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