The question is not, “how best to teach mathematics?” The question, educator, is “how best for YOU to teach mathematics?”
The debate over what constitutes “good” math instruction flares up over and over again. We are over 25 years into “reform” mathematics curricula, which started with the publication of the National Council of Teachers of Mathematics standards (1989), and has its deeper roots in the “‘new math” after Sputnik was launched in 1957. In recent years, in Canada you may recall the 2015 CD Howe institute report, which gives recommendations for educators on how best to teach mathematics, including the idea that teachers should consider a balance of 80% direct instruction, versus just 20% what they call “discovery” instruction. The debate is usually characterized as being between “back to basics” advocates, and “reform” or “discovery” mathematics advocates, and that’s partially true, although the real truth of modern math classrooms is by no means as binary as newspaper articles on the topic would have you believe.
Here’s an image that might depress you.
This little pamphlet was written in 199 (the former Peel Board of Educaton)! It would seem that, 25 years on, we are still having some of the same debates. It might seem like we aren’t moving forward, but I would argue we are leaps and bounds ahead of where we were a generation ago.
But 25 years later, the question still remains, for North American educators, how best do we teach mathematics in K-12 classrooms? What methods workk best? How do we balance “basic skills” and concepts”? Should they be set opposite to each other? Are these even opposites at all? The intent of this short blog piece is not to attempt to unravel terms like “discovery”, “inquiry”, or even “direct”’ or “explicit” instruction, or even “procedural” and “conceptual” instruction. (On that last pair, fifteen years of work by Bethany Rittle-Johnson and her co-researchers is showing that procedural and conceptual understanding iterate on each other, in a constant back and forth. In short: it may not matter which type of instruction comes first).
I would hope that all educators make up their own minds. There are very personal choices to be made, based on one’s own strengths as a teacher, the strengths and weaknesses of your students, and the type of math being taught. As a general rule of thumb, there are very few math concepts that I believe can’t be introduced in an intriguing and interesting way. No, I don’t believe algorithms should be taught first, either. But algorithms are one piece of the puzzle, one we shouldn’t discard or scoff at. A lesson on how the standard multiplication algorithm works doesn’t take much time, and shouldn’t be avoided. Show your students how the algorithm pulls together partial products, adding them together, and giving one neat number. It’s not magic- it’s mathematics. I think students should get an Algorithm Licence- show how it works, and you can use it forever after, no questions asked.
Many big ideas in mathematics can be introduced through simple and interesting investigations. Think Pythagorean theorem, or “finding” a value for pi, for example. Of course, good investigations are nothing without powerful teacher talk- deciding when to intervene, when to give whole class instruction, and how to consolidate the activities. Kids should never be left to just “discover” math on their own-teachers have an important role to play in constructing understanding of powerful math concepts. Kids need us.
You can probably agree that, for the most part, your instruction is somewhere in the middle of this line, depending on the topic and lesson. Both fully guided and unguided instruction are equally disastrous. (Fully guided instruction is less disastrous, in my opinion, but I don’t believe it’s optimal in this day and age). The real trick is knowing where and when to let students “discover”, and where and when to tell or explain (yes, telling and explaining is still part of our jobs!)
We must make up our own minds about how best to teach mathematics. My friend Tim Boudreau said the following of me in a tweet:
Even then, it depends on the lesson. For myself, I believe at times I swung out way too far toward unguided instruction. The evolution of my practice was realizing that those precise and short and specific 5-10 minute mini-lessons to the whole class went such a long way. But my big evolution over time was realizing how kids talking to other kids about mathematics needed to be the mainstay of my classroom. Both are needed!
Others will have a different optimal balance. Some years your class will be more suited to exploring more, or will need more direct instruction, and that’s fine. Professionals need to know their own strengths and weaknesses, and teach to their strengths, while tempering their weaknesses. Classroom practice is a daily grind. Know yourself, educator, and know your kids. Make purposeful instructional choices. Explore powerful and interesting mathematical ideas with your kids. Don’t be afraid to explain concepts to them when they need it. There is no one single “best practice”- there are better and worse practices, though. More importantly, there are better and worse practices for YOU. Choose wisely and well!