The Power of Small Ideas

William McCallum, IM President

Big ideas are popular in mathematics education, and you can find many lists of big ideas on the web. Some are more thoughtful than others, and I can see how some might be useful for organizing a curriculum. But few of the ideas I see in these lists really get me excited, or really capture what I love about the subject. I am a big fan of small ideas; like intricate joints in a fine piece of carpentry, small ideas often evade the eye, but are crucial to the beauty and structural integrity of the finished product. I’d like to mention a few of my favorite small ideas.

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Making Sense of Distance in the Coordinate Plane

Linda Richard, Curriculum Writer

I used to teach my students a catchy song to memorize the distance formula. We all had fun goofily singing this song. My students hummed it to themselves during tests and successfully calculated distances. I was pleased with this outcome—but what did my students actually understand about distance in the coordinate plane? In retrospect, very little.

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Designing Coherent Learning Experiences K-12

Kristin Gray, Director of K–5 Curriculum & Professional Learning

One challenge in curriculum design is considering all we know and believe to be true about math teaching and learning and translating that into realistic and actionable pieces for teachers and students. Our recent post about the K–5 curriculum focused around our belief that each and every student should be seen as a unique person with unique knowledge and needs. And while that post centered on elementary materials, to truly design around this belief we must look past K–5 to consider each student’s unique K–12 mathematical journey. A journey that, for most students, looks very different as they move from elementary to middle to high school.

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Rigor in Proofs

Tina Cardone, Geometry Lead, & Gabriel Rosenberg, Curriculum Writer

There is no doubt that proof plays a central role in the human endeavor of mathematics, but there remains much debate on what role it should play in high school mathematics. At least two standards for mathematical practice in the common core focus on this concept. Certainly MP3, “Construct viable arguments and critique the reasoning of others”, is about the need for students to be able to write their own proofs and to analyze the proofs of others. MP6, “attend to precision” goes deeper, though, by noting the need for precision, including the use of clear definitions, when communicating their reasoning. This is what we mean by rigor in mathematical proof.

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Presenting IM Algebra 1, Geometry, Algebra 2

Kate Nowak, Director of 6-12 Curriculum

When I was teaching high school mathematics, my local colleagues and I spent a whole lot of time creating problem-based lessons. We were convinced that this style of instruction was a good way to learn, but the textbooks in use at our school simply contained definitions and theorems, worked examples, and practice problems. One day I was talking to my dad about how much time I had been spending lesson planning. His response was, “People have been teaching geometry for, what, 3,000 years? Shouldn’t the lessons be, like, already planned?”

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