How might we deal with the mismatch?

Last week I heard Kat Holmes talk about her new book, Mismatch. I learned how designing for "normal" can miss the mark. As a result, many of us encounter mismatched experiences in our physical and virtual spaces. Effectively dealing with these mismatches requires inclusive design principles: 1) recognizing exclusion; 2) learning from diversity; and 3) solving for one - extending to many. In order to illustrate the second principle, Kat introduced us to Victor Pineda who made the following point:

There's a triangle of three different things that have to come together to really unlock human accomplishments for people with disabilities. And those involve assistive technology, personal assistance—somebody that's aware, understanding how to support you, and three is coping strategies. And so these three things sort of create a variety of tools.

Of course, I started connecting these ideas to teaching.

A curriculum is often designed for the normal/average student.

In reality, a student and the curriculum are typically mismatched.

To help the student to connect with the curriculum and be a contributor, the teacher might need to offer personal assistance, assistive technology, and/or coping strategies. And we can ask the student (learning from diversity, or as Dr. Emdin writes - co-teaching) to participate in the design.

Having planned for one student, the teacher must consider how to extend the design to many students.

After Kat's talk, I had the opportunity to watch a student-teacher deal with the mismatch between her students' experiences solving problems involving scientific notation [8.EE.A.4] and the curriculum used in her school. We talked about using a think-aloud (personal assistance) to create an anchor chart (assistive technology) that students could refer to (coping strategy) while solving 8.EE.A.4 problems. 

In the next lesson, she tried these ideas out. The lesson started with her making her thinking visible while solving an 8.EE.A.4 problem. Next, she asked students what they noticed in her thinking and added it to an anchor chat. She happened to put the anchor chart in the back of the room so it was obvious later in the lesson how many of the students were using this tool as they turned in their seats to see it. Because of her efforts, students were able to successfully connect to the curriculum.

I'm still processing a lot of this and would appreciate you sharing your thoughts in the comments.

[This blog post was written with the help of the Innovators' Compass. Check out my planning.]

What does it mean to do mathematics? IIc

Previously, we were introduced to the Doing Math Anchor Chart task (here). Then I shared a recent exemplar that used the metaphor of riding a bike to communicate the preservice teacher's vision of what it means to do mathematics. In this post, I will offer a teacher's more traditional concept map representation.

Chartists Statement

My Anchor Chart is in the form of a cycle or a process because I see the act of doing math as a cycle with the central goal of deepening our understanding.  When we do math we begin with a problem that we want to solve.  Using prior knowledge and problem solving skills that we have developed (which could include using representations) we work on the problem to get a response.  We then must evaluate our response and evaluate the process we took to get the response.  If our method isn’t working or we don’t feel our response is correct then we go back using this knowledge we gained to implement a different strategy or to see if it’s possible we responded to a different question.  If we don’t need to go back to rethink about our question or our problem solving method, we make conjectures or generalizations about the responses we obtain.  Many times we have questions about the conjectures about whether or not they work for all cases.  This gives us a new question that we may want to explore.  Regardless of exploring new conjectures we somehow share our thinking and responses with other people to gain their insight.  Because we share our responses, conjectures, and thought processes, we allow other people access to these things so they can ask questions and do math themselves.  In addition, the conjectures we develop may allow us to make connections to problems in other contexts or we can use the problem solving skills we developed in other aspects of life, so the cycle of doing math is not closed, and because we go back and retry different problems and form new questions based on the work we are doing the cycle does not go one way. 

Key

I choose to do develop my chart without specifically writing down the Process Standards in any area because in my cycle of doing math, different aspects of each standard are included in different steps of this cycle.  Here I will explain how the different standards fit into the chart and where. (Note: the numbers correspond to the labeled boxes):

1.     Communication:  the problem we have to solve maybe to analyze and evaluated someone else’s work.

Content: The problem we are working on is based on the content we want to learn, for this unit that that would be Algebra.

2.     Problem Solving: The problem that we have to solve requires that we implement a variety of strategies that are appropriate to the context of the problem.

Reasoning and Proof:  If our problem is to prove something then the strategies we implement will be the different methods of proofs and determining which proof method is most appropriate for the problem presented.

3.     Problem Solving: Implement a variety or strategies determining which one will work best as we work towards a response.

Connections: Recognize similarities between the problem that’s presented and previous problems we have solved.

Representations: Use representations to help you think about a problem and translate between the representations to help solve problems.

4.     Problem Solving: Reflect on the strategy we used.  Did the process we used allow us to effectively find a response?

Connections: Think about how the problem we are solving connects to and builds on other mathematical ideas.

Representations: Use representations as a way to organize our thinking.  Look back that the ones we used when solving problems and think about how they helped us.

Metacognition: interpreting our response and strategies require that we reflect on our process and how we thought about the problem. 

5.     Reasoning and Proof:  Make conjectures about the response we obtained, this may occur though connecting our work with previous work we have done.

6.     Problem Solving: It’s possible that the knowledge we gained and would like to share is the skills we used to solve the problem.

Reasoning and Proof: We can share a conjecture that we have developed or we can share our thinking in the form of a proof.

Communication: Clearly expressing the response and the process to others either formally through writing, possibly a formal proof (Reasoning and Proof), or through discussion with others. 

Connections: It is possible that we use examples and make connections to other problems to help our audience understand what we are communicating.

Representations: Maybe we choose to organize our thinking into some form of representation as way to communicate our thinking with others.

Metacognition: We may decide to share our thinking process; inorder to do this we must think about our own thinking. 

7.     Reasoning and Proof: Throughout the entire process of doing math, we are building an argument.

Connections: We build on the knowledge we gained in order to solve problems in the future.

8.     Connections:  We can apply the knowledge we built through this process to contexts outside of math. 

What does it mean to do mathematics? IIb

Previously, we were introduced to the Doing Math Anchor Chart task (here). In this post, I share a recent exemplar that used the metaphor of riding a bike to communicate the preservice teacher's vision of what it means to do mathematics. The next post will offer a more traditional concept map representation.

Artist's Statement

Doing Mathematics is like riding a bicycle. It requires all pieces to work together to move forward. When riding a bike one must stay balanced. This is the same for mathematics. We must employ all the processes to complete a problem and understand if fully.

Pieces

Algebra = Bike Rider

As a student, doing mathematics means doing algebra in some cases. The student must employ all of the tools (the bicycle) in order to perform - just as the rider must pedal as the wheels move and control using the handlebars to ride the bike.

Pedal 1 = Reasoning and Proof

When doing mathematics, reasoning and proof is central to the process. When riding a bicycle, moving the pedal is essential to moving the bike along. When we are reasoning we are investigating a problem and developing arguments. We can also select how we want to reason, like we can change the pace at which we are pedaling.

Pedal 2 = Problem Solving

Problem Solving and Reasoning and Proof go hand-in-hand - just as the two pedals work together to move the bike forward. When problem solving you are building on mathematical knowledge and working to use appropriate strategies (like the appropriate pace of pedaling).

Bell = Communication

The bell on the bicycle is used to communicate with others around. in mathematics we use communication to talk with others in a clear fashion about our work. There are precise signals one can use to tell others you are oncoming when using the bell just as mathematicians must use precise language. Also, the rider must evaluate when the best times are to use the bell and evaluate if others will run into them before using the bell - like we evaluate others' thinking in mathematics.

Gears = Connections

The gears of the bicycle work together with the pedals and the wheels to move the bike forward; they are also the pieces that keep the whole process of riding a bike continuous. The fluidity of a bike is similar to the fluidity of mathematics in which we can find connections and then apply them to doing mathematics.

Wheels = Representations

We use representations in mathematics to communicate or record our ideas. Essentially representations are what help us solve problems through their application. Without the wheels on the bike we would go nowhere, thus we need representations to model mathematics like a bike needs wheels to move.

Handle Bars = Metacognition

When riding a bike we balance on the handle bars. While you can take a hand off now and then, we find we are most balanced with both hands resting on the handle bars. In mathematics, we use metacognition to think about and communicate our thinking. It controls the steps we take when working as we analyze what we have done or what we need to do. The handle bars control in which direction we go. The brakes are also located on the handle bars. At times we may get stuck; this is when we stop our work and think about our thinking once again.

What does it mean to do mathematics? IIa

One of this blog's most popular posts describes how a group of preservice teachers envisioned doing mathematics. They combined elements of concept maps with metaphor to create an anchor chart that expressed their views. This activity is typically untaken at the end of the first unit in Teaching and Learning Middle Grades Mathematics - a unit that focuses in on the NCTM's Process Standards. It seemed like a good time to share some more recent exemplars. 

First, here's the workshop:

Schema Activation:  What will it look like?

• Look back over your work from previous Teaching Math Workshops as you determined what was important in the NCTM’s Process Standards. Note any patterns you see in your journal.
• “Students entering a classroom that visually represents the mathematics being studied are more likely to share in that enthusiasm and be willing to create and share their work (Ennis and Witeck, 2008).” So what will your math classroom look like in order to show what it means to do mathematics and encourage learners to do the same?

Focus: Anchor Charts

The following description comes from Debbie Miller’s (2002) Reading with Meaning:

…I do create “anchor charts” after lessons from which I want children to remember a specific strategy or concept. I write a note of explanation at the top of the chart and note snippets of conversation, individual comments, and statements that reflect our work together.

Anchor charts make our thinking permanent and visible, and so allow us to make connections from one strategy to another, clarify a point, build on earlier learning, and simply remember a specific lesson. (p. 57)

An anchor chart is one way to communicate to learners your expectations regarding what doing math will look like in your classroom.

Activity: Create an Anchor Chart for Doing Mathematics

1. Reflect on the Anchor Chart Rough Drafts completed in class.
2. Develop an anchor chart called "Doing Mathematics."

Reflection:  What’s important

Review your “Doing Mathematics” anchor chart. Write an "artist's statement" that highlights what is important in your chart.

*****

Two exemplars from Fall 2012 are found in the next two posts. (It was turning into a really long post.) The first leans heavily on metaphor as she associates doing math with riding a bike. And the second uses a more traditional concept map to communicate her vision.

Now what? Part II

In the prior post, I introduced a problem written by a seventh-grader as both an example of what middle school students could do when deciding what comes next and an opportunity for preservice teachers to develop and explore their own "Now what?" questions. This was the student-generated problem:

In the Community, you get two pets. The Elders pick the pets for each family. There were six choices of pets to have: dog, cat, fish, snake, bird, and hamster. What was the probability of getting a dog and a cat?

Typically, the preservice teachers came up with an answer fairly quickly. After all, since it comes after a unit on combinations the solution method seems obvious. Still, I ask them to explore the problem further by using one of the extension questions we collect over the course of the semester.

It is sometimes difficult for the preservice teachers to consider alternative answers, however, because of their own experiences with math problems having a single correct answer and the fact they think this problem is so cut-and-dry. Fortunately, I have examples of alternatives to their expected answer of 1/15 that were identified in previous classes. If no one comes up with these alternative answers in the current class, I offer them as other possibilities we ought to consider. I say, "A group came up with an answer of 1/30. Another was pretty sure that it was 1/21, although they also considered 1/36 after they hear the 1/30 rationale."

The preservice teachers' initial reaction is, "Those answers are wrong." I remind them that as educators we must consider that learners are not wrong but they may have answered a question different than what we expected. (I wrote about this here.) Therefore, the natural "Now what?" question that a teacher can consider is, "What question does this answer?"


To be continued...

What does it mean to do mathematics?

My MTH 329 class just finished the first unit of Teaching and Learning Middle Grades Mathematics. The focus of this unit is on the NCTM Process Standards. Yesterday we spent a considerable amount of our class period reflecting on what we’ve been doing so far.

We start the workshop with a schema activation/connection that asks the learners to review a simile survey taken the first day of class. One part of the survey says, “Choose the simile that best describes doing math and explain your choice.” The similes to choose from are: climbing a mountain, conducting an experiment, cooking a meal, reading a book, working a puzzle, or playing a game. In reviewing their original choices, I want my learners to consider how their view of doing math has changed and why.

The focus/concentration for the workshop is to create an anchor chart representing what they think it means to do mathematics. This chart might hang in their future classroom as a constant reminder to their learners what it means to be a mathematician. (My colleague shares his experience with anchor charts here.)

During the activity/construction phase of the workshop, groups of four work together to develop a rough draft anchor chart. The first group builds on the simile idea and develops a chart showing doing math as working a puzzle. Each piece of the puzzle represents a different aspect of their vision of doing math.

Group two decides that they want to make a chart that would appeal to middle grade learners. They use graffiti as the theme and begin searching Urban Dictionary for terms they could use for the Process Standards. At one point they discuss whether or not the chart is appropriate (culturally sensitive) and decide that it is because it is intended to engage and not to mock.

The final group considers the anchor chart from a graph theory perspective. They had just been discussing a discrete mathematics assignment and it seems to find its way into the representation. That’s the Green Arrow up in the Algebra block. The color of each arrow has some significance, I think. (These are works in progress.)

These are rough drafts that I post on our classroom Blackboard site. The future teachers will make adjustments to whichever one they choose and make it their own for inclusion in the course portfolio. This is the reflection/consolidation portion of the workshop.

So what do I think it means to do mathematics? It involves collaborating with others in an effort to solve problems, making and revising representations for/of our thinking, trying to connect various ideas, and communicating the reasoning behind our thought process to others. In other words, yesterday doing mathematics involved making anchor charts.