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Follow-up:  I found my error.   Apparently 6-2 = 4 not 2.   So yes.  Distance is the area under the curve.  So in this case, one can use the formula for the area of a trapezoid (just use the correct height) or one can calculate the integral.

Do you see my error ?

So some questions for teachers:  How do you grade a student who makes this type of error?  (Submits just the top portion?  Submits both portions?) What messages do our grading practices send to student? How do we encourage students to move past the answer?   To realize the importance of understanding the concepts and to make sense of the process?   And to not give up when something goes wrong?   (Please share your thoughts in the comments.)

So I already have a change of plans.   I think math is all about getting dirty.   Struggling with the tough problems.   Discovering the boundaries and/or working within them.

Me sharing my mostly correct work on calculus review questions does not fit this description and really is not that interesting. (However, I am interested in exploring questions like those listed above.)  So I am going explore math tasks from all grade levels.

To begin with I want to share my son’s work on an Inside Mathematics 2nd Grade Performance Assessment Task

Sheep and Ducks.

Here are the questions:

1. How many legs on 1 duck?
2. How many legs on 4 ducks?
3. How many legs on 5 sheep?
4. Next to the barn is a pen with 2 sheep and 3 ducks?  How many legs altogether? Show how you know your answer is correct.
5. One of the farmer’s pens has a high fence around it.  He can see 32 legs under the fence.  How many sheep and ducks are in this pen? Show one way to have sheep and ducks with 32 legs in all.  Show another way to have sheep and ducks with 32 legs in all.

Try the question yourself.

Where do you think your students or children might struggle?

According to Inside Mathematics the following are areas of difficulty:

• Organizing their work.

• Showing their work.

• Working backwards from a given number of legs to create correct combinations of sheep and duck legs together

Here is my son’s work.  You will see most of these difficulties in his work. 

I was surprised that I had to push him to explain/show his thinking.  He was quick to write the an answer, but did not want to record his thinking as he solved the problems.  To get me started, I recorded what he told me about the process.

I was also surprised how quickly he wanted to quit when he first encountered a problem.  In problem number four he very quickly concluded that the answer was 10.  When I  had him record how he solved the problem he was very disappointed when the answer was 14 not 10.  We had a conversation about how the process was helpful, but he still felt defeated by the fact that he was initially incorrect.  He carried this with him into the next question and did not want to try to solve it.   When I suggested he draw the legs he regained his confidence.  But once he had one answer, he did not want to consider (draw) other possibilities.  I asked him if he could have all ducks and he used the picture to explain why this was a possibility.  He was also able to explain why there could be all sheep.

So this little conversation takes me back to the questions above.  I think it really is about building those mathematical practices as habits.  But I want to hear from elementary teachers and parents.  How do you help students develop these habits?  How do you know if you are successful?  How do you help them embrace getting dirty?