Last weekend, a group of math teachers on Twitter formed what I believe to be the world’s first Common Core State Standards Mathematics draft (#CCSSMdraft from here on out). To read about how the #CCSSMdraft came about, click here.
Here is how the draft went down:
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Back in the fall, I introduced my seventh grade pre-algebra students to three-act math tasks with Dan Meyer’s Taco Cart. We had not yet investigated the Pythagorean Theorem, so the students were measuring with rulers and setting up proportions to answer the question “Who reaches the taco cart first?” It took them quite a bit of time as they had to refine their measurements, carefully record more data than should have been needed if we were using the Pythagorean Theorem, and struggle through the task in general. Every day we worked on the task, they would beg me for the answer, even though they had not got to that point yet. I let their curiosity fester as they plugged away for 15 minutes per class for a week. By the end, some had almost given up. Let’s just say it was a bit tedious.
Fast forward to late April.
We spent last week deriving the Pythagorean Theorem from the diagonals of squares and rectangles. This week we worked on formalizing the Pythagorean Theorem and practicing solving problems. Today, I brought back Taco Cart, Part Deux. Students instantly identified that using the Pythagorean Theorem would save them a tremendous amount of time in completing the task. In fact, it took the majority of the class about 15 minutes in total this time around. Even though they were excited to see Act 3 the first time we did Taco Cart, they seemed much more satisfied this time. Something about learning a skill that made their life easier.
Yep, the powerful motivator known as tedium.
Recently, Michael Pershan conducted an investigation into student intuition of exponents. Specifically, he was targeting whether students fall back on the heuristic of multiplication when faced with exponent expressions with which they have not had explicit instruction. Namely, rational exponents, negative exponents, and the exponent 0. You know, those pesky exponent expressions like 4^-2, 10^1/2, and 5^0. I’m really interested in his idea of students falling back on concepts with which they have confidence, such as multiplication or addition, when they are not confident with the problem in front of them. Hopefully, these investigations/studies will result in an index of common student misconceptions and errors, and teachers can use this resource as a planning tool for their own lessons.
I decided to steal borrow his exponents survey and modify it for use with my students. This is what I used with my students, students who have had instruction in positive integer exponents, but nothing further. If you’re interested in the 96 results, you can view those here.
While analyzing the results, I was not surprised to see the common error of -16 for 4^-2 and 5 for 10^1/2, or the inability of any student to answer problem 4 (which was only included as a baseline for student confidence). What did perplex me was some of the student answers to 2^3 and 10^1/2, specifically 2^3 = 6 and 10^1/2 = 50. On face value, you could look at these mistakes and easily see how the student made the error (2^3 = 6 by adding 2 three times, and 10^1/2 = 50 by multiplying 10 by 5). That’s the easy part. What I’m interested in is why these students made those mistakes. What in their conceptual context and understanding of mathematics led to these errors? I think I’ve formed a hypothesis, so here it goes.
| HYPOTHESIS: Students fall back to the the procedure with which they have confidence concerning their conceptual understanding. |
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Let me explain this hypothesis first with respect to the math mistake 2^3 = 6. This mistake was most common among my lower-ability students (you may notice the “Math 7″ at the top of some of their surveys); they generally have very low number sense. These students have no difficulty multiplying 2 x 3 and getting the correct value of 6. But I am not convinced that this ability is because they have a strong conceptual understanding of multiplication. For example, these students were asked (after the survey) to model the multiplication expression 2 x 3. The most frequent response was a variation of this:
* * x * * * = * * * * * *
While the students are modeling quantity, they are not modeling multiplication. This, to me, shows linear thinking. I was hoping to see a model along the lines of an array, which would lean towards an understanding of multiplication as area. I have a suspicion that a lot of multiplication instruction in the primary grades is still relegated to memorizing multiplication tables and learning to apply algorithms to problems. So 2 x 3 = 6 is a memorized “fact” for the students. 2^3, which through my discussions with students is viewed as 2 x 2 x 2 (because, you know, exponentiation is just “repeated multiplication”), becomes a multiplication expression which students have not memorized. Without a conceptual understanding of multiplication, they revert to the procedure that they do have an understanding of, addition. So 2 x 2 x 2 becomes 2 + 2 + 2, which obviously has a value of 6. This would make sense to the student who has been drilled that multiplication is just “repeated addition.” And since addition is a way to think linearly, which is a lower level of cognition in mathematics, it becomes the go to for students who are conceptually confused. Now I can’t be 100% certain that this is the reason for this type of math mistake, but it sure is a good place to start.
I have some ideas about the math mistake 10^1/2 = 5 or 10^1/2 = 50, but I’ll save that for part 2 of this inquiry. Meanwhile, if you have your own lines of thinking about this, or want to comment (or take issue with, I’m a big boy and can handle it), please do so in the comments below. I look forward to driving this conversation forward, so we can start to plan for these misconceptions ahead of time.
Before my complete investigation into Birds Eye Foods marketing strategy is complete, I wanted to give a little update on our current estimates:
I’ve been thinking about the marketing strategy of Birds Eye Foods every time I use this particular product:
