Construct Viable Arguments and Critique the Reasoning of Others

Common Core Standards for Mathematical Practice. They sound really cool, but also kindof really scare me. Something new I’m working on to push SMP.3 – Construct viable arguments and critique the reasoning of others. One challenge I’ve had this year is that I ask students to explain their answers all the time (actually my highest score in the student survey I gave earlier this year). However, I haven’t seen a marked improvement in explanations from many of my students–particularly students that have a tougher time with math. I got this idea from a Doug Lemov post on “The Art of the Sentence”. Instead of just “explain your answer”, I require students to explain their answer in exactly one well-crafted sentence, pushing them to be both concise and clear. This is a difficult skill, and providing a sentence starter as Lemov does works wonders for pushing my students to articulate more precise thinking. We’re working on 3-D geometry in class right now, and I posed this question to students last week, from Five Triangles. Take a second to try it on your own first.

cornered cube


The figure above is created by taking a cube (the dotted lines) and cutting a triangular face off of each corner.

1. How many squares are in the new figure?
2. How many triangles are in the new figure?
3. How many edges are in the new figure?
Explain your answer to question 3 in exactly one well-crafted sentence. Use the sentence starter below.
I think there are ______ edges. I found my answer by counting …..


This was a hard question for my students, but I was really impressed with the reasoning they put into it, and the quality of their writing. In fact, one student surprised me with a much more efficient method of counting edges than I had found! (She realized that each edge borders exactly one square and one triangle. There are 6 squares, so 6 squares times 4 edges each = 24 edges.)

Anyway, my basic protocol here:

Ask students a conceptually challenging but accessible question (conceptual questions from our Common Core books work well because they are really difficult, but their multiple choice structure makes them more accessible for everyone, and writing about them takes away some of the misery of multiple choice questions).
Give them a sentence starter and time to write.
Circulate and read sentences as students write.
Pick three students to cold-call to read their sentences (try to get a range of conciseness and clarity). Take volunteers if more students want to share.

Here’s where I’m unsure. I’ve gone two directions. One is to let students talk about which answer they like best until they come to a conclusion. The other is to have them talk about which explanation they like best, pushing them to articulate that longer (and more information) is not necessarily better. Students generally want the correct answer, but I got some great discussion out of analyzing explanations one day–this works best when the question is easier or the answer is trivial to check, and I chose three high-quality but different explanations.

My biggest challenge so far has been to hold students accountable for writing quality sentences–I haven’t graded any of these, and time is an issue as students need time to work the problem and then write, and this is wildly divergent student by student.

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