In a classroom where students speak to each other about mathematics, the ideas of those students are valued instead of ignored or potentially marginalized. This gives students agency in their learning. It also allows new ideas the students learn to extend from the existing ways they understand the world.

Supporting students speaking to each other means that mathematics is much more likely to become a way of knowing and being rather than just a body of existing knowledge (although the value of mathematics as a set of tools that have been developed over time should not be marginalized). As students develop their understanding of what mathematics is and what it is useful for, they are more likely to insert themselves in the role of the mathematician rather than imagining this to be someone else, potentially from another culture. They can see themselves being part of a mathematical community.

In order to completely understand the language we know, we have to use it, either in writing or ideally in conversation, and hear other people using the same language. So from a practical perspective, students need to talk in order to develop their use of language (mathematical or otherwise), and rather than students talking in serial, one at a time mediated through a teacher, it is far more efficient for them to talk in parallel, to each other.

We remember what we think about. When students construct ideas and communicate them to each other, they necessarily have to think about these ideas, which means that they are building memories. While this occurs no matter what students do, the focus is more likely to be on the thinking with student discourse rather than the activity (eg. completing a task).

Finally, students talking and writing to each other also provides their teachers with more information about the ways they are thinking which makes it easier for the teacher to orchestrate productive whole group discussions and to plan activities that respond to the ways students are actually thinking. It is difficult to plan lessons that build off of student knowing if you don’t know how and why students think the way that they do. When students talk to each other, their teacher can gather formative assessment information about not only **what** they understand but ideally **how** they understand it.

This should not diminish the importance of students having *independent time* to work quietly on mathematical problems *by themselves*. Students are better positioned to work together when they have had time to think about ideas themselves first. Also, some students find working with other students really difficult for a variety of reasons, so in some cases the benefits of students working together may be outweighed by the challenges some students face with this activity.

What would you add as reasons students should talk to each other in a mathematics classroom?

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If these four people come to talk together, they will find it challenging to have a conversation since the way they are teaching is so different from each other. Each person potentially has a valuable perspective but they may have so little in common that it is difficult for them to talk to each other.

Now imagine instead these four teachers are all focused on working on the same instructional activity, perhaps even with the same mathematical tasks within the activity. Their teaching is still different as they still teach in different schools in different contexts but their conversation about teaching becomes much more coherent as they have far more in common to talk about. Instead of talking about their individual teaching they can talk about *teaching practice*.

Over the last two weeks we launched the instructional activity, Contemplate then Calculate with 100 or so teachers for this exact purpose; to make our shared conversations about teaching focused and coherent. After two days of professional development, which included rehearsing the instructional activity together, virtually all of the teachers indicated that they were excited to try out this activity and then come back together in October to talk about its impact on students.

At one point during the two weeks, some participants and I ended up in a whole-group discussion about when exactly we should annotate our students’ strategy sharing. It was the most specific conversation I’ve ever had with a group of teachers and I feel fairly certain everyone understood the point that was being debated and why it might matter one way or another.

All of these teachers are going to go back to their individual schools and teach according to what they know with their individual contexts and their individual students but now, hopefully, at least one aspect of their teaching will be similar enough that they can come back and talk about the differences.

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- The NY Hall of Science has recently published a series of science and math apps for the iPad.
- Motion Math has some great low-cost math apps.
- DragonBox has 4 really high quality math applications, all of which I have personally played and tested with my son. These are the most expensive apps on this list but also some of the best apps available for learning math.
- BrainQuake’s math puzzles will present a challenge for people of all ages while being accessible to young children.
- Geogebra is cross-platform and a must-have for people interested in constructing and exploring their own interactive math activities.
- Desmos is a fantastic online graphing calculator which can be installed and used on many different types of devices.

As I learn about more companies publishing apps for the iPad (and as I have time!), I’ll update this list. I know there are lots of apps that I’m missing but this is what I had time to put together this morning.

]]>It turns out that Twitter is a really good medium for testing these activities because most people don’t have access to pencil and paper (a key component of Contemplate then Calculate) and there are a lot of people from a wide variety of backgrounds willing to try out the tasks.

What I do is first share the task on Twitter, kind of like so:

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```What is the value of the missing number? How do you know? What did you notice? #mathchat pic.twitter.com/Xltfxm8fpm

— David Wees (@davidwees) August 4, 2015

Then, I wait. Usually, I get a few responses right away and a few more responses over time.

Notice how I follow-up to those responses with questions to find out what people were thinking. If my objective with the task is to engineer opportunities for students to think about mathematics, it’s helpful to know in advance how they might think about any task I give them. A key benefit to Twitter is that a variety of different ways of thinking about the task emerge quickly and in many cases have led to me modifying the original tasks.

Some tasks get more responses than others. I try my best to respond to everyone who takes the time to try the question, but sometimes I miss people.

How many circles are in this shape? How do you know? What are all the ways you can figure this out? #mathchat pic.twitter.com/2p0neLZvhF

— David Wees (@davidwees) August 3, 2015

I also work to make sure that share back the work of the community back to the community.

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```@CDawson18 Contemplate then Calculate. Here’s the lesson plan: https://t.co/uUIEN4elbl and the slides: https://t.co/A1g5YzW9Uc

— David Wees (@davidwees) August 3, 2015

All of the tasks I’ve been working on, and so far we are up to about 30 of them with many dozens more in the pipe-line, are going to be shared, with a Creative Commons license, back to the math education community. Stay tuned for the URL.

I wrote four blog posts for NCTM’s Mathematics in the Middle School blog on using student work to understand and plan around student ideas. Each post is about using student work to make inferences about how they understand mathematical ideas and then using those inferences to help you plan. Note that these posts are actually relevant for mathematics educators at all levels.

- What Knowledge Do You Need to Plan a Unit?
- Categorizing Student Work
- Using Student Work Meaningfully
- The Mathematics of Students

While these posts focus on qualitative information about student learning, this doesn’t mean quantitative information isn’t useful. I have just found that quantitative information abounds while qualitative information seems to be rarely used in systematic ways.

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- If you want to walk away from the conference with something different about your practice, focus on one or maybe two ideas during the conference and only go to sessions that will support you in learning and revisiting those ideas.On the other hand, if you don’t know what you are hoping to learn at the conference, you might be better off going to many different sessions in the first couple of days at least before deciding however note that this may something you can learn from the conference planner rather than attending a scatter-shot of different sessions.

No one or two hour session is likely to lead to any significant change in your practice. 20 hours thinking about one part of teaching, especially in the different ways different presenters think about the idea, on the other hand may. - Skip a couple of sessions and take the time to reflect on your learning and make a plan for implementing the new idea(s) from the conference into your teaching. If you don’t eventually come up with a plan for implementing new ideas in your teaching, you will never try them out. The sooner you have a plan, the easier it will be to come up with that plan while the ideas are still fresh in your mind. You should also make a note of questions you still have.

- Find someone with whom to share and discuss your learning experiences at the conferences. In my case, I went to NCSM & NCTM with my colleagues, but if you end up going alone, try and arrange a buddy or two to spend the week with early in the week. This is one potential powerful use of Twitter; if you are active on Twitter, chances are good you will know other people at the conference.

It’s worth noting here that not every presenter at NCSM and NCTM is equally good at communicating what they know and so having someone to talk after attending a session is useful to fill in whatever parts of the story or idea you missed but maybe someone else caught. - Take the time to network with other people at the conference, even if this means you may miss a session or two.

- Be strategic in the exhibition hall if you visit it at all. I find it overwhelming and draining. I typically take 30 minutes and scan through the entire hall quickly to see if there are any
*types*of products I don’t know about and then come back to follow up later if necessary.

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Unfortunately, the video tracking is not great so much of the annotation I was doing of the participant ideas is not easy to see as I am doing it.

The instructional activity itself is called Contemplate, then Calculate and was developed by Grace Kelemanik and Amy Lucenta. The slides, script, resources, and references are available here.

Things I would do differently:

- Set up the projector screen in advance of starting the activity so I don’t have to fiddle with it during the strategy sharing.
- Not wing the recording of noticings and strategies because I ran out of time to prepare before this talk, but take the time to make a template for collecting data.
- Record the initial noticings of participants about the problem as they were happening.
- Bring my own markers so that I can ensure I have access to more than one colour when recording the student ideas.

Things I decided to do or not to do somewhat deliberately:

- I did not focus on student to student discourse during the full group portions mostly in the interest of time. My meta-objective for this activity was to share the overall structure in a somewhat limited amount of time.
- I did not enforce participants writing using the prompts mostly because I knew I had little to no relationships with participants and I wanted to make sure no one felt alienated during this portion of the talk.
- I did make sure that when I was recording student strategies that I tried not to impose, as much as I could, much interpretation of those strategies. One of the participants actually came up after-ward and said she really noticed that I was making an effort to write down representations of what participants were saying rather than filling in too many of the gaps based on my understanding of the problem.
- I also focused on having participants share multiple strategies to solving this problem rather than attempting to focus participants on seeing one particular strategy.
- I decided to summarize participant ideas at the end rather than take the time to have them share out to the room, mostly in the interest of time.

What else about what I did do you have questions or feedback about?

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The audio isn’t terrific unfortunately but it is manageable. The slides, script, references, and resources are available here.

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Here is the activity I gave my students to do:

Here’s what I anticipated they might do:

Here’s a sample of what they **actually** did:

This student seems to understand that if you want to shade 1/4 of something, you want to split it into four equal parts and then shade one of those parts. The triangle drawn is more challenging to divide, so the student likely estimated their partitioning of it.

However, if one wants to shade 1/3 of an object, it looks like this student thinks that they need to split the shape into four unequal parts and shade the smallest of those parts, maybe because they think that 1/3 is smaller than 1/4?

This student told me that they know that 1/4 is the same as “half of a half” and so they divided each of the shapes in half, and then divided one of those halves into a further half, and then shaded it.

When it came to shading 1/3 of a shape though, this student looks like they divided the shape into four equal parts and then shaded three of those parts. My guess is that they thought that fractions involve dividing into four parts, and then used the denominator of the fraction to choose how much to shade? I’m not sure that they noticed the inconsistency in this procedure and what they did in the first part.

This last student seems to have done pretty much the same as the second student, but seems aware that when you want to shade 1/3 of a shape, you want to first divide it into three equal parts and then shade one of those parts. I asked them about it and they said they just weren’t sure how to cut the shapes into three equal parts.

With this gap between what I was expecting to see and what I actually saw, I was left without knowing an effective way to respond. I decided that this task had little entry points for someone who really did not know what 1/4 and 1/3 meant (one of my student’s work was blank, which I am not showing since I do not have permission).

With the worry in the back of my head that I was probably reducing the cognitive demand for students, we split into two groups with my student-teacher leading one group and me leading the other group and we tried a different task, one which offered some context through which one might make sense of what 1/3 means.

“If you had a pizza and you wanted to share it equally with three people, how would you cut it up so that each person got 1/3 of the pizza?”

With this prompt, students in my group almost all produced work similar to the following:

With one student drawing this:

This second student, by the way, said that pizza comes cut into ten-slices and so the best we could hope to do is to give each person 3 slices of the 10 and then give away the last slice to someone else.

The knowledge that would have helped me better anticipate student understandings here is what I call “the ways children typically understand mathematical ideas” and is the kind of knowledge that is rarely explicitly taught before starting teaching. Every time I teach a new topic, I notice that the ways students think about the ideas are different than what I expect, and over time, I learn to anticipate student thinking better as I get feedback from working with them.

If you had students showing these kinds of responses to this task, what would you do during your next lesson on fractions to support them?

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