# A Snapshot

First, a little background.

The theme of our spring unit is always financial literacy. As teachers of students with varying degrees of need, strength, and interest this means different things for different groups of students. One of my groups is working on selling tickets for our school play, Alice in Wonderland.

We sell tickets at two price points. An adult ticket costs \$10 and a child/student ticket costs \$8. This is partly my doing, because having two different prices sometimes allows my students to investigate more interesting mathematical questions. Today was one of those days.

Show-goers are also able to purchase play tickets in one of three ways: cash, check, or online with a credit card. My students record the type of ticket and the method of purchase for each order in a table. Students then represent this information visually using graphs. We will use these tables and graphs later on to reflect on the trends and patterns in the ticket sales to make suggestions to our play directors for future ticket sales initiatives. But that’s the bigger picture and I promised you a snapshot. So here it is.

I realized I had been giving my students too much information. As they recorded the total amounts of cash, checks, and credit, I was also telling them the type of ticket. Today we began our routine of using math to figure out the type of tickets using our knowledge of the ticket prices and total amount of money. I gave them this problem as a warm-up:

# 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.

# 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.

# 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.

How many dots?

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…

# 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.

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.

Encouraging students with disabilities to think deeply about mathematics has always been one of the goals of this blog.  But since the audience of this blog is mainly teachers, the goal is really to encourage teachers to encourage students with disabilities to think deeply about mathematics.

Duh! You’re thinking, “I asked 35 questions today! Numbers 1-35 on the multiplication fact fluency worksheet were math questions. This guy!”

But, the questions I’m referring to come after you ask those initial questions.  Sure, you proposed a math problem to your students or even better they proposed one to you based on some mathematical situation you presented, but then what happened?

Andrew Stadel recently wrote about and collected questioning strategies from the MathTwitterBlogosphere.  His focus was on strategies for asking questions before and after the launch of the day’s mathematical problem, task, lesson, activity, etc.  My focus has been on post-launch questioning strategies.  The stuck/unstuck questions and questions to explore student misconceptions.  In an NCTM article, which discusses warning signs of instructional moves that generally lead to taking over student thinking, the alternative teacher moves are also focused on asking questions when a student is stuck or has a misconception.

# In Which I Give A Survey About Math To My Colleagues…

Justin Lanier gave a fun, beautiful, challenging, and useful talk at the Global Math Department on Tuesday.   His talk centered around teacher’s views of mathematics and how they can affect their student’s views.  Please take sometime to watch Justin’s presentation.  It’ll make the rest of this post make much more sense!  Or at least visit Justin’s blog where he issues a call to action.

I took Justin’s call to action and gave a google survey to my colleagues.  I sent it in an email to every staff member at my school.  This included administrators, math teachers, non-math teachers, related service providers, para-professionals, etc.  In other words EVERY staff member at my school had the opportunity to answer Justin’s question.

This is what happened…

# Unscaffolding Math Problems

Several months ago, the new NPR show Invisibilia did a broadcast about expectations.  The main theme of the program was that the expectations others hold of an individual can effect the outcomes of that individual, either positively or negatively.  If you’d like to know more about this idea please listen to the radio show, its great!

I wanted to incorporate the show’s theme into a blog post about special education math classes, but was unsure how until Alex Overwijk, a teacher from Ottawa, sent me the following comment about my post about scaffolding

Expectations…

This led me to consider how the over-scaffolding of mathematical tasks and problems for special education students creates an atmosphere of lowered expectations.  Both Alex and I agreed that students with disabilities need a certain amount of scaffolding to be successful. What we didn’t know was to what degree and when this scaffolding should be provided.

Thinking more deeply about this question, I believe the degree to which scaffolding is provided to students with disabilities is a very individual, personalized process.  Great special ed teachers who understand their student’s learning pathways will be able to determine the appropriate level of scaffolding for them.  But the timing of when scaffolding is provided can show students what a teacher’s expectations are for them in math class.  If scaffolding is implemented too early in a lesson or unit, students may feel a sense of lowered expectations which according to Invisibilia would result in lowered outcomes as well.  You can’t get much earlier in a lesson or unit than the pre-assessment, so let’s start there.

# Financial Literacy: Shopping

One of the major components of our school’s mathematics curriculum is our spring financial literacy unit.  The reason we focus on this for so long is that for our students to become successful independent adults they need to be able to use money effectively in the community.  The three components of our financial literacy unit are shopping, banking, and budgeting.

Our students have learning and developmental disabilities that impact how they relate to the outside world.  It is often hard for our students to transfer what we teach in class for use in the community.  Students who seem to have mastered a skill in class may not be able to demonstrate this mastery when needed in the community.  The theory and problems with the transfer of learning have been well documented (here and here).  This is why we go out into the community as part of our financial literacy unit.  We want to see what the students can do when faced with using these mathematical skills in the “real world.”

I have written in the past about how we are integrating MathTwitterBlogosphere (#MTBoS) resources for our financial literacy unit.

Thanks Graham, what a good idea!

This will be the first in a series of posts about our financial literacy unit.  The focus of this post will be shopping.