Simple Prompts Can Lead to Complex Mathematical Thinking

This post is inspired by chapter 8 of Steve Leinwand’s book Accessible Mathematics.  If you haven’t read this book, do it!  Leinwand is a leading voice in the push for math instruction that makes sense to students and will lead to longer lasting mathematical understanding.  Chapter 8 is entitled, “How Big, How Far, How Much?” and in it Leinwand encourages this instructional shift:

Tie the math to such questions as How big? How much? How far? to increase the natural use of measurement throughout the curriculum.

He goes on to say that measurement as a mathematical skill is often a “skipped chapter,” but is also one of the most pervasive life skills in the mathematics curriculum.  Leinwand goes on to encourage teachers to incorporate measurement as “an ongoing part of daily instruction and the entry point for a larger chunk of the curriculum” (p. 46).

The game shelf!

The game shelf!

Now you may be saying, “But I have a million other goals and standards and expectations and test prepping and whatnot that I have to do before I teach the kids to use a ruler!”

Well, lucky for you the Standards for Mathematical Practice also have you covered.

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#MTBoS Mod: Estimation 180 Edition

As I have become more involved in the MathTwitterBlogoSphere (#MTBoS) it has been a pleasure to share the really great material created by the MTBoS with other teachers in my school.  Our school has three sites.  There is a grammar school (elementary and middle), a high school and a post-high school which assists students in the transition from school to “the real world” in a more targeted, vocational way.  Earlier this year I was able to share sites like Estimation 180 and Would You Rather with teachers at all three sites.  One of the middle school teachers has integrated Estimation 180 and the work of Andrew Stadel into her classroom culture.  What follows is her reflection on this process in the form of an edition of the #MTBoS Modification Series.  You can also read the first edition that featured Mathalicious.

#MTBoS MOD: Estimation 180 Edition

Students in one math group have been using Estimation 180 as a starting point for further exploring the concept of estimation.  They know that to estimate means to make a guess based on known information.  When you are asked to estimate it is often within a context and you must use any relevant information (known or given) to guide you.  After several months of answering Mr. Stadel’s prompts on the Estimation 180 website, students created their own original estimation projects. They were encouraged to research a topic of interest and provide enough information through facts or visuals so that classmates could make a reasonable guess.

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There’s More Than One Way to Skin a Task

Teaching is really only as valuable as the learning experiences and opportunities created for the students.  Crafting and choosing which experiences and opportunities are available to students should be at the forefront of any teacher’s mindset when planning lessons or units.  So let’s delve a little deeper into crafting and choosing learning experiences.

One of my classes has been using the geometric study of area in order to practice and apply multiplicative thinking.  Crafting and choosing specifically rich tasks that engage the students has been one of my major goals for this unit.  So I’d like to just investigate different iterations of reasonably similar tasks that apply multiplicative thinking in the geometric context of area.

You could just start and end your search with:

Draw a rectangle with an area of 56 units.

This certainly gets the job done.  Students have to think about factors of 56 and which make the most sense.  The great thing about this task and others like it (check out Open Middle for more) is that there are a multitude of correct answers.  Having many “right answers” leads to deeper group work and richer class discussions.  The drawbacks to this task are the lack of visual support and the separation from a concrete representation or real world context.  Also the language processing comprehension of this task is high, so students who struggle in this area would require additional supports.

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Automathography: My Path with Math

What follows is a piece I wrote as an assignment from Justin Lanier‘s smOOC called “Math is Personal.” If you are just starting out as a math teacher or are a seasoned vet, I highly recommend participating in Justin’s course. You will think more about your personal views about learning, teaching and mathematics than you normally allow yourself during the school year.

So here goes…

This is the story of a boy. This boy loved reading and writing. He loved reading books from “Goosebumps” to “Roll of Thunder, Hear My Cry” to “Of Mice and Men”. He read and he wrote as much as he could. When things were good, he wrote about them. When things were bad, he wrote about them. One day in sixth grade, he had gotten mugged on the way to school and he told his teacher about it. His teacher told him to write about it. So he did, and it made him feel better. Writing was his life and as he grew older it became more and more important to him.

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#MTBoS Mod: Mathalicious Edition

First, I’ll give you a little background.  In the past I have discussed my use of curriculum created by the online math education community known as the MathTwitterBlogoSphere (MTBoS).  I have already written about the use of and modifications for Fawn Nguyen‘s Visual Patterns and Christopher Danielson‘s Which Shape Doesn’t Belong.

So, I decided that what I can add to the community is to share how our classes are using the material from the MTBoS and how we are modifying it for students with disabilities.  This is the first post in the series I am calling #MTBoS Mod(ification).  The first subject of the series is the wonderful curriculum development website called Mathalicious.  If you haven’t seen or used the work of this website, please take a minute to go to the link and poke around, but then come back here to see what modifications we made!

So here goes…

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A Student’s Eye View

Ashli Black, over at Learning to Fold, recently posted this little bit of wit and whimsy.  The post essentially recounts her experience in algebra classes and compares it to the experience of contestants on an extremely confusing, quite vague, and thus hilarious math game show.  Ashli makes the point that, “As that kid without conceptual understanding in algebra, this skit is pretty much exactly what it was like in class for me. Confusing, almost no stated rules I understood, and at any moment the scene might change or I might be shoved in a box for not achieving Wangernumb.”  Ashli considers the difference between teaching for conceptual understanding and teaching for procedural understanding in her post, but it got me thinking about my own students.  I often think my students are holding their breath, waiting for me to tell them their answer was in fact “Numberwang.”


My goal as a special educator is to communicate the day’s lesson or task so the students will be able to access, understand, and apply the mathematical content.  This often leads to accommodation, modification, and differentiation of everything for everyone.  When one thinks of accommodations the first things that come to mind are standardized testing accommodations.  The general list usually looks something like this:

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To Calculator, or Not To Calculator, That is a Question

When it comes to planning lessons in special education, or general education for that matter, the goal is for all students to be able to access, understand, and be able to successfully apply the content to show evidence of full understanding. The application can take many forms: performance assessments, formative assessments, summative assessments, teacher observations, etc…

But how do you get to that final assessment piece? This post is about the planning process that goes into successful lessons for all students.  Let’s begin with Universal Design for Learning (UDL). UDL is a model for planning lessons and units that creates access to the content for all students.  Here is a cartoon that embodies the philosophy of UDL.


One major component of planning in special education math classes is prioritizing the mathematical goals and the needs of the students to access the mathematics in a lesson.  A Teaching Children Mathematics article from 2004 suggests the following steps for beginning to plan a successful math lesson for students with disabilities:

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Humans vs. Zombies! (or How We Learned About the Coordinate Plane)

This week began our study of the coordinate plane.  I used the first lesson of Transition to Algebra’s unit 6 as a pre-assessment.  It proved that I needed to take a couple steps back and address many of the basic concepts relating to the coordinate plane (axes, integers, ordered pairs, quadrants, etc…) in a more direct way.  Our class goals are pulled from the Common Core State Standards Initiative:

Screen Shot 2015-02-03 at 8.23.00 PM

First, I used this game as an anchor for plotting ordered pairs, then the students did some individual practice on worksheets.  Today we played another game…

Humans vs. Zombies!

My very crafty assistant teacher, Ms. Avellino, took a game from a website and turned it into this… Continue reading

A Tale of Two Tasks

Differentiation is a widely accepted (and debated) strategy for meeting the needs of a diverse range of learners, especially in special education classrooms.  According to Carol Tomlinson, “a differentiated classroom provides different avenues to acquiring content, to processing or making sense of ideas, and to developing products so that each student can learn effectively.”

But what does it look like in practice?

First, let me describe our class setting to give you some background.  I teach at a self-contained special education high school in Manhattan.  The learners at our school range from students with learning disabilities or speech and language delays (which effect academic performance, but do not generally effect their physical appearance or how they react in social situations) to those with autism spectrum disorders or down syndrome (which effect socialization and communication as well as academic levels.)  Our math classes are mixed grade (9th graders with 10th graders and 11th graders with 12th graders) in order to create groupings that can best meet each student’s academic and social/emotional needs.  There are three concurrent math classes, which means our class groupings are no bigger than 8 students with a head teacher and an assistant teacher.  This leads to collaborative in-class groups that can be as small as 4 students to 1 teacher.

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A Better Shapes Class

Christopher Danielson recently released “A Better Shapes Book” for free on his blog. Before you read on, go take a look at it, download it, and enjoy!


Our Hallway Display

Since some of my classes are studying geometry this trimester this was a fortuitous release.  My students, who are in self-contained special ed classes, can identify benchmark shapes (squares, rectangles, triangles and circles), but we are currently investigating how squares and rectangles relate as quadrilaterals.  This book was the best way for our students to explore shape properties without having to read, write and remember a lot of vocabulary.  We were able to discuss what they saw and critique the arguments of classmates in a safe space, because all arguments were valid for one reason or another.  The elimination of the potential to be flat out “wrong” created a safe space for my student population.  As long as there was some semblance of justification, you were “right.” The students liked that!

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