In early October, I presented at NCTM Hartford with the incomparable Heidi Fessenden (@heidifessenden / blog). Our session was called “Strategies for Cultivating Mathematical Thinking for All Learners.”
Slides available here.
The first blog post, contextualizing these strategies and providing some research that inspired us, was published on Monday.
The second blog post, published on Tuesday, detailed the first of three strategies from our presentation: rewording the standards for mathematical practice (SMPs) in kid-friendly language, or “mathematician statements.”
The third blog post, published on Wednesday, discussed the important practice of assigning competence.
In order to implement the first two strategies, there need to be opportunities for students in your classroom to do math. Using high-leverage routines can provide these opportunities, with significant benefits for both students and teachers.
- Use high-leverage instructional routines
to give students opportunities to engage with tasks mathematically — focusing more on the content and less on the (already internalized) structure of the routine
Heidi and I modeled a Which One Doesn’t Belong, by Simon Gregg (@simon_gregg) and Pam Wilson.
Participants had so much to say! They were fascinated by the notion of square numbers, and the colors, and the partitions…
We talked about how this routine is more than just fun, but it can provide entry points into content across all mathematical domains (e.g. number & operations in base 10, measurement, geometry, etc.)
Geometry has to be one of my favorite domains to use WODBs.
I shared a story about a special education program within my school. It’s the district-wide program for students with language-based learning disabilities, which means that students with significant needs are bussed from across town to participate in our program. These students receive substantially separate ELA. Sometimes, their IEPs say that they should receive substantially separate math instruction, too. The elementary special educator and I work very hard to get as many students as we can back into the general education classroom. Sometimes this takes a few years.
We were working with some fifth grade students who had been pulled out for years — all of which had focused on number operations. They had not done any geometry in years. (I’m stating this mostly as a matter of fact for context, and not a judgment on their past experiences. Their teachers worked so hard to help them understand number concepts more conceptually.)
Students in this language-based program do not do well in the typical geometry unit, which has dozens of vocabulary front loaded, and flying at them at 100 mph. We still wanted the students to develop the language needed to talk about these geometric ideas. We also wanted them to develop their spatial reasoning and justify their thinking.
We realized that we could teach almost all of the geometry standards through two routines: Which One Doesn’t Belong & Quick Draw. The teacher decided to use them 4x week, for 5-10 minutes at a time, to build up geometric understanding over time, in place of an intensive unit.
We started with some #WODB like this. Students attended to notions of “straightness” — some figures were all curves. Some had straight lines. One wasn’t closed. Students also talked about lines of symmetry, and parallel lines. From there, we were able to build our definition of polygon.
It was an amazing experience. The teacher was able to focus on developing the content in a coherent way, rather than having to plan many discrete activities. The students were able to attend to the mathematics, instead of the procedure for an activity, because they had quickly mastered the structure of the routine.
Students continued to work on more and more precise language.
Look at how this student developed his idea, eventually reaching the idea of ‘parallel lines,’ with minimal teacher support. This student is an English language learner who is also in our language-based learning disability program. He transferred to the program when he was in third grade, after he had spent enough time in the US to diagnose a disability (and not just that he was new to the language). As a third grader, his only strategy for adding two numbers was to count all — fingers, tally marks, etc. I was particularly proud of his development over the course of his time at our school, especially this WODB experiment. He is now in 7th grade, in general education math. He achieved a proficient score on our state test last year — his first ever.
STUDENT: The one on the bottom left is the fattest.
ME: What do you mean by ‘fattest’?
STUDENT: It goes out like this. (he extended his arms up and down, parallel, and then bent them outwards, away from his body)
ME: Oh! Hmm… how can we say that in words?
STUDENT: It doesn’t have those parallel lines.
I nudged him forward, but I never cued him.
He played the starring role of the mathematician.
The special educator also built up a math word wall in her room. The strength in the word wall was not necessarily the diagrams, or the definitions, but that it was student-driven.We always started with the informal language, and introduced words as they came up organically — as students saw a need. Students noticed the phenomenon of differently sized angles. They needed a word to describe it. So: angles was added. Then there were some special angles we noticed on some shapes… again and again, they had these nice, neat, square corner angles: right angles was added. The word wall grew and grew.
We encouraged students to use the word wall when crafting their arguments. “Stop. Think to yourself for 10 seconds. Is there a word on the word wall that can make your argument more clear?”
When students felt stuck — which happened regularly — we would sometimes point to the wall. “I notice the word ‘parallel’ is on our word wall. Can we use ‘parallel’ to describe why one of these shapes doesn’t belong? How? Or why not?”
The word wall was dynamic — it was constantly changing, and we were constantly interacting with it — and it all became a part of our classroom routines.
How do we help all students identify as mathematical thinkers?
- Reword the Standards for Mathematical Practice using student-friendly language
- Assign competence
- Use high-leverage instructional routines
- …and that’s just the beginning.
How do you envision using these strategies in your own classroom or school? Tweet us @jennalaib @heidifessenden