Five Design Patterns for Digital Math Curricula

The Center for the Study of Mathematics Curriculum invited me to give a talk last week on digital math curricula. I described how print curricula limit the experiences we can offer our math students and then I made five recommendations for designing experiences digitally:

  1. Show, don’t tell.
  2. Introduce the task as early and concisely as possible.
  3. Climb the entire ladder of abstraction.
  4. Crowdsource patterns.
  5. Prove math works.

Any questions or criticism, please don’t hold back in the comments. I also have limited availability for consultation on these kinds of projects. Drop me a line at

2012 May 1. Here’s the feedback [pdf] from the academics at the conference.

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. Impressive talk!

    Especially exciting are your ideas about the lower levels of abstraction — you convinced me that the NCTM statement (that technology allows students to work at higher levels of abstraction) is “exactly half right.”

    I am salivating over your examples of aggregating and displaying student responses. I hope you elaborate on these ideas in the future, along with the many other productive themes you routinely hit.

  2. I really enjoyed your talk. In fact, I initally estimated that I would skip through it in 5 minutes but the estimate couldn’t be backed up by a mathematical tool which would allow me to model how long I would watch a video for and therefore my estimate was 30 minutes short off the mark. :)

    I liked what you said about the fact that textbooks often describe a problem in words when a picture (or any visual phenomena) would paint a much clearer picture. It’s interesting that after marking some IB coursework tasks today I found myself getting frustrated when one of my students described everything with words when she could have cut her essay at least by half using visual representations. She must have been educated into this…I hope it wasn’t by me!?!

    Throughout the talk I was pondering whether it is an important skill to be able to pick out information from a long question and use the relevant parts to solve a problem. I think it is important and I’m sure you would agree that there will be times in which students should have a chance to do this. However, I agree with you that this is very intimidating (especialy for EAL students) and I also agree that it takes most of the fun and excitement out of mathematics.

    Assuming that you did indeed work through the airport question on behalf of your parents,

    I’m sure that everyone is more enthusiastic about applying mathematics to a problem which directly affects a decision they have to make.

    How realistic is it to think that we can educate students to ask their own questions and get them to utlize your 5 step process above? A curriculum in which this 5 step process was used for the first few years of high school and then the students are expected to find there own questions to do this with and present their findings would be great in my opinion. Imagine a student filming something or taking a picture of something and then going through the steps above with his or her classmates – awesome! I wonder whether it is reasonable to ask students to come up with questions in an nono-naturalistic way? Some of the best questions are the ones that come at us when we least expect it? Should we be encouraging students to be open-minded to questions arising in their daily lives which they could then bring into the classroom? I guess I still have many questions about this. All these questions also relate to your 101questions blog – I’d like to see some students putting the media up there.

    Anyway, I think your 5 step process is great for modelling tasks and once again I thoroughly enjoyed your presentation.

    Dan Pearcy

  3. Enjoyed the talk, I’ll show it to the rest of the department at my school.

    Finding the fact we depend on so literacy skills, quite telling.

    Wonder if the assessment will ever change? Because that seems to be the biggest driver of change in the UK education system.

  4. @Jerrid I can think of two major ways: Digital media is limiting as compared to having physical manipulatives which allow different types of exploration of objects.
    They are limiting in the “film vs. book” way that you are presented with one representation. If a word problem says “polygon”, it could be any number of sides, convex, concave, regular or not, etc. but an image would be a specific shape, instead of the generic concept.

  5. I absolutely loved your video and your 5 points for structuring a math learning experience. It made me reflect on a real-world problem that I posed last year for my 10-grade math class in Boston (50% African-American; 40% Hispanic; 10% Asian/Caucasian). I didn’t have your 5 points to guide me then, but here’s how it played out, looked at through your 5-point “learning lense”.

    First, the context:
    a) I was their new math teacher.
    b) I look younger than my age.
    c) I told them that in life, it doesn’t matter where they start; it only matters where they end.
    d) I challenged each of them to measure themselves on their level of progress from where they are now, and that math could help.
    e) To jump-start the self-assessment process, I did pushups to failure in front of the class, put my “this week’s best effort” number on the board as my baseline for a future repeat assessment, and invited other students to do the same.
    f) Five boys and one girl chose to do pushups in front of the class and post their numbers. I congratulated them for enduring good-natured laughs from their non-participating classmates and reminded the others that they could do this in the privacy of their homes.

    Given the above context, a simple math problem surfaced in which the process by which it was solved turned out to be astoundingly more interesting than the actual answer.

    I’ll share the problem, the process and the solution as well as I can in terms of your 5-point structure. Note: It seems to me that Step 3 (climb the entire ladder of abstraction) and Step 4 (crowdsource patterns) happened in reverse order, which I’ll show below. (Suggestions from readers for reframing my experience for future classroom use are welcome.)

    1) Show, don’t tell.

    They all watched me do my pushups on top of one of the tables (so they could see me). One of the students promptly asked: “How old are you?”

    2) Introduce the task as early and concisely as possible.

    I asked each of them to write down their best guess to the question: “How old is Mr. Foster?”

    4) Crowdsource patterns.

    I asked each student to share their guess with the class. We wrote all of their 22 guesses in a column on the board.
    Lowest guess: 42
    Highest guess: 72

    3) Climb the entire ladder of abstraction.

    I said to them: “Most of you know that the “range” for a group of numbers is the space between the lowest number and the highest number. I asked them: “What’s the range of your guesses about my age and how big is that range?” (They did the computations.)

    I then said: “Aside from me giving you the answer to my age, how might you figure it out from the data we have on the board?” A pregnant pause…
    Then I posed a philosophical question: “What if it’s true that all of you working together are more intelligent than any one of you?” Another pregnant pause…

    5) Prove math works.

    I asked them to compute the average of the class’s guesses.
    They set about doing that and reported their answer: “61.1”
    Another long pregnant pause…”OK MR. FOSTER, HOW OLD ARE YOU REALLY???”

    The answer: “61”!

    Only a couple students had guessed correctly, but the class as a whole nailed it! In fact, as I computed after the class, (we should have done this in-class), the average of their guesses when computed to the fractional year was exactly 6 weeks from my birthday! Not only that, but I had another class of 14 students guess my age, too, and their average was also 61.

    I asked the class: “Given the intelligence that you just demonstrated when we combined your answers by mathematically averaging them, what does that say about the intelligence of a population of people in a democracy who cast votes for or against a proposed law or a candidate for office?” “Does the wisdom of the masses work, mathematically?” “Is there a time when it wouldn’t it work?”

    I later read that if a crowd of a 100 people all guess the number of jelly beans in a large jar, the range of guesses will be wide, but the average will be very close to the actual number.

  6. I love your work, Dan. I keep encouraging my preservice elementary/primary teachers to watch and read your materials, and show bits and pieces in class.

    This video is no exception; you have very clearly set out your thesis regarding the problems with print resources, and illustrated it brilliantly with examples of both print and digital materials.

    My question at this point is, “Will this work with younger students?” I am thinking that if you are teaching place value, basic operations, manipulating fractions and so on, it may be harder to come up with the engaging sort of extended problems you are able to conjure up for algebra and geometry topics.

    I am happy to be shown to be wrong; this is just the question I have reached so far.

  7. Hi Dan,

    I’ve been spreading the word regarding your work and this latest film has further inspired me.

    We have a big problem in the UK at the moment regarding the available funding in schools for interactive media to support the use of Mathematics digitally, never mind the development of resources.

    This, tied in with the big behaviour problem we have in schools in the UK adds to a double-whammy of stagnancy in the classroom.

    I would be interested to know your thoughts on whether a more creative approach such as yours results in more engagement/better behaviour – or is there a need for the right attitude from students before one takes the great leap forward like you have?

  8. Going to have a textbook burning ritual this summer. I’ll blame you if anyone asks.

    The last math PD I attended was painfully un-inspiring (boring), yet your brilliant stuff is right here, right now. We are a small one-school district, but I bring up your name and site whenever and wherever I speak math. What I appreciate most about your work, including the 5 design patterns, is its simplicity – it makes sense at the gut level.

    @Amir I really believe behavior problems arise when the lessons are not engaging. Even I, an adult and a teacher, WANTED to misbehave when I’m bored at a 45-minute workshop.

  9. Nice Talk. I have to comment on a very small piece that I almost missed. That was when you mentioned Bret Victor ( Ever since I saw his Kill Math project last year and more specifically the Scrubbing Calculator (!/ScrubbingCalculator) I have been waiting for him to make something like that available. Perhaps I haven’t checked in on his sight recently or maybe I just missed it but Tangle is pretty cool (!/Tangle). Now all I have to learn to do is program Java. I think I saw the same talk that you were referring to (Inventing on Principle –

    and probably should have checked on his site. Thanks for the heads up.