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”.

7 thoughts on “A new series: Teaching traditional math

    • Thanks; I’ve heard Levitt before on this and don’t agree with him. Data science, or statistics, are fine, and should be offered as an elective in high school. But the algebra/trig/calc sequence is important for preparing students who wish to go into the sciences, math or engineering. On the one hand, there are people saying the US should be more like Singapore, China, Japan, Russia in terms of math proficiency. Those countries teach algebra, geometry, trig and calculus. Why denigrate them because Jo Boaler and Steve Levitt seem to think they’re unnecessary? Yes, some students do not go into STEM fields, so there should be an option to just take algebra and geometry.

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  1. Stanislas Dehaene is a neuroscientist who was awarded the 2014 Brain Prize – the Fields Medal of neuroscience. In his 2020 book How We Learn, he writes this (page 183):
    “Research is clear on this point: learning works best when math teachers first go through an example, in some detail, before letting their students tackle similar problems on their own. Even if children are bright enough to discover the solution by themselves, they later end up performing worse than other children who were first shown how to solve a problem before being left to their own means.”
    He contrasts this approach to “discovery learning.”
    Is the process he describes in part the process you will be advocating?
    Is a key part of the debate whether to teach a worked example before assigning extensive problem solving?

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  2. Yes, I advocate worked examples; I consider it an important part of how I teach math in a traditional manner, which is what this series is about. I’m also not presenting a debate. This is a series that describes how I approach traditional math, so people see more what it is about, rather than carry a mischaracterized interpretation of it. It also serves to provide assurance to other teachers who use a traditional approach that they are not crazy, and there’s no need to feel guilty for teaching in a style that ed schools and the ed establishment says “has failed thousands of students”.

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  3. Barry,
    This really goes nicely with how I try to convince the kids to “learn” the material, not just regurgitate it.

    I am fine with regurgitation – yay, they at least studied and have the foothold to take the next step.! But the thing is, when you teach AP classes, you need to do the example problems for two reasons: 1) so they know how to do the basics, and there will be AP problems on that, and 2) to set them up for the conceptual questions that are sure to be on the AP test.

    My experience is that people who have taught AP tend to structure their tests so there is some element of “do you understand all that stuff you did?” There are a few teachers who know what is coming down the pike, but my experience is that they are few and far between. Most teachers are comfortable “just teaching their curriculum.” And I think this explains a lot about why we have middle-school kids who can’t do basic arithmetic – “I taught my curriculum.” Yes, you did, but without any understanding of how important what you taught is when they get to Xth grade. In other words, most teachers don’t know what their students will be seeing next and how their teaching fits into that puzzle.

    And how do you handle that by the way, off-topic question. You clearly know what you are setting your students up for, but how does that get communicated to new hires?

    We start every class – even AP Calculus – with timed drills. Depending on the class, the drills vary wildly, You might give an adding-fractions drill to algebra 1, but an AP Calc drill might be solving logarithmic equations.
    Math teachers – more than any other content area – really need to have a sense of where on the spectrum their stuff feeds the next class. And having read the CC standards so many times my eyeballs hurt, I don’t think the authors of the CCS really did that. I mean, yes, the stuff is sequential, but they spend 4 years drawing pictures then in 5th grade you-know-what hits the fan! All of a sudden they are supposed to be masters of fractions and decimals and the relationship between them AND place value AND area/volume, all without having had to master the addition algorithm until the year before! It is just insane.

    Another example of the insanity: CCS has mastering exponent rules in 8th grade math. But if you look at the ACT standards, this is actually a much higher skill – A512, which is applying properties for positive exponents, and that is in the 24 -27 score range!

    I suppose the answer is more vertical teaming among elem, middle, and high schools but that isn’t always possible. I am in a parochial school, so we get those opportunities. I wonder how other schools and districts address this challenge.

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    • Yes, good points.

      One has to recognize that there are levels of understanding. The level that is most productive in early years (K-6) is procedural understanding. And while some may poo-pooh it as “algorithmic thinking” and rote memorization without understanding, it requires mathematical reasoning. One has to know the context of the procedure in order to apply it to solve problems. Multi-step problems such as those found in SIngapore’s math books are indicative of how procedural understanding provides the reasoning needed to solve problems such as:

      The average weight of John and Alice is 55 pounds. John weights 60 pounds. How much does Alice weigh?

      This is taught by explicit instruction: First find the total weight of John and Alice. Since the average is the total weight DIVIDED by 2, how do we undo this to find the total weight? We multiply by 2, to obtain 110 pounds. Then subtract John’s weight from 110, to obtain Alice’s weight of 60 lbs.

      This worked example can ultimately be extended to apply the underlying (procedural) method to a more complex problem: The average cost of 3 books is $4.50. The average cost of two of the books is $3.90. Find the cost of the third book.

      Or: Jeremy has averaged 89% on his last three math tests. What score does he need on the fourth test to increase his average to 90%?

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