How’s this for a prickly real-world math problem: In the United States, only about one-third of eighth graders are proficient in mathematics. Just half of high school graduates taking the ACT are prepared for college-level math, and algebra remains a barrier for many community college students—hindering their progress both inside and outside school walls.
And consider the following situations that aren’t often discussed, but are just as troubling: I’ve talked with fourth graders in accelerated and gifted programs who are earning A’s in math but can’t solve 302 – 298 without a pencil. I’ve met far too many thoughtful, successful adults who can’t explain what y = mx + b means, let alone when it could be useful to them. And many mathematics majors and doctoral students secretly worry that they aren’t actually good at math.
How is it that so many students across such a wide range of ages and achievement levels fail to understand and be confident in math? My conjecture: there is too much “instructional” time in mathematics.
Mathematicians rightly insist that we define our terms at the outset of any proof or argument. “Instruction” is the key term here, defined as (1) knowledge, mandates, or information imparted, (2) the act of furnishing with authoritative directions, and (3) an outline of how something is to be done. In regard to computers, “instruction” means a command given to a computer to carry out a particular operation.
Using these definitions, one could translate “excellent math instruction” to mean furnishing students with clear mandates and imparting perfect directions for exactly how particular math operations are to be done.
As I look back on my own experiences as a student, that statement describes the majority of the math lessons I encountered—and probably most of yours, too. I believe this widely-used lesson design is a significant contributor to the difficulties students have with math and algebra because, by definition, “instruction” alone cannot help students develop the key transfer outcomes that are essential for success both in and out of school: the capacity for independent critical thinking and problem solving in new and unfamiliar situations.
Given the rapid growth and development of math education technologies, I am continually disappointed to see that many edtech products disregard research-based pedagogical principles and instead merely digitize the exact same “instruction” that is already creating the problems I described earlier. For example, technologies like PowerPoint and YouTube have made it much easier for mathematics to become a spectator sport where students “sit and get” lessons in an impersonalized learning environment. There is a flawed pedagogical premise behind the use of those tools. Simply put, we need to stop believing that “more and better math instruction” is the appropriate prescription for the problem of student achievement in mathematics.
From Kindergarten through Calculus, I had caring mathematics teachers and experienced enough success that I chose to pursue a mathematics degree in college. But given that the high quality math “instruction” during my K–12 education had—unintentionally—simply trained me to carry out particular operations in response to predictable prompts, it’s understandable why I was terrified of the first simple algebraic proof I encountered during my freshman year of college in a Foundations of Mathematics course: Prove that the sum of two even numbers is always even. At the time, I didn’t remotely know how to begin constructing an argument and proof that would satisfy my professor, and no authoritative directions that prescribed how I should proceed had been provided.
This situation was an example of how my professor—whether intentionally or not—was using a principle of deeper learning. In contrast to “instruction,” deeper learning empowers students to design their own solutions to complex problems. It requires that students be given more time to make sense of problems, experience productive struggle, and think for themselves rather than merely trying to remember someone else’s complicated procedures, confusing mandates, and obscure formulas.
Here’s one example of how deeper learning principles can be practically used in the classroom, compared to the more “instructional” lesson plan.
With an “instructional” approach to designing lessons about the surface area (SA) of rectangular prisms, students are told exactly which variables to set up for length, width, and height. They then hear an explanation of how those variables are used to represent the areas of the six faces of the prism in a typical formula like this one:
As I have detailed in a book about mathematics intervention, there are alternative lesson designs that use the principles of deeper learning to engage students in devising their own formulas—and they learn much more by doing so (pdf). In one case, we observed two students as they collaborated to independently develop their first rudimentary version of a generalizable formula:
They used t and b to represent the areas of the top and bottom of the box (instead of l×w). And because they were also still learning how to communicate using the conventional order of operations, they treated the l + w in their expression as if it were in parentheses. A more conventional way of writing their initial formula is SA = 2h(l+ w) + T + B.
This solution was student-designed, student-owned, and therefore student-understood. With additional teacher facilitation, these students later derived more conventional forms of the surface area formula for rectangular prisms. But because they engaged in deeper learning experiences from the beginning as well as over the course of several class periods, these students developed conceptual understanding that equipped them to solve non-routine test problems—while also ensuring surface area concepts and formulas wouldn’t just become more things to eventually forget.
As demonstrated in this example, one quick way to get a sense of whether students may be involved in deeper learning is to gauge their level of confusion when trying to respond to an unfamiliar, thought-provoking scenario. Rhett Allain, Associate Professor of Physics at Southeastern Louisiana University, rightly points out that confusion is the sweat of learning; deeper learning could be considered an invigorating mental workout.
These days, even though teachers often have less class time for mathematics and higher stakes resting on standardized test scores, I believe everyone benefits when much of the “instructional” time in math class is replaced with time spent on deeper learning experiences. Because many standardized tests increasingly require students to solve complex problems and justify their solutions, time spent on deeper learning will improve student achievement on both standardized and classroom-based assessments.
Two years after my freshman professor gave me the opportunity to struggle with that even numbers proof, I recall designing my own solution to a challenging new proof on an advanced Algebraic Structures test. When that professor returned my test, he had written, “Nice proof! I hadn’t thought to prove it that way.”
How often are math teachers able to give students such feedback because they were intrigued by a student’s truly original idea? If our lessons are mostly “instructional,” then we shouldn’t expect to be surprised by a student’s thinking. But when we spend less time on “instruction” and use more time for deeper learning, students are empowered to surprise us with their mathematical brilliance.
