Shall we close up shop?

I don’t think it’s time to close up shop, because of the long-term purposes of a site like this. I believe the passage suggests closing up shop only if the goal is simply to admire the problem of these mistakes and misconceptions. But I think the conversation has been, and is productive, and the purpose is to start looking for causes instead of trying to document more symptoms. If we’re to generate research concerning the “evolution of expert understandings” as the passage indicates, then looking at these mistakes yields some key insights into where things are going wrong in students’ mathematical development.

I generally classify student mistakes as either “naturally occurring” or occurring because the student is trying to do something they were shown or told but never really understood. For example, I worked with a 4th grade student in a gifted program who answered 83 x 24 = 192 on a placement exam. His work revealed he was trying to use the multiplication algorithm, and he clearly didn’t recall the steps correctly. Which means he didn’t understand them. What’s worse, he didn’t bother to use any number sense to throw up a red flag that his answer wasn’t anywhere close. I don’t think this mistake was “naturally occurring.” Therefore we can use this mistake to better examine the learning pathways and progressions that resulted in this student’s mistakes. My guess is that he was given the algorithm prior to understanding the distributive property, and that can inform future curriculum decisions.

Sharing and discussing these mistakes is very helpful for learning about students’ mistaken intuitive/natural notions (like the Veritasium science resources) as well as the notions students acquire from the sequencing and instructional choices professional educators make. Either way, I see value in this work that can really help students.

I think that that statement assumes that mathematics teachers are all universally aware of the misconceptions of students – which, by the way, they are not. Your site fills a valuable space for teachers to discuss these ideas from a concrete perspective – the mistakes that we see in students.

Granted, I agree that we need a lot more discussion and research on long term impacts of teaching as nearly every study I have ever read is focused on units of time like “a year” or “a unit” and nearly none of them looks at the generational affects of teaching.

We also have another site, for which I would like more submissions if possible, that is attempting to document student thinking when it is more successful. See http://maththinking.org.

I notice that there are many useful specific repeated errors, and that your posting the errors, particularly with the grouping, is helping me determine what the basic problem is. When I teach, I get groups of answers, but rarely enough with enough time for me to “see” the problem: oh, kids don’t get how addition works with base 10. What if I teach them with base 5? (pennies, nickels, quarters)
I don’t teach elementary math, but I do notice, based on SBG information, that there are a bazillion (roughly) skills if we break things down small enough: this project lets us break the errors down in similar fashion. I am grateful to you.

Interested in another opinion related to the work here. I had an ed prof when I was an undergrad who was opposed to ever putting anything on the board that was not correct. His reasoning was that students might only remember that they saw something on the board. I have long felt that dissecting certain misunderstandings with my students – even if it meant I wrote those misunderstandings on the board – is a crucial part of our learning process. This site has alerted me to a number of mistakes that might be emerging in my students’ minds and I have used them as examples in class to spark discussions of why something is not correct and, more importantly, why someone might think that it is correct.
I say keep on!