Book Review: Mathematical Thinking and Communication

Sometimes having a blog pays off and you get advance copies of upcoming publications. This is one of those times…

The good people at Heinemann sent me a book entitled, Mathematical Thinking and Communication: Access for English Learners by Mark Driscoll, Johannah Nikula, and Jill Neumayer Depiper. If you are regular reader of this blog (and why wouldn’t you be?) you may be thinking, “English learners? I thought this blog was about students with disabilities, why are we talking about English learners?” That is a good question, faithful blog reader, so I’ll address it first.

There is a well documented disproportionate representation of English language learners in special education. Amanda Sullivan notes this may happen because”both underreferral and overdiagnosis occur because of misunderstanding of the educational needs of students identified as ELLs (Case & Taylor, 2005), poorly designed language assessments (MacSwan & Rolstad, 2006), and weak psychoeducational assessment practices (Figueroa & Newsome, 2006).”

ELL special ed disproportionate table

Taking this into consideration, the following review is going to focus on the pedagogical framework highlighted in Mathematical Thinking and Communication: Access for English Learners. Even though most English learners do not share the same cognitive challenges that some students with disabilities do, they do share challenges relating to expressive and receptive language and communication. Thus the strategies to create access for communication of mathematical thinking can be shared by teachers of all learners.

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Productive Struggle vs. Frustration

This past Friday, I gave an introductory presentation on the educational ramifications of new brain and psychological research, specifically, Carol Dweck’s Mindset. What came out of the discussion during the session, was that our school already does a fairly good job of inherently implementing most of the underlying themes in Dweck’s research.

What we realized we still needed to work on as a school, was allowing students to struggle productively. Robert Kaplinsky recently posted his ignite talk from the Northwest Mathematics Conference. Kaplinsky gives a very accessible account of the differences between productive struggle and what he calls, “unproductive struggle.” In our school, “unproductive struggle” is frustration.

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An Inch Wide and An Inch Deep: A Call To Action

One of the most popular ways to critically describe mathematics education in the United States is “a mile wide and an inch deep.” The TL;DR is that most mathematics education focuses on too broad an array of topics with a lack of emphasis on conceptual understanding and critical thinking.

My worry is that most special education math classes are an inch wide and an inch deep. I ran across this chart from Browder, Spooner, Ahlgrim-Delzell, Harris, Wakeman (2008).


Demonstrated here is a clear focus on an extremely small amount of topics and the only one investigated in any kind of depth is financial literacy, which admittedly is an extremely important topic for students with disabilities. For students with disabilities to be successful members of their communities they need to be financially literate. But this need should not preclude students with disabilities from exploring other mathematical topics.

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Contemplate then Calculate: Dominoes

The math department at my school is working to implement instructional routines in our classes for several reasons. The first reason is to increasingly apply the standards for mathematical practice on a daily basis. The second reason is to improve our communication as a department by creating common language centered around shared routines and activities.

One of the instructional routines we plan to implement is Contemplate then Calculate (#CthenC on twitter). Created by Amy Lucenta and Grace Kelemanik at the Boston Teacher Residency, #CthenC is a highly structured routine that simultaneously allows for open mathematical thinking and problem solving. Contemplate then Calculate highlights looking for and making use of mathematical structure in problem solving.

In an effort to assist the implementation of Contemplate then Calculate at our school, I created this activity to do with my class as a model for implementation and discussion.

Here is the task:


How many dots?

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Hollywood Doesn’t Understand Teaching

In 2013, Dan Meyer released a video which edited together all of the negative comments about math made in Hollywood movies. It was called Hollywood Hates Math. Recently, two television shows premiered that center their stories around the professional lives of teachers.

Here are a couple of clips:

Art class in Teachers on TV Land

Spanish, History, and Gym classes in Those Who Can’t on TruTV


Besides filling the stereotypical gender roles of elementary and high school teachers, these scenes provide a litany of examples where teachers are portrayed more as “babysitters” than professionals.

Finally, here’s a bonus clip from a show you may already be familiar with, Girl Meets World. I’ll leave you with this question, what was the lesson plan?

Reimagining Inclusion with Positive Niche Construction

Every classroom is a bustling ecosystem of voices, ideas, and inside jokes. Every member of the classroom, teachers and students, is working together to form a cohesive learning environment. Each classroom is different based on the unique members that make up its ecosystem.

The broad range of experiences and perspectives brought to school by culturally, linguistically, and ethnically diverse students offer a powerful resource for everyone to learn more—in different ways, in new environments, and with different types of people. Every single person in this enormously diverse and ever-changing system has the power to serve as an invaluable resource for all others—students, teachers, and the community as a whole. Rather than constituting a problem for students and educators, the growing diversity in U.S. classrooms necessitates and encourages the development and use of diverse teaching strategies designed to respond to each student as an individual (Saravia-Shore, 2008).

The diversity of the classroom is at the epicenter of the learning environment. Diversity, however, can come to mean a variety of things.

Recently, I have been reading about the concept of Neurodiversity. Essentially, neurodiversity is based on the idea that brain diversity is similar to cultural diversity and the diversity of ecosystems.

We don’t pathologize a calla lily by saying that it has a “petal deficit disorder.” We simply appreciate its unique beauty…Similarly, we ought not to pathologize children who have different kinds of brains and different ways of thinking and learning (Armstrong, 2012).

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Numberless 3-acts

On Wednesday I posted a 3-act task that I planned to use in my class the next day.

Here’s how it went…

Act 1 asked students to watch a video and record what they noticed and what they wondered. As a usual modification for my students I had them make a T-chart graphic organizer in their math notebooks to record their observations. We watched the 1 minute video 4 times, twice for noticing and twice for wondering.


One student’s math notebook

Students noticed:

  • The plastic bag was being filled
  • There were some cheerios
  • There were some chex
  • There were some M&Ms
  • There was 1 raisin

Students wondered:

  • How many trail mix pieces altogether?
  • Why didn’t I add more cereal?
  • Why the whole bag on M&Ms was being put in the trial mix?
  • Why only 1 raisin?
  • How many cheerios, chex, and M&Ms?

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3-act Task: Trail Mix

Student engagement is a funny thing.

On twitter I’ve been pretty critical about using extrinsic rewards to increase student engagement.

Today was our 100th day of school (as calculated by our students!) To celebrate we made 100 piece trail mix. Our trail mix included: cheerios, chex, raisins, and M&Ms. Candy! Talk about extrinsic student engagement! Before we dove in to the rewards food, I gave my class the following problem:

We have 4 ingredients to make trail mix. How many different combinations of ingredients can we have if our trail mix only has 100 total pieces?

The students persisted through their work on this word problem, until they arrived at various solutions based on their calculations and personal taste. For instance, one student is allergic to nuts and could only eat the cheerios and raisins, so that impacted his work on the problem. The students worked diligently and happily ate the trail mix once they had arrived at a reasonable solution.

However, after class I channeled Graham Fletcher and Dan Meyer to try to make this mathematical experience a more rich one for the students. So, here is a preview of the 3-act task we will be doing tomorrow in class…

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Scaffolding For Executive Functioning

Over at Reason and Wonder, Michael Fenton is exploring the possibilities for using Alex Gendler video puzzles in the classroom. Michael’s wonderful take on these rich resources, reminded me of one of the main goals of this blog, to show how students with disabilities can access rich mathematics instruction.

As we began this school year, my goal was to model how our class valued perseverance and sense-making over answer-getting. I did this for a couple of my classes by using Gendler’s Zombie Bridge Problem video. The video is long and there are a lot of details to account for before you can come to a reasonable solution. This requires quite a bit of what is called executive functioning. Executive functioning includes (but is not limited to) the abilities to initiate a task, make a plan, prioritize information, organize information, think flexibly about strategies, and self-monitor (i.e. check your work). Sound familiar? My students tend to struggle with executive functioning skills and this is often where my scaffolding is targeted.

To help scaffold my student’s executive functioning while solving the Zombie Bridge Problem, I used EDpuzzle. EDpuzzle allows a teacher to modify an already existing youtube or uploaded video by cropping it, including voiceovers and adding questions. Here is how I used EDpuzzle to scaffold the Zombie Bridge Problem.

Screen Shot 2016-02-29 at 10.19.25 PM

First, I cropped the video to exclude the solution. As anyone familiar with 3-acts knows, the solution is vital, but should come after students have had time to explore first! So it was gone.

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3-Act Task: The Doubling Die



CCSS.MATH.CONTENT.7.SP.C.5 – Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Act 1

What do you notice? What do you wonder?

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