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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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.

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Why Count Your Money, When You Can Estimate?

The end of spring break means we are in the midst of our school’s spring financial literacy unit.  This is always a favorite of both students and teachers.  We ground our work in very concrete community related activities, such as going to the bank and going to the store.  Students love spending money and teachers love going on community walks in the spring and early summer. Everyone wins!

Before we start going on trips, I wanted to do some number sense work with my classes relating to the counting of money.  As we left for spring break I tweeted about an estimation idea inspired by counting money.

Right on cue, Graham Fletcher, who writes his own amazing blog here, gave me some sage advice.

I took Graham’s advice and ran with it.  As much as we like our money math standards to relate to identification of coins and bills and getting accurate counts on prices and change, estimation is a key skill in any “real world” financial transaction.  When was the last time you stood at the supermarket register counting out the entire pile of change you got from the cashier? Generally, we look at the coins in our hand and make a quick estimate as to whether we think it is the correct change or not.  So I used the idea from my tweet, took some inspiration from fellow math teachers Andrew Stadel and Joe Schwartz, and turned it all into a financial literacy lesson.

Here’s how it went…

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Noticing and Wondering About Scaffolding

Often I wonder how much to explain or define for my students before engaging in the problem solving process.  Proponents of sense-making in mathematics classes like Dan Meyer and The Math Forum encourage presenting a perplexing scenario to students and letting them develop the questions to be answered using math.  This is a very enticing proposition.  Who wouldn’t want a math class which uses the Socratic method to solve problems as a community.  I do!  Professor Ilana Horn recently wrote a piece investigating the merits of this pedagogical philosophy with other popular options like Doug Lemov’s Teach Like a Champion.

Some students, however, need more scaffolding, language support, cultural background, or skill reinforcement before they are ready to grapple with a truly perplexing situation.

Vygotsky's Zone of Proximal Development

Vygotsky’s Zone of Proximal Development

For instance, what if your students view their zone of proximal development much differently than you, as the educator, do?  What if the student views every problem as lying in the outer ring, but it truly lies in the middle or inner ring according to your professional opinion?  Which leads into my question about problem-based learning.  How much do you scaffold for students who need it before you set them free to make sense of a great, perplexing mathematical scenario?

This is a major question for special education math teachers.  How much scaffolding is too much so that the process of solving the problem is taken out of the hands of the student?  One area where this comes up is when teachers are deciding what order in which to present information to students during the problem solving process.  As an example, here is a problem I have been developing in which there are two components.  Which of these components should go first in a truly problem-based classroom?  Maybe you can help me figure it out!

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Daily Math Routines

It all started when Dan Meyer tweeted this Which One Doesn’t Belong prompt…

The MathTwitterBlogosphere had a ball using logical and comical thinking to conceive of ways to not pick the muppet.  I said Daniel Craig didn’t belong because his hands were showing.  Then I decided to send a response…

But all the fun and games got me thinking about my own classes and how I could leverage this fun for my students.  Enter Mary Bourassa and her new #MTBoS site Which One Doesn’t Belong?  Inspired by the work of Christopher DanielsonSteve Wyborney, and Chris Hunter, Mary has created a wonderful new website meant to spark mathematical conversations and debates among students and teachers.  It joins a list of other #MTBoS inspired websites that provide prompts for beginning of class activities. Beginning of class activities are called many different things: do nows, openers, bellwork, warm-ups, but I like to refer to them as daily routines.

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#MTBoS Mod: 3-Act Edition

One of the missions of this blog is to take the work of the amazing online community of math teachers known as the MathTwitterBlogoSphere (MTBoS) and to show what modifications are made for students with disabilities.  I call it the #MTBoS Mod(ification).  You can read the first two editions here and here.  This edition is about the lesson structure created by Dan Meyer known as a 3-Act Task.

#MTBoS MOD: 3-Act Edition

The 3-Act math task I chose was created by Graham Fletcher called It All Adds Up.  I chose this because in our spring trimester we focus solely on financial literacy.  As a teacher of students with disabilities we spend a great deal of time on the adaptive mathematics that is often over-looked or just simply considered  a “real world context” in the classes of typically developing students.  In the world of special education these tasks are known as Instrumental Activities of Daily Living (IADLs), which are complex skills needed to live independently.  IADLs are not to be confused with the Activities of Daily Living, which are basic self-care tasks.  At my school we call these skills the Mathematics for the Instrumental Activities of Daily Living.

As a pre-assessment for our “money unit” (as the students call it) I used “It All Adds Up.”  The goal was to see how comfortable the students were with identifying coins and counting different combinations of coin denominations.  I launched the task with three of my student groups.  The task is great for students at different computation levels.  At the simplest level the students can solve it by adding coins together to equal $1.00.  At a more complex level students can look for patterns that can help them solve the problem more efficiently as well as reflect on the possibility of multiple solutions to the problem.  I gave this task to groups of students with a variety of different needs and modes of processing.  I’ve broken the three groups into the three stages of the Concrete-Representational-Abstract method of instruction.

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The Importance of Implementation

A recent NPR article entitled, 5 Lessons Education Research Taught Us in 2014, seems to have a lot of definitive answers about our currently polarized educational climate.  The article mentions a research paper which encourages the use of teacher-directed, explicit instruction of mathematical computation skills for procedural fluency with students with mathematical difficulties.  To me this read as favoring explicit instruction and direct modeling of mathematics for students with disabilities over the project or problem-based, hands-on (manipulatives), collaborative (student-led), investigative style instruction that makes up some “reform” mathematics curriculum.

As a counterpoint to the NPR article, the National Council of Teachers of Mathematics lists procedural fluency as just one part of what is referred to as Mathematical Proficiency.  In chapter 2 of the book, Achieving Fluency: Special Education and Mathematics, mathematical proficiency is discussed as including the following four components: procedural fluency, conceptual understanding, strategic and adaptive mathematical thinking, and productive disposition.  Together these four components lead to mathematically proficient students which lead to mathematically proficient adults, disabilities or not.

1. Procedural fluency involves using basic skills such as facts, procedures, and formulas efficiently (i.e., quickly and accurately). It also entails knowing when to use them and, if necessary, how to adapt them. In other words, procedural fluency is skill in carrying out routines appropriately and flexibly as well as efficiently.

2. Conceptual understanding is knowledge of facts, generalizations, or principles underlying the comprehension of concepts (categories), relations (between categories), or operations (actions or events involving categories).

3. Strategic competence involves the ability to formulate, represent, and solve mathematical problems, and adaptive reasoning entails the capacity for logical thought, reflection, explanation, and justification.

4. Productive disposition entails believing that mathematics makes sense and is useful, that learning it requires diligence, and that everyone is capable of significant mathematical learning.

Since the NPR article about educational research only references one paper specifically about mathematics instruction, you only get one point of view.  This NCTM book provides another viewpoint of what are important goals for mathematics lessons with struggling students.

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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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