Where was I using the Process Standards?

MTH 110 is an Intermediate Algebra course that explores topics typically found in Algebra I and II. Most of the students have seen the content before but for some reason it did not stick (nobody's fault, just a fact). Instead of simply re-teaching the topics, we are using the content to extend our understanding of what it means to do mathematics.

One of the ways we are attending to the mathematical process is by making our thinking visible is through Metacognitive Memoirs. Because this is a new approach to many of my students, I spend a good portion of Flight School (the first three weeks of the semester) developing the idea of how we can go beyond showing our work. This past week, I facilitated a workshop that demonstrated one way to share our thinking.

Schema Activation: Predict how many small cubes are in Step 43

from Visual Patterns

Focus: NCTM Process Standards

As I share my thinking, please keep track of where I am attending to the bullet points associated with these processes

Activity: Metacognitive Memoir Demonstration

Reflection: Where was I ...

• Problem Solving
• Reasoning
• Communicating
• Connecting
• Representing

It might help to see my notes.

Please share your thinking (where I made the processes visible and opportunities I missed or messed up) in the comments.

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.

When will it end?

[In the United States] Teachers act as if student interest will be generated only by diversions outside of mathematics. (The Teaching Gap, p. 89)

Schema Activation: Journal Jot

Describe the process of converting three-nineteenths to a decimal

Focus: Changing the Script

We have read in The Teaching Gap how teaching mathematics in the United States typically focuses on preparing students to perform prepackaged procedures. What if we tried to change the focus from mathematical procedures to mathematical practices?

In this workshop, try to refocus your efforts away from simply following a mindless process and explore what new knowledge is waiting to be learned. As you work on the following problem, please keep track of how you are (or aren't) engaging in these practices.

Activity: Find the exact decimal representation for three-nineteenths

[The follow represents how learners have engaged with this activity.]

This seems like a fairly straightforward exercise. Why not just plug the numbers into a calculator? The converting-a-fraction-to-a-decimal procedure requires us to divide the numerator by the denominator until the decimal either repeats or terminates. Three divided by nineteen - easy. Except, the quotient displayed on the calculator screen does not provide enough information since it does not show enough places.

Maybe using a different calculation tool would help. What about an Excel spreadsheet?

This seems to suggest that three-nineteenths terminates. And it matches the calculator's answer. A problem solver would not simply accept these and be done - would she?

What other strategies could apply?

• Work the division out by hand
• See if another related fraction might shed some light on the decimal
• 1/19 since 1/19 multiplied by 3 is 3/19
• 16/19 since this would be the complement of 3/19
• Find x such that 19x = 100 and then calculate 3x
• Multiply 0.157894736842105 by 19 to see if the product is 3

[Learners often explore these approaches with various levels of success. Having monitored their own progress, the learners often turn to other strategies or decide to seek a solution to other related problems. They do not want to be stuck trying the same approach over and over again.

WolframAlpha has added a wrinkle to this activity as it provides an exact decimal representation for three-nineteenths. Originally, we thought this would be problematic as it gives a clear answer to the original question. It is interesting, however, that focusing on a problem solving approach generates similar alternative/extension problems whether or not the answer has been found.]

New problems:

• Which fractions repeat and which terminate? Why?
• Can we predict the period of the decimal representation of a given fraction?

Partial table created by learners in order to look for patterns

[Answers to these alternative/extension question rarely are answered in the time available to the workshop. Learners are encouraged to continue exploring the problems if they are interested. Even though they have not come to any firm conclusions, they usually have some interesting answers to the reflection questions.]

Reflection: Mathematician's Chair

• What did you do as it relates to the mathematical practices?
• So what was important about this work?
• Now what might you do the next time you encounter a problem?

Which representations will help?

A few years ago, John Golden and I co-taught Teaching and Learning Middle Grades Mathematics. During the first unit on Doing Mathematics, we noticed that the preservice teachers were struggling to unpack how they used various representations to support their problem solving. We wondered what would happen if we limited the representations they could use while looking for a pattern in a sequence. That is how this workshop came to be.

Doing Math Workshop (Representations)

Schema Activation: What representations did Carl use?

We have been talking about writing Metacognitive Memoirs as a way to make our thinking visible as we solve problems in mathematics. Here is an example from Carl that we found on the internet. Please identify all the representations he uses in his write-up.

Focus: Representation Standard

This comes from the NCTM's Principles and Standards for School Mathematics:

In order to make our think alouds explicit, as it relates to this Process Standard, we need to highlight where we are using these algebraic representations:

• real; pictorial; verbal; written; numeric; tabular; graphical; and symbolic (recursive and explicit relationships)

These are the images we use to understand and communicate mathematics. This is analogous to mental images used to enhance reading by immersing the learner in the subject.

Activity: Limiting Representations for Thinking

The students are asked to avoid using any of the representations not explicitly identified on the worksheet. In this first case, they are limited to using only pictorial and symbolic representations. After a couple of minutes, I allow them to use verbal so that they can share with one another their efforts thus far.

The next worksheet limits the students to using verbal, written, and symbolic representations. This semester, one of the students expressed a great deal of discomfort with the fact they she was not going to be able to use the representations with which she was most comfortable. We talked briefly about how this might mirror how students who struggle could feel regarding the representations we expect them to use in our classes. Again, I added another representation, this time manipulatives (real), after a few minutes.

On the last worksheet, the students are limited to using tabular and symbolic representations. One student said she was relieved to be back in familiar territory. When I offered to add another representation of their choosing, they asked for verbal so that they could talk over the problems. It turned out that tables were not enough to help them feel confident in their answers.

Reflection: Journal Jot

How did limiting the representations you could use affect your thinking?