The term “traditional math” or “traditionally taught math” is fraught with images and connotations. 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. They do so because they believe it incorporates pedagogical methods that have been proven to be effective, like direct/explicit instruction, worked examples, and scaffolded problems.
This series is not about the pedagogy of traditional math. It is also not a message to teachers who teach math traditionally that “You’re doing traditional teaching wrong.” If you are teaching math traditionally you feel guilty enough already–many books and internet posts are constantly harping about how wrong traditional teaching is.
That said, in this series I do not advocate against certain practices such as inquiry-based and student-centered approaches, nor how to arrange seats in the classroom. Rather, I clarify through examples how I teach math in the traditional manner. I discuss how I explain particular topics, the type of worked examples I use, and how I keep previously covered topics fresh so that students remember the procedures when they come up again—and they do come up again. In all the topics I cover, my descriptions implicitly demonstrate what I mean by explicit/direct instruction, as well as how I incorporate some amount of inquiry and activity in the process. Most importantly, I show how I teach not only the procedure but the conceptual understanding behind 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 seventh and eighth grades (the latter including the regular Math 8 as well as eighth grade algebra). The limitation is for several reasons, the principal ones being: 1) The foundational aspects of math in the lower grades can be summarized concisely in an introductory chapter with reference to books such as Singapore’s series; and 2) seventh and eighth grades are for many students the last chances they get to revisit and synthesize the foundational procedures they have learned but not necessarily have mastered from K-6.
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.
My approach in this 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, I 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, I believe 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 be made to identify the constant of proportionality (or variation) or express 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, I cover it but do not obsess over it.
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. I go around the room. If there are many mistakes, I make a game out of it, writing down the answers on the board until the correct answer comes up. This has worked well, and I haven’t seen students become unduly depressed with such approach.
Another method I use, 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: Seventh grade, and an intro to algebra
For an in-class look at traditional math, check out “Out on Good Behavior: Teaching Math While Looking Over Your Shoulder”.