abysmal way to set the question, more likely to damage understanding than enhance it. $\endgroup$ – Bob Tway Nov 28 '16 at 21:41
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    $\begingroup$ I got an answer of 16 - four rotations of the example triangle around each of the four inner dots. $\endgroup$ – Xavier Nov 28 '16 at 22:50
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    $\begingroup$ Like @user52673 I first thought of 16, and I think (without actually picking up a pen—I wasn't able to get anything useful out of the Flash app at the bottom of the page) I can make a case for 24. Where do they get 9? Is the solution on line anywhere? $\endgroup$ – David Moles Nov 28 '16 at 23:26
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    $\begingroup$ I've found 32 so far. It would be interesting to see what the folks over at Puzzling.SE would make of this question. $\endgroup$ – Mark Nov 28 '16 at 23:28
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    $\begingroup$ @DavidMoles: If you click on the solution link (in the sidebar), you'll find that they were asking for congruence classes of triangles. $\endgroup$ – user797 Nov 29 '16 at 1:06
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Having taught gifted math to elementary school students for over 25 years, I was delighted to see that your 6 year old daughter was given this question. However, as I read further, I discovered that the teacher had given this for homework and was focused on the correct answer of 9.

Student need to be taught how to solve problems like this. We want student to develop strategies for solving problems which is why the answers focus on strategies.

One method for assigning this question is to give it in class for children to work on in groups. The teacher could listen in and give hints to those that are stuck and/or clarify any questions that arise. Note that one hint is to work with a partner. In this setting the ambiguities won't matter as much. After the groups work by themselves, the groups should come together as a class and discuss their answers and their assumptions. The next step would be for the teacher to clarify ambiguities. Groups could then reassess their answers. Only at this point would it be be reasonable to discuss the correct answer.

If you approach the teacher which I think you should, I suggest you approach the teacher as a partner in your child's education instead criticizing the question. Don't raise your questions (did you mean equilateral triangles?). Instead tell the teacher your daughter's questions and ask how you should have handled the ambiguities. Ask how students are supposed to develop strategies for such explorations which are different then the typical word problem. Did the teacher teach students how to figure this out and what strategies can you reinforce at home. If needed to make your case, you might also point out that the nrich website seems to focus on strategies as much as the answer and suggests under hints that students should work with a partner. You might also ask if the students worked on other problems like this in class. I hope you'll report back on your conversation with the teacher.

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    $\begingroup$ 9 is most definitely not the right answer. I can get at least 44 $\endgroup$ – Kevin Dec 1 '16 at 22:21
  • $\begingroup$ The solutions page shows the 9 triangles. From the solution shown at the bottom of the page we see that the translations, rotations and reflections aren't counted. This isn't clear in the question which is definitely one of the problems. $\endgroup$ – Amy B Dec 2 '16 at 7:23
  • $\begingroup$ Yeah, they definitely need to put in something like "different triangles" $\endgroup$ – Kevin Dec 2 '16 at 13:34
  • $\begingroup$ @Kevin They did ask "How many different triangles with one dot in the middle can you draw?" But I still don't think that's clear enough. $\endgroup$ – Waxen Dec 2 '16 at 19:34
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    $\begingroup$ @Waxen, sorry, meant "different shaped triangles" $\endgroup$ – Kevin Dec 2 '16 at 20:35
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Just for amusement: an advanced formulation of the problem is that of reflexive polygons. "Reflexive" means one lattice point on the interior. A nice expository paper about this problem is found here

https://www.math.hmc.edu/~ursula/notes/reflexivepolytopesarticle.pdf

Up to GL(2,Z) equivalence there are 16 classes of reflexive polygons in the plane. Five of these classes are triangles. See figure 10. These come in two pairs of dual triangles, and one self-dual. You will see these among the 9 in the answer to the original problem. So some of the 9 in the solution are equivalent under GL(2,Z). So from this formulation, the answer is "5 triangles up to GL(2,Z) equivalence."

Note that by shears transformations, you can generate reflexive triangles that need arbitrarily large grids to fit. See figure 8.

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    $\begingroup$ +1 for being insane in a good way.:) (I'm considering the fact the daughter is 6 if she was 12 I figure this would obviously be a reasonable explanation.) $\endgroup$ – DRF Nov 30 '16 at 14:09
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    $\begingroup$ As much as I appreciate learning about reflexive polygons, I don't think it really answers the question in the OP, which was about the posing of the problem. This answers the mathematical question the teacher asked, not the math education question the OP asked. Better as a comment. $\endgroup$ – mweiss Dec 1 '16 at 21:11
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The question is:

How many different triangles with one dot in the middle can you draw?

I don't know where is the middle of a triangle --who knows (exept, maybe, an equilateral triangle where the centre seems to be a good candidate). As NiloCK says, you can't draw an equilateral triangle on this 4x4 grid.

So my answer is 0: no triangle have one dot in the middle.

Sure my 6 years daughter answers differently to the question.

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    $\begingroup$ Sure, it should have said "interior" not "middle". $\endgroup$ – Ben Voigt Dec 2 '16 at 0:00
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This is an interesting exercise, as the many different responses show. Sure, it's ambiguously worded (I thought they meant all the different reflections/rotations which is certainly hard). With the right attitude, this is a wonderful chance for a student to explore and have a good time with you, the parent. She can find some, you can suggest other avenues and she can find more. Then she can go to school and find out there were perhaps more you did not consider.

The problem is when the teacher (whether tired, overworked, or just incompetent) makes this about the "correct" answer of 9. There may be only one answer if the question is worded correctly, but when it's not worded correctly the teacher is just starting a pattern of mental rigidity.

When my son was 5, he brought home a worksheet from his teacher, and asked for my help. There was a picture of 2 frogs in a pond. Next to it, was a picture of 4 frogs jumping out of the pond. The question was, "How many frogs are left in the pond?"

My son had watched Cyberchase, a TV show on PBS that is wonderful, but intended for older children (maybe 10). He was precocious, with a good visual and spatial sense. He turned to me, and asked "-2?". He knew that made mathematical, but perhaps not physical sense. He did not know quantum physics yet.

I laughed, and said "I can't believe that is what they are asking in a kindergarten class, but put that down and bring it to your teacher. I think your answer is amazing."

It turns out that we were supposed to consider the two pictures together so that 6 frogs minus 4 = 2. This was completely unobvious to both of us. But the teacher, rather than say "That's wonderful! They didn't mean that, but your answer shows that you understand negative numbers, bravo!" just wrote an X on his paper, which he then brought home.

I was the one who had to tell him that it's ok if they were not looking for that answer, that his answer made sense in its own way, and that he was wonderful, and should keep thinking about math and having fun with it.

Not blaming her (entirely), but today, his interest in math is largely burned out. It's sad because he showed such amazing early promise.

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