It’s been a busy school year, as always, and my blog has been quiet. I have been working on some deep thinking about how students learn mathematics, and haven’t quite known what to post here about that thinking. I have, however, given two presentations this year on ways that I am incorporating spatial and visual reasoning tasks in my high school math classes, so perhaps that is a good topic to post here as well.

Frequent readers will be familiar with my post on “seeing” mathematics, which was inspired by this book I read last spring. That led me to some exciting new connections in academia, who have been very supportive as I develop my ideas for building spatial reasoning in my secondary math students.

The way these ideas have affected my math classes is subtle, but they have become a key component in my intentional re-framing of class time this year. While traditional math classes follow a fairly standard routine (discuss homework questions, direct instruction of new topic, independent practice), and progressive classes may include more inquiry, discussion, or investigation activities, there is often not an explicit effort to teach spatial visual reasoning skills. Small changes to class routines can fit any teaching model, though (and my classes involve many types of instructional methods). To that end, I read a few good books over the summer:

Creating Cultures of Thinking, by Ron Ritchhart

Syllabus, by Lynda Barry

Using the ideas around routine and developing creativity from these books, I have added a few small routines to intentionally bring spatial visual reasoning skills into my classes this year:

- We draw more. Exit tickets are often drawings with annotations. This gives me a quick view into student thinking and understanding, and requires that students create multiple representations as a way to improve their mathematical thinking.
- We share more. I have always encouraged multiple solutions methods in math class, but now we discuss them more as a group. Students engage in metacognition as we name different types of reasoning (proportional reasoning vs algebraic reasoning vs visual reasoning, etc). In class it might look like putting 3 different solutions on the board and discussing why they are the same, how they are different, etc. Like this picture:

I have big plans for getting students to a level of blended cognition, where they can fluently move between multiple forms of representation, that is similar to what some of my advanced students produced a few years ago (showcased in this post on visible thinking). I suspect that scaffolding the size of the task throughout the year would help, and my next iteration will be to build this into my unit planning.

For now, the ideas continue to percolate, and I am already enjoying the effects of these small, intentional changes to my classes. I have more of a window into my students’ mathematical thinking, and get to celebrate their individual strengths much more frequently! I am now very familiar with who prefers to think visually, or who reasons numerically, and the occasional student who actually visualizes the act of writing mathematics when mentally solving problems. It fascinates me, and also makes it easier for me to help them construct meaning as they learn.

@rutheichholtz I do hope you keep “Deviating from the Mean” active forever. Your insights and classroom reflections are truly valuable to the teaching community both year and far. This post is one of many examples of deep reflection that models the kind of thinking that all school leaders strive to fill their schools with. Snif Snif…. we are going to miss you Ruth!

Ruth – a challenge: To honour your recognition of different thinking styles and strengths (and to celebrate your own gift of the captivating gab), could you also turn this post into a 3-minute video?

@adamcaplan challenge accepted! I’ll work on that… in May? Since you didn’t stipulate that my face has to appear in the video, I’ll try and put something together with examples that I can talk over. Sound fair?

Ruth –

How lucky are your students that you have taken the time to celebrate their individual strengths! Your objective of having all students reach a level and awareness of blended cognition is truly admirable. You have much to share with others who are also searching for that all-important connection with every student and their learning.

@jancampbell how lovely of you to leave such encouraging comments. Thank you!

@rutheichholtz your mention of encouraging multiple solutions serves as an excellent reminder to me to encourage this more often in my classroom. Inspired by you, I have slowly been transforming my math studies classroom into a more inquiry-based learning environment. I love the idea of having students share and discuss multiple solutions with one another. That is something I will need to try in the coming weeks!

I’m wondering if you might be able to provide an example of an exit drawing with annotations you have students do. Possibly one that you do in math studies? Thanks for this lovely post!

@sregli I will show you some on Friday. I realized when I wrote this that I should have taken some pictures of the visual exit tickets. I have really liked using them, and want to incorporate them even more often than I do, because every time I get SO MUCH out of them (and I think the students do also). I hope you try them out too and let me know how it goes!