## Overcoming Confirmation Bias with the 2, 4, 6 Puzzle

The promise of mathematics is that it will teach students to think.

Sadly, there’s been no solid evidence that math class actually succeeds, statistically speaking, in achieving this end. A pessimist would conclude it’s a hopeless project. But far more likely is that math classes, statistically speaking, haven’t been spending much time doing tasks that lead students to think, or learn to think.

The researcher Peter Cathcart Wason came up with a series of games and tests in the sixties to explore common failures in human reasoning. The first of these was the 2, 4, 6 Puzzle.

Happily, the 2, 4, 6 Puzzle is a delightful and devious way to play with kids (and adults!) in your life, be they students or children. Our writeup of the lesson is here.

The brilliance of the game is that it demonstrates to the player how staying in their comfort zone, avoiding “errors,” and sticking with safe guesses prevents them from solving the puzzle. It teaches inductive reasoning, and why avoiding confirmation bias is important.

In other words, it teachers students how to think.

Equally delightful is that the math students can get into as they explore the puzzle. The more math people know, the more they tend to do as they play.

Update: my 9-year old just schooled me. Gave me 16 2/3, 31 1/3, 50 as his pattern and then started laughing deviously. His pattern is below. pic.twitter.com/9pCRWQAAaB

— Robert Kaplinsky (@robertkaplinsky) October 15, 2017

Math can teach students to think. Now we just need to make sure our math classes to the same.

3

## Poetry, botany, arithmetic, memorization

It’s probably a coincidence, but last weeks NYTimes had two articles in the Sunday Review on the value of memorizing relatively arcane knowledge.

One encourages readers to commit poems to memory, and recite them to each other; the other encourages us to learn the names of the trees that surround us, and start paying attention to their active, wild lives.

Two things jump out at me as I read these articles. The first is that memorization, so often maligned in educational circles, has a place in learning. It serves a function: to focus attention, to help us notice the invisible. There is no question that the role memorization plays in education can be overstated and overplayed. In math education especially, entire subjects are reduced to incoherent laundry lists of isolated, irrelevant facts that we’re made to memorize for some reason. Still, I worry that saying memorization serves no role at all is to remove a meaningful learning tool.

The second thing I notice is my own state in reading these articles. My mind quiets and fills as I think of the poetry I know, and want to know. I want to organize a party where everyone shows up and recites a poem to the group. Even though my own knowledge of botany is laughably poor, the invitation to spend time with a tree is welcome, and while I may not seek out the opportunity to learn the names of trees, I’d certainly be happy to if the opportunity arose.

Memorization is one of the great bugaboos of mathematics learning. As someone who usually didn’t have a great memory—acquiring foreign languages, remembering biological and chemical structures, and so on was always tricky for me—math was always appealing because it didn’t require much memorization. Or if it did, it was a kind of memorization that fit my love of stories. I can remember lines in a play if I understand what the characters want, and what motivates them; in this case, their words make sense. Math is like this for me too: the unexplained pattern cries out for explanation, and the facts and formulas are short hands for the arguments that make the structure sensible. On the other hand, memorizing all the rivers in Europe was an empty exercise when I had to do that in seventh grade. Those river names are all gone from memory now. I bet if we had learned the stories of history at the same time, I would have been more interested, though; I still remember the Arno, in Italy, because in Arieti’s historical novel The Parnas the characters have to decide whether or not to flee to the river, which marks the dividing line between the Allied and Axis forces in Pisa. I care, so I remember.

The point of all this is that context matters. Unmotivated memorization is one of the great ways to kill interest. But even in an age where information is everywhere, I maintain that memorization serves a function in learning, as long as it acts as an invitation to see something new.

For example, I find being able to know and be fluent with some basic arithmetic helps you see new ideas and patterns in mathematics, just like knowing the names of trees helps you see the natural world alive around you. I once led a class where we interested in the areas of tilted squares on a grid.

What is the area of this square?

There’s no simple formula to calculate the area of this square (well, there is, in a way, but my students were young enough not to know it).  And memorizing formulas to calculate the areas of geometric shapes is usually a waste of time and energy. “Memorize arguments, not formulas,” as a professor of mine used to say. Which is to underline the point that most – probably 95% – of the memorization we ask students to do in math class is a waste of time, and counterproductive.

But there still is that 5% that’s worth it. And in this case, after we’d spent several days finding the areas of squares and organizing our lists in various ways to make sure we had them all, we had discovered this list of numbers, representing the areas of tilted squares:

2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, …

What in the world is going on here? This is the kind of pattern that can leave us bewildered unless we actually have enough earlier patterns memorized to see structure in the noise. And this is precisely the kind of pattern that emerges in math sometimes, where there’s a deep, strange connection that reaches in from a completely different section of mathematics.

On the one hand, who cares. But on the other hand, exploring the interplay of patterns in the sequence above leads to a truly tremendous breakthrough known as quadratic reciprocity, which in its own right inspires vast new explorations in mathematical thought. But unless you have some previous patterns and arithmetic in your working memory, finding this kind of pattern is pretty much out of reach. This may seem like a peculiar example, but I think this is common in math: instant recall of certain objects, patterns, and algorithms helps us to see what can be invisible otherwise.

Forcing students to memorize something is like making them put something in their backpack before you hike up a mountain. It works best if they want to climb the mountain, and believe this tool in their backpack is going to be useful. And some tools can be picked up on the way! However, if they have to stop and go back to grab a missing tool every time they take a step, they never experience the beauty of the hike. That, to me, is the experience of the student who has to pull out a calculator when they are subtracting one and two digit numbers; they are constantly distracted from the beautiful experience of finding meaningful patterns.

So memorization has a place, but we have to be very selective. The moment the preparation goes too long, the experience becomes about being loaded up with a heavier and heavier backpack, filled with junk we don’t need, to climb a mountain we don’t want to go up anyway. Confusing memorization with learning is a huge error. Recognizing that memorizing a small body of selective knowledge at the appropriate time in order to aid learning and reveal beauty in the mathematical (or real) world is something else entirely.

Confusing memorization with learning is a huge error.

Still, I get a little nervous defending memorization at all. I recently had a back and forth on twitter with Eugenia Cheng, discussing if there’s any merit to memorizing in math at all. The pendulum has historically been so far in the direction of memorizing that the impulse to push it back is strong. If we had to memorize poems we didn’t care about (or that were in different languages we didn’t understand) for 12 years, we’d quickly come to say that memorizing poetry was pointless too. And yet, I can’t shake the sense that we are in danger of going too far; that teachers see the value in the selective use of memorization for students, and are afraid that it’s so strongly counter-indicated by experts that they should never do it.

So even as I emphasize the journey up the mountain, and all the reasons for taking it, I’ll keep reassuring teachers that they are allowed to help students memorize a select body of terms, ideas, and algorithms that will make the journey richer for them.

But keeping your eyes on the real point of learning – that’s wanting to take the trip – is always primary. As a sign off on that note, here’s a quote from Paul Lockhart’s new book Arithmetic: “There are a lot of people who hate arithmetic (far too many to count!), and it makes me sad. Usually it’s because they were made to do something they weren’t interested in doing. Let’s not have that be you.”

#### Postscript

One thing I realized in looking this post over is that I didn’t really define what memorization means to me. I tend to find context-less memorization (mnemonic devices and the like) useless or worse than useless. For me, useful memorization is the act of going over facts and ideas that you understand to keep the details fresh, and which might slip away from you otherwise. It’s memorizing lines in a play where you have a sense of the motivations of the characters, as opposed to the useless memorizing lines in a language you don’t understand. Sometimes definitions and vocabulary need to be memorized, but I usually like those to come from a genuine need to name something as well. So I guess there’s much more I need to say on this topic to articulate my own thinking.

Ben Orlin has some nice articles on the evils and value of memorizing in learning. (Interestingly, I don’t agree entirely with his “memorize on Monday” idea, but both articles are still definitely worth reading.)

1

## Math Teacher Circles

In the run up to the upcoming year of Math Teacher Circles, we decided to offer a kickoff workshop for 4th – 8th grade teachers to drum up some excitement for the upcoming year. (The “we” in this case is myself, from Math for Love, and Jayadev Athreya of the Washington Experimental Math Lab, run through the UW.) This partnership (M4L + WXML) led a series of eight Math Teacher Circles last year, for K – 8th grade teachers, and they were great. The model is one of my favorites: teachers opt in to meet once a month to study topics in math and pedagogy, then try things out in their classroom between sessions. The circles are a fantastic way to meet and collaborate with colleagues from other schools, focus on fascinating and critical issues in math teaching, and energize your practice. They’re also free! Sign up now for elementary or middle school math teacher circles.

This year, we’re hoping to put together resources that will allow others to take this model and adapt it around the state/country/world. To that end, I wanted to summarize the last week’s kickoff workshop, especially because it was a remarkably fun workshop to be a part of.

Part 1: Introduction and Motivating Examples

We had about 35 teachers in attendance, teaching math for 4th – 8th graders, plus a few mathematicians. The workshop began with conversations among groups about people’s math autobiographies. What were the most memorable experiences in people’s mathematical histories, positive and negative?

Some of the big ideas that came from these discussions:

• Math is emotionally charged. We feel a lot – pride, shame, satisfaction, etc. – in the course of doing math.
• One teacher can have a huge impact on how someone views themselves mathematically.
• Those moments where the math is meaningful creates a sense of empowerment; alternatively, those moments when math is empty and we’re working without understanding undermines that sense of personal power.

After the discussion we showed Robert Kaplinsky’s mind-blowing How Old Is the Shepherd video. This video always baffles me, especially because I suspect that younger students wouldn’t respond the way these 8th graders (they’re eighth graders!) do.

There’s an idea that certain institutions create a mirror image of the very outcomes they’re trying to prevent: there is a certain kind of sickness that exists only in hospitals, and a certain kind of ignorance created only in schools. We need to take this seriously. If we’re not careful, we estrange students from their own sense of knowledge, and they end up not being able to make the obvious observation: that something doesn’t make sense.

Part 2: Conjectures and Counterexamples

I’ve been developing a structure to help students to help invite students into a genuine mathematical process, starting with their own understanding. We call it Making and Breaking Conjectures. In the next section, we talked about this structure.

First, what are conjectures? What are counterexamples?

Conjecture. A mathematical hypothesis. A guess of the underlying structure or pattern based on what we know so far.

Counterexample. An example that proves a conjecture false.

Mathematics as a field progresses by way to conjectures and counterexamples. The good news is, we can use them even with very young kids.

We played the game Counterexamples to get a sense of how this works. (Lesson plan PDF here. Online here.) The game is super-simple: the teacher makes a false conjecture, and the students prove it false with a counterexample.

Teacher: All pets have four legs.

Students: No! Because birds have two legs!

Teacher: Okay – refined conjecture: all pets have two or four legs.

Students: What about a snake?

Teacher: That’s a pet with no legs. So I’ll refine my conjecture again. All pets have at most four legs.

Students: What about a spider?

And so on. So compelling is the game that we almost got off track when we considered conjectures about area and perimeter. But there was work on conjectures to be done first.

To warm up our observing/noticing/wondering/conjecturing muscles, we started with a paperclip.

This is a classic exercise, and Kindergartners tend to beat the pants off of adults. After spending ten minutes or so, groups had come up with everywhere from 20 to 65 uses for a paperclip. That was just the warmup, though, so we didn’t go too deep into what the uses actually were; the main event was still on its way.

Considering this question was trickier, and required a little more discussion to draw out observations and questions. By the time the conversation was done, however, we had a tidy collection of questions to consider.

• How many rectangles can be formed by connecting dots with straight lines?
(Refinement: diagonal lines vs only horizontal and vertical)
• How can you connect all the dots with connected straight lines drawn in a continuous path? How many lines does it take?
• How many sides could a polygon have on the grid? Could it be all 16 dots?
• Can you make a square with any number of dots on its perimeter?
• What grid came before? What grid comes after?
• Maze: how many different paths are there from the bottom left to top right? Refinement: direct route or can you loop back?
• If you can define “1” differently (length or area), what it the area of the whole grid?
• How many different angles can you form if you connect three points?
• How many different areas can you get if you form a triangle with three points on the grid?
• How many lines of symmetry can you make? What if the lines don’t have to be on the dots?
• How many symmetric shapes can you make with corners on the dots?
• How many lines would there be if you connected every dot to every other dot?
• Can you find two triangles with the same area that are not similar?
• How many different lengths could you find by connecting two dots?
• Can you make a triangle with area 1/3? (If the whole is 1)
• How many fractions could you represent on the grid?

Take that list in for a minute. There are weeks of beautiful, high level problems to explore here. For each, you can start by casting around. Soon, you’ll find you have conjectures. Once you have a conjecture, you can try to break it right away by looking for a counterexample. Refine and repeat until you end up with something that seems to be true, and then you can put together an argument, with luck and a little insight, into why it actually might be true. We’re really doing mathematics!

With all these questions to consider, we wanted to provide some guidelines. So we posed a choice: teachers could choose the problem that inspired them most and work on that one. Or they could work on the question of how many different areas a triangle could have if its corners are on the grid. (Having a default question is something we’ve found can help prevent groups from becoming aimless. With an actual classroom, you might want to skip some of these steps, and start from a more tightly focused, teacher-chosen task in the first place. Still, opening up the entire thinking process can definitely be worth it, if you’re ready to take the step.)

And with that, we sent the groups out to work on their own. However, we had one more idea ready at hand to ensure things went as well as they could go.

Aside: Thinking Classrooms

Peter Liljedahl has been developing a series of concrete steps to change the classroom to support genuine thinking in mathematics. The first, central step is to use tasks, that is, make real, meaningful mathematical experiences the heart of what you ask students to do. But you can help students actually transition, and the steps Liljedahl has determined to encourage this transition are bizarrely simple to employ. Here’s how you begin, according to Liljedahl:

• Use vertical, erasable surfaces (i.e., whiteboards)
• Assign groups in a visibly random fashion

It seems almost too easy, but the room was electric with thinking energy. Here’s what it looked like.

It’s hard to overstate how getting everyone standing and working together impacts the quality or thinking and engagement in the room. One of Liljedahl’s recommendations is to use just one marker per group. You can see the value of that suggestion in the video below, as the carrier of the marker is drawn back up to the board to write in details of someone else’s computation.

Don’t have enough whiteboards? You can write on windows too!

With so many teachers considering such a range of problems, we decided to let groups pair up to share what they had discovered with each other.

All in all, we were totally energized by the workshop, and I think the teachers were too. They’ve got a whole bunch of great questions to pose about grids, and the motivation to use task, whiteboards, and visibly random groups in their own classrooms.

And we’ve got a goal for this year’s math teacher circles, especially for the middle school level: create a series of PD sessions that will help everyone turn their classroom into a Thinking Classroom, where students use Conjectures and Counterexamples to power genuine mathematical experiences.

You can sign up for our 2017 – 2018 Math Teacher Circles now:

Part 3: Epilogue – the Global Mathematics Project

There was one more takehome point for the teachers: October will be the launch of the Global Mathematics Project! We shared another grid-related resource from the GMP.

Not only is this kind of problem a super-fun way to launch an exploration with students, it’s also grid-related! And it’s also just a taste of the kind of math that’s going to be highlighted during the Global Math Project in October, 2017. We’re looking to give 1,000,000 a genuine math experience the week of October 10. Interested? You can learn more and sign up here.

So, a long post, but hopefully just a beginning of what will be a framework to help others lead their own Math Teacher Circles, and help making authentic and beautiful mathematical experiences the center of math class.

2

## Math Games to Play at Home & NMF2017 Retrospective

I’m hosting a math night for parents and students tonight at a local elementary school tonight. These are always fun events, because

1. I get to preach how games and puzzles are some of the best ways to support math understanding at home
2. We get to play said games and puzzles and have a great time.

See the bottom of this post, where you can see my handout of puzzles and games to play at home, and other resources to check out.

In addition to my other favorites, I think I’m going to share the “Polka Loop Puzzle.” This is a classic puzzle we included as one of the puzzles to play in Tiny Polka Dot.

Last weekend I took this puzzle, among others, to the National Math Festival in DC. It was, not surprisingly, a blast. My only sadness is I was so busy sharing the puzzles and games I brought that I didn’t get to see all the other spectacular presentations.

I got to spend a lot of time with the folks from ThinkFun, from the Julia Robinson Festival nationwide, from Gathering for Gardner, and others, since we were all in the same section of the event. One big theme: the values of games in creating a math-positive culture at home. Everyone basically had their own variation on the sentiment that “What books are to reading, games are to math.” (One variant I was struck with: “What books are to writing, games are to math.”

Students would start by solving the puzzle from 1 to 5, and then add on a 6, then a 7, until they had it solved up to 10. One student really showed me something new when she concocted, and solved, a totally new variation on the puzzle I’d never seen before. Instead of putting one card on the bottom of the deck, you put one card on the bottom for each letter in the number that’s coming next.

Want more math games to play at home? Check out the handout I’m passing out at the Math Games Night tonight. And let me know what great games and puzzles I should add!

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1

## Raymond Smullyan

The logician and puzzle-maker Raymond Smullyan died yesterday at the age of 97. After reports circulating on twitter, the news was confirmed, and articles in memorium have begun to appear.

Smullyan was a brilliant designer of puzzles, and his books, especially The Riddle of Scheherezade, had a big impact on me. His idea of coercive logic, in particular, impressed me deeply: with the right construction, and the stipulation that someone will answer you truthfully (or even, truly or falsely), you can compel them to do whatever you want.

The New York Times published this obituary today, which is worth the read.

The Times also published a small sampling of Smullyan’s puzzles here. It’s interactive, and a nice place to start remembering the contributions of a prolific and wonderful puzzle designer.

Reposting our last email newsletter to the blog. If you’d like to sign up on our email list, click here.

#### Loss: A farewell to Sid the goat

The year 2017 began on a sad note for us. After an abrupt decline, our beloved pet goat Sid passed away on January 3. Sid was a dear friend; he slept indoors, and has accompanied us, in his storied, 12-year existence, on hikes, cross-country car rides, adventures and abundant wonderful moments.

We will miss him, and remember him.

He is survived by us, and by his companion Myshkin. I wrote a blog post several years ago considering the two of them as problem-solvers, and it feels right to call back to it now.

If you’re interested to know about what we learned from Sid as a problem solver, you can read Goat River Crossing.

I highly recommend the videos.

#### Birth: Tiny Polka Dot arrives

In happier news, our new game, Tiny Polka Dot, has now been delivered to almost all of our Kickstarter backers, and is available on Amazon.

Photos and reviews are starting to roll in, and it’s fantastic to see the response. We’re hoping to see Tiny Polka Dot helping 3-8 year old kids fall in love with numbers in homes and classrooms around the world.

#### Julia Robinson Math Festival, Feb 25

Registration is open for Seattle’s 6th annual Julia Robinson Math Festival, February 25th. Get your spot now before it fills up!

Registration is open for 4th grade and up (i.e, 10 year old and up). Tickets are \$10 – 15, sliding scale. Email dan@mathforlove.com for info about bringing groups.

If you’re someone who has a love of math to share, we’re looking for volunteers too!

#### Math for Love Classes start Jan 22 – register now!

There are still a few spots open for our Winter Math Classes, running at the PNA Sundays, Jan 22 – Feb 12:

For 1st & 2nd Graders
Mathematical Games and Puzzles
For 3rd – 5th Graders
Geometric Puzzlers

All classes take place Sundays at the Phinney Neighborhood Association.
Instructor: Paul Gafni

#### Problem of the Moment

Imagine walking in the following pattern:
1 block North
2 blocks East
3 blocks South
1 block West
2 blocks North
3 blocks East
1 block South,
and so on, repeating 1, 2, 3, 1, 2, 3 and N, E, S, W.

Do you ever get back to the spot you started? If so, how many blocks do you walk before you do? If not, can you prove it?

Try with other sets of numbers and directions.
What patterns do you find?

## A Math Magic Trick

I’ve been meaning to write this lesson up for a long time. Finally, here it is, in all its glory. You can check it out below, or find it here on the lessons page.

This trick is an extraordinary introduction to the power of algebraic thinking, and a reminder of what makes algebra so awe-inspiring. When I was in middle school, I used to make up

A video Launch for this lesson is available in two parts. PDF of the lesson is below the videos.

Part 1. When using as a class launch, stop at 2:45 and let the class come up try to find their own counterexamples.

Part 2 – the solution to the magic trick.

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## Warm Ups: Number Talks and More

In my last blog post, I wrote about creating Doing-Math and Thinking-Math classrooms. One small but important ingredient I’ve found helpful for both is a good warmup activity.

The goal of the warmup activity is to get students thinking and active right away. The barrier to entry should be extremely low, so that everyone can able to participate. It’s a chance for students to limber up their minds and possibly get an early success under their belts.

There are lots of great warmups out there, including games like Don’t Break the Bank and simple arithmetic challenges like Target Number. One teacher we work with has been leading some amazing Broken Calculator challenges as warmups, which has been fantastic to see.

But today, I want to talk about a classic warmup, and one that provides one of the best returns on investment in terms of time and energy required: Number Talks.

Number Talks are one of the best bangs for the buck routines you can incorporate into your math classroom. They’re build number intuition and fluency while giving you insight into how your students think; they support the idea that math makes sense, and you can explain what you see to help it make sense to others.

If you’re not familiar with Number Talks, our how-to guide is here. In a nutshell, the idea is this: pose a simple arithmetic or counting question, along with the prompt to not merely answer the question, but to come up with as many different ways of answering the question as you can. That minor tweak makes the activity more challenging, interesting, and sparks conversations that the students actually get invested in. It takes a few days to help kids learn all the routine, but once they do the benefits are massive.

From teachers who are familiar with Number Talks, the number one request I get is where to find sequences of questions to pose. We’ve written up a bunch of these lately, starting with counting collections of dots, and moving into solving math expressions. If these would be useful for you, please take them for a spin.

There have also been some exciting developments on Number Talk technology recently. My favorites:

• Fraction Talks (for upper elementary, middle school, and high school)
This is a fantastic resource with tons of ideas for extending classic Number Talks into higher grades with fraction-based images.
• Visual Patterns (for upper elementary, middle school, and high school)
These tend to be accessible as warmups only after students have had some practice with how these growth patterns work. For upper middle and high school students, however, these are an effective warm up, and a place you can see the huge payoff in how the numeracy encouraged by doing Number Talks regularly at the lower grades connects to algebraic thinking in upper grades.

• Would You Rather
Leading a Number Talk using a simple comparison question (“Would you rather have 16 dozen dollars or \$150?”) can motivate estimation. Using questions like the ones at the link, where certain aspects may be more poorly defined, can be an interesting way to connect mathematical thinking, common sense, and logical argument.
• Unit Chats
Unit Chats are a new innovation of Christopher Danielson. These are in a nascent stage of development, as you’ll see if you click the link, but I’m very excited about them. The main idea is to show a picture that contains different choices for units (i.e., avocado halves or avocado wholes vs pits?. You’re not just giving an answer + your strategy; you’re giving a unit as well, which has the potential to change the question and the answer.

I recommend finding a warm up that works for you and then making it a habit. Keep it short—5 to 10 minutes is optimal, in my opinion—and make sure it gets students thinking immediately. Try to do it 2-3 times a week at least, or every day if you can. It’s a small adjustment to your teaching routine, and one that can pay off in a big way.

One last thought: most curricula start with an instruction for teachers to “show students” how to do such and such a problem. You can run these teacher demonstrations as Number Talks instead. By giving students a chance to think about the problem first, they’ll be that much more primed to learn when the answer—and the explanation—arrives.

## The Doing/Thinking/Loving Math Classroom

This summer, we had the opportunity to draw up and institute a wholesale program from scratch, using our own lesson plans and providing the PD and support. Following that program, we began some fascinating conversations about how to articulate our vision of excellent math classrooms in more detail. From 30,000 feet, the direction we want everyone to move feels clear: our goal is to help design experiences that give everyone a chance to fall in love with math. But I feel like we’re due for a new articulation of where we’re aiming.

So what does this loving-math classroom actually look like?

1. The Loving-Math Classroom is a Doing-Math Classroom
The central activity in the class must be the doing of mathematics. Our metaphor here is brain as muscle, and classroom as gym. The instructor needs to launch the activity so that everyone can get started (and won’t injure themselves), but fundamentally, the actual “working out” is done by the students. The students must encounter activities that are difficult enough to be challenging but not debilitating. Most of class time (50 – 80%) should be spent actually doing math, preferably individually or in groups. This might mean playing math games (i.e., Pig), working through complex math tasks (i.e. The Power of 37), and also story problems and more plebeian math worksheets, though it can be problematic to rely too heavily on these.
2. The Loving-Math Classroom is a Thinking-Math Classroom
In addition to the actual doing of math, which is primary, it’s vital to find moments to take a step back and reflect on the process of doing math. What type of tools are useful to solve problems? How should we organize our data to best find patterns? What are good problem-solving techniques, and which whens are most useful for which kinds of problems? What do we currently know, and what questions do we still have? These times for reflection and discussion are critical for encouraging a depth of thought and development of the habits of mind that are, arguably, the true goal of math education.

What does this look like in real life? Our Summer Staircase classroom had a four-part structure:

1. Warm-up (5-10 min) — keeping with the gym metaphor, a quick game or exercise to get the mathematical thinking started. Number Talks are a prime example.
2. Launch (5-10 min) — this is the only time in lesson structure that doesn’t explicitly include doing or thinking math: the time when the teacher is explaining something new. In general, the teacher should streamline all explanation to the minimum required to allow students to work on their own. Sometimes the launch is time for a mini-lesson, or to pull together ideas from previous lessons. Sometimes it’s simply to demonstrate a new game or activity. In general, the goal of the Launch is to get them to the starting line, not the finish line.
3. Work (30 – 45 minutes) — this is the period when the students actually do the math. For the summer, we used a station model, since younger kids usually have trouble focusing on a single activity for a long period of time. Typically, stations included a math game, a new problem or task, and an activity aimed at helping kids practice to master a technique or get comfortable with a mathematical representation. Starting in 3rd or 4th grade, and certainly by middle and high school, it’s more reasonable for students to be able to focus on a single, more complex task for the duration of this time.
4. Wrap Up (5 – 15 minutes) — this is when the doing-math class has the opportunity to become a thinking-math class. The Wrap Up allows students to articulate their conjectures, counterexamples, arguments, and questions from class. It’s also where the teacher can underline deeper lessons learned, in line with the Common Core Math Practices, for example.

We found that virtually everyone could create a doing-math classroom. All it takes is the teacher belief that student activity is the central function of the classroom, combined with good materials, which we provided. Creating a thinking-math classroom is trickier; it takes more artistry and more practice.

______________________________

Aside: Last year, we wrote an introduction for our 2015 Summer Staircase curriculum that outlined what we dubbed five principles of extraordinary math teaching that later become my TEDx Talk:

2. Give students time to struggle
3. You are not the answer key
4. Say yes to your students’ thinking and ideas
5. Play!

It’s clear that these are all urging teachers toward a doing/thinking/loving math class as well. Principles 1-3 are designed to get kids doing and wondering as quickly as possible, and to avoid short-circuiting the doing/thinking process. Principle 4 is about expanding from doing to thinking. Principle 5 is about creating the atmosphere that actually makes it all jell.

______________________________

So here’s the goal for us now: we need to articulate and fully formulate a vision, first of the doing classroom that teachers can make their own by a series of small, manageable steps, and second, of a pathway from the doing to the thinking classroom. Here’s a small beginning to that process:

Doing-Math Classrooms ideally involve a Warm Up, as brief a teacher Launch as possible, and then most of the class time devoted to the students working on a deserving problem or task.

Thinking-Math Classrooms are Doing-Math Classrooms that additionally include Wrap Ups for reflection and development of higher order skills, in line with the Common Core Math Practices.

Loving-Math Classrooms are Thinking-Math Classrooms that, at least occasionally, blow kids’ minds.

We’ll try to go forward with sketching a fuller vision of all of this. In the meantime, I’d love to hear your feedback.

## App Review: Exploring the Core

I just downloaded a new app that I think will be helpful reference! It’s called Exploring the Core. (At last check, it was \$2.99.) It is, essentially, an easy-to-use encyclopedia of Common Core math content standards. It includes a list of the standards, by grade, and a glossary to help you understand what all the terms mean. So, handy, though not much added value there.

But what is really special is the skills section of the app, which gives a map by major content area of the skills students will need to know. Say we click on Operations and Algebraic Thinking.

We’ve got an easy-to-scan map of skills by grade and type. And the types are easy to understand: word problems, fluency, equations, graphing, etc. Click on one and you get a page that shows you the target, the standards that match it, and mathematical images that explain it, along with the conceptual knowledge beneath it. And here’s the real beauty of Exploring the Core: the examples are mathematically rigorous, clear, connected to a solid conceptual understanding, and feature useful diagrams/manipulative ideas to use when teaching. How easy would it have been to just say, know your sums and differences up to 10, and apply up to 20 with this standard? By applying the images, the app suggests the kind of thinking numerically fluent students should actually be doing to build fluency. If they don’t have everything memorized yet, you can use ten frames and manipulatives or pictures to help get your kids there.

And that’s what makes this app so useful. I think Exploring the Core will prove helpful for me and the teachers I know, as a reference for the CC standards that fills in what’s between the lines with mathematically accurate images.