Schoenfeld divided his talk into three parts.

1. Students are currently not engaging in mathematical sense-making.

He gave the example of the following problem posed to 97 first- and second-grade students.

There are 26 sheep and 10 goats on a boat. How old is the captain.

76 students (78%) tried to solve the problem by combining those numbers using various operations.

He explained that what you test is what you get (WYTIWYG). Current tests are skill-oriented but Common Core demands more, which it tries to explain with the mathematical process standards. He says that of course content matters (obviously students have to learn the skills of math), but that the real action is in the practices. You can’t just put these on a list and check them off.

2. There are tools we can and should use more often in the classroom.

He advocated more formative assessment more often. Scoring formative assessments rather than just giving feedback ruins their utility. Students see the grade and never look further. I can’t even tell you how many times a student has thrown a test away after checking the number on the top without looking once at the notes I’ve made.

Schoenfeld also recommended letting students experience material and using questions to elicit productive mistakes. These mistakes then lead to formal mathematical conversations that can build a student’s understanding.

He recommended the Shell Centre’s formative assessment lessons as “diagnostic situations” to help introduce students to these concepts.

3. There are tools we can use to help us reflect on our teaching.

Schoenfeld showed us two videos:

1. A TIMMS video from a 1995 study.
2. A video showing students explaining their work in different ways, including students who had gotten the problem wrong in the first place. (Link unavailable.)

It was obvious which video we should emulate and not emulate.

Schoenfeld claimed that often teachers deny their students the right to engage their mathematical muscles. I agree with this. It’s often a struggle, at least at first, to get students to explain their reasoning. However, I do think that once you push past that struggle, it will ultimately benefit the students. He posed these questions, which are a useful assessment for our math instruction:

• Was there honest-to-goodness math in what students and teachers did?
• Did students engage in “productive struggle,” or was the math dumbed down to the point where they didn’t?
• Who had the opportunity to engage? A select few, or everyone?
• Who had a voice? Did students get to say things, develop ownership?
• Did instruction find out what students know and build on it?

LeeAnn Allen blogs at DIY: Math PD.

On April 19, 2013, the third day of NCTM’s annual meeting in Denver, Uri Treisman gave a forty-minute address on equity that Zal Usiskin, director of the University of Chicago’s School Mathematics Project, called the greatest talk he’d ever heard at the conference in any year. Stanford math professor Keith Devlin would later call it our “I have a dream” speech. At least one participant left in tears.

I’ve personally seen it three times. I got the video feed from NCTM and the slides from Treisman. I then spent some time stitching the two together, resulting in this video. His message is important enough that I’d like to use whatever technical skills I have, whatever time I have, whatever soapbox I can stand on, to help spread it.

You should watch it.

If you’re interested in equity, you should watch it.
If you’re interested in teacher evaluation, you should watch it.
If you’re interested in school reform, you should watch it.
If you’re interested in charter schools, you should watch it.
If you’re interested in understanding which student outcomes teachers can control and which they can’t, you should watch it.
If you’re interested in the trajectory of math education in the era of the Common Core State Standards, you should watch it.

If none of those conditions apply to you, well, I can’t imagine the series of misclicks that brought you to my blog. Watch it.

Here’s a fair enough summary from Treisman himself:

There are two factors that shape inequality in this country and educational achievement inequality. The big one is poverty. But a really big one is opportunity to learn. As citizens, we need to work on poverty and income inequality or our democracy is threatened. As mathematics educators … we need to work on opportunity to learn. It cannot be that the accident of where a child lives or the particulars of their birth determine their mathematics education.

That was his destination and the talk took only three stops along the way:

1. What did education reform groups like Achieve, the Gates Foundation, et al, recommend in their “Benchmarking for Success” document in 2008?
2. How does TIMSS and NAEP data contradict or clarify those recommendations?
3. What should we actually do about equity, as teachers and citizens, if those recommendations prove unfounded?

Highly Quotable

• [A]s math people we know that if we’re going to work on a problem, we have to formulate it clearly. And as math people are wont, we need to swaddle ourselves in the numbers and the data because that’s what gives math people direction, strength, and courage.
• Let’s look at “Benchmarking for Success” and see its analysis of the problem. Then let’s look at the data and see how it actually lines up with what we know today. And then let’s see where we need to go to really enact the vision of NCTM for equity.
• So the notion was: “Let’s focus on teachers as the central driver of reform and rethink how we evaluate teachers.” They had the view that teachers were the single most important in-school factor in student achievement. And math people know that was just an artifact of the way they modeled the problem.
• I’m now going to show you two graphs that I don’t believe anyone in the math community has seen. It’s the PISA data disaggregated by child poverty rates.
• About one half of students who go from high school to college are referred to remediation and mostly developmental math. Fewer than a quarter of those students will ever get a credential. Those students are more likely to end up with debt than a credential. [..] Those remedial programs are burial grounds for the aspirations of students. And it’s mostly math that’s the key trigger. 35,000 students in California two years ago repeated a developmental course for the fifth or greater number of times. So no one can say those students don’t have persistence.
• So states – where you go to school – are a profound influence on what you actually get to know.
• Low income student scores in Texas were the top in the country in 2011. It’s really good for Texas to be the top of the country. Because whenever Texas does something well, everyone else is positive that they can do better. When Massachusetts is at the top, people go, “Ah, it’s just Massachusetts.”
• Again, two and a half years difference in opportunity depending on where you happen to go to school. This is something that, as a math teaching profession, we can influence. Poverty is something we need to work on as citizens. Opportunity to learn is something we need to work on as math educators. That’s a core message for this talk.
• So you would think that charters would fix this. Almost all the charters in Texas produced 0% of students who are college-ready. There are a few of them – one KIPP, one YES Prep, one IDEA, one Harmony – that are pretty good. Most of them are well below the public schools. So this theory of Achieve, NGA, CCSSO, Race to the Top, that charters were the answer? Not so clear when you actually climb into the numbers. The reverse looks true.
• When you visit most math classrooms it’s like you’re in a Kafkaesque universe of these degraded social worlds where children are filling in bubbles rather than connecting the dots. It’s driven by a compliance mentality on tests that are neither worthy of our children nor worthy of the discipline they purport to reflect. That is the reality. That’s something that we as math educators can control.
• What this shows is that the current theory about school improvement – that charters, Common Core, value-added measures of teaching are going to solve the problem – is profoundly wrong. That doesn’t mean we can’t use the Common Core powerfully to reboot our systems but it’s not the solution to the basic problems of schooling.
• Guess what? Poverty really sucks. It’s incredibly hard. All the lifespan studies going back to the 1920s show that poverty and youth is a very hard force. We need to build fault-tolerant schools and systems if we’re actually going to address equity.
• Just think about it. The great majority of our children finish our schools positive that there’s a whole list of things they’re not. They come out of schooling believing they’re not mathematical, they’re not artistic, they’re not philosophical, they’re not athletic. And these self-imposed beliefs undermine your sense of personal freedom, the font from which all freedoms come.
• You have to remember that when the Common Core was created, they didn’t come to NCTM. They got David Coleman to write it and he brought his friend Jason Zimba to do the math. They did not come to NCTM. It’s time for us now – the professional societies – to talk about what standards should be and how to reshape the Common Core so that it reflects our best practice knowledge of schooling. Hard message, but a necessary message.
• What is the determinant of whether you have a high-skill job in the US? Overwhelmingly it’s mathematics. It’s the single biggest factor in upward social and economic mobility. It’s our beloved subject. It would be wonderful if it were music instead of math. Think how great the country would be if everyone were striving to learn to play an instrument instead of factor quadratic equations but the fact is it is our discipline that is the primary determinant.

Dan Meyer is the editor of MathRecap.com. He blogs at dy/dan and tweets @ddmeyer.

As you can see from the picture, it was a packed house! After waiting in line for fifteen minutes, I was so lucky (and excited) to get a seat to hear Jo Boaler speak, even if my seat was in the next-to-last row.

Jo opened the presentation with Carol Dweck’s research on mindsets. “In the fixed mindset, people believe that their talents and abilities are fixed traits. They have a certain amount and that’s that; nothing can be done to change it. In the growth mindset, people believe that their talents and abilities can be developed through passion, education, and persistence.”

Jo said that the fixed mindset contributes to one of the biggest myths in mathematics: being good at math is a gift. She referenced her book, The Elephant in the Classroom and showed the audience various television and movie clips that perpetuate this myth.

Jo then moved from Hollywood to the science behind learning. She briefly discussed brain plasticity, the capacity of the brain to change and rewire itself over the course of one’s lifetime. When learning happens, synapses fire and create connections. These synapses are like footprints in the sand, that if not used, wash away. To illustrate this plasticity, Jo showed the variation in two child brain scans, one child from a loving home and the other living in extreme neglect. At this point, the neuroscience has me completely transfixed.

Jo went on to discuss the London “Black Cab” Drivers. To become a Black Cab driver, one must pass a test called “The Knowledge” consisting of 25,000 streets and 20,000 landmarks. I had to Google it to find the image because I thought DC was bad.

Brain scans have shown that Black Cab drivers have a larger hippocampus after studying for and passing this test, demonstrating neuroplasticity, the brain changing/rewiring as new things are learned.

She shared a letter written by a high school math department arguing against using Algebra II as a graduation requirement. The letter, in so many words, implied that certain students can’t learn, whether because they are minorities or due to lack of maturity, and would not be able to pass this requirement. The reasoning in the letter goes against brain research that shows that every child can excel in math. I am so impressed with Jo’s use of research to dispute the comments we hear all too often, even at the elementary level. Research shows that every learning experience changes one’s “ability,” yet we used fixed ability language often. (eg. “high kids” and “low kids.”)

Jo read a quote by Laurent Schwartz: “What is important is to deeply understand things and their relations to each other.  This is where intelligence lies.  The fact of being quick or slow isn’t really relevant.  Naturally, it’s helpful to be quick, like it is to have a good memory.  But it’s neither necessary nor sufficient for intellectual success.” I think that needs to be a poster every classroom wall!

So how does mindset impact how students view themselves? Jo shared 7th grade data in which students with a growth mindset outperformed fixed mindset students. Growth mindset students demonstrated more persistence in challenging situations and the gender gaps were eliminated in SAT levels.

Jo posed the question to the audience, “What do you think encourages a fixed mindset in a student?” As we discussed our thoughts, I checked out Twitter only to find there were a few folks tweeting about this particular session, so we shared our ideas:

Jo suggested that student grouping, assessment, grading, and the math tasks we use in our classroom all contribute to creating a fixed mindset in a student. She presented this block pattern to the audience:

Typically, teachers ask how many blocks will be in a certain figure number, leading to an input/output table response. Jo suggested asking students, “What do you see happening? How do you see it growing?”

She showed video of a group of students working together for over an hour, sharing how each saw the pattern growing and changing. They were engaged, following different pathways through the problem, creating arguments, and persevering. She suggests that when tasks are open and engaging, a growth mindset is developed.

I have to confess, I was reading some tweets about Jo’s session from other #NCSM13 participants at this point. I heard Jo mention Gauss and Cathy Humphries, so I jotted them down to check out later.

My attention was quickly drawn back in when Jo said, “Grades are not that important.” She said that diagnostic feedback of classroom observations leads to higher achievement in students. Then the popular topic of timed tests arose. According to neuroscience, math should never be associated with speed. She shared numerous honest, yet sad, student reflections regarding timed tests. A fourth grader said she felt, “nervous because I am scared I will not finish or make a mistake.” A second grader said she felt “that I am not good at math.”

Mistakes are good. Mistakes grow synapses and yet students are pressured not to make them. Why? Jo said that students have been brought up in a performance culture, not a learning culture. Jo ended with the message that teachers and students should be encouraged to have a growth mindset. How we teach will then impact our students’ mindsets.

Kristin Gray blogs at Math Minds and tweets @MathMinds.

Dockterman is an adjunct lecturer of education at Harvard Graduate School of Education, and works for Scholastic Education as well. He walked us through definitions of a “fixed” mindset and a “growth” (or “incremental”), based on Carol Dweck’s research. He referred to one particular quote from Dweck: “The brain is a muscle. Giving it a hard workout makes it stronger.” He then emphasized that shifting students from a fixed to a growth mindset was critical to improving their learning.

He shared how we need to shift our language around mistakes. “You learned that quickly, I’m sorry I wasted your time” and “You really struggled to understand that, but you persevered and got it – can you tell me how you figured it out?” are quotes that both support the growth mindset rather than making failure a dead end.

He discussed how gaming environments are often growth mindset-driven, with rewards for effort not just completion and highlighted one example of an app that rewards effort: Slice It, which focuses on geometrical figures, and symmetry.

He then took session attendees through a quick multi-question quiz to determine how much of a growth mindset we have. He said that the questions and the assessment was available at MindsetWorks.

He then spent some time talking about Math 180, a Scholastic product. “Digital games, simulations, visual models provide immediate feedback and can be incredibly failure tolerant”, he said. He also said that telling our students that a topic is important for them doesn’t work, and can lower performance. He recommended having students share or write why something is important to them – this provides meaning and context and makes the learning “stick.”

Tanis Thiessen is a veteran educator who blogs at Joy of Education and tweets @tjthiessen.

In this session several teachers from Indiana shared their experience using problem-based learning in their classroom.

The first example that one of the teachers gave was a PrBL which challenged students to consider how long they would have to grow their hair in order to donate it to Locks of Love. The students were required to research the necessary data, plot a graph of hair growth, and answer the question using mathematics to defend their opinions.

The students had to research the following questions:

• What is the “starting point” of their hair now?
• Where would this point be on a graph?
• What is the rate of growth for their hair? How does this affect the graph plotting hair growth?

The students made predictions of the rate of hair growth based on a video depicting the hair growth of a young man who grew out his hair for “Locks of Love”. Students measured their own hair and researched how long hair must be in order to donate it.
Students then used the data to create a graph and to model it using an algebraic equation. They then gave a presentation on their findings and defended their predictions and mathematical calculations.

Another teacher provided an example PrBL which challenged students to create a new snack using chocolate and peanuts which could not exceed specified conditions regarding weight and fat content. This project required students to apply systems of inequalities to a real-life scenario.

A third teacher provided an example PrBL which asked students to use trigonometry to analyze the “biorhythms” of the two quarterbacks in the Superbowl to predict which quarterback would win the game.

The teachers provided the following suggestions about implementing PrBLs:

First, student grouping and collaboration is important for driving discussion and investigation. They suggested allowing students some amount of choice in who they work with so they are enthusiastic about working together. Students must be willing to discuss ideas in order to work together to achieve the expected results.

Second, the teacher should provide students with a rubric so that they have a goal to aim for. Students find it frustrating to work on a project if they are confused about what the teacher wants.

Third, the teacher should ask questions to guide students’ thinking. Instead of telling students “that’s wrong,” they should challenge the students to defend their ideas and use mistakes as a base for learning.

Finally, authenticity is extremely important. Students have to be “hooked” to get passionate about what they are doing. PrBLs are excellent for developing 21st century skills and helping students understand how to use math in their lives.

Resources

• Project handouts
• Slides
• Project-Based Learning Online

Benjamin Graber blogs at Math Explorers and tweets @mathexplorers.

a.k.a. Experience before Formalization

This session was task-oriented, which was a great way to start the conference. It introduced several exploration activities meant to be the introduction to a unit or topic. These activities were mostly taken from a textbook, although the teachers did say that they created their own tasks and activities occasionally. These teachers, who were all at the same high school, would use these activities to introduce students to a particular concept, hopefully emphasizing patterns in the activities and highlighting the intuition students might already have, whether correct or incorrect.

We started with an activity to find the center of a hexagon without tools. This eventually moved into developing an area formula from number of triangles to one based on perimeters and apothems. Make another leap and it is easy to reason the area of a circle formula. I tried this with my class the next week and I saw students remember the triangle splitting and how to get area from that a couple days later.

The other activity that I found most useful was one that involved the unit circle and conceptualizing sine and cosine graphs. We started by making a unit circle (tracing CDs) and using protractors to make various triangles within the unit circles. We did just the standard 30-60-90 and 45-45-90 triangles, but in the actual classroom, the presenters had their students make the triangles in increments of 10°. Then we used sticky notes cut into strips to measure the height of each triangle.

The presenters asked their students what the height represents. (A: sine.) Then, on a different piece of paper, there was a horizontal axis labeled in increments of 10°. Students plotted the sticky note representing the height of each triangle gets on the corresponding angle measurement on the axis. It ended up forming a pretty beautiful sine curve. Et voila, repeat with the x measurements for a cosine graph.

A big takeaway for me is that these activities are not meant to replace formal math instruction, but rather they serve to incite students’ thinking about them. At first I thought this model was in opposition with a successful of problem- or task-based model. I eventually reconciled the two, thinking that PrBL uses a problem to teach the formal math concept while simultaneously demonstrating to students how and why they would use it. This session and these activities are dealt with pure mathematics (ie. numbers out of context, instead of starting with a situation) and are used only as an introduction to a particular subject. Whether or not these two instructional models can or should be used simultaneously, is a question that I’ll leave for debate.

LeeAnn Allen blogs at DIY: Math PD.

This was a great session that was almost completely hands-on. We started by building a cylinder around a sphere and then moved into proving that the sphere’s volume is 2/3 the volume of a cylinder. The other activities were fun, but not ones that were as easy to replicate. We made snowflakes, which then connected to the unit circle. We turned envelopes into 3D shapes and found their surface area and volume. I’ve attached his slides (which has directions for all the activities) and the worksheet for the sphere/cylinder activity.

Resources

• Handout
• Slides

LeeAnn Allen blogs at DIY: Math PD.

Preliminary note: This recap cannot capture Karim’s charm and rapport with the large crowd. My apologies for the inadequacy.

Karim opened with his thesis — We should use real-world topics to teach math in a different way — and he promised to show us what this looks like through a series of Mathalicious lessons.

He invited us to introduce ourselves to a neighbor and to ponder the question, How fast is the Earth spinning?

I turned around to talk with two teachers from Barcelona (yes, that Barcelona, and they were in the country for this conference!) After a few false starts involving km v. miles and radius v. diameter, the three of us settled on a bit less than 1000 km/hour at the equator.

Karim revealed an answer not too far off from that (in miles/hour, natch). My Spanish friends and I patted ourselves on the back while Karim pushed the task further. How fast is the Earth spinning at the Tropic of Cancer? The Arctic Circle? The North Pole?

In one sense (miles per hour), the Earth is not moving at all there.

The route to these computations is trigonometry. A cross section of the Earth reveals a 23° angle between the equator, the center and the Tropic of Cancer. We can use that angle to compute the circumference of the T-o-C, and voilá!

After whetting our appetite for out-of-the-ordinary questions, Karim cited his go-to statistics: (1) that a majority of students would rather take out the garbage than do their math homework (which bodes well for our nation’s competitiveness in the waste-hauling industries, does it not?), and (2) that teacher job satisfaction is at an all-time low.

Karim offered two characterizations of standard school mathematics (what Hung-Hsi Wu would call Textbook School Mathematics, or TSM):

1. Math is seen by students as a bunch of random skills to regurgitate, and
2. Math is seen by students as a recipe for something no one wants to eat.

This brought us to the central question of the session: Why Math?

Karim began answering this question with an example of what the answer is not. Why Math? is not answered by So you can write the equation for a line through two points in the plane. On this particular matter, he offered an analogy to the complicated rules of baseball. There was something about Bryce Harper and tagging up on a fly ball to center, the point of which was that the complex and seemingly arbitrary nature of the rules and conditions for a particular play in baseball is an awful lot like the complex and seemingly arbitrary nature of the rules and conditions for the moves involved in writing an equation for the line through two points in the plane.

On matters of Bryce Harper and tagging up, your humble correspondent is analogous to a struggling eighth grader who has asked his teacher three times Where did the b come from? and received three straight rounds of the same unsatisfying answer, at ever increasing volumes (yet decreasing pace), and who is seriously reconsidering even bothering asking a fourth time. Which is to say, Karim’s analogy was apt but is being done little justice here.

Karim proceeded to offer an alternative.

He began with Domino’s Pizza, which he claimed has improved their quality measurably. Again, your correspondent offers no expertise in this area. But that’s not the point. The point is that you can assemble your pizza on the Domino’s website, and it will spit back a price. A two-topping pizza costs $13.57, while a 4 topping pizza costs $16.95. If we make certain assumptions, we can reason out out both (1) the cost per topping, and (2) the cost of a plain cheese pizza (i.e. a pizza with no toppings).

Suddenly those complex rules for writing an equation of a line through two points have meaning.

And in the process of asking and answering a wide range of interesting questions about the Domino’s website, Karim noted that we have touched on a large number of Common Core standards.

Next up was dating.

There is a rule that evidently everyone but me has known for years pertaining to appropriate ages for dating. Namely, you can calculate the age of the youngest person you can date by multiplying your own age by one-half and adding 7 to the result.

We plotted some points, and asked the question both ways: (1) What is the youngest person that a person of age a can date? and (2) For a person of what age is a person of age b barely datable? These are the sorts of questions that sound more natural with a live audience. I hope you get the idea.

Once again, a variety of interesting and natural contextual questions facilitated some really nice mathematics. I am speaking of inverses and systems of inequalities here. ON that latter point, the region of acceptable datability has a name: The Romance Cone, or the RoCo to those in the know.

Karim reminded us of our central question: Why Math? Answer at this point: So you can tell your 11 year-old they’re not allowed to date.

Is Wheel of Fortune rigged?

How long would it take to burn off the calories in a Big Mac? And—correspondingly—should restaurants be compelled to list minutes of exercise for their offerings instead of calorie counts?

These are the sorts of questions Mathalicious uses to answer the question Why Math?

Due to technical difficulties, Karim sang to us the 8 second suicide of Inspector Javert.

Yes, we can use combinatorics to determine the number of different kinds of shoes you can design on the Nike ID website, Karim implored, but who cares about the precise number? What matters, he argued, is that this much choice has negative consequences for our happiness and well-being. When we have too many options, we cannot make a satisfying choice.

And that got to the heart of the presentation, which remains the heart of Mathalicious. In short, math matters only insomuch as we can use it to inform our lives. Karim and his Mathalicious colleagues have made a certain kind of peace with the mandates that are rooted in content standards; they strive to use this content to begin conversations in classrooms about things that matter to the people who are there.

Photo Credits

• Roco photo from Robin Mathews.
• Globe photo from Ashli Black.

Christopher Danielson is a professor of mathematics education at Normandale Community College. He blogs at Overthinking My Teaching and tweets @trianglemancsd.

Frank Lester was the editor of the 1234-page Second Handbook of Research on Mathematics Teaching and Learning. This was an interesting talk, not so much for any specific content but for how it was put together. Lester began the talk by demoing Algebra Touch, an iOS app that promotes fluency with symbol manipulations in solving equations. He asked, “What will the math classroom of the future be like?”

Then Lester went into problem solving, something he feels has slowly slipped out of most mathematics curricula. Problem solving, says Lester, is “What you do when you don’t know (or aren’t sure) what to do.” That leads to the teacher’s role in the classroom:

The teacher’s job is to make students better at being productive when they don’t know what to do.

From here, Lester showed some of his favorite problem solving tasks. The first is probably familiar to most of you:

A snail is at the bottom of a well that is 10 meters deep and it wants to get out. Every day it climbs up 4 meters. It then slides back 2 meters when it rests at night. If it does this day after day, how many days will it take the snail to reach the top of the well?

As part of the discussion, Lester related his problem solving heuristics, harkening back to Polya’s How to Solve It. Suddenly a talk that began with a discussion of technology and the future was using a (good?) problem that felt like it was from the 1980s and (good) strategies that were from the 1940s. Lester’s choice of heuristics to apply here were “Draw a picture/diagram” and “Be skeptical of your solutions,” since many initially reason that the snail reaches the top of the well in 5 days.

Lester then looked at finding the square root of 12,345,678,987,654,321. His heuristic — one he called a “super heuristic” — was to look for a pattern. I couldn’t help but feel like this was a trivial problem with a trivial answer.

Things got better with the next problem: “On a European river cruise, 2/3rds of men are married to 3/5ths of the women. How many men and how many women are on the cruise?”

Lester joked that this problem predated talk of same-sex marriage, and I found it to be a bit out of touch. Lester said the problem could be adapted to involve pairings of animals or objects. Another heuristic here was “make reasonable guesses, not as final answers, but to get you started.” After discussion of this problem, we moved on to one more: “A club has 500 members. At the Spring dance, tickets for new members were $14 but $20 for longtime members. All of the new members attended but only 70% of the longtime members attended. How much ticket revenue was collected?” It seems like there isn’t enough information, but solving this plays off the fact that $14 is 70% of $20. That might elicit some interesting reasoning, but again I think this trivializes the problem.

Lester returned to technology at the end, mentioning strategy games like Math Dice and Rush Hour. He advised that teachers have an important role to play when students play games. Prior to the play, teachers need to help students be clear about the rules for playing, model how to play, and discuss special situations. I think that depending on the game and the goals, this could turn into too much guidance. During student game play, teachers need to watch students play, attend to their thinking, help and point out misunderstandings. It’s important, says Lester, to not suggest strategies for playing. Kids should be left to figure those out for themselves. After gameplay, reflection is important, just as with other classroom activities.

Resources

• Slides

Raymond Johnson is a graduate student at CU-Boulder. He blogs at MathEd.net and tweets @MathEdNet.

Dan Meyer covered the tweeting duties during the talk:

My takeaway? Math teachers need to push for more and better collaboration. No longer can teachers just teach what they enjoy, or pretend teaching is mostly improvisational. If we are truly professionals, we need to do serious work around our new standards and curriculum, including critiquing the teaching of colleagues, reviewing and refining lessons over time, and recognizing the body of knowledge about teaching mathematics that can be built upon and further contributed to. But listen for yourself.

Resources:

• Slides

Raymond Johnson is a graduate student at CU-Boulder. He blogs at MathEd.net and tweets @MathEdNet.