And Like That They Invented Mathematics

I asked them to pull out their notes and write down “New Hampshire.” They did. Then I told them to write down five more state names.

I should have ratched this up to fifteen but five was annoying enough for most. They grumbled and I gave them permission to abbreviate the names in whatever way made sense to them.

Most students balked at “Mississippi.” They abbreviated every state name but that one. “Too many states start with ‘MI,'” they said. We talked about how tricky it is to decide on a rule for abbreviating, how it can lead to confusion later.

You see where this is going, right?

I had them write down the number “5,449,203,159,204,210,” which they did. Then I had them write down more numbers and I gave them permission to abbreviate again.

The next part happened quickly but required a lot of encouragement because students have been trained to treat numbers like so many sacred little statues. (“Do not touch the numbers! Do not feed the numbers!”) We asked ourselves, “which is the most important digit here?” After that, students started coming up with variations on the same theme:

5, 15

From there it was a really quick shuffle step to 5.45 x 1015 through this slide here.

Too many of my students have decided that math is a weird irrelevant game with arbitrary rules that are known only to strange old people whose hands are stained by dry-erase marker. From my experience, nothing works quite as well to disabuse them of that impression than putting them in a place to accidentally invent that game themselves.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Hey Dan,

    You may already have this somewhere on your website, and if you do please direct me there. But it would be great to see, in action, a full video of one of your WCYDWT lessons. I do remember seeing a clip from your cost of various liquid lesson, but I’m pretty sure it was just a clip.

    I say this because I’ve been doing project-based lessons for quite some time now, most of which are very closely related to the idea behind WCYDWT. And in trying a few of your WCYDWT lessons I feel that I’m pressing too much with the students, providing too much direction/encouragement/pushing, and the lessons are not developing as we educators see them in our heads when we read your blog. I don’t feel that this happens with my various project based lessons, and I’d like to see your WCYDWT lessons in action to understand what’s missing.

    I’m going to send you an email about a recent project-based lesson I’ve done. Probably too long to put here in a comment section, but please feel free to post any parts of it on any part of your blog for dissection/discussion.

  2. Dan,

    I appreciate your enthusiasm for letting students invent the math. I found your blog via a video of you that Ezra Klein posted on his WaPo blog. Reading your blog, I noticed in an older posting that you weren’t sure whether to be satisfied by your final exam results.

    I don’t see much reference on your blog to the research about discovery learning, and the last few years have seen some major developments in the research of this topic. I thought you might like to see what I think are the best recent articles on the subject:

    Practice Enables Successful Learning Under Minimal Guidance, Journal of Educational Psychology (2009), available at

    What Needs to Develop in the Development of Inquiry Skills?,Cognition and Instruction,26:4,512 – 559. A free version of the article is available at

    The references in both articles provide ample material for further reading.

  3. I love this! I am totally going to steal it. Thank you for translating math into English for the kids and sharing it with us!

    Also, for a fun clip on the misuse of exponents. Sorry for the whole episode – I haven’t figured out how to import and edit clips – yet!

    Summary: Kirk tells the crew that the computer enabled everyone to hear the heartbeats by amplifying the heartbeats by a degree of “one to the fourth power”. It is at the very end of the clip (last 10 minutes).

  4. Dan,

    I read your blog every day, and I don’t teach, at least not in a classroom – I do private math tutoring, which doesn’t really allow for the same experiences and activities that you employ. I did teach, for one year, and hated it – it wasn’t the thing for me, and sometimes people ask me if I’ll ever go back. I always say, “No, not a chance,” and that’s still true, but every time I read some lesson idea on your site, like this one, that’s just terrific, part of me says, “But what if I could do it like THIS?” So what I’m saying is that, for someone who really doesn’t want to teach, you make me think, “Just maybe,” which, if you knew me, you’d realize was the biggest compliment I could give. Bravo.

  5. Hi Dan,

    I stumbled on your TED talk the day after my husband and I had a two-hour discussion about why math education is so bad and how anyone could possibly change it. We have been discussing homeschooling our kids, and I thought it was interesting that with all the cool things families are doing with homeschooling science and literature, almost all of them revert to a pre-cooked textbook and worksheets when it comes to math.

    I rambled on a lot about how real math has very little to do with the arithmetic and rote skills children learn early on in school, and how our methods of teaching almost guarantee that very few kids will want to study higher mathematics later. The best analogy I could come up with was this: “The way we teach math would be equivalent to English classes consisting only of spelling and grammar, with the promise that when kids go to college they might use those skills to read a novel. Someday.” How many kids would willingly pick up a book under those conditions?

    But then I found your talk, and here was someone doing exactly what we’d been trying to figure out how to do ourselves! If you have never read John Holt’s book, “How Children Fail,” you might be interested in his sections on realizing how much more his 5th-graders “get” math when he allows them to discover the process for themselves, very much the strategy you are taking.

    Math is just as much a part of nature as any other branch of science, only we reduce it down to symbols and give it special rules. Kids are always interested in the world around them and how it works. As I pointed out to my husband, when our two-year-old holds up two pieces of granola, says “Two oatmeal,” then eats one and says “One oatmeal,” he already has the concept of subtraction. The symbols are just the next step.

    Thank you so much for what you’re doing. As a math major myself, it’s exciting to see someone finally breaking free of the boring stranglehold math curricula have created for generations of students. Keep up the fantastic work!


  6. Dan,
    I have been reading your stuff now for about a year now and have enjoyed it immensely. You are insightful and refresshing. What I would like to know is, how you stay motivated? Just curious.

  7. Hi Dan,

    I had an idea to piggy back on this idea about scientific notation. How about instead of starting with state names and abbreviating them, you instead start with typical text speak? Like: ttyl, lol, pre, etc.
    Keep up the good work by the way :)
    (who likes having a network of equally keen math teachers to chat with)

  8. Hi Dan!

    I found your blog through the TED talk. What are the chances of finding my very own words on some blog on the internet? I agree with everything you say. (Well, “my own words” but in English, I’m from Mexico).

    I just wanted to ask about the last slide… What’s the purpose of ending the numbers in the same way? I didn’t fully get it.

    I hope I could share some of my material (I love videos and flash animations), and get your approval. Well, there’s one video I like, but its not linked with the topic of the classwork done

    Thank’s for sharing.

  9. Amanda,

    I woke up with the same thought about the text abbreviations this morning (I get my best ideas at 6am). I like it bc teens make so many of them up!

    I also thought about other common abbreviations, ave, rd, st, or dr. mr. mrs, bc, ad, and www of course.

  10. Dan,
    I love seeing the way that you use discovery to help kids learn how to “play” with numbers. I’m a fourth grade teacher, working inside a very rigid old-school math curriculum, while doing my best to draw my students out into larger pools of thought.
    Math is far more interesting when students learn that it can be played with and manipulated for whatever they need. Love you site!

  11. I don’t think there’s really much of an analogy to be made between state name abbreviations (or lexical abbreviations in general) and scientific notation. To be useful a state name abbreviation needs to be unambiguous. That’s why “MI” is problematic: it doesn’t distinguish MIssissippi from MIchigan. Scientific notation, on the other hand, is inherently ambiguous. It indicates an approximation for a range of possible values. And the number of significant digits used in scientific notation is not chosen to distinguish between separate values, but to indicate a range of possible values as determined by the accuracy which is available from measurement or required for application. Also, while scientific notation is usually shorter to write than other representations, that isn’t a primary criteria. Note that “102.3” is shorter than “1.023×10^2” and “12×10^9” is shorter than “1.2×10^10”, but they aren’t in scientific notation.

  12. From here it’s a good transition to logs and log scales. I didn’t think of this myself, but a log scale is a way to view numbers strictly in terms of their relative magnitude. So your “times 10 to the x” concession to hugeness has applications later on as well.

  13. Yes, brilliant, but I’ve known this to be true from the time I was forced to learn math in the abstract. It was a complete failure on the part of the public school system I attended on many levels, needless to say, I didn’t graduate high school. I later gained a deep appreciation for math as I became more and more enthralled by theoretical physics and as I was required to learn complex math problems for the type of work I found myself doing. I gained the philosophy of “Lessons learned the hard way, are the best lessons learned”. To learn something from your own experimentation is the best way and for some people like me, the only way.
    I know find myself facing the same dilemma I faced as a child and failed miserably with my own daughters, well mostly one of them. She is quite intelligent but is handicapped by attention span deficits as I was too, only this time around we have treated her for it with some good results. She is fascinated in all things science and I can see her taking her place amongst other females in my family as a scientist but her struggles learning math are threatening to kill her motivation, if it hasn’t already. I have tried to teach her or at least get her to listen to examples of how I use math in my job but her attitude is already bad to the whole idea, like you said “trying to sell a product no one wants but has to have”. Now that I have seen how you have presented it, I get a feeling that it may really help her. I cannot let my girls fall victim as I did to the public school system here, I will be looking for resources locally to attempt to teach her math in this fashion and I will be researching this methodology in more detail.
    Good Job

  14. Brilliant, Dan! I work for an educational assessment company, creating evil math tests, and am forwarding the link to your TED talk to my coworkers tomorrow. There’s a trend toward highly interactive “performance assessments” right now, and I think this will really motivate all of us in education to think in new ways about how we teach math and assess mathematical ability. Kudos!

    One thing on this particular blog post, though … is there a typo? I don’t see how the 5.23 x 10^15 corresponds to any number on any of your slides.

  15. Andrew: One thing on this particular blog post, though … is there a typo? I don’t see how the 5.23 x 10^15 corresponds to any number on any of your slides.

    Yeah, what’s that about? I think I have myself fixed now.

  16. This activity is exciting to me, because it is a step toward “inviting students to do what you do” that I brought up in your previous post

    The next step would be to ask each student or a group of friends to come up with some other things they could abbreviate in math, and share with the group what they could do with it ;-)

  17. Hi Dan

    I’m a bit of a techno-idiot so forgive me if you’ve received my email twice already (but I wasn’t sure if it sent right, as I didn’t get a carbon copy). So I thought I’d post here just in case.

    I’m a TV documentary producer from the UK from a production company called Fresh One Productions working on a major educational series filming this summer.

    I’d love to chat to you about it if you’ve got a few minutes? I thought you’d be perfect for it.

    Hope to hear from you!

    Thanks so much

    Storm Theunissen

  18. Speaking of the numbers on the slides, where did they come from? Are they real world numbers? What might they represent?

  19. maureen mason

    May 21, 2010 - 9:06 am -

    You have hit on the critical elements that my experience tells me leads to learning that is rich, deep and generative: curiousity, open discussion, constructed experiences to extend thinking, and the application of new understanding in real contexts. The magic bullet.

    I have work with this concept in building general curriculum for the “hard to engage” but always struggled with how to an inquiry based model could work in math. Student teachers that I work with would often ask, “What about in math?” and until now I have struggled with how to answer this. My answers always seem hypothetical and constructed and uninspiring against the way I have seen exploratory models works in other curricular areas. Student teachers are reluctant to subscribe to a framework that does not pass the, “ya but’ test for math. Your work has changed everything in this regard.

    I loved that you were courageous enough to push against convention and give over power to the student. What a leap of faith! What a glorious pay off for you and for your students.


  20. Long time fan, first time commenter. Stealing this for Monday’s lesson.

    Thinking of adding a wrinkle: What about really small numbers? The negative exponent thing throws a lot of kids.

    (@Zeno, I’m pretty sure the ambiguity you’re referring to is the entire point.)