Thursday, February 28, 2013

Do they know what it looks like?

I find it easier to see the hidden figure when I know
what I am looking for. (from)
When I ask teachers to fill out an action plan identifying their current challenge in teaching, I sometimes get comments like these:
  • Students are not engaged in the lesson;
  • Students do not take good notes;
  • Students fail to do quality work;
  • Students struggle to apply what we are doing to new situations;
  • Students do not engage in productive discussions; or
  • Students cannot work well in groups.

My initial response is usually to ask, "Do they know what this looks like?"

This happened recently around the last issue, group work. When I asked the question, the teacher responded as they typically do, "I don't know. I haven't addressed it explicitly." Herein lies the problem; too often we expect students to come to class with certain skills that they do not possess.

As a middle school math teacher, I spent considerable time at the beginning of the school year focusing on fostering skills associated with working effectively in groups. For nearly two weeks the students took part in "plays" representing productive and unproductive groups. After reflecting on these experiences, we created expectation around working in groups that were referred to throughout the rest of the school year. This was time well spent as it allowed me to work with individuals and small groups confident that the remainder of the class knew what was expected of them as they worked together. (It also served another purpose. Near the end of the second week, someone inevitably asked, "When are we going to do math?" Before my two-week bootcamp on groups, students rarely begged to do math.)

Unfortunately, it is not the beginning of the year, and the teachers was looking for something to do now. We brainstormed things that would support the development of productive group work through explicit action. For example, the teacher was tired of constantly reminding the students to get back on task, so I suggested setting a timer that went off at regular intervals as a reminder that they ought to monitor whether or not they were being productive. If students start ignoring the timer, stop the activity and identify some of the positive behaviors exhibited during the activity. Next time an activity involves group work, remind the students of those positive behaviors and try again. Once they complete an entire activity, set the timer for longer and longer periods of time until the students begin to self-monitor their efforts.

What are some examples of ways that you have ensured that your students know what group work (or some other classroom expectation) looks like? I would appreciate your thoughts in the comments so that I can share them with the teachers I am coaching. Thank you in advance for you participation.

Thursday, February 21, 2013

Should I use elimination or substitution?

Be forewarn, this is probably not the post you are looking for.
A teacher I was coaching sent our this tweet after a recent observation. The class was looking at solving systems of linear equations using elimination and substitution and we had talked about how he could use data from recent Big Ten basketball games to set up scenarios that might encourage them to use the different techniques. Early in the lesson, several of the students (the class is all boys) made it clear that they were huge basketball fans.

Here are a couple of examples I came up with:
Against Wisconsin, Trey Burke scored 19 points. He made 1 free throw and 2 three pointers. What would his box score look like for scoring?
Trey Burke scored 18 points against Michigan State. He made 1 free throw and 7 shots from the field. What would his box score look like for scoring? 
Christopher Danielson has thought more and deeper about this topic than I, so I defer to his expertise (here). Sorry for the bait-and-switch, but I want to think about how this experience helped me to re-evaluate my homework policy.

About two years ago, I wrote about how overwhelmed our novice teachers are during their first teaching experience. We do this somewhat intentionally so that they can experience firsthand all that is expected of teachers, and so that they can become better at managing their time. Part of that time-management is learning to prioritize what is important and learning how to say, "No!" -- in other words, elimination. (I know that we aren't talking about mathematical elimination anymore, but this is where my mind goes. Pity my wife and my co-workers.) In that post, I went as far as to say, "It is okay to skip this assignment."

Looking back, I think that suggesting this technique, elimination, was a mistake in this situation. Simply skipping work does not match the reality of teaching. In order to prepare them for that reality, what I want them to be good at is substitution. If something else on their plate is more important than one of our assignments, then they ought to make a substitution. However, as they do their other work, I want them to recognize they are doing it on our time and look for ways to synthesize the concepts and skills we have been exploring into their "more important" work. Hopefully, this mindful effort to synthesize across classes will provide practice in attempting to meld ideas in their teaching practice.

Again, my apologies for bringing you here under potentially false pretenses. And for making you wade through the ramblings of my mind. As I tweeted Christopher recently, I need an editor. If you have your own thoughts, please add them to the comments.

Monday, February 18, 2013

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. 

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. 

Sunday, February 17, 2013

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.

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.

Saturday, February 16, 2013

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.

Friday, February 8, 2013

Can we follow the trail?

It snowed last night - a lot.

from NOAA
This morning, as I began the process of digging us out, I began noticing how the snow kept a record of my progress.

This got me thinking about assessment and monitoring the progress of learning. Too often, it seems that the assessment information we collect provides a picture of where learners are at but not enough about how they got there. It's as if they magically appeared in the snow.

But moving through the snow forces us to leave tracks. If the tracks are clear enough, we might be able to follow the path. One of the reasons I have learners write metacognitive memoir is that it leaves a trail for me to follow - it makes their thinking visible.

Sometimes the trail can be confusing, however. It's not always clear the path that was taken or even which footprints belong to the traveler and which were left by someone else.

That's why I often like to interrupt learners in the middle of their efforts -- to catch them in the act of thinking.

It is also the reason why I like Ewan MacIntosh's idea of engaging learners in problem finding rather than problem solving. Perhaps a pristine area of snow will make it easier to follow their path.

Thanks for indulging me this metaphor. It was a great day to play in the snow and think about assessment.

Tuesday, February 5, 2013


Have you ever been reading a book where things are getting tense and you wonder how the hero is going to get out of the crisis he finds himself in, only to be surprised by some solution that seems to come out of nowhere? This is a writing device called, deus ex machina (god from the machine), and I find it to be a very dissatisfying way to address plot problems. Unfortunately, this same method is used frequently in math lessons.
From Science Cartoons Plus
I have written before how I like to view teaching as storytelling. What I have not shared is the story I used to tell early on in my career to my middle school classes whenever a student asked, "Why?" It goes something like this:
Do you know what the largest animal on Earth is? <brief pause, but not long enough for anyone to answer> It's the blue whale. The thing is huge! But its throat is less than a foot wide. Do you know why? <another brief pause> Because that's the way it is. And that's why we invert and multiply <or some other mathematical rule>, because that's the way it is.
There you have it - my own version of deus ex machina in my teaching. In retrospect, I am sure it was as dissatisfying to my students as it has been to me in my reading, but I needed to move the plot onward and did not want to get bogged down in details.

I am not proud of this chapter in my teaching history, but like so many others I taught the way I was taught. Consider this a cautionary tale. So the next time a student asks why we subtract one from the exponent in the explicit formula of a geometric sequence but not in the exponential function, do not just respond with, "Because that's the way it is." Help the students to come up with a better reason. If you do, they are more likely to stick with the story of math instead of discarding it because it doesn't make sense.