Conceptual vs Procedural Mathematics - Challenges from the field
I recently received correspondence about supporting a math initiative in Florida. A sense of nostalgia rushed over me as I thought about my time overseeing the development of the middle school math Florida Standards Assessment while working with the American Institutes for Research as a Math Test Developer. I even went as far as pulling up the practice test portal and worked through a little more than half the questions for the grade 7 practice test. While doing so, I came across a few items I wrote. One of them even had my name in it. As I was going through the practice test, I thought about the challenges teachers have preparing students for an assessment like the ones we were creating. I remembered it was one of the reasons I moved on from test development to supporting initiatives to improve education outcomes as a consultant.
Today, I work with school districts across the nation, in part, providing instructional support to math teachers. One of the greatest challenges I have is helping teachers transform their instructional practices to be more 21st century student centered and less like the way we were taught math. The five main challenges I face when supporting educators through this transition are issues with self efficacy, content knowledge, pedagogy, new resources, and motivation. I'll talk a little about each one and how they have been demonstrated through my experiences working with educators.
Self-efficacy is the belief in one's own abilities to achieve a goal. Some psychologist believe self-efficacy is more important than one's own ability. Henry Ford once said, "If you think you can do a thing or if you think you can't do a thing, you're right." Everyday in Math classes across the nation, teachers have a goal of teaching all their students something about a Math concept and/or checking to see how well students learned what was taught. However, there are several factors that lead to many educators feeling like it is a hopeless endeavor instead of David excited about becoming a giant slayer. Having previous success, seeing someone else have success, being coached up, visualizing success, and/or a person's psychological or emotional state are factors that influence self-efficacy. Many low-performing schools or schools where performance outcomes are on a steady decline easily result in decreases in educator self-efficacy.
Content knowledge relates to an educators mastery of the subject matter they teach. Federal education policies such as Race to the Top ushered in more streamlined, rigorous, and conceptual math standards which required a deeper understanding of mathematics than what was required before No Child Left Behind. A challenge for many veteran educators is the lack of content mastery in today's educational landscape. So they teach the steps of solving a Math problem instead of the concepts
Pedagogy is the way educators get students to learn. It is the how in education. How a teacher gets students to learn a math concept is important because of the diversity amongst learners in any given classroom. People are different and the acceptance of those differences are embraced more than ever in the U.S. The internet allows people to cultivate an entire social world where who they are can be celebrated and their learning experience can be customized. How do teachers account for these differences in a traditional lecture note taking model? Engaging students in learning styles that are unconventional is challenging for many educators not familiar with using web 2.0 tools in an educational landscape. Which brings us to resources.
Resources often enter school districts as education initiatives that have been forced upon educators as a new policy to improve or enhance learning outcomes. Most times they are done to the person instead of for the person with an individual having little to no input into the decision. Resources can be overwhelming because the time required to make sense of what it is and how to use it encroaches on personal and family time. With this in mind, some educators inadvertently look for resources that will teach students the math or the missing mathematics required to get a kid caught up to grade level. However, most resources don't account for the variety of ways students can engage with mathematics. Some of the questions we created had nothing to do with procedural calculations but pure conceptual understanding demonstrated through math practices.
Lastly motivation is a huge factor because there has to be a spark to make someone want to put in the required effort to master the mathematics at a conceptual level and transform their instructional practices. During a professional development workshop for a math resource product, I recently had an individual tell me, "I have six years to retirement. I'm not changing a thing about what I do. Also all this technology stuff is making it so that kids don't talk to each other because their heads always in a phone." So what does this mean for the next six years of students that come through this teacher's math class?
In closing, choosing to watch an old VHS movie with tracking issues wouldn't make much sense if you could stream it on Netflix in 4K. Playing a cassette tape of your favorite song while exercising at the gym would be pointless when you have an iPhone. Using the same instructional practices to teach math from the era of cassettes and VHS's is obsolete. In a world where the shift in mathematics education is like the change in complexity from playing a cassette to streaming a song on a mobile device, the ways to learn math have to change with the times like the ways to consume entertainment. Blockbuster is closed and Bestbuy won't sell tapes anymore. However students are still forced to learn multiplication fact tables in elementary school, when the learning standard states:
Represent and solve problems involving multiplication and division.
CCSS.MATH.CONTENT.3.OA.A.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
So using this
to teach students to be prepared for this
is as effective as a boom box on an elliptical.