Drawing heads, tails, knees, and toes – useful?

Drawing a useful picture or diagram is an important problem-solving skill.  From teaching different groups of elementary school children, I have noticed that some students are very reluctant to draw anything whereas others will draw elaborate pictures with lots of extraneous details.

It can then be tricky to push both types towards the happy medium of drawing just enough to help one solve the problem.

My first graders so far have mostly worked on problems where the pictures are drawn for them and they just have to interpret them.

Last week, I decided that they were ready for a little less hand-holding.

For my first experiment, I chose the following problem:

How many legs and tails do 5 cats and 4 birds have altogether?

I chose the numbers to be big enough so that they wouldn’t be tempted to do it in their heads, but not so big that the drawing would get overwhelming.

The students did not disappoint.  Their solutions included a full range of drawings – from none to very detailed.  Everyone got the correct answer, although it took some several attempts.

First, there were the students who drew very elaborate animals and spent much more effort on the pictures than on counting the legs and tails.


Then there were those that still clearly drew out all the animals but in a much more minimalist way.


Then there was the student that just drew lines for all the items that needed to be counted.  What impressed me the most here is that not only were the correct number of lines drawn, but they were also counted correctly from the first try.  I’m not sure that I’d be able to do that!  Oh, and then the student went on to draw an elaborate farm scene, but only after solving the problem.


Finally, there was the student that didn’t draw anything at all but computed everything in their head.  I did try to convince them several times that drawing would help with keeping track, but the student either didn’t believe me or wanted to prove that they could figure it out without those silly pictures.

The student obtained the correct answer from the third attempt but then was able to explain it by saying that 5 fives is 25 and 4 threes is 12 which together make 37.


We have since solved several similar problems and whereas the tendencies of the students have remained the same, I did notice ever so slight movements towards that happy medium.

We had nice discussions with the students about whether pictures help and whether it is important to draw that fancy mane on the lion to convey that it has 4 legs.  They laugh and agree that it is not necessary to draw that mane.

But they draw the mane anyway.  Because it’s just too much fun.  And I let them.  At least for now.


About aofradkin

I enjoy thinking about presenting mathematical concepts to young children in exciting and engaging ways.
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