What's your next move?

Our presentation from Math in Action 2017

Description: 

Games are an effective way to engage students in learning. Participants will experience how to support the development of pre-adolescent mathematicians through purposeful play. [Grades 3-5]

Consider what you think it means to effectively teach mathematics. Now take the Simile Survey provided below. What are the characteristics of your simile selection that relate to good mathematics teaching?

A while back, Dr. Doug Fisher introduced me to another teaching simile: Teaching is like being an expert commentator. During the lesson, the teacher highlights important aspects of the "routine" that the student might otherwise overlook. In cases where the action moves too quick, the teacher might need to "rewind and show it in slow motion" in order to clarify some move. Here is an example from the 2016 U.S. Olympic Trials that demonstrates these characteristics. So what does this look like in math class?

Imagine we are in a 3rd-grade class playing BINGO. If the students are fluent in reading number symbols, there's not much to the game. So let's break it - add another dimension by allowing players to decompose the number that's called.

If you were in a 5th-grade class, they might ask why they can't decompose the called number into more than two addends ... or use operations other than addition. Then the challenge might be, "Can I get a BINGO with just one number called?"

After (or during the game), what sorts of things would you want the students to notice? What would you highlight and maybe have to slow down? It depends on the game and our players.

• If I was playing the regular game of BINGO with young kids still struggling with number recognition, I might be sure to call "thirteen" and highlight ways to tell the difference between 13 and 31.
• If we are decomposing, I might want kids to recognize that 38 can be decomposed into 30+8 or 31+7 and highlight the concept of compensation.
• For 5th graders, I might show how "thirteen" can be written as 13+8/(9-7)-4 and highlight an important property of zero in our number system. [To demonstrate another important property of zero, ask students if they could cover the entire board if "thirteen" was called.]

It is important that teachers have the opportunity to play games before using them with their students. That way the teachers can consider possible modifications (ways to "break" the game) that would meet their students' needs. It also gives them experience playing the games that can lead to insights into important mathematical aspects encountered while playing that the teachers might want to highlight for their students.

Game Centers 

Number and Operations - Fractions 

Grades 3-5 

• Fraction Go Fish 
• Fraction Memory [pdf] 
• Cookie Fractions [page36]
• DecimalPickle 
• Pattern Block Fraction Game
• Fraction Track [app]

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Other Game Resources

• Adventures with Mathematics [MCTM]
• Gameswith Fraction Strips and Fraction Cards
• Rational Number Resources [NRICH]
• Investigations Games

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After playing the games, we reflect on our experiences using Math Teacher Chair:

• What games did you play?
• So what mathematical ideas would you want to highlight?
• Now what would you do to break the game or slow down the play so students would benefit mathematically from playing?

Thanks for your participation. You can reach us using the following contact information. 

"Rocket science is child's play compared to understanding child's play."

~ Unknown

If you are attending the upcoming 2017 NCTM Annual Meeting and Exposition in San Antonio, we will be presenting this session again. 

We promise it will be better next time thanks to the feedback you've provided on your session evaluations (or in the comments below).

How does a home workshop work?

This past week, Esther Billings, John Golden, and I presented Making Workshop Work in Mathematics at the NCTM Regional Conference in Chicago. One of the problems with presenting at these conferences is that the session description is due almost a year before the presentation. So while the program says, "explore several mathematics lessons and assignments that use the workshop model," we decided that the session would be more meaningful to participants if we worked through a single workshop, highlighted the workshop phases and research, and discussed how the workshop structure could support exploring the Standards of Mathematical Practice.

Although we provided a few minutes for participants to reflect on how they might apply what they learned about workshop to their classes, we were not explicit about using the approach on assignments. I hope to remedy this oversight by sharing a home workshop I used recently in my Teaching and Learning Middle Grades Mathematics course. This comes near the end of a unit on rational numbers.

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Doing Math Workshop (CGI Grouping Stories)

Objective: The learner will use representations to support their thinking and make their thinking visible to others as they find solutions to grouping stories involving rational numbers.

Needs: One hour and a copy of CGI Grouping Stories

Schema Activation: Reviewing Grouping Stories (no more than five minutes)

Recall that we used Grouping Stories from Cognitively Guided Instruction to provide context to multiplying and dividing integers. Review the different stories provided below and consider what it might look like as learners use the given contexts to compute their answers.

Buschman, L. (2001). Using Student Interviews to Guide Classroom Instruction - An Action Research Project. Teaching Children Mathematics, 8(4), 222-227.

Focus: Goldilocks Problems (no more than five minutes)

As you read through the CGI Grouping Stories, you will notice that the values in each story have been left for you to choose. The goal is to select the row of values that is not too soft (so easy that it does not require any thought on your part) and not too hard (so difficult that you would not be able to make progress without a significant amount of help). In other words, find the "just right" numbers. In the space on the right, record the representations you used to support your thinking so you can make it visible to others.

Activity: CGI Grouping Stories (no more than forty minutes)

Reflection: What? So What? Now what? (at least ten minutes)

As you look back at your efforts, pick the one record that best demonstrates your ability to use representation to support and share your thinking.

• What support did the representations provide as you worked on this story?
• So what would you want others to see as you share your efforts?
• Now what does this mean for you as a teacher - as you consider designing rational number computation lessons?

*****

The next class period, we usually have a math congress where some of the students share their work with their peers and discuss their reflections. If there is any interest, I will try to provide some generic examples of these in a future post. Please leave your interest or your questions in the comments.

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?

Why take the road less traveled?

I love being in the woods. It is nearly always an adventure. Even if it's a path I have been on before, the possibility of a surprise around the next bend is exciting to me. I never know for sure what I will see or hear. That's not to say that I am always comfortable with this state of uncertainty. A few weeks ago, I heard two coyotes howling up ahead as I was walking on a logging road in the Upper Peninsula of Michigan. Or there was the time I encountered an overwhelming smell that I can only describe as wet dog as I walked alone along the nature trail at Seney National Wildlife Refuge. In cases like this, I try to find a big stick to "support" me as I walk.

What keeps me coming back to these roads less travel are the sights and sounds that come from exploring something new. For example, while walking down one of the access roads at Seney, I came across this beautiful Trumpeter Swan that, true to its name, announced my presence to others in the area.

I have found that this metaphor of taking the roads less traveled also applies to my teaching practice. When I first began asking my middle school students how they would solve a problem before showing them the right way, I was quite uncomfortable with the idea that they might share something new - something for which I was unprepared. Fortunately, I had mentors that assured me it would be alright and encouraged me to explore.

The first time I recall letting my eighth-graders lead the way was when we began a section on dividing fractions. I put three-fourths divided by one-half on the board and asked for a volunteer to share how they thought we ought to calculate the result. Here's essentially what happened (requires ShowMe log in).

Because the student had not used "invert-and-multiply" to calculate the quotient, I considered the effort incorrect and tried to determine what went wrong. First, the student was finding common denominators, so there was a chance the student was confusing this procedure with the one for adding and subtracting fractions. Then the student divided across, which resembles the procedure for multiplying fractions. Clearly, the student was mixing up the various rules for fraction computation and I would need to be explicit about the differences. But something was bothering me. The answer the student came up with using this mish-mash of methods was correct. I attributed it to the numbers I had selected and decided to try the method with another pair. It worked again. I was at a loss.

I don't remember what I did with the students, but I do remember exploring this approach more and finding out that it always works and why. This experience hooked me. Now I am not suggesting that the student was doing anything other than trying to apply various rules and happened upon a solution.  But I might not have ever heard or seen this approach if I had not been open to following this unfamiliar path.

If you are still uncomfortable with exploring the "wilderness" with your students, then take solace in knowing that you do not need to go too far off the regular roads to experience something wild. Simply being alert to students' thinking while covering familiar ground can allow for new ideas to be uncovered. Just like when I was able to film this Peregrine Falcon on the bike path near our house, it only takes being aware of something new and prepared to see where it goes.

Which tool makes sense?

Confession of a control-freak: I want lessons to run smoothly (exactly the way I envision them). I have written about this issue before and my efforts to give learners more control in the classroom. If the goal of my teaching is learners who possess phronesis, then I need to provide them with ample opportunities to practice making and evaluating choices. In this post, I give another example of my efforts to turn over more responsibility to my learners.

During a recent lesson, my preservice teachers were relearning what it means to add fractions and the role manipulatives can play in supporting understanding. In the past, I would have: 1) put out one manipulative (perhaps the pattern blocks); 2) explained the rules of using the manipulative (2 yellow hexagons represent a whole); and 3) provided them plenty of practice in using the manipulative to represent fraction addition. Then, I would have them put away the pattern blocks and grab some fraction circles and go through the process again. As I said, controlling. I came to realize that my management of the tools and the rules was disempowering my learners. This time I simply put out a variety of manipulatives and asked them to explore.

Workshop

Schema Activation: All learners need time to explore the tools at their disposal. Please take five minutes to play with any of the manipulatives at your table.

I find that it is important to provide learners, regardless of their age an opportunity to play with manipulatives (they are going to anyways, I might as well embrace it). The learners spend the time building, organizing, and comparing the shapes. At the end of five minutes, I ask them to put the manipulatives back into their separate containers. This provides a break that I find helps learners to shift their vision of the manipulatives from toys to tools. I make sure to make this point explicitly.

Focus: Each of you is now going to pick a manipulative and consider what happens when you add two unlike shapes. For example, If I combine the blue rhombus and the red trapezoid from the pattern blocks, then what would I get? How did I get it? Why does it work? When does it work? And what if I combine other shapes, will my thinking still hold or do I need to adjust it?

This is usually where the manipulatives' rules are shared. Instead, I want to provide a framework of questions for them to keep in mind as they consider combining shapes. Not having done this workshop before, I am unsure this will provide them with enough structure. If there is any uncertainty, I am prepared to model how my wife had thought of combining the two pattern blocks by focusing on their side lengths: 4 units (rhombus) plus 5 units (trapezoid). There is no need, however, as the learners get right to work.

Activity: Learners work in groups on the task.

This is an opportunity for me to conduct some formative assessment. As I walk around, I try to focus on asking questions that check for understanding - any teaching/leading questions can wait until later. Based on the data I gather, I organize the remaining part of the lesson.

One group is working together with a single manipulative. They are discussing different ways to view the result. Another group has split the manipulatives between them and are seemingly working separately. But every so often, one learner shares with the others her conjectures and the others provide their feedback. From these observations, I ask two learners if they are willing to share their thinking with the rest of the class. They agree.

Reflection: Mathematician's Chair - learners share their thinking and any struggles with the rest of the class in order to expose their work to the larger learning community.

The first person I ask to share was working with her group to combine two shapes from the fraction circles. They had chosen to combine the two pieces shown at at the right. These seem pretty basic but the group had gotten into an interesting discussion about what represented the whole. The gist of the discussion, which she recounts for the class, was, "If we let the white circle be the whole, then the result is one-and-a-half white circles. If the orange half-circle is the whole, then the result is three orange half-circles. It all depends on what you denote as the whole."

Next, a learner shares what she found using the pattern blocks. It goes something like this. "I found it easiest to break down each shape to the smallest shape. So the blue rhombus is two green triangles and the red trapezoid is three green triangles. That means together they are five green triangles."

At this point, another member of this learner's group interjects. He had taken the idea and tried to apply it to the Cuisenaire Rods. He essentially points out, "It really comes down to names. I can break down any of the sticks to the smallest one, the white block, and then combine them because they have the same name."

Teacher's Reflection on the Workshop

I am pleased with the workshop. The learners were able to recognize the importance of identifying a whole and the need to find a common name (denominator) in order to combine fractional pieces. These are big ideas that are often reduced to rules without reason for students being instructed about fraction addition.

There are a few things I will do differently next time. First, I need to make even more explicit my decision to give them choice about which manipulative they would work with. Much of the work of teaching is invisible and I worry this was the case in this workshop. Next, I want to have a larger discussion about the wise use of manipulatives in math class. This might be a conversation around Deborah Ball's article, Magical Hope. Finally, I want to further expand on the idea raised during the reflection regarding naming. Using the Cuisenaire Rods, we could explore different names such as those shown below.

While this lesson did not go perfectly (they rarely do), I see real progress in my learners and myself. This experience will provide a pivotal experience that we can return to in subsequent lessons as we talk about appropriate use of educational resources. It also provides data that I can use in future classes. It is impossible to say how learners will respond, but now I have examples of how learners have responded. If necessary, I can always share these examples with as models of how others thought about the task. Often, one of my most effective teaching moves is to ask, "Would you like to see how other learners thought of this problem?"

What did you see/hear?

Last week, I attended the Michigan Reading Association's Annual Conference where I went to a session lead by Doug Fisher. He was presenting on Response to Intervention from a gradual release of responsibility perspective. It was the gradual release model that I was most interested in - especially since his book with Nancy Frey presents a slightly different version of this approach than the one I use. But I will leave that discussion for a later post. Today, I want to share my current thinking about what makes for an effective demonstration lesson (what Doug labels Purpose & Modeling).

From Doug Fisher's Michigan Reading Association Presentation

The demonstration I am sharing comes from a workshop for my Teaching and Learning Middle Grades Mathematics [TLMGM] course (a combination content and methods course that our secondary majors take prior to student teaching). This workshop focuses on comprehending the purpose behind a lesson that a teacher might encounter in an unfamiliar curriculum. Many of our student teachers find themselves using curricula that typically require a significant amount of professional development. Because our student teachers have not had this support, they sometime struggle to implement the lessons effectively. Therefore, I hope to share an approach I might use to better understand the rationale behind a lesson.

I used the ShowMe App to take a picture of a number string from a lesson found in the Context for Learning Mathematics series and then added my thinking. ShowMe records both my whiteboard annotations and voice to create a video that I can share with others.

A key piece to any demonstration is the debriefing: What did you see and what did you hear? This reflects Cambourne's perspective on who is ultimately responsible for learning: "Learners need to make their own decisions about when, how, and what 'bits' of information to learn in any learning task." By asking the learners what bits of information they attended to during the demonstration, the teacher is gathering important data that can inform future instruction (formative assessment). If the observers did not attend to something important presented in the demonstration, the teacher can highlight the missing points by saying, "I noticed that I was also..." The resulting list provides an anchor chart that learners can refer back to as they take more responsibility for employing the approach.

After I shared this demonstration with my learners in TLMGM, they noticed that I was trying to understand the string by:

• making connections between the expressions;
• considering different representations that might further my understanding;
• recognizing that computing the answers might help but wasn't enough;
• thinking about ways to put the expressions into a context; and
• analyzing my options before jumping into any plan.

Satisfied that they had attended to the major points of the demonstration, I provided them with further resources related to the Ratio Table unit and had them work collaboratively. They tried to look at the strings from the various angles in order to understand the purpose of the lessons.

During the workshop's reflection, I provided the opportunity for them to share what they had uncovered through Twitter. This provided everyone who wanted to contribute a voice and me with another artifact that I could use for formative assessment. Here is a sample of the discussion:

UPDATE 3/21/12
Reflecting back on this post, I have come to realize that if I were going to embrace the idea of "flipping" my class, this is what it would look like. I would use ShowMe to create demonstration lessons and have my learners watch them. When they finished, I would ask them to Tweet those things that they noticed in order to assess what they paid attention to and create an anchor chart for the class.

I know that my learners (college students) may differ from your's if you teach in a K12 school. But what do you think? Would this work for you?

How can we communicate our thinking?

The math teachers I work with often express frustration with their learners' inability to communicate their thinking when it comes to solving problems. If learners are stuck, they often struggle to articulate what they have tried. If the problem has been solved, learners have trouble explaining their efforts. Too often, learners respond to a teacher's question about their thinking with, "I don't know." This does not present teachers with the assessment data necessary to evaluate what learners can do or are trying to do, which makes it difficult to plan what comes next.


I encountered this same problem while working with fifth-graders on fraction computation. They were practiced in giving answers and even showing work but when I asked them to share their thinking they often said, "I don't know." This provided me with an opportunity to try a response suggested by Ellin Oliver Keene in a session at the 2009 MRA Conference: "Pretend that you did know - what would you say?" The fifth-graders found that this framing supported their communication efforts by freeing them to take a risk because they were "pretending."


While this got them talking about their thinking, they still needed more support to organize their efforts. I thought it would help to demonstrate what a reasoning recount might look like. I was introduced to the recount text form through Margaret Mooney's book, Text Forms and Features. Here is the model reasoning recount I wrote based on the prior efforts of the class to think about adding fractions.

Recently, I have been collaborating with Jennifer Brokofsky via Twitter and email about ways to connect reading, math, and writing. The figure below represents our current thinking.  I hope this vignette further demonstrates the link. We recognize that this is work in progress, and your support would be appreciated. Please share your thinking in the comments.

When does it work?

I have been sharing my experience teaching a group of fifth-graders how to problem-solve around fraction computation and using it as an opportunity to demonstrate the Teaching-Learning Cycle in action. Previously, I wrote about how I planned for a problem-solving lesson and then described my instruction during the following lesson. In this post, I want to discuss how I used assessment and evaluation to monitor the learners' progress and inform future planning and instruction.


We were using the clock model as a context for adding fractions and I wanted to gather data about whether or not the learners could determine when this model was an effective approach. I used an existing set of textbook items and asked the fifth-graders to: "Look at the expressions shown below - circle the ones that you think you could use the clock model to solve and place an 'X' through those you could not."

from Scott Foresman – Addison Wesley Math [5^th Grade]

Once the kids had completed this task, I asked them to solve one of the problems they had circled. As they worked, I gathered data on whether or not they were able to determine when the clock model could work.


In analyzing my observations, I noticed what the fifth-graders could do and what they were trying to do. First, they all recognized that fractions involving ninths and sevenths were poor candidates for the clock model. Those who chose to solve #2, #9, and #10 were also fluent in applying prior experiences with the time context. Some learners thought eighths could work (circling #4) and others struggled to see that fifths could work ('X'ing out #3, #5, and #8). These last two areas of approximation gave me some ideas about what to focus on next.


The last assessment I gave was intended to gather data about how the fifth-graders might apply the idea of context to a problem that could not be easily solved using the clock model. As a ticket out the door, I asked, "Now what could you do to solve a problem you put an 'X' through?" Based on my evaluation of these assessments, I was prepared to plan for future lessons.


What would you do next?