This is part 2 of a series that will eventually be a book by the same name as this blog. Enormous changes to what you read here will likely happen at the last minute before publication, so enjoy this raw version while you can.
I believe strongly in how math should be taught and even more strongly in how it should not be taught. Nevertheless, when I am involved in teaching it as I believe it should be taught, (which happens to be in the traditional mode) I feel vaguely guilty, as if I am doing something against the rules and perhaps even wrong.
I have heard from other teachers who identify and empathize with this. Traditional math teaching is vilified by education schools, education consultants and other educational rent-seekers that have pervaded in the profession over the last three decades. Teachers feel bad for teaching using a method that has been proven to be effective. Practicing math problems is disparaged as “drill and kill”. Whole class instruction is thought to be ineffective because it doesn’t promote collaboration.
The picture that many have when hearing the term “traditional math” is a classroom in which seats are arranged in straight rows, the teacher stands at the front of the room and lectures non-stop for the duration of the class, students learn all procedures and problem solving methods by rote, and no background on the conceptual underpinnings of same are presented. Word problems are dull and uninteresting and students do not feel any desire to try and solve them. Problem sets (commonly called “practice problems”) are repetitious and do not present any challenge.
The traditional mode of teaching math has been slowly and steadily displaced over the past three decades by reform methods and are seen mostly in the lower grades. High school has remained somewhat impervious to this displacement; middle school less so. The methods for teaching math in the traditional manner is rarely if ever taught in schools of education. The result has been that newly arriving teachers from ed schools have been steeped in the math reform methods and are taught that inquiry- and activity-based student-centered, collaborative teaching is superior to explicit and whole class instruction.
Defenders of the reform methods will insist that teachers use both explicit and inquiry methods, and therefore teachers on both sides of the aisle are saying the same thing. We are not. Yes, some amount of inquiry and other methods co-exist within explicit instruction as are activities and group work. Just not to the same degree.
In this series we will therefore discuss what it is we are explicit about when we teach math explicitly. What is it that we say in terms of explanation, and what questions do we ask of our students? With that, let’s look at what goes into a typical traditional approach in a classroom.
Traditional Classroom Approach
The following is a typical daily approach in traditional math teaching. There are other aspects of traditional teaching that in the interest of brevity and staying to the topic of math education, I don’t go into here. If you are interested in the general principles of traditional instruction, however, you may wish to read Tom Sherrington’s book “Rosenshine’s Principles in Action”, which describe Barak Rosenshine’s principles of instruction.
Warm-ups: These are four or five problems that students work on in the first five to ten minutes of class. Some of the problems are from previous lessons to keep old material fresh. Others are what has just been covered. And still others may be problems that lead into the day’s lesson. For example, if a class has learned how to factor trinomials in which all the terms are positive, such as x^2 + 5x + 6 there might be a problem where they are asked to factor x^2+x – 6 some students can make the leap; others have questions for which I provide hints. When it comes time to go over the warm-ups, this last problem will then set the stage for the lesson to come which focuses on trinomials where the signs are not all positive.
Go Over Homework Problems: I have provided answers to homework from the previous day so students have checked their work. Therefore, I spend this time going over problems that they find difficult—usually three or four. I may have a student who has done the problem correctly explain it; otherwise I explain.
The Lesson and Start on Homework: I then go into the lesson. This series will go into what is talked about explicitly, which includes the worked examples that are part and parcel to the instruction. The pattern followed is the “I do, we do, you do” technique, in which students are given problems to solve after a few initial worked examples that are done together.
I leave enough time (approximately 15 minutes) at the end of the lesson for students to start working on their assigned problems. This allows me to answer questions and provide help. This helps prevent the situation of students not knowing how to do the problems because they may have forgotten how to proceed. Starting the homework problems in class allows for practice and the learning that comes from it.
On the topic of worked examples, a paper by Liljedahl and Allan (2013) sheds some light on the topic that I believe is useful to understanding where I stand. It talks about what they term “Now try this one” problems and states “These are the problems assigned, usually one at a time, by a classroom teacher immediately after s/he has done some direct instruction concluding with some worked examples. We recognize the rather traditional approach in this method of teaching and, although we would not ourselves approach the teaching of the topics in this fashion, we make no judgement about it here.”
Although deferring judgment one can guess how Liljedahl and Allan really feel about traditional classrooms. This is made a bit clearer by Pershan (2021) who doesn’t hold back in his book “Teaching Math with Examples”. He states: “Some of the dullest teaching on the planet comes courtesy of worked example abusers. These are the math classes that consist of a steady march of definitions, explanations and examples, one after the next. Practice (and learning) happen out of the classroom hours later, while students work on their homework.”
As I said, I start students on their homework in class, not hours later, and as you will see in this series, the explicit instruction is not a steady march of definitions and explanations. There are questions, and—dare I say it—even some inquiry along the way. But as far as examples, with some exceptions discussed below, Liljedahl, Allan and Pershan have pretty much nailed my traditional “Now try this one” approach.
For those who find such approach offensive, you may stop reading at this point. For others who are curious, I will say that my traditional approach does not quite fit the definition of the “worked example abuser” much as some would like to believe, nor is it worthy of what I imagine is the negative judgment that Liljedahl and Allan have deferred. I offer problems that are scaffolded and ramped up in complexity and difficulty so that there is in fact learning involved in doing the examples as you will see if you stick with this series.
I will say here that Pershan’s book does in fact offer good advice and methods for presenting examples and is worth reading. I would also add that this series on traditional math teaching is also worth your while and which I hope will dispel mischaracterizations and myths about traditional math.
Liljedahl, Peter and D. Allan (2013). In Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 1., Kiel, Germany: PME.
Pershan (2021). Teaching Math with Examples. John Catt Educational, Ltd., Woodbridge UK; Clearwater, Florida