Our Students and Their Mathematical Ideas

  • Our Students and Their Mathematical Ideas

    By Zachary Champagne, posted August 14, 2017 —

    I have the pleasure of working with and learning from groups of teachers from around the country. When I collaborate with teachers in a professional development setting, I always begin the session with this statement: “Kids have important mathematical ideas.”

    It’s a starting point of sorts for me. I like to communicate to teachers that this is the most fundamental belief I have about the teaching and learning of mathematics. Every student that walks into our classroom has important mathematical ideas. They are certainly not all correct mathematical ideas, but they are important. They are so important that I believe it’s our role as educators to find out what those important mathematical ideas are and to make instructional decisions on the basis of what we find out.

    But how do we find out what those ideas are? And what do we do when the conceptions that students hold aren’t necessarily the ones that we desire them to have mathematically? I’ll attempt to answer those two ideas separately.

    First, we must find time and space to talk (and much more important, to listen) to our students. If we can all agree that each one of them has these important mathematical ideas, then we have to find avenues to determine what those ideas are. Sometimes that can happen through their written work, but more often, we should be listening to their responses to carefully crafted tasks (both written and verbal) and making sense of what those ideas are through conversation with each and every student. Let me be clear, I respect the limitations that exist in a mathematics classroom and one-on-one conversations. Therefore, I’m not suggesting that teachers should be attempting to talk to each and every student in a one-on-one setting every day, but I am hoping that we’ll try to do a whole more of that than we currently do. 

    Second, when students hold conceptions about mathematics that don’t align with the conventions that we want them to know, we should give them opportunities to challenge their existing conceptions. We should provide space for them to make sense of these ideas through intentional experiences with those mathematical ideas. Sure, sometimes this takes some “teaching,” but more often than not, it takes each student making sense of the mathematics on his or her own.

    This short post certainly doesn’t have the space to go into exactly what that looks like, but if you are interested, you should check out Mike Flynn’s book Beyond Answers, Tracy Zager’s book  Becoming the Math Teacher You Wish You’d Had, the Cognitively Guided Instruction series, NCTM’s recently published Taking Action, Cathery Yeh’s Reimagining the Mathematics Classroom book, the CPALMS Mathematics Formative Assessment System, and Skip Fennell, Beth Kobett, and Jonathan Wray’s Formative 5, to name just a few.

    I hope as you are thinking about this new school year approaching, you will consider your role as a listener and learner in the classroom. And to do this well, we must all start each day with the understanding that each and every student we teach has important mathematical ideas, and it’s our job to find out what those ideas are and to help them make sense of them. 


    Zachary Champagne is an assistant in research at the Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM) at Florida State University. He previously taught for thirteen years as an elementary school teacher with a specialization in math and science. During this time, he received many state and national awards for excellence in teaching, including the Presidential Award for Excellence in Mathematics and Science Teaching (PAEMST), Duval County Teacher of the Year, and Finalist for Macy’s Florida Teacher of the Year. He is the current president of the Florida Council of Teachers of Mathematics (FCTM), and is currently interested in learning how young students think about mathematics and how to help them understand that mathematics makes sense. He tweets at @zakchamp.


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    Glenda Frutiz - 4/23/2020 8:23:29 PM

    As I read your article, 

    I feel that we need to listen and hear the students ideas in order to understand what they are thinking. Listening to the students thoughts and ideas then giving them the oppurtunity to challenge their thinking when solving a problem is a great way for them to learn because they can learn from themselves and learn the different strategies they have tried. Once you observe and listen, the teacher is able to have an eye-opening understanding about the child's thinking. As educators, sometimes you have to let the students try it out first and see what they will do and what strategy can they use that will help them accomplish the problem. Thank you for this article, listening to our students can educate the teachers of a students thought process. 

    -Stephanie Escobedo 

    (Professor Cutler, I apologize for using Glenda's Frutiz account. I can't afford MyNTCM membership during this tough time so Glenda was able to let me use her account. Sorry for the inconvience).

    Ashley Bilbo - 2/29/2020 12:25:19 PM

    Thank you for writing this article. I appreciate your thoughts about not only talking to our students but also LISTENING to our students. I can also respect the time restrictions of the school day. However, I believe that if we spent more time listening to our students we will learn much more about their thought process. I also appreciate that you state every child has important mathematical concepts. As educators, we often feel so rushed that we rarely take the time to nurture a childs thought process. Spending time letting children work through concepts and problem solve will inevitably teach them more than a rote memorization lesson.