Traditional Math (2): Prefatory Remarks, Disclaimers and Warnings

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.

References

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

A new series: Teaching traditional math

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 educationists. 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. Topics are presented in isolated fashion with no connections with any other topics, so that students are prevented from seeing how one mathematical idea may relate to another. Word problems are dull and uninteresting and students do not feel any desire to try and solve them. They have no bearing on any aspect of students’ lives, and all information needed to solve the problem are contained within the problem itself.  Problem sets (commonly called “practice problems”) are repetitious and do not present any challenge.

I could go on, but I’m sure you’ve heard the many ways that traditional math is mischaracterized. And yet, despite the slings and arrows that are hurled toward traditional math, there are teachers who continue to teach math in a traditional manner. Traditional math teaching incorporates pedagogical methods that have been proven to be effective, like direct/explicit instruction, worked examples, and scaffolded problems and, despite the claims that such methods have failed thousands of students, has produced successfully achieving students.

This series is not about the pedagogy of traditional math. Rather, it provides examples of how math is taught in the traditional manner. It discusses the type of worked examples used, and how previously covered topics are kept fresh so that students remember the procedures when they come up again—and they do come up again. In all the topics covered, it makes explicit what is meant by explicit/direct instruction; that is, what is made explicit for each of the key topics. It also will show how some amount of inquiry and activity is part and parcel to explicit teaching.in the process. Most importantly, it shows not only how procedures are taught, but the conceptual understanding behind it while recognizing what levels of understanding students will likely retain–without obsessing over it.

The series has two purposes; 1) For approaches that are similar to what you already do, it may give you the assurance that you are not the only one who does these things and that you are not crazy, and 2) It may give you some new ideas.

The series is focused on key topics covered in K-6, as well as seventh and eighth grades (the latter including the regular Math 8 as well as eighth grade algebra). The foundational aspects of math in the lower grades will be discussed with reference to books such as Singapore’s and other useful series. For all grades, the book will also provide a list of books one can obtain from the internet or other sources that can supplement the textbooks that a school has adopted. Which brings me to the topic of textbooks.

Textbooks

Textbooks are handy things to have because they contain a sequence of topics and a breakdown of what gets taught within each one. Unfortunately, many textbooks are poorly written. There is very little explanation of how a procedure works, and frequently two or three sub-topics are embedded in a single lesson. For example, in one algebra book I saw, there were two types of word problems presented in one single lesson: 1) Mixture problems, such as: “How many liters each of 20% and 50% sulfuric acid must be mixed to obtain 30 liters of a 45% sulfuric acid solution” ; and 2) Wind and current problems such as “A plane flies with the wind for 2 hours and travels 360 miles; when flying against the wind, it takes 3 hours to cover the same distance. What is the speed of the plane in still air, and the speed of the wind?” While the problems are set up similarly when solving with two variables, it is a lot of information to present in one lesson. I would break it up into two separate lessons.

The approach in this series (which will ultimately become a book) is to provide workarounds to the various shortcomings of textbooks. This frequently involves supplementing the textbook with material in other books, which I will recommend. For seventh grade and regular textbooks I have usually had to reorder the sequence of topics for a more logical flow, add some material within each topic area, and provide problems from other books. It does not involve designing a new curriculum from scratch. For eighth grade algebra, my solution has been to use a 1962 algebra book by Dolciani. While this book is very good, its availability on the internet has decreased to the point that they are now very expensive. There are other books that may be used, however and I will provide a listing.

For all courses, the series will describe how to address some of the topics that are included in Common Core, but which, in my opinion and others with whom I’ve consulted on such matters, do not need the degree of emphasis which is typically given in textbooks. For example, in seventh grade, proportions can be presented simply as they have in the past. Students should be given the opportunity to solve many types of problems using proportions. They do not need at this point to spend a lot of time identifying the constant of proportionality (or variation) or expressing the proportional relationship as an equation. They will do this in algebra where they learn about direct variation and how to represent such relationships as equations. It will make more sense then because they will have mastered the foundational aspects of proportions. i.e., they learn about direct variation, and how to represent directly proportional relationships as equations.

I recognize, however, that end of year state testing may include questions on this and other topics, so I am careful to explain it, show how it is done, and put a question or two on the quiz or test and give extra credit points for those students who can do it.  In short, it is covered but not obsessed over.

A quick example of how this series will unfold

As an example of how this series will proceed, here is a discussion about how to handle the situation when a student gives the wrong answer in front of the class.

I frequently hear that it is not a good idea to tell a student that they are wrong, even though some advocates say that making mistakes should be a goal of teaching math. It is not a goal for me, but mistakes will happen. I don’t shy away from telling a student that they are incorrect. Some people use mini-whiteboards that students write their answers on, and then hold them up. 

My method, when a student says the wrong answer, is to say, “Not what I got”. If I can see what the mistake was I will sometimes say “Oh, it looks like you multiplied instead of divided”, or whatever the mistake happens to be. Or I may ask the student to show how he or she obtained their answer which provides insight into what the mistake was, leading to how to do it right. I might go around the room. If there are many mistakes, making a game of it, writing down the answers on the board until the correct answer comes up. These methods have worked well, and I haven’t seen students become unduly depressed with such approaches.

Another method, usually when introducing a new topic is to preface the problem with “I am willing to bet $100 that the answer you give me when you hear this problem is going to be wrong.” This serves as a dare, and students will rise to the challenge. One problem that I use when introducing algebraic equations to solve word problems is: “John and his sister have $110 between them. John has $100 more than his sister.  How much money does each person have?”

Usually the first answer I hear is $100 and $10.  I show quickly why this cannot be right. Students then resort to guess and check and finally hit upon the right answer: $105 and $5.  This serves as a segue to how using an equation is a much more efficient way to get the answer—something I’ll get into more in a later chapter. It also serves as a way for them to say an answer out loud without feeling embarrassed if it is wrong.

Next chapter: Early grades and what should be covered.

For an in-class look at traditional math, check out “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder”.