Excerpt from the Introduction
I believe strongly in how math should be taught which happens to be what has come to be known as the traditional method. Nevertheless, when I am involved in teaching math in this 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. (For the record, J. R. Wilson, my coauthor, is not one of them and I’m sure there are others.) The term “traditional math” is fraught with images and mischaracterizations: 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. Students are viewed as “doing math” but not “knowing math” and have no conceptual understanding of what they are doing. Topics are presented in isolated fashion with no connections with any other topics. Word problems are dull and uninteresting and students do not feel any desire to try and solve them.
Despite the pervasive criticism of traditionally taught math, there are teachers who continue to teach math in this manner. They do so because they believe it incorporates pedagogical methods that have been proven to be effective, like explicit instruction, worked examples, and scaffolded problems.
I taught seventh and eighth grade math as a second career, spanning about eleven years, after retiring from a career in environmental protection. I majored in math in college, used it in my first career and realized I wanted to teach in my retirement. I have written articles about math education over the years and chose to teach it the way I was taught: i.e., via explicit instruction with worked and scaffolded examples.
This book provides a glimpse of what explicit instruction looks like in the classroom for grades K through 8. The book seeks to inform readers of our approaches to traditionally taught math. This has two purposes: 1) For approaches that are similar to what you may already practice, it provides some assurance that you are not alone, and 2) It may give you some new ideas.
Part I of the book provides discussions of key topics in math for grades K-6 and the typical approaches used.
Part II covers regular seventh grade math (Math 7) and accelerated Math 7. Accelerated Math 7 covers all the topics in Math 7 as well as those in eighth grade math (Math 8). A sample of topics from both courses are in Part II. Part III includes a sample of topics in eighth grade algebra.
The example lessons in Parts II and III illustrate what explicit instruction looks like in the classroom. They show how particular topics and procedures are explained, the types of worked examples used and how previously covered topics are kept fresh so that students remember them when they come up again. Most importantly, the lessons show how traditional math teaching addresses not only the procedures but the conceptual understanding behind them.
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 of today’s textbooks are poorly written with very little explanation of a procedure, and frequently two or three sub-topics embedded in a single lesson. My approach in this book is to provide workarounds to these shortcomings. This frequently involves supplementing material that is in the textbook with material in other books, which are named throughout.
For seventh grade textbooks I have usually had to reorder the sequence of topics for a more logical flow, add material within each topic area, and provide problems taken from other books. It does not involve designing a new curriculum from scratch. For both grades, I have sometimes crafted homework worksheets using problems from a variety of sources, and ordered so that the problems start with easy problems and progress in difficulty. Present day textbooks I have had to work with often start with more difficult problems, as well as problems for which no examples or discussion has been provided.
For eighth grade algebra, I have used a 1962 algebra book by Dolciani et al. While this book is very good, its availability on the internet has decreased to the point that they are now very expensive. Also some of the topics are presented very formally—indicative of the set-theoretical approach that was in vogue in the 60’s New Math era when it was written.
The following is a list of the books I draw upon which are also mentioned throughout. (Publication information for these are in the References at the end.)
Brown, Smith, Dolciani (1990) “Basic Algebra”
Mary Dolciani et al. (1962): “Modern Algebra: Structure and Method”
JUMP Math (2015): Assessment and Practice, 7.1 and 7.2
Paul Foerster (1990) Algebra 1: Expressions, Equations and Applications)
Singapore’s “Primary Math” series (Singapore Math, 2003)
Many of the problems included in the example Warm-Ups in Parts II and III come from Singapore’s Primary Math series, and Dolciani’s algebra 1 textbook.
Classroom Routines
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. The latter types of problems help set the stage for the day’s lesson. I provide hints and guidance while they are working the problems, and then go through the answers after time is up.
Go Over Homework Problems: I provide the students with answers to homework 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: The lesson examples in Parts II and III of this book provide information on what is talked about and how it is delivered explicitly. The pattern followed is the “I do, we do, you do” technique. This technique employs worked examples that are part and parcel to the instruction, followed by examples that students must do independently, with guidance from the teacher as needed.
I leave enough time (approximately 15 minutes) at the end of the lesson for students to start working on their assigned homework problems. This allows me to answer questions and provide help and guidance. It also works to prevent the situation of students not knowing how to do the problems when they get home.
Some students will work with others on the problems, but I limit this to no more than two people. I monitor those working with a partner to make sure they are on track. Starting the homework problems in class allows for practice and the learning that comes from it. Most importantly, I try to stem struggling with a problem and aim to have students be successful. My guiding principle is that struggling to learn the breast stroke is not the same as struggling to keep from drowning. The latter doesn’t teach you how to swim.
When Students Make Mistakes. 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.
I have several methods for dealing with mistakes. First and perhaps most important is what happens when I make a mistake—which I do. I tell the class the first day that if a student catches a mistake that I’ve made, they will receive a package of Goldfish crackers. Although showing students that even teachers make mistakes helps them see that it is part of doing and learning math, this doesn’t stop the shyness and fear of being wrong in front of one’s peers.
If a student responds to a question I ask with the wrong answer I might 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 and call on others. 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. I make a point of coming back to the students who made a mistake so they can try again.
In general, students in middle school can be quite guarded about mistakes. One technique that has worked well is to have students work a problem in their notebooks. I walk around and will offer help or guidance for the weaker students, and once seeing they have solved the problem correctly have them present the answer and how they worked it to the class. Knowing they have it right fills them with the needed confidence to do this. (Some teachers rely exclusively on mini-whiteboards. I prefer notebooks for the reasons stated above, but also because my eyesight is not all that great, and mini-whiteboards can be hard for me to read.)
I also encourage students to try to solve challenging problems by offering a Kit Kat bar for the first student to come up with the correct answer—and show the class how he/she did it. This has proven to be unusually effective.
Checking for Understanding
Much is written about “checking for understanding” and “formative assessment”. Without getting too far into what constitutes understanding, in this context it means absorbing what has just been taught and applying the basic procedure or concept to answer the teacher’s questions. Teaching is a combination of providing instructions and asking questions. Questions may address material just stated, and other times require students to make a conjecture. This last is done to provide a segue to the next part of the explanation as well as a check on whether and to what extent students are following. Students are kept on their toes knowing they could be called on. It also helps teachers know if the explanation is working as is or needs to be altered.
Instructions and questions take many forms. The Warm-Ups provide a check on material presented in past lessons, and also front-load material covered in the upcoming lesson. Worked examples are a source of checking and reinforcing instruction. Extra credit and challenge problems provide a way to evaluate ability levels and identify students who may need more of a challenge. Tests and quizzes also provide information on students’ understanding and to that end can serve as a formative assessment even if a test is meant to be summative. It provides a window into what areas students need more practice and instruction. In short, there is no one way to check for understanding; methods will differ for every teacher.
One Last Word
The book is designed so that you can pick and choose which topics or lessons you want to read about. It is a reference to what traditional teaching is—and isn’t. In Parts II and III of the book in which examples of lessons are provided, I have depicted some of the reactions and responses of students to the teacher’s questions in what I feel is somewhat realistic and only slightly exaggerated. It was not my intent to show that students do not try hard—quite the opposite. When encountering a topic for the first time, new information can be overwhelming and students can experience various degrees of “brain freeze”. To prevent this, I present new information incrementally but even so, students are going through the balancing act of incorporatingnew information on top of what they have learned previously. I try to represent when it becomes too much, as well as show how to get beyond this inertia.
We hope you enjoy this book and wish all teachers well in their teaching.