Organizing by math topic has its advantages, but it masks important things about students, because it's not organized by either the nature of the mistakes or by the student thinking that generates those mistakes.
Consider these three items that were listed as preconceptions.
To me, each of these is strongly related, despite the fact that they are listed in the table under the different topics of fractions, distributions, and non-linear functions.
In the first example, the student multiplies by two everywhere they can, which is top and bottom. In the second example, the student multiplies in one place when need to multiply in everywhere. In the third example, they square both terms, but in doing so, they miss the operations of x*y and y*x. Although the topics and grade levels are different, something about nature of the mistake is similar. Building a table of preconceptions solely around topics masks those similarities.
Next, consider these examples:
“You can’t divide smaller numbers by larger numbers.”
“Division always makes a number smaller.”
-10 > -6
1/3 > 1/2 (because 3 > 2)
-10 > -6
1/3 > 1/2 (because 3 > 2)
These preconceptions were listed under the different categories of "integers", "exponents", "fractions" and "inequalities". But to me, all of these are related by the fact that they seem to stem from ideas that students are have around whole numbers and whole number operations. Children build up a lot of intuitions, ideas, reasoning, and procedures around whole numbers, and we are seeing here that students try on and rely on those ideas in a variety of ways as they learn about fractions and integers and exponents.
Placing the difficulties in different categories masks the fact that they seem to have a common origin-students' prior understandings and thinking around whole numbers.
What's the point?
I think there are lots and lots of potential connections and stories to tell by looking across the difficulties and ideas students have. I believe that a list of student difficulties is only useful if we do the work of trying hard to make sense of it by looking for connections and telling stories that help us see links between students' thinking and mathematical thinking.
Often it begins by asking, "Why would a student do this?" or "What does this mistake imply?" I often ask, "What good ideas do they have that would lead them to do this?" or "What could a student be trying to figure out what to do?" This often leads me to ask, "What ideas seem to be in place?" or "What ideas are they missing?" In the process, I often come to see my students and the discipline in a different way.
I'm curious to hear from other, what connections or stories they can see from the list?
