Speaking Mathematically

David Pimm:

We name things for reference, and hopefully for ease of reference, to draw attention to the thing named. But naming also classifies and hence causes us to look at the named thing in particular ways, the chosen symbol stressing some and ignoring other attributes of the named object. Naming something gives us power over it, particularly in algebra, as we can transform and combine expressions involving the unknown — to find out more about it (p. 127).

This is the strongest case for algebra. Your ability to speak, think, and use variable notation makes you powerful — particularly when you interact with computers. But how often do students think of variables in math class and feel powerful? Those experiences aren’t simple to devise.

I read Pimm’s excellent book over the holiday in preparation for my dissertation proposal. I’ve pulled out several pages worth of quotes and supplemented them with a) my analysis and b) some details about my upcoming study. Comments are turned on in the Google doc, so let’s talk about it.

Featured Comments

David Lloyd:

Perhaps using words as the descriptor (“number of songs”) instead of using X (as in “Let x = the number of songs on Dave’s iPod”) would be a step in the right direction? The same level of rigor without the confusion of what X equals.


How often do students name their own variables?

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. Hello Dan
    1. Best wishes with your dissertation.
    2. I not only agree with Pimm, I was agreeing with him before he wrote (in practice, that is).

    I really hope that this, and yours, reach a wide audience. It does look to me that the CCSS experts missed this one. Or they didn’t understand it.

    Do you know David Hilbert’s book “Geometry and the Imagination”? So much mathematics and so few symbols !!!!!!!!

    Regarding algebra, the question “What does it say ?” usually produces the answer “why equals two ex plus three”, which is a very sad state of affairs.

  2. There is a reason why naming ideas and concepts has always been a central function of human thought. It comes up in every discipline I have ever studied. Thank you for connecting it to what I’ve been thinking about lately and good luck with the dissertation proposal.

    – Elizabeth (@cheesemonkeysf)

  3. sheldon lebowitz

    January 9, 2014 - 6:56 am -


    The following piece is from Wikipedia:

    Morrie’s formula
    From Wikipedia, the free encyclopedia
    Morrie’s formula is a name that occasionally is used for the trigonometric identity:

    cos(20)*cos(40)*cos(80) = 1/8.

    Richard Feynman’s friend Morrie Jacobs sent it to him when they were both school children. Feynman was so taken by its sheer simplicity that he remembered it for the rest of his life, it is almost magical in this respect.

    Feynman also frequently commented on the “most beautiful” equation he ever came across:

    e^(i*pi)= -1

    Again, simplicity and succinctness over ride anything else.
    I feel that this concept can turn children on and motivate them in mathematics.

    Good luck in your upcoming tests and future.

  4. Hi sheldon, I’m curious why either of those formulas would motivate students. Rather, which students would they motivate and under what circumstances? Why? What’s the theory of motivation there? That simple things are motivating? What are the students supposed to do with either of those propositions? They’re just … equations.

    I’d like to know more.

  5. Mark D. James

    January 9, 2014 - 4:54 pm -

    Great excerpts, great thoughts, and great comments by so many. Late to the MTBoS party though I am, and though mostly still lurking…I am really feeling inspired and energized by what has become the most meaningful PLC in my 14 years of teaching mathematics. Thanks Dan!

  6. Hi Dan,

    Been reading your blog for a long time. Amazing journey.

    Comment on “premature symbolization of school mathematics”

    I have been teaching a grade 10 course for applied learners (many at risk for various reasons) for years now through activities only. Early in the course students are given linear relation situations to solve in a context. Think linear patterns. Something like a string of toothpicks in the form of a square with a common side. T=3S+1 where T is the number of toothpicks and S is the number of squares.

    Most of these students, when left to their own devices (this means we have not discussed linear relations-not even seen them in the course yet), will look at the pattern of numbers and extend it until they reach the answer. Even the students that can symbolize the context tend to answer the questions using logic rather than the symbols. Knowing the number of toothpicks they would say” well I have to subtract one and then divide by 3″.

    What am I getting at?
    These students in this context do not need the symbolization. They need a voice. They need to explain how they got their solution. They need to understand that they can do math.

    They do not need the symbols. I do not force it on them like I might of years ago. (I would not of even introduced linear relations like this years ago – I would of “taught” them it)

  7. Alex:

    These students in this context do not need the symbolization. They need a voice. They need to explain how they got their solution. They need to understand that they can do math.

    Thanks for your thoughts here, Alex. I hope you find a minute to dip into the Pimm quotes. I think you’ll find a lot that resonates. He’s pretty emphatic about the power of symbols, though. Both because you can manipulate them but also because it can be faster and clearer to express “well I have to subtract one and then divide by 3” as “(x – 1)/3”.

    What do you think would happen if you showed your student the symbolic form after she wrote it in words?

  8. Dan,
    Thanks for the post. There is quite a bit in there to think about and process, and I’ve left some comments in the google doc. One quote which has resonance:
    “The teacher may be too concerned with the form of what is being said at the expense of the meaning which the pupil is trying to convey.”
    Coming at it from an elementary school perspective, I see teachers who are uncomfortable with the concepts and the material themselves. They become overly dependent on their manuals and keys to provide them with the “correct” vocabulary and language and answers. When what the student says and what the “book” says does not match, then the student is wrong. Sometimes they cannot interpret what the student is trying to say, and they do not have the confidence and the experience (and experience is a big problem in this new age of attrition) to go “off message” and let the students take the lead expressing mathematical concepts in age-appropriate language.

  9. I agree with you all! :) As a middle school teacher of 10 years, primarily to remedial students, they are so much more capable of working with patterns when you take away the “math”. If you asked remedial students to write the algebraic expression for a linear toothpick pattern, less than half would succeed. If you asked them simply to write a paragraph describing how they would find the next figure, then the 100th figure, most would succeed. Fighting this culture of math-phobia is hard, but I’ve found that teaching the variables and “correct” math form should come way at the very end, after the students have found their voice in describing a concept and become convinced of it in their own mind.

  10. I too feel as Eric and many of you, that we should let students encounter the process first, then the formula. I am currently teaching at a high needs school and have many students struggling to learn math. Both the students and the other math teachers are caught up in teaching math formula first, pushing the memory tricks, rushing through the lessons (2-3 days per standard) and testing to death. They moan over the lack of progress. The students are slow to explore, ask questions or get curious. I know my colleagues think I’m addled as I push lessons that explore, expect the students to become engaged, and take a little more time in exploration with my class. And it’s not that they aren’t good, caring teachers. They are! They are simply stuck teaching how they were taught and teaching the way they learned to teach. And it is hampering progress for our students. Thank you Dan for continuing to spread this information- and for helping us recognize that these ideas are not new to us – and that they are really better for our students!

  11. Clara’s comments resonate with me so very much. At the start of my career, the 7th graders I taught received this type of exploration-based experience with little worry from me about the #1 enemy (TIME) because, at that point in history, 7th graders did NOT have state-mandated standardized testing in Pennsylvania – only grades 3, 5, 8 and 11 had to learn a year’s worth of math by the big test in March.

    It saddens me that now I feel arms pulling in opposite directions constantly, between s l o w i n g down to let students explore, smell the mathematical roses and construct their own meaning, versus “getting through the content in time”. All this to say, perhaps some teachers are nervous about taking this type of teaching plunge because of the time factor and pressures to “get through curriculum”…? After all, it sure is quicker to just tell students what to do, isn’t it? (tongue-in-cheek)

    I wish I could say that I always make the “right” choice here for my own students, but sometimes time-pressures win.

    Thanks for sharing, Dan, and best wishes on your dissertation!

  12. Well said Cathy. Here’s a quote from an interview with Howard Gardner that I’ve been thinking about recently:

    “Another obvious implication, one that only a few people have begun to take seriously, is that we’ve got to do a lot fewer things in school. The greatest enemy of understanding is coverage. As long as you are determined to cover everything, you actually ensure that most kids are not going to understand. ”


    That’s from 20 years ago.
    I worked on our fourth grade curriculum last summer, getting into alignment with the common core. We took out almost 15 days worth of lessons and replaced them with 15 days to dig deeper and explore fewer things in greater depth, and it’s a little better but it’s still not enough time.

  13. Knowing why we teach algebra is key. Algebra is the language we speak, we read and interpret as well as communicate with. Younger Ss should be exposed to growing patterns (The queen of this has to be Fawn Nguyen-Math talks). Kids translate what they see into algebra. They communicate in words what they see. They translate it to expressions that make sense to them. The opposite can also be done, if p=2s + 2 what could I be describing? Older Ss make other types of data match to algebraic expression. Algebra comes from the matching of this data to a particular function type. The expressions are the sentences with which we communicate and these sentences can sometimes be written in simpler forms. We then lose the visual of what we saw but is allows us to do other things. Solving for x at this point has meaning.
    Transitioning from reading, to writing, to translating, to interpreting, to using this language is a beautiful thing!