Representations and Manipulatives and Tools and Things

One of the things I think about is the relationship between teaching math and the physical stuff that goes along with teaching math. This relationship gets distorted sometimes.

For a while, the elementary world got all kinds of swept up in manipulatives. All lessons became “hands-on” because somehow “hands-on” led to “minds-on.” Deborah Ball’s classic Magical Hopes article does the best job I know exposing the flaws in this stance. If you haven’t read it, by all means, click that link and read it right this second.

People sometimes get hung up on tech in the same way. Recently, I had the chance to share some outstanding work Kristin Gray got from her students when she asked them to take out their notebooks and write down what they were wondering about doubling and halving. They’d been working on 14 x 25 = 7 x 50. Check out these conjectures:

I mean, so great.

Both times I shared this work, people oohed and aahed, and then asked the same question: “Could you use tech to do this? Maybe a google doc?”

I have to confess, I don’t understand this question. With paper and a pencil, students were able to shift back and forth between words and numbers effortlessly, much faster than 11-year-old kids can type. If they’d wanted to make a quick sketch or doodle (perhaps an area model, in this situation), they could. They didn’t have to lose their train of thought while hunting through their device’s symbols for ÷ (an obelus, for fellow #wordnerds). The only apps I know that allow students to think and write so freely are apps that turn tablets into $800 notebooks by letting you write on a screen with a stylus.

I kept wondering, what’s the value added there? What’s the rationale for adding tech? What can it do for you that cheapo paper notebooks can’t?

That’s the question I ask myself about tools, in general, whether they require charging or storage in a plastic tote. What will they do for the mathematical teaching and learning here? Sometimes, the answer is not much. Other times, A LOT.

This year, I did something new in my school. My principal and I made it a priority for me to work with our paraprofessionals. These colleagues are overworked, underpaid, undertrained, and almost never supported to go to PD. Yet they’re responsible for educating about 20% of our students–the neediest 20%. So, this past year, my principal worked out a schedule where, one week per month, I had all the paras in two different shifts on Tuesday and Thursday afternoons. Over the year, we did a book study of the second edition of Children’s Mathematics. I chose it because it’s the foundation of everything in elementary math. The book and its embedded videos teach educators how students think about the operations, and how that thinking progresses. They also teach educators to listen closely–and quietly–to students while they think through problems. Given that the paras had been jumping in too quickly to tell or spoonfeed or direct kids, I wanted them to start learning how to tolerate wait time, how to trust that their students have mathematical ideas, and how to listen as a core part of teaching mathematics.

[Update, June 30 2017. I received a note from one of my colleagues–a proud paraprofessional–today. It was a hard note to read, but I am so grateful that she wrote. She taught me a ton. I am leaving the blog intact so you can see what I wrote, and what I learned. Let’s take a look at that paragraph again, but I’ll fix it, and then make changes throughout:]

This year, I did something new in my school. My principal and I made it a priority for me to work with our special education team, including certified teachers, therapists, and paraprofessionals. These colleagues are overworked and underpaid, and there are many demands on their time because they frequently have trainings about specific disabilities and student needs, as well as IDEA compliance, not to mention the meetings required for IEPs. undertrained, and almost never supported to go to PD. Yet they’re These colleagues are responsible for educating about 20% of our students–the neediest 20%–but both the literacy coach and I get less time with them than we do with the classroom teachers. So, this past year, my principal worked out a schedule where, one week per month, I had all the paras the full special education team in two different shifts on Tuesday and Thursday afternoons. Over the year, we did a book study of the second edition of Children’s Mathematics. I chose it because it’s the foundation of everything in elementary math. The book and its embedded videos teach educators how students think about the operations, and how that thinking progresses. They also teach educators to listen closely–and quietly–to students while they think through problems. Given that I’d observed that some of the paras had been jumping in too quickly to tell or spoonfeed or direct kids, I wanted them to start learning how to tolerate work on their wait time (which we all always need to work on). I’d spent a few years working with the classroom teachers, reading the same book (Children’s Mathematics), encouraging them how to trust that their students have mathematical ideas, and  how helping them learn to listen as a core part of teaching mathematics. I wanted the special education teachers, therapists, and mainstream coaches to have similar opportunities to work on their listening.

I also knew that digging into students’ mathematical ideas would allow me to get the parasspecial education team digging into the mathematics itself. One para loves math and I knew she’d be game, but many were reluctant and some were downright hostile to the idea of this year-long focus on mathematics. I had my work cut out for me last fall.

Fast forward to June, and I gotta tell you, I loved my time with the paras special education team. I think it’s some of the most important work I’ve done in the building and I can’t believe it took me so long to get there. We built a safe space and strong relationships and, most of the time, they got more and more willing to try new ideas, wonder about why things worked, and make sense for themselves. I hope they understand that I’m sharing this story partly to encourage my fellow math coaches to think about how they can support their special education colleagues, partly because it helps me make my point about tools, and partly because I’m so proud of the growth I saw over the year, and want to share it. I hope I’ve done the story justice, and I hope, if I haven’t, they’ll tell me. Nothing matters more than the trust and safety we’ve built together.

One observation I’d made over the year was that several parascolleagues openly despised both the area model and the related partial products strategy for multiplication. They didn’t understand why anyone would do that, and were resistant to multiplying any other way than the standard algorithm. As our year drew to a close, I wanted to devote time to multiplication of double-digit numbers and see if I could get anywhere with this animosity toward these two essential multiplication strategies. I knew that if I just drew or recorded the strategies with equations on chart paper, I would lose them. I’d learned that lesson the hard way, and wanted to avoid the shut down I’d caused before. So I needed something. I needed a tool that would unsettle our typical patterns.

In this case, I reached for graph paper. I handed them each a rectangle of graph paper and asked them, “How many squares are there?” Note, I did not use the word “multiply.”

Nobody shut down. Everybody got to work, and I got a great range of strategies, including the ones above. (The graph paper was 14 x 21, if that helps.)

Man I love a variety of strategies. It’s just the best. Now I had a whole range of decisions to make about where to go from here. (If you want a thoughtful discussion about that decision-making process, you need to read Intentional Talk. It’s had a huge impact on me.) There were a number of things I could do, and a number of competing goals in my head. A few of them:

  • I needed to explicitly connect what they did with the graph paper to multiplication.
  • I needed to get them more comfortable with representations of what they did with the graph paper, both in pictures and numbers. Optimally, they’d be the ones recording, not me.
  • I needed to expose some rich mathematics by digging down into one of these, or by drawing connections among a few of them. Which ones?
  • I needed to take this opportunity to highlight the fresh mathematical thinking from some parascolleagues who have had negative histories in math, who started out the year reluctant but dove into this problem bravely, and who still needed support to see themselves “as math people.”
  • I needed them to explain their strategies to one another so they could put words to their own thinking, and listen to and try to follow their peer’s thinking. (This was an ongoing goal all year and we’d made a lot of progress.)

So here’s what we did. Each person explained their strategy. While they did, I asked for volunteers to come up and record on paper what their colleague had done on our chart paper.

I stayed quiet while the person with the marker recorded, and then naturally turned to their colleagues and said, “Is this what you did?” or “What was your equation again?” We were getting somewhere.

In each group, someone had surprised me with their strategy. In the first group, J said she looked at her rectangle and thought, “Way too many to count.” So she folded her rectangle in half:

She thought, “Still too big,” and folded it in half again:

She looked at that and thought, “I can count that and then multiply by four.” The thing is, one of the factors was odd, so quartering led to a fractional result. She didn’t bat an eyelash:

She had a 7 x 10 rectangle, which yielded 70 squares. She then combined 6 half-squares to make 3, and had 1/2 left over. Each quarter had 73 1/2 squares. To figure out the total number of squares, she pinched together the 4 half-squares into 2 whole squares, multiplied 73 x 4 to get 292, and added them together to find 294 squares. She told us that the folding grew out of her comfort with sewing, and she was completely in command of her strategy.

I did the recording on J’s work. Not beautiful, but everyone agreed that what I drew matched what she did.

In the next section, L also began by folding her rectangle in half, but the other way:

Then she groused at me, “Oh, you made it not work out evenly!” A moment later, she said, “That’s OK, I love thirds.” Look how pretty:


A perfect square!

When she unfolded it, she had this:

Now we see one power of a tool. If I had explicitly asked them to solve the multiplication problem 21 x 14, I would have had almost all identical column multiplication solutions, which aren’t ripe for rich discussions. But this little graph paper rectangle yielded a wide range of approaches, including two strategies that made beautiful sense, visually, but had almost no chance of emerging if we’d only worked with numbers:

21 x 14 = 4 x (10.5 x 7) and

21 x 14 = 6 x (7 x 7)

Not only that, both J and L were in the spotlight for innovative math, when both J and L have historically not been so keen on the subject. They were the experts on these strategies, teaching me and the rest of their curious peers. If I could have bottled that moment and given it to them for safe keeping, I would have. It was a highlight of my year.

That brought us to the end of Tuesday. I knew I’d start with these on Thursday. Because J and L were in different sections, they hadn’t seen each other’s solutions, although word spread quickly throughout the staff and I heard them comparing notes after school (yeah baby). We began Thursday by marveling over the two strategies, comparing and contrasting the difference it made to want to avoid fractions or not be bothered by them. I then focused us on L’s strategy, written numerically. When we looked at the piece of paper, we could all see that there were six 7 x 7 squares. But when I wrote the equation:

21 x 14 = 6 x (7 x 7)

there was a lot of wondering about it. They all agreed that, if they’d been working with numbers only, they never ever would have transformed 21 x 14 into 6 x (7 x 7). They wondered, where did those numbers come from? Especially that 6? How could a 6 come out of 21 x 14? Eventually, there was recognition that 7 was a common factor of both 21 and 14. That insight led us to write:

21 x 14 = (3 x 7) x (2 x 7).

The 6 was in there somewhere, starting to become more visible, but this is a place where nobody was sure about the rules they’d learned once. What could you do with that 3 and 2? Add or multiply? Five or 6? We went back to the paper:

Do you see a 2 x 3 array there? A 2 x 3 array of 7 x 7 squares? Six 7 x 7 squares? Holy smokes, there it is. They saw where the 6 came from.

We played a little with the quartering strategy in the same way:

21 x 14 = (10.5 x 2) x ( 7 x 2) = (10.5 x 7) x 4

We wrote them up, talked some more about the associative property and what happens when you break factors up by multiplication. Now, to be clear, I am not saying everyone in the group would be able to recreate this logical flow independently yet. But I am saying everyone in the group was following along. They didn’t shut down at the formal math vocabulary, at the symbolic representation, at the diagram.

And that’s why I was glad to have a tool. In this case, the tool made it possible for everyone to access the mathematics here. It helped me gather a variety of solutions so we could make connections among them. It made tangible what had been abstract. It allowed my colleagues to bridge from something intuitive to something a little out of reach. And it made us talk with each other about the mathematics more, not less.

I’d call that added value.

I don’t get fussed over whether tools are high tech or low tech. I love and use them all. But I do take care to use them thoughtfully, not for the sake of using tools or tech, but for the sake of the mathematical learning and conversation they’ll allow me to engineer.

In early drafts of my book, there was a chapter called Mathematicians Use Tools. I was planning to get into all of this stuff. I cut it for the sake of length–it was already a huge book–and I thought tools had been written about a lot elsewhere. They have. I decided, instead, to showcase thoughtful use of tools throughout the book, which wasn’t hard because effective lessons often involved the strategic use of tools. Probably the right call.

There are times I regret not taking the deep dive into tools, though. I see so much tech-for-the-sake-of-tech, tools-for-the-sake-of-tools. I also see teachers still afraid to use tools for fear of mess or noise or lack of control or time or organization. I’d love to explore when and how and why we reach for them–or don’t.

Maybe this is the space.

My Criteria for Fact-Based Apps

Disclaimer: I do not work for any edtech companies and I have no desire to work for any edtech companies. I have no skin in this game. I make absolutely no money off any apps and plan on keeping it that way. This disclaimer means I have nothing to disclaim and you are getting my straight-up opinion here. 

People often ask me what apps I can recommend for multiplication and division fact fluency. This question usually puts me in a tough position because, while eventual automaticity matters, I care about building conceptual understanding of these operations. My buddy Graham Fletcher recently described one path the progression of multiplication can take in this outstanding video. As Graham points, out, let’s not rush past all that great work!

And yet the requests come. This week, I fielded one from a teacher I needed to answer, so I threw the question out to twitter to see what’s new on the market. I’ve been looking around since, and the big money math fact app world is enough to send me into despair. It’s almost all awful. As I looked at them, I noticed I use three baseline criteria, and I’m unwilling to compromise on any of them:

  1. No time pressure. In some apps, there is a giant timer counting down. Or you have to answer before the sun sets. Or the context is such that the whole experience feels like an anxiety nightmare.

    These screen shots came from the Arcademics game “Meteor Multiplication.” They’re basically embedding flash cards in the great Atari game of my youth, “Asteroids.” Except now, blasting asteroids isn’t fun. Now, if I can’t think of my facts fast enough while the meteors close in on me, I feel like I’ll be crushed to death. I know my facts. I found it hard to think when facing impending doom. This isn’t surprising. Studies like the one from Ashcraft and Moore have shown that:

    “Overall, math anxiety causes an “affective drop,” a decline in performance when math is performed under timed, high-stakes conditions, both in laboratory tests as well as in educational settings. This means that math achievement and proficiency scores for math-anxious individuals are underestimates of true ability. The primary cognitive impact of math anxiety is on working memory, particularly problematic given the important role working memory plays in math performance.”

    I don’t have math anxiety, but many of our students and teachers do, and timed drill is a leading cause. My 1st-grade daughter’s classmates told me they have stomachaches from the “Speed Round” in Addimals. Under no circumstances will I recommend any apps that involve time pressure or speed rounds. This criterion strikes the vast majority of apps because the premise of so many of them is that speed races are “fun.”

    Screen Shot 2016-01-05 at 9.26.31 AM
  2. Conceptual Basis for the Operations. I don’t want to see naked number drills, especially not for 3rd graders. Flashcards embedded in silly or glitzy contexts are still flashcards. I want to see mathematical models like arrays, groups, hundreds charts, and number lines. Most of the apps put their money into developing exciting or humorous graphics to lure in customers, but what kind of mathematical sense does this make?

    This screenshot is from SumDog’s Junk Pile, where the answers to math problems inexplicably turn into trash piled up in a junkyard. In this example, my correct answer to 80 – 10 turned into garbage. I mean, I’m trying not to be too sensitive here, but what kind of message does it send when we turn math into garbage? Literally! The first question I answered turned into a toilet bowl.

  3. Mistakes Must Be Handled Productively. The first thing I do when I trial a game is I make mistakes on purpose to see what will happen. The overwhelming majority of apps give some form of a Family-Feud-style giant red X. Sometimes with flames, molten lava, or puffs of smoke, because I guess the X is too subtle?
    red x

    I’d estimate that, 9 times out of 10, most apps just move to the next problem after a mistake. The kids don’t get to figure out where they went wrong, don’t get to learn from the mistake, and don’t get to try again. Talk about missed opportunities. And then there’s the way mistakes affect scores and the endless parade of stickers, stars, and tokens in these games. Screen Shot 2016-01-04 at 9.28.22 PM

    Screen Shot 2016-01-04 at 9.30.39 PM

    In this game, we played tug-of-war in tractors, because I guess that’s a thing. Anyway, it was a race. In the second round, we did some tandem tractoring where I was partnered with a stranger on the internet and we raced two computers. I was Player977, and I answered 35 questions/minute. That’s a much faster rate than we can ever expect students to recall facts, even with automaticity. I made one mistake on purpose and guess what? We lost. I can’t tell if I lost because of the one mistake, or I lost because my internet-anonymous partner, Player807, needed a little more time to answer. Am I supposed to think of Player807 as deadweight? I mean, come on. However I slice it, I lost through some algorithm that only counts speed and correct answers and discounts mistakes as bad. Little surprise that most of these app web pages have correct answer counters, like McDonald’s does for burgers.

So what apps can I recommend? The list is painfully short. One program rises right to the top. Dreambox is my preferred app, without a doubt, by a mile, far and away from the others. Students have plenty of time to think. Cathy Fosnot’s landscapes are built right into the structure of the app, so students are often working on a variety of interrelated ideas and models. For example, Daphne is in 1st grade, and she’s working with rekenreks, number lines, and tens frames.

The modeling is just off the hook. I took these screenshots while Maya (3rd grade) was working in a context for place value. She needed to think of 4 different ways to pack 703 objects among cases of 100, boxes of 10, and loose 1s.

Screen Shot 2015-11-08 at 4.32.58 PMScreen Shot 2015-11-08 at 4.35.36 PMScreen Shot 2015-11-08 at 4.37.55 PM

Given the poor conceptual underpinning of my kids’ curriculum, I sprang for dreambox at home, so I get to peek over Maya and Daphne’s shoulders now and then as they play. When they make a mistake, the response is, “Oops. Something doesn’t seem quite right.” The kids are expected to try again. If they’re stumped, the app supplies really good support, layered in like a teacher would.

This is straight talk. My kids love dreambox and ask to play regularly. I’m down with it because, on top of it nailing all three of my criteria, the story contexts are fun – pirate ships, dinosaur fossil excavations, amusement parks – and the messaging about math is aces from both equity and growth mindset vantage points.

What else is out there? I’ve heard good things about Wuzzit Trouble from Keith Devlin. I haven’t tried it myself, though. I plan on giving it a go.

h/ts to John Golden and Paula Krieg for pointing me toward Bunny Times this week: It’s actually free. It isn’t as glitzy as the other apps and doesn’t produce teacher reports or have a dashboard. It’s also focused singularly on multiplication using arrays. But, my friends, the math is right.

Screen Shot 2016-01-05 at 10.07.30 AM.png

They build in some nice decomposition work as the problems get bigger, so kids start to build foundational understanding of the distributive property.

Screen Shot 2016-01-04 at 7.23.55 PM

Over time, a fog rolls in, which discourages counting by ones.

Screen Shot 2016-01-04 at 7.21.17 PM

And check this out. Look what happens when you make a mistake.

Screen Shot 2016-01-04 at 7.22.37 PM

You see that feedback? See that carrot there? And do you see that lovely sign, right in the foreground? The one that says, “TRY AGAIN?” So we did.

Screen Shot 2016-01-04 at 7.23.10 PM

I wondered what Maya would think of this app. She adored it, and I watched her do some really good thinking. For example, she’s still a big fan of doubling. At the beginning of her bunny time, she’d think about a problem like 8 x 12 as 2 x 12 = 24, 24 x 2 = 48, 48 x 2 = 96. After some time with the bunnies, I saw her solve 8 x 12 with (8 x 10) + (8 x 2). That’s a nice development. She started thinking strategically about breaking the dimensions into smaller numbers, using more 5s and 10s and place value-based thinking.

So, I was saved from my despair by the lovely design at dreambox and the solid math of the bunnies.

My message to most app makers? Try again.

Clarification, Jan 5 2016. 

I think I need to clear something up. Dreambox and the Bunny Times aren’t comparable. Dreambox is a full program. Soup to nuts, as they say, A huge K-8 progression, thoughtfully organized and designed, that develops both understanding AND fluency, together. That’s why I like it so much. And, I’m sorry to say, that’s why its pricepoint is much higher.

The Bunny Times has nothing like the scope or scale of dreambox. It’s not trying to be more than it is. It’s an app tightly focused on one part of developing multiplication. But when they decided how to structure it, I think they did a nice job thinking it through.

And they met my criteria of no time pressure, a conceptual basis, and handling mistakes productively.

I personally think that’s not a high bar. I mean, those three criteria are my bare minimum. That’s why it’s stunning that so few apps clear it.

People have been tossing apps my way all day both on twitter and in the comments. I’m happy to take a look. But I’m telling you right now, if it has time pressure, no emphasis on understanding, or poor handling of mistakes, it gets a fail from me.

Clarification #2 Jan 7 2016

Looking through comments and twitter conversation about this blog, it seems there are two things I didn’t make nearly clear enough.

  1. This list is not exhaustive. I certainly didn’t try everything out there. I shared dreamBox and Bunny Times as two examples at two price points and two scopes that meet the criteria I outlined. And I shared that I’ve purchased dreamBox at home because I think it’s great.
  2. Lots of people have been asking about other factors and attributes about games, apps, subscriptions, programs, etc. This is a great question and the main thing I want to clarify. There are lots of other characteristics I look for when evaluating games and apps. Here’s a partial list, the best brain dump I could do before caffeine this morning:

Is the game engaging? Do kids like it? Are they motivated to use it? Does it encourage students to interact with each other? Does it help build mathematical intuition and sense-making through things like prediction, estimation, revision, reflection? Do students get multiple iterations to solve until they decide they’re satisfied and pleased? Does it reduce barriers to entry? Does it break down or contribute to stereotypes about who can and can’t do math (students of color, multilingual students, students with disabilities, gender)? Is the mathematics it addresses important? Does it contribute to societal myths about what mathematics is, or unseat them? Does it adapt to students’ performance? Does it yield useful formative assessment for teachers (not just “data”)? Do the problems have any openness (beginning, middle, end) or is it all answer-getting? Do students get to write? Are there options for student creativity and initiative? How are the graphics? Is there mathematical beauty and delight? Do the kids ever get to choose among representations and models or create representations and models themselves, or is everything provided? Do the kids decide their strategy or do they execute what the program chooses? Does it portray mathematics as inherently interesting, or something that needs to be dressed up in gimmicks and rewards?

If you’re wondering what program could possibly do all that well, I’ll send you to Have a look around. Enjoy.

But here’s the deal. In this particular blog, I wasn’t trying to list all the things I consider. I was trying to give my 3 non-negotiables. As I said above, I won’t compromise on any of them. I don’t actually care how great the graphics are, how good the teacher reports are, or how many people tell me “but kids like it!” if the program has time pressure, no conceptual basis, and handles mistakes negatively. 

That’s what I meant by baseline criteria. If an app meets those three criteria, then I’ll look at the rest. But when I first look at an app, these are my three questions:

  1. Is there time pressure?
  2. Is it conceptual?
  3. How does it handle mistakes?


Update July 11 2016

Thanks to Julie Wright and Dan Meyer for recommending Sumaze! Super strong. Thumbs up. And free!