Word Problems and the Problems with Words

Yesterday, I posted a new 3-act task on the blog. In the tradition of digital mentors like Graham Fletcher, Andrew Stadel, and Dane Ehlert, I will rarely post an activity on the blog that I don’t intend to use in my own class with students. Today, we did Make It Rain.

Here is what my students noticed during Act 1

  • There’s a lot of money
  • There are 20’s, 10’s, 5’s, and 1’s
  • There are more 20’s than 10’s

And here’s what they wondered…

  • How much money is there?
  • Why did it go from greatest to least?
  • Why was it being spread out?
  • What kind of bills were in the pile?
  • How many of each bill is there?

My students are used to analyzing their questions collaboratively. Some of the students noted that we couldn’t answer the “why?” questions without asking the person in the video, who we did not have access to (even though it was me!)

So, then our wonderings looked more like this…

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One Moment, One Decision

Teaching is hard.

As Magdalene Lampert notes in her book Teaching Problems and the Problems of Teaching, “One reason teaching is a complex practice is that many of the problems a teacher must address to get students to learn happen simultaneously, not one after another (2).”

Teaching is hard.

As Max Ray says in his 2014 NCSM ignite talk, “Teaching isn’t Rocket Science. It’s harder.” Max goes on to say that teachers make a litany of educational decisions on the fly based on deep knowledge of content and their students as learners.

Teaching is hard.

As Ball and Forzani write in The Work of Teaching and the Challenge for Teacher Education, “The work of teaching includes broad cultural competence and relational sensitivity, communication skills, and the combination of rigor and imagination fundamental to effective practice. Skillful teaching requires appropriately using and integrating specific moves and activities in particular cases and contexts, based on knowledge and understanding of one’s pupils and on the application of professional judgment (2009).”

Teaching is hard.

As Jose Vilson relates, “We’ve known for decades that building relationships is a central part of our work, but this has even larger implications when we work with disadvantaged students. The teacher-student relationship has so many subtle nuances across race, gender, and class lines that opening our eyes to these nuances would make us better educators.”

So teaching is hard, because reasons.

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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:

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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).

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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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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.

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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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Accessibility and Mathematics

Disclaimer

I’ve tried to write this post many times. Each time I write the opening sentence, it seems to pale in comparison to the grand scope of what it should encompass. Access and equity is a huge topic, not only in math classes, but in education at large. Often equity is discussed in terms of gender, socio-economic, racial, or sexual orientation. These conversations are also vital, but this post will focus on equity for students with disabilities through access to rich mathematics curricula. However, writing a post about access for students with disabilities in robust math classes is still a daunting task. Since I believe in the importance of this topic I’m going to just begin, though I’ll probably regret how I began once I’ve finished.

Preface

When one considers how to create an accessible math class for students with disabilities it is generally done through deficit thinking. “My students can’t do _____, so what interventions can I implement to fix their deficits?”

At one level, the evolution of deficit thinking in special education stemmed from beliefs that, although some individuals functioned in ways considered “subnormal,” they were still humans and deserved to be educated. A review of the history of the development of programs for children with mild disabilities reveals that, in the early 1800’s, advocates of the child saving theory attempted to determine the etiology of students’ symptoms that resulted in learning and behavior problems.

These psychologists, physicians, and educators developed therapies and instructional interventions designed to improve the educational outcomes and quality of life of individuals with disabilities (Trent, Artiles & Englert, 1998).

Unfortunately, the idea of intervention is inextricably linked to deficit thinking and the belief that students with disabilities are not “normal.” I can’t help but disagree with this. Concepts like neurodiversity and presumed competence provide a much more equitable stance on how students with disabilities should be viewed and treated in the school environment. With this in mind, here are two effective lesson planning guides to increase access to rich mathematics for students with disabilities in your classroom.

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Ask More Questions!

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.

So here goes…Ask More Questions!

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.

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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...

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.

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