What does it mean to be smart in mathematics?

In the last two posts, I discussed the idea of status. First, I talked about why status matters, then I talked about how teachers can see it in the classroom.

Sometimes, after I have explained how status plays out in the classroom, somebody will push back by saying, “Yeah, but status is going to happen. Some kids are just smarter than others.”

I am not naive: I do not believe that everybody is the same or has the same abilities. I do not even think this would be desirable. However, I do think that too many kids have gifts that are not recognized or valued in school — especially in mathematics class.

Let me elaborate. In schools, the most valued kind of mathematical competence is typically quick and accurate calculation. There is nothing wrong with being a fast and accurate calculator: a facility with numbers and algorithms no doubt reflects important mathematical proclivities. But if our goal is to address status issues and broaden classroom participation in an authentically mathematical way, we need to broaden our notions of what mathematical competence looks like.

Again, my naysayers roll their eyes and groan, assuming that I want to “soften” mathematics or dilute the curriculum. But I claim that broader notions of mathematical competence are actually more authentic to the subject.

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Let’s cherry pick some nice examples from the the history of mathematics. We see very quickly that mathematical competencies other than quick and accurate calculation have helped develop the field. For example, Fermat’s Last Theorem was posed as a question that seemed worth entertaining for more than three centuries because of its compelling intuitiveness. When Andrew Wiles’s solution came in the late twentieth century, it rested on the insightful connection he made between two seemingly disparate topics: number theory and elliptical curves. Hyperbolic geometry became a convincing alternative system for representing space because of Poincaré’s ingenious half-plane and disk models, which helped provide a means for constructions and visualizations in this non-Euclidean space. When the controversy over multiple geometries brewed, Klein’s Erlangen program developed an axiomatic system that helped explain the logic and relationships among these seemingly irreconcilable models. In the 1970s, Kenneth Appel and Wolfgang Haken’s proof of the Four Color Theorem was hotly debated because of its innovative use of computers to systematically consider every possible case. When aberrations have come up over the years, such as irrational or imaginary numbers, ingenious mathematicians have extended systems of calculation to encompass them so that they become number systems in their own right.

This glimpse into the history of mathematics shows that multiple competencies propel mathematical discovery:

  • posing interesting questions (Fermat);
  • making astute connections (Wiles);
  • representing ideas clearly (Poincaré);
  • developing logical explanations (Klein);
  • working systematically (Appel and Haken); and
  • extending ideas (irrational/complex number systems).

These are all vital mathematical competencies. Surprisingly, students have few opportunities to recognize these competencies in themselves or their peers while in school. Our system highlights the competence of calculating quickly and accurately, sometimes at the expense of other competencies that require a different pace of problem solving.

Evaluating people on one dimension of mathematical competence will rank students from most to least competent. This rank order usually relates to students’ academic status, and students tend to be aware of it. One way to interrupt status is to recognize multiple mathematical abilities. Instead of a one-dimensional rank order, we create a multidimensional competence space. Although some students may have multiple mathematical strengths, more places in which to get better surely exist. Likewise, a student who ranks low on the hierarchy produced when we focus on quick and accurate calculation may have a real strength at making astute connections, working systematically, or representing ideas clearly. We cannot address status hierarchies without emphasizing multiple mathematical competencies in the classroom.

A multiple-ability classroom represents a dramatic shift in the topography of mathematical ability. Instead of lining students up in a row in order of smartness, a multiple-ability classroom has students standing on different peaks and valleys of a hilly multidimensional terrain. No one student is always clearly above another. This structure may unsettle students who are used to being on top, as well as those whose vantage points and contributions have been presumed less valuable. In other words, challenging the status hierarchy by developing a multiple-ability view can provoke strong emotions from students, positive and negative. Teachers should not be surprised to see this response in their classrooms.

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23 thoughts on “What does it mean to be smart in mathematics?

  1. I enjoyed this post and I certainly benefitted from the quick calculation hierarchy that I grew up with. Being the fastest at doing math problems always kept me in the highest ability classes. This held true until the second semester of calculus in college. Because I had (nearly) a 100% average in Calc 1, I went to an Honors Calc 2 – so I could be around peers of “equal stature”.
    In this class I suddenly discovered a new hierarchy. Every student in that class was good at calculations and algorithms. But some could visualize ideas mathematically. It scared me, to be honest, because I was never pushed to think mathematically. I was just pushed to calculate numbers rapidly following set guidelines and constructs. I finished my math degree but I always felt like a lesser student than others getting their degree.
    I’m not sure, in this test-heavy country of the USA, we will ever get to a point where we can get away from the quick calculations mindset. But it would be good for teachers to embrace the idea of allowing students to explore mathematical concepts in ways other than number crunching.

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  2. In classrooms I have been in where multiple kinds of smartness are truly valued and used, the kids who are the traditionally successful because of their facility with calculations benefit. They have opportunities to slow down, see things from other perspectives, and deepen their understanding by making the critical connections that they may have otherwise missed. I can’t help but wonder if you would have developed more strategies for visualizing mathematics if you had a chance to be exposed to that way of thinking before you got to the Honors Calc 2 class.

    Thank you for sharing your experience.

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  3. 1. Love all of these.
    2. One of the things that stuck with me when I visited a CI school (@cheesemonkeysf’s current school) is the teacher pointed out that he has to be very intentional about the types of rich tasks he does in class to make sure each student has a chance to be the ‘smart’ one. Since then I’ve realized how much I privilege certain types of thinking in my class and so tasks I choose or design will tend to reflect my values and I need to be mindful.

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    • Thank you, Jason.

      I liked to use posters in my class describing different kinds of smartness. This way I would remind myself and my students of them. I would love to see the list in science — I am guessing it is similar. Maybe something about observing carefully to?

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  4. I really enjoyed this blog.

    I’ve been working on this question with a group of 1st and 2nd graders all year as part of research for a book I’m writing. We’ve been asking the kids what it means to be good at math, and the answer we get is the same answer I get everywhere:

    Someone who is good at math answers the teacher’s questions right, fast.

    I take issue with every part of that definition: the answering, the right, and the fast. We ended up building a mini-unit introducing students to mathematicians so they could figure out what mathematicians actually do, and learn what’s actually valued in mathematics. When we first asked what mathematicians do, we got responses like, “They answer questions on papers.” After the mini-unit, we got responses like, “They wonder.” “They notice.” “They ask questions.” “They play.” “They figure.” “They make mistakes and learn from them.” And so on. So much better!

    The experience crystallized what I’d been thinking for a long time: we need to define the discipline for students. We talk about what science is, and what writers do, and the strategies great readers use. But in math, in elementary school, we just start doing it. Students draw conclusions from their experiences, and their experience tells them that math class is all about speed and right answers and worksheets. We can do better!

    When a second-grader told me she wasn’t good at math because she didn’t really “get subtraction,” I asked her to tell me more. She said, “I can do it, but I don’t really understand what it means.” I told her she was thinking like a mathematician. Mathematicians are not content to do it without understanding it. Her desire to really understand is a big part of doing math. Her jaw dropped. Ever since, she’s been much more involved in math. Broadening the definition of what it means to be good at math can have huge impacts.

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  6. Thank you for sharing, Ilana. You raise a great question – What does it mean to be smart in math and are *we (Adults in Education space) unfairly advancing a narrow vision of what math ‘smarts’ or thinking is all about?

    Should our primary goal as educators be to cover content and show ‘mastery’ of understanding through one math (computational) lens? Or should our goal be to develop the young mathematicians, in every sense of the word, we are lucky enough to have in our classroom/school/region/country?

    Here is a short post, which elaborates upon my thinking.

    https://chalkboardinquiries.wordpress.com/2014/03/11/balancing-knowing-math-basics-with-developing-mathematical-thinking/

    Thanks again,
    Paul

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  9. I am looking for an effective way to share these ideas with students who have grown used to the one dimensional view of “smartness” that you describe. How do I help them understand why I am not doing direct instruction in my classroom and that I am not looking for just quick and accurate calculations? How do I change the definition of smartness in my classroom? How do we (in general) include their voice in this dialogue?

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    • Great question, Matthew. If you use activities that require different kinds of smartness, you can catch them showing these smartnesses and name it and praise it. It becomes a lot more convincing once you have shown them in action. What grade level/topic are you teaching? I can think of a few good examples of tasks, but I may be off base with your students.

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      • I couldn’t agree more! I love how complex instruction disturbs the ability hierarchy schools have created. I am teaching calculus to grade 12. All students are taking the course regardless of their (perceived) “level.” This is only the second year our school has required students take a 12 grade math course. I fought to eliminate regents prep courses (New York State) at our school and I am trying to change students perceptions of their own ability. I would love to hear your ideas about good tasks for calculus.

        My rationale for this course is that I have heard so many people say they’ve taken calculus, but they have no idea what it was about or why they took it and then they say they don’t remember any of it. I believe the big ideas of calculus are very accessible but rarely taught. In the first week of school I have caught many students in the the act of being smart. Many of whom aren’t used to the idea of being smart, which is great. But I still feel the need to work through this idea of smartness in math with my students and I just don’t know how to do it.

        I would like my students to read something engaging or view something that we can discuss to help them develop a new perception of smartness. I also don’t want to do one lesson on smartness and never address it again. I am looking to make this a continued dialogue with my students. We do a lot of talking about this stuff as teachers at my school, but I feel like often leave students out of the conversation.

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  12. Hi Ilana, I am a student and I hope that you can answer my question here. Basically, it is about a longstanding problem I have with mathematics. I consider myself quite good at it so far though never quite as good as my brother and my friend, and that crushes me because I value mathematical ability above all other academic skills. So, the idea that you present in this post really encourages me, though I still have some doubts, and I hope you can clear them up for me.

    I have some problem viewing mathematical ability as multidimensional, because well, in my class those who are good in math are generally better in all areas of math tested. I am quite good in math as compared to my peers but I still lose out to my brother and my friend, who are both boys. Generally, I value problem solving ability much more than computational ability, because the numbers can always be crunched by the calculator. Anyway, both my brother and my friend are better than me in all those other mathematical strengths you mentioned above in this post. So, that got me to thinking that mathematical ability is innate, and if you have it, you are generally better in math overall. I can never be as creative as them in coming up with solutions to novel problems, nor can I think as logically or as rigorously as they do. My brother also grasps mathematical concepts quicker, and has a deeper understanding in them.

    And finally, if the fact remains that they are indeed just simply better than me, can I ask you for some advice to cope with that fact? I know that I may come off as silly or unreasonable but I have difficulty seeing other abilities as equal to mathematical ability. Everyone around me views mathematical ability as the ultimate measure of smartness, so my ‘status’ in my classroom depends on this, so to speak. Anyway, by looking at the job market, math related professions earns the most, so I think this will translate to a higher status in society.

    I wonder if I am rambling, but I do hope that you can help me out with my problem.
    Thanks.

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    • Hello! I think you are officially the first student to find my blog, so thank you and welcome!

      I have several thoughts. First, it is important for you to realize that math class =/= mathematics as a field. I majored in math, and my advisor got his PhD in Theoretical Algebra. When people would ask him to calculate stuff, he would say, “I did my PhD research and I didn’t see a number the whole damned time!” I have known Logicians who say similar things.

      The moral: Your mathematical talent, unfortunately, may not be the kind that is valued most in school math, but it doesn’t mean it is not mathematically valuable.

      Second thing: there is evidence that traditional teaching styles tend to depress girls’ achievement — particularly that of bright girls. Jo Boaler studied 300 students in two English schools and found that the performance of high achieving girls in the traditional setting fell over the three years of her study. If the comparison between you and your brother reflects this general pattern, biases in school math may particularly make your mathematical strengths look a little less dazzling. (You can read a paper about this here: https://ed.stanford.edu/sites/default/files/jo_gender_mtl_paper.pdf.)

      Thanks for writing. And keep your chin up. You seem like a very smart and thoughtful person.

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      • Hey… just want to say thank you for the reply…and for that interesting paper. I really appreciate it.
        Many thanks again.

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