The Math Student’s Manifesto

[Feature photo above by Texas A&M University (CC BY 2.0) via Flickr.]

Note to Readers: Please help me improve this list! Add your suggestions or additions in the comment section below…

What does it mean to think like a mathematician? From the very beginning of my education, I can do these things to some degree. And I am always learning to do them better.

(1) I can make sense of problems, and I never give up.

  • I always think about what a math problem means. I consider how the numbers are related, and I imagine what the answer might look like.
  • I remember similar problems I’ve done before. Or I make up similar problems with smaller numbers or simpler shapes, to see how they work.
  • I often use a drawing or sketch to help me think about a problem. Sometimes I even build a physical model of the situation.
  • I like to compare my approach to the problem with other people and hear how they did it differently.

(2) I can work with numbers and symbols.

  • I know how numbers relate to each other.
  • I’m flexible with mental math. I understand arithmetic properties and can use them to make calculations easier.
  • I’m not intimidated by algebra symbols.
  • I don’t rely on memorized rules unless I know why they make sense.

(3) I value logical reasoning.

  • I can recognize assumptions and definitions of math terms.
  • I argue logically, giving reasons for my statements and justifying my conclusion.
  • I listen to and understand other people’s explanations.
  • I ask questions to clarify things I don’t understand.

(4) I can model real-life situations with math.


  • I recognize joining, separating, comparing, and sorting situations and can describe them with mathematical expressions or equations using addition or subtractions.
  • I recognize proportional, grouping, or sharing situations and can describe them with mathematical expressions or equations using multiplication or division, or with fractions.
  • In algebra or geometry, I know how to recognize typical function or shape relationships.
  • I can make assumptions or approximations to simplify a complex situation.
  • I always ask myself, “Does this make sense?” and try to make my mathematical models better.

(5) I know how to use math tools.

  • I can make a chart, graph, data table, or diagram.
  • I can use a ruler, protractor, or compass.
  • I know how to use a calculator when I need it. I never copy down all the digits on my calculator, but round numbers to the appropriate degree of precision.
  • I like to experiment with online graphing tools.
  • I know how to look up information online and how to recognize a trustworthy website.

(6) I communicate my ideas clearly.

  • I know how important it is to define my words and symbols.
  • I don’t misuse the equal sign.
  • I’m careful about units of measurement.
  • I label my graphs and diagrams.

(7) I look for patterns and use them.

  • I know that patterns can make math easier to work with.
  • I use common number patterns to simplify arithmetic calculations.
  • I use common algebra patterns to simplify equations.
  • I use common shape patterns to simplify geometric and trigonometric puzzles.

(8) I make generalizations and justify them.

  • If I see a new pattern, I don’t automatically trust it. I always ask, “Does it make sense?”
  • I ask myself, “Will the pattern always work? Or does it only work in special cases?”
  • I look for ways to explain the pattern in general terms.
  • When I find a true general pattern, I use it to help me solve new problems.

[Based on the Standards for Mathematical Practice, translated into conversational English.]

6 thoughts on “The Math Student’s Manifesto

  1. I might add the following which I’m sure overlap with things that you have already
    1) It is better to understand than to memorize
    2) I will be OK with “failure” and know that with effort I can learn new things.
    I’m sure you can wordsmith those better than I did.

  2. Thank you so much for sharing my blog today! Reading your Kindle book, “Let’s Play Math!” was such great inspiration for me to change my approach to the subject, and now I am never looking back!

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