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Why Mathematical Practices Matter as Much as the Content
January 16, 2013 | José Vilson
I have a confession to make: at some point this year, I realized that there’s a difference between the teacher I would love to be and the teacher I currently am.
Most teachers want to do interdisciplinary projects, project-based learning and every other education phrase with the words "exploration" and "project" in it. Despite evidence to the contrary, their reality of having to teach directly to a standardized test (ultimately affecting their municipality's perception of them) casts a longer shadow on them than even the bravest of us want to admit.
In math, the need to stand in front of the classroom particularly rings true, not just because of the stakes, but because of the long precedent set by previous math teachers to do exactly that.
Yet math teachers simultaneously know that, in order for students to solve problems on their own, we have to teach them not just the "what" but the "how." "What" equals content, but I’m expanding the definition of "how" beyond simply learning skills and procedures. The "how" should be about how to help students think more critically about the problems in front of them.
In other words, the approaches and uses of the tools students learn in math matter just as much as the topics and situations in which they apply.
An Expanded Skillset
The Common Core State Standards seem to address this well with their seven mathematical practices, but if the practitioners of the CCSS do as we've done in the past, then we've missed another opportunity to fortify students' knowledge of math. Whether they're learning fractions, exponents or the distributive property, they have to learn how to approach mathematical problems.
For instance, would I rather my students learn how to find the slope of a linear relationship or how to make sense of the answer after they've figured it out? One might say you can't make sense of the resulting quotient without actually finding the quotient, but I submit to you that, without making sense of the slope (either by explaining it or showing another representation), they won't know whether the response they got actually made sense. Students ought to have the skills to self-correct, or at least think twice before looking at a response and moving on.
Simply finding the answer is only one part of math.
Now, unlike the topics (or content) we teach in math, teaching students how to approach a problem will feel more like a soft skill to teachers, in the same category as recognizing when a kid needs to go to the restroom or getting a student a tissue five seconds before you know they'll sneeze all over the pencil you lent them. Teaching students how to disagree with another person's argument carefully (and factually) or how to move on to the next problem without constantly checking with you (I'm still working on this) demands a certain dexterity from educators, and a consistent eye on making sure those sorts of behaviors flourish.
Rather than just trying to sift through our 800-page textbooks by chapter or blaze through a curriculum map, let's focus on developing mathematicians. Many of the things we take for granted, like teaching students how to ask better questions or picking apart word problems, actually make students better at math.
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Comments (10)
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One of the most important
One of the most important components of being able to teach "HOW" is that the students have suffient "mental reserve" to be able to hold the necessary content in short term memory. This is only possible if the students have acquired basic number skills and are capable of performing them automatically. In other words they have acquired "automaticity'
I have a simple test I give to all of my middle school students to find out which need more basic work so that they can learn the "how" of advanced concepts. I call it the 100 addition facts.
I provide a test all of the addition facts from 0 to 9 randomly listed on one sheet of paper. ie 1+3, 0+6 etc. It is a timed test in which students are not allowed to skip questions. They are given 60 sec from the time they turn the paper over. Each should be able to write 60 answers in 60 sec --- if they have automaticity. I have had several 8th graders who answered less than 20. Try it on your class. I hope you are pleasantly surprised.
Thanks, Paul. An honest
Thanks, Paul. An honest mistake, but one I caught only after submitting. As you alluded to, though, the point still stands: it's about the how and the what.
Science teacher, the point of
Science teacher, the point of that paragraph, nay, the entire piece is that we must pay attention to both sides of this thing we call math. 32 years of experience might have also taught you that having a clear sense of approach (and not just procedure) actually helps students become better mathematicians. Hope that helps the understanding.
Well said, Ms. Garcia. Having
Well said, Ms. Garcia. Having some sort of connection matters lots.
I went to a training this
I went to a training this past Saturday on CCS for Math, and I discretely asked my favorite math teacher at my school what a quadratic equation actually does. I told her I could solve for x, but I had no idea why I'd ever use it. After talking, we started to share ideas on how to make our content more real for our students. As masters in our own content, it can be difficult to see how the "other side" thinks and reach all of our students.
I always loved the math teachers who explained the WHY and the HOW of each equation.
Math teacher
Teaching Honors, and Concepts classes are two different things. Honors students have no problem following along as demonstrations are done in front of the class. However Concept students, or divergent students is completely different, and they need alternative methods of instruction. It is difficult to keep them interested in the topics, and frequently their fundamental knowledge is so weak that they need remediation in even the most basic areas.
Would you rather your
Would you rather your students know how to find the slope of a linear relationship or that they can make sense of the answer once they find it? You want both. If they don't make sense of the answer, they won't remember how to find it after the test. I know this from experience, because as a high school science teacher, the slopes of many graphs are important, and I know that middle school math teachers went through the motions of teaching what slope is, but I always have to reteach it before we begin to interpret the data that is graphed after lab. If you teach the thinking and analysis part, their standardized test scores won't go down...they will actually go up. 32 years of experience has taught me that.
Jose, you have stated very
Jose, you have stated very well the change that, difficult as it may be, we must pursue if we are ever to improve our student's math proficiency. However, one minor note - there are eight mathematical in the CCSSM, not seven. These are closely aligned with the Process Standards that NCTM has been advocating for over 20 years.
If the CCSSM are to be implemented properly, then it is really about how we teach math as much as the what.
Constructing Arguments
I like how you describe some of the practices as 'soft skills'. I believe you are absolutely correct in stating that the practices are as important or more important than individual content topics, and agree that if a student can't make sense of a slope that they are unable to know if they are correct or not.
Thanks for sharing your post. I will be forwarding this on!
Thank you for your candor.
Thank you for your candor. Each class of students is unique, as each child is unique, and I appreciate your thoughts on helping kids develop rather than just help kids test.