- A Parent’s Guide to the 1st Grade Math Common Core - February 13, 2013
- A Parent's Guide to the Kindergarten Math Common Core - January 29, 2013
- First in Math and Reflex Math: A Program Comparison - November 5, 2012
- Procedures versus Concepts: A Mathematical Dilemma - October 18, 2012
- The Mathematical Workshop Model: How Data, Differentiation, and Classroom Management Combine in an Elementary Classroom - October 12, 2012
There have been a lot of articles lately debating procedural teaching and concepts-based teaching in the classroom. As an elementary school teacher, this topic is of particular interest as mathematical reform models are sweeping through our curriculum. Whether you are a Common Core Standards state or, as in Virginia, simply “aligned” with Common Core, mathematical reformists argue that teaching the way of the past simply does not work for our students of today. The old “drill and kill” model, reformists argue, simply do not teach our students what math means in order to partake in higher level student discourse. On the other hand, traditionalists argue that one of the reasons for the slow improvement in mathematical education in the US is the fact that procedural based teaching is almost taboo in today’s classroom. Meanwhile, intermediate and middle school math teachers are wondering why their students still do not know their basic facts. So what do we do?
I argue that a healthy dose of procedural based instruction mixed with conceptual based learning is what is needed in the mathematics classroom. When teaching basic facts, our students should be able to discuss that repeated addition is the same as multiplication. They should be able to tell us that 5 x 4 = 20 because 5 + 5 +5 +5 = 20. They should be able to draw an array and other representations that depict this equation and even discern a multiplication word problem over an addition word problem because they have identified the word “product” or “factors” over “addends” or “altogether.” However, when students are asked to multiply 1,245 x 623, do we really expect them to draw a picture? Do we want them to add 1,245 over and over until they have done it 623 times? Hopefully your answer is a resounding NO! This is where the procedural teaching takes place. Despite the fact that in the “real world” we would simply take out our iPhones and open the calculator application to solve this problem, our students for some strange reason are asked to do this by hand. We can teach them lattice multiplication, area model multiplication, or even Egyptian multiplication style of halving and doubling if you like. At the end of the day, these are all procedural methods to help the students actually solve the conceptual problem.
This seems like a rather simple and mundane answer to a relatively simple issue. The problem is, reformists argue that concepts based mathematics should come before procedural based teaching. In fact, this is the dominant point of view in the state of mathematics. States have designed their curricula around it, textbooks are written for it (see Mathematical Inquiry) and administrators are looking for it when they come into the classroom.
For math teachers, all of this creates a sort of cognitive dissonance. They know what needs to be done, but are worried about being caught teaching procedurally. Let me make a medical parallel here for a minute since that is often how educators like to compare their profession. Teaching concepts before procedure or in lieu of procedure is like a doctor knowing where a heart is located, why it beats and how it beats, but has no idea how to perform heart surgery. I don’t know about you, but I would rather have a doctor that knows how to perform heart surgery but doesn't know how a heart beats than the other away around. Although having one that knows both would give me the greatest confidence in their abilities.
Stephen Wilson, a math professor at John Hopkins University, recently stated “the way mathematicians learn is to learn how to do it first and then figure out how it works later,” at a conference held in Winnipeg. (Thank you Barry Garelick, writer for Education News for this information). To the chagrin of most of his colleagues, Stephen Wilson is echoing what many teachers are feeling in the classroom. We understand that concepts are important and that high-level questioning and open-ended responses will create a well-rounded young mathematician. What we also understand, however, is that unless they know how to do the math, that the concepts will simply become abstract ideas that are grounded in nothing.
What this means for the mathematics teacher is that we must be confident in our abilities. We must understand both the content and the pedagogy behind mathematics so that we can speak to what we are doing and why we are doing it in the classroom. We should be able to explain to our administrators why we are teaching procedurally to one group of students while others are grounded in concepts, number sense and theory. All educators understand that a single initiative for any content area will not be the panacea, but rather taking resources from a variety of areas to meet the diverse and unique needs of our students is the best way to reach them. Whether you believe in concepts or procedures, my argument is to believe in both.