My Day at Ed Camp

Originally appeared in Education News , Nov, 2015; and is included in “Math Education in the U.S”

I attended an “Ed Camp” recently. This is one of many types of non-professional development and informal gatherings where teachers talk about various education-related topics. The camp I attended was free of charge and took place at a charter school that prided itself on a student-centered approach to learning. In keeping with the school’s focus, the camp also took a student-centered approach which it boasted about in its announcement, calling the event an “unconference”. It stated that the Ed Camp “is not your traditional educational conference; sessions will be created by attendees.”

And that’s exactly what happened. Participants wrote ideas for sessions on Post-It notes which were placed on a whiteboard. The conference organizers then put the Post-It notes in categories that formed various sessions which were then led by whomever wanted to lead them.

The topic suggestions were placed into nine separate categories/sessions. For each of three one-hour periods, there were three sessions that participants could choose from.  I chose “Motivation,” “Feedback in lieu of grades” and “The balance between student-centered and teacher-centered in a classroom.”

A few decades ago there was a mix of opinions on what are considered “best practices” in teaching—some of which included traditional methods. The older generation of teachers, however, has been almost entirely replaced by the new guard. This has resulted in a prevalent new group-think which holds that traditional teaching is outmoded and ineffective. The participants at Ed Camp were of the new guard; mostly people ranging in age between 20’s and 40’s. A few people were in their 50’s or early 60’s, but were subscribed to the same group-think. From what I could tell, I was the only traditionalist present.

Motivation Session

All participants at this session generally agreed that motivation was important and that if classrooms do not have focus, there is loss of attention. They also agreed that students did well in a structured environment and that set routines and clear expectations were motivators. These two consensus items were uttered with the same somnambulant automaticity with which many say grace before chowing down a meal.

Participants then went to town describing various motivating/engaging activities including having students spell out words using their bodies to shape the letters (though I have forgotten what this had to do with whatever was being taught).  After a few more suggestions, someone pointed out that no matter how engaging the activity, the novelty of it wears off, so you can only do it a few times before students are bored — which, I suppose, leaves teachers with the option of more traditional approaches like a warm-up question and then teaching the class.

The issue of group work came up. Group work ranks high on the group-think spectrum as something worthwhile for all students.  So when a teacher said that group work may be difficult for students who are introverts, the feeling of cognitive dissonance was distinctly present in the room.  But the dissonance was quickly dispelled by the same teacher who brought it up. “Well, think of it this way,” he said. “How many times have you gone to professional development sessions and the leader says ‘Now turn to your neighbor and discuss such and such’ and you go ‘Oh, no! Do I have to?’” General agreement ensued.

“But,” he went on, “You kind of think to yourself, ‘Well, OK, let’s get this over with’ and pretty soon you’re doing it and it isn’t that bad. So I think maybe we just have to get kids to think beyond themselves and just go with it, and they’ll see it isn’t that bad.”

I’m fairly certain most of the attendees had been through—and probably hated—professional development sessions that were group-work oriented.  But if there was any disagreement with what he said, it was not voiced.

There was consensus that students responded well to competition.  Teachers noted that students like to see high scores posted or go for extra credit assignments or questions on tests. Such agreement was surprising given that it goes against the trend of the “everyone is special” movement in which all students win awards or graduating classes have multiple valedictorians. Unless one includes competition as being an integral part of collaboration and working in teams and groups, competition would seem to be its antithesis.

Another unexpected result was related by a second grade teacher who taught at the school where the Ed Camp was held. She had assigned her students to groups and arranged her class in clusters of desks as many classrooms are these days. One day her students asked her, “Can we be in rows facing the front of the classroom?”  She tried to reason with them, explaining that when she had been in school she always had to sit in rows and would have loved the opportunity to sit in groups. They told her that it was easier to be in rows because they wouldn’t have to twist around to see what the teacher was doing at the board.  The students assigned themselves numbers randomly so the teacher could put them in straight rows according to their numbers. Since this was a student-centered decision at a school that valued student-centered activities, the teacher reluctantly went along with what they wanted.

Think Pair Share: Harbinger of Things to Come?

The initial premise of the next session I attended—feedback–was that students should be given guidance rather than interim or even final grades. This is not a new concept, as evidenced by a recent comment I saw on a popular education blog: “When numerical/letter grades are king, real learning is kicked to the curb, along with meaningful assessment.”

Like many educational ideas, this one sounds like it ought to be superior to a system of grading that many have accused of being unfair for years, until you get into the details—things like subjectivity and how students will be assessed. The moderator—who did most of the talking in this particular session—said that in guidance-based regimes, students should be told whether they are doing a task correctly or incorrectly and that the key to completing a task was to ensure that students had an appropriate process. I couldn’t be absolutely sure, but it sounded like process trumped content.

He brought up math as an example and said, “I like to give kids problems they don’t know how to do.” This is not the first time I’ve heard this. While I agree that students should be given challenging problems, I also believe that they need to start from a place that they know and advance bit by bit to variants on a basic problem structure to be able to take on non-routine problems.

Such process is known as scaffolding, but modern purveyors of education theory hold that scaffolding should not be used and that flexible thinking –applying prior knowledge to a new and unfamiliar problems or situations—comes with repeated exposure to such problems. Supposedly this develops a “problem solving schema” and “habit of mind” that is independent of acquired procedural skills or facts. But to pull off what this teacher wanted—having them solve something totally different than what they’ve seen—students are given feedback. The feedback is in the form of questions to motivate them to learn what they need to know and ultimately to solve the problem in a “just in time” basis.

The notion of supplying feedback in the form of guidance seemed to this moderator to be a new and cutting edge thing, and in fact announced that the activity of “Think-Pair-Share” was antiquated and should be abandoned. “Think-Pair-Share” has been around for at least 10 years.  The first time I heard about “Think, Pair, Share” was in a course I took in ed school. Briefly, students work together to solve a problem or answer a question, discuss the question with their partner(s) and share their ideas and/or contrasting opinions with the rest of the class.

But now it was considered passé, the main problem with it being that students didn’t know what to say to each other about whatever it was they were to discuss. And that was likely because they had little or no knowledge of the subject that they were supposed to talk about, and which was supposed to give them the insights and knowledge that they previously lacked.

Did this mean that perhaps there was now some evidence that direct and explicit instruction could have beneficial educational outcomes? No. Feedback and guidance was the new “Think Pair Share.” Student-centered and inquiry-based approaches are still alive and well. And in closing, the moderator added that students need good solid relationships with one another and with the teacher. To this end, the moderator said, putting students in straight rows will not build such relationships.

I was tempted to bring up the story of the second grade class that insisted they wanted to be in rows but we were out of time. In fact, we ran over and I was late to the last session on the balance between teacher and student in a student-centered classroom.

Defining Balance—or Not

The conversation in the third and last session of the day was already underway with some talk going on about how effective student-centered communication is fostered using something called “Sentence frames” or “word moves”. These are a set of certain phrases students are encouraged to use when engaging in dialogue, such as, “One point that was not clear to me was ___”, “I see your point but what about ___”, “I’m still not convinced that ___”.

The discussion was in the context of procedures used in conducting student-centered classes. I didn’t know how much about balance they had discussed, and although it is not my habit to interrupt a discussion, I did inject myself using the following sentence frame: “So what do you think is the balance between teacher-centered and student-centered instruction?”

The responses I received were immediate:

“Oh, I just talk at the students forever and go on and on,” said a youngish woman. Another teacher chimed in, “Yes, I tell them that it would be so much easier for them if they just listened…”  This went on for another few seconds, and though I was tempted to use a sentence frame like, “I see your point but what about___?” the one I chose was a bit more aggressive.  “Is that your answer to my question?” I asked. “You think a teacher-centered classroom is all about lecturing with no room for questions or dialogue?”

The woman who first answered me said, “No, I was just being funny.” The conversation turned serious once again with the answer to my question being that the teacher-centered portion of a student-centered classroom is, “teaching the students to be student-centered successfully.” That, roughly translated, means giving them instructions and guidance to do their student-centered inquiry-based assignment.

Example: “In ten minutes, you will complete an outline of what you are going to investigate. Go.”  Ten minutes pass, teacher spot checks various outlines. “Now one person will be the lead investigator, another will be the note-taker, the third person will write the conclusion and the fourth person will do the presentation.”  And so on.

The conversation turned to “student outcomes” and “growth-mindset.” This last phrase, a concept made popular by Carol Dweck, is the theory that students can develop their abilities by believing that they can do so. The term has taken hold as its own motivational poster in classrooms, professional development seminars and Ed Camps across America.  Someone remarked that the idea of growth mindset itself is a student-centered concept. I suppose it is, if you combine belief in yourself with hard work, instruction, and practice—things I don’t hear much about when I hear about growth-mindset.

“Growth-mindset” led into students’ beliefs in themselves, which led to how grades are bad and rubrics were better. A middle school social studies teacher lamented that he was stuck giving students grades because the school district required them, though most of the teachers in his schools used rubrics not to grade, but to provide feedback to students.  (The charter school at which this Ed Camp was held did not give grades, but rather student reports. After the social studies teacher’s lamentation about grades, one teacher who taught at this school cackled “I’m so glad I don’t have to keep grade books anymore!”)

The social studies teacher said that what used to be an A under the old grading system was now a C in his class using his rubric. He didn’t go into details about his rubric except to say that he bases grades on it, and “meets expectations” would be a C.  “I tell parents that I have no problem with a student who gets a C in my class, because that means he or she was meeting expectations. If a student wants better than a C, they can go over the rubric with me to see what is required.”

This struck me as strange. If you give tests and assignments that cover the material and take some effort to do well on them, then maintaining an average of a 90% or more would assure some mastery of the material.  Or does he consider that to be “middle school stuff” and to get an A under his rubric now requires—what?  I never found out. Classes I’ve seen that use rubrics have several: rubrics for group work, presentations, collaboration, essay analysis, presentations and so forth, and there are many categories – like this one for a project presentation in a middle school social studies class . How does one differentiate between “strong student creativity” and “exceptional degree of student creativity” under the “Originality” category? I suspect it’s a matter of “I’ll know it when I see it”.

As time grew shorter, discussions cascaded onto each other, culminating in a discussion about homework. The social studies teacher said he didn’t assign homework, and this turned out to be the practice of most of the teachers in the room. Some of the teachers did report that they received pressure from parents about lack of homework. Parents who ask their kids what they do in school and get the usual “Not much” often follow with “Well, what’s your homework?” and were dismayed to find that the student had none. Parents confronted various teachers, arguing that not assigning homework will not prepare students for the real world. The social studies teacher who was emerging as de facto opinion leader for the session said that in the real world you didn’t have homework, so why should we expect it of our students? This was a bit confusing given that teachers do a lot of work at home. In fact, in many professions it is not unusual to have to do work at home.

But he went on. “And if the real world is high school and college, first of all, not all students go to college. And show me the evidence that homework in high school prepares them for college.” This is the type of argument that seems beguiling if you practice saying it in front of a mirror with an audience applause track playing in your mind. Or alternatively, saying it at Ed Camp sessions like these.

“It is not preparation for the real world,” he repeated, and then clarified that he viewed homework as largely drill and practice activities which in his view held absolutely no value, and certainly, in his opinion, is not something done in the real world. (I should note that I was the only math teacher in this session, but I decided to keep quiet given the reaction when I asked my question at the beginning of the session.)

With parents spotlighted as detractors from how teachers conducted their student-centered classrooms, the session ended with one teacher lamenting how one parent complained that, “This education of my child is becoming my job.”  The teachers all identified with having heard that before. “Gee, sorry to hear that being a parent is so tough” was the general response in the room.

Having been in the position of a parent raising a daughter subjected to student-centered classrooms, I think what that parent meant was not so much, “Why should I be involved in my child’s education?” but rather: “I’m doing a lot of teaching at home that should be going on in the school.” Many parents have complained that students are not being taught grammar, math facts, and other necessities of education, but which teachers of student-centered classrooms consider “drill and kill” and “drudge work.”  That may account for the popularity of learning centers like Sylvan, Huntington and Kumon, which all focus on these things.

The Group-Think of Teaching

Driving home from the Ed Camp, I was reminded of a movie I saw long ago called “The Wicker Man,” in which a deeply Christian, Scottish police officer investigates a missing child on an island in Scotland that practices paganism—and in the end is burnt to death as a human sacrifice to the islanders’ gods. A key point of the film was that the officer’s religion counted for nothing in the midst of different and prevailing beliefs. The winners in such conflicts are those who by virtue of numbers have the means to enforce their beliefs.

I wondered whether in ten years’ time more parents would accept the inquiry-based and student-centered approach more readily as a result of having been subjected to such techniques themselves? Or would there now be a permanent split: parents who came through the system who are happy with their kids being taught as they had been, and parents who had benefitted from the more traditional techniques used in learning centers or from the dwindling number of schools who practiced them?  Would the ideas and techniques discussed at Ed Camp be viewed as outmoded, just as “Think-Pair-Share,” so popular a few years ago, had fallen out of favor? Or would they be replaced by a slight variation of the same thing? Whatever the outcome, it was fairly clear to me that any new educational techniques would be portrayed as a measured and informed decision, a step in the right direction and, of course, progress

Traditional Math (3): Scope and Sequence of Topics for Seventh Grade Math

This is part 3 of a series that will eventually be a book by the same name as this blog. While this is labeled Part 3, it will likely occur much later in the book, after presentation of topics for lower grades (K-6).

The math textbooks in use today have a dearth of good explanations, as well as word problems. Examples usually take the place of any kind of extended explanation, and while explicit instruction relies on worked examples, a textbook should have some additional explanation to go along with them. Indeed, the explanations provided in the teachers’ manuals for these books are instructive and should be included in the students’ textbooks as well.  In addition, today’s textbooks generally include too many topics in one lesson, and the problems at the end of the lesson sometimes go beyond what was discussed in the lesson itself.

Compounding these difficulties are that many of these books tout a “balanced approach”. This usually means that preceding the lessons so described above there is an activity or “inquiry” for students in which they are to discover the general principles behind whatever the topic might be. For example, preceding a lesson on multiplying negative numbers, there may be thought problems on how much weight you would lose if you lost four pounds per week for five weeks, leading to the multiplication of -4 by 5, resulting in a loss of twenty pounds, or -20. The next day, they then have the more straightforward lesson made up mostly of more examples with the rules of multiplication of negative numbers stated formally.

While it is sometimes worthwhile to set aside a day to front load key concepts prior to a lesson, many times these inquiries can and should be incorporated in the main lesson. In a later chapter I will discuss the best way to proceed through such “balanced approach” textbooks.  This chapter focuses on the sequence of topics that I and other teachers with whom I’ve worked believe make the most mathematical and logical sense.

This chapter also discusses textbooks and materials with which to supplement the textbooks that schools have adopted. These additional resources can be obtained from the internet and provide a source of better explanations as well as problems.

What order/sequence makes the most sense?

In one school where I taught, a sixth grade teacher who was hired at the same time as I noted that the math textbooks start off with ratios and proportions. She felt that this might be confusing to students. I noted that the seventh grade textbook did the same thing, but that I had reordered the sequence of topics to make more sense. I felt that starting with integers, and how to operate with negative numbers and then going into rational numbers would be a better way to start. We both agreed that students need some grounding in fractions to better understand ratios.

I’ve noticed many math books for seventh grade math begin with ratios and proportions including the US edition of JUMP Math. I believe this order is followed because it is also the order of topics in the Common Core Math Standards for that grade. There is nothing in the standards that prescribes that order, and I feel certain that teachers will not be terminated for using a different sequence than what appears in the Common Core standards, or in most textbooks that are aligned with the Common Core.

Similarly, I feel that an introduction to algebraic expressions and equations is necessary prior to ratios and proportions. Since ratios and proportions are one of the largest topics in seventh grade math, students need the appropriate background and tools with which to understand the concepts and operate with them to solve problems.

Scope and Sequence for Math 7

What follows is the scope and sequence for a regular seventh grade math course (Math 7), and an accelerated one. For the latter, the additional topics are introduced in a regular eighth grade math course (i.e, Math 8, not Algebra 1).

Key topics for which this series provides descriptions of their presentation in the traditional manner are: Integers, Rational Numbers, Expressions and Equations, Inequalities, Ratios and Proportions, and Percents

I. Integers

A. Integers and absolute value

B. Negative numbers (Adding, subtracting, multiplying and dividing)

II. Rational numbers

A. Rational numbers (general definition; ordering)

B. Adding rational numbers

C. Subtracting rational numbers

D. Multiplying and dividing rational numbers

III. Expressions and Equations

A. Algebraic expressions

B. Adding and subtracting linear expressions

C. Solving equations using addition or subtraction

D. Solving equations using multiplication or division

E. Solving two-step equations

F. Multi-step equations (as necessary; recommended for accelerated classes)

G. Word problems

IV. Inequalities

A. Writing and graphing inequalities

B. Solving inequalities using addition or subtraction

C. Solving inequalities using multiplication or division

D. Solving two-step inequalities

E. Word problems

V. Ratios and proportions

A. Ratios and rates

B. Proportions

C. Writing proportions

D. Solving proportions

E. Slope

F. Direct variation

VI.  Percents

A. Percents and decimals

B. Comparing and ordering fractions, decimals, and percents

C. The percent proportion

D. The percent equation

E. Percents of change: increase and decrease

F. Discounts and markups

G. Simple interest

H. Compound interest (for accelerated classes)

VII. Constructions and scale drawings

A. Adjacent and vertical angles

B. Complementary and supplementary angles

C. Triangles

D. Quadrilaterals

E. Scale drawings

VIII. Area

A. Review of area of triangles and quadrilaterals

B. Circles and circumference

C. Perimeters of composite figures

D. Areas of circles

E. Areas of composite figures

IX. Surface area and volume

A. Surface areas of prisms

B. Surface areas of pyramids

C. Surface areas of cylinders

D. Volumes of prisms

E. Volumes of pyramids

X. Probability and Statistics

A. Outcomes and events

B. Probability

C. Experimental and theoretical probability

D. Permutations

E. Combinations (for accelerated classes)

F. Compound events

E. Independent and dependent events

F. Samples and populations

G. Mean, Median and Mode

H. Comparing populations

Extra topics for accelerated classes (these are covered in Math 8)

XI.  Transformation (Translations, Reflections, Rotations)

XII.  Graphing and writing linear equations

XIII. Real Numbers and the Pythagorean Theorem

XIV. Volumes of Cylinders, Cones and Spheres

XV. Exponents

Additional Resources with Which to Supplement Textbooks

JUMP Math

Singapore’s Primary Math Series (U.S. edition)

Pre-Algebra: An Accelerated Course (Dolciani, Sorgenfrey, Graham); Houghton Mifflin Company. 1988.

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”.

Prevention Equals Treatment, Dept.

NOTE: For those interested in math education issues, the Science of Math organization has been formed to do for math education what valid research has done for the science of reading. Consider becoming an affiliate of Science of Math to support the efforts of this organization.

A recent paper by Sweller discusses how and why inquiry based approaches harm student learning and is largely ineffective. This is not the first paper that Sweller has written about the subject (not to mention those written by Kirschner), but it may be the most definitive so far.  The paper has sparked interesting observations. In particular, I was intrigued by a friend’s remarks that the prevailing mindset in schools and districts is that only students with learning disabilities need direct instruction and that it functions primarily as a student services support.

In math education circles direct (or explicit) instruction has been painted as the prevalent currency of traditionally taught math. And those who seek to reform math education tend to deride and mischaracterize traditionally taught math as  1) consisting solely of direct/explicit instruction with no engaging questions or challenging problems, 2) focusing on rote memorization and no conceptual understanding, and 3) failing to teach math in any complexity. In fact, traditionally taught math employs some inquiry based approaches, while reform math teaching generally relies more on discovery/inquiry approaches than direct. (A paper by Anna Stokke (2015) addresses what an appropriate balance of direct and inquiry-based instruction should be and states: “One way to redress the balance between instructional techniques that are effective and those that are less so would be to follow an 80/20 rule whereby at least 80 percent of instructional time is devoted to direct instructional techniques and 20 percent of instructional time (at most) favours discovery-based techniques.” This was corroborated in a paper by Adam Jang-Jones (2019) which quantified the “sweet spot” between inquiry and discovery based approaches.)

Reform approaches in math in the lower grades (K to 6) have steadily grown over the past three decades. I had long wondered whether some diagnoses of math learning disabilities (MD) of students were in fact incidents of low achievement (LA) due to lack of access to effective instruction. In other words, did the inadequacies of reform math mimic cognitive deficits?

I was therefore surprised and delighted to learn, when I took an Introduction to Special Ed class in ed school, that there was no beating about the bush when talking about students with learning disabilities and other disorders. We learned that students with learning disabilities have been shown to overcome their learning difficulties when given explicit instruction along with other methods. This was mentioned in our textbook (Rosenberg, et al., 2008) and often repeated by our teacher—a tremendously kind teacher named Carmen. (Interestingly, this course was one that was not required as part of my certificate program.)

Of particular interest in this course was the topic of “Response to Intervention” (RtI). RtI is a procedure in which struggling students are pulled out of class and given alternative instruction which includes direct instruction and other evidence-based approaches supported by randomized control trial studies. If they improve under RtI, then the student is presumed to not have a learning disability and is returned to the normal class. If they do not improve, that is an indication that they have an underlying learning disability. (The procedure was established under the Individuals with Disabilities Education Improvement Act (IDEIA) passed in 2004).

I recall the discussion we had about RtI in which I posed a hypothetical to the teacher. “Suppose someone is doing poorly in a math class that relies on an inquiry-based math program,” I said. “And they pull the student out and give him RtI using direct instruction and other techniques, and this student does well.  What happens next?”

“Then the student is placed back to the class during math.”

“But then suppose the student does poorly again? Wouldn’t that indicate that he needs more direct instruction rather than inquiry based approaches?”

“It doesn’t work that way,” Carmen said.

“So he’s stuck with a program in which he doesn’t do well.”

“Right,” she said.  “But if he did poorly in RtI, then that would be evidence he has a learning disability.”

What Carmen was telling me was the Catch-22 aspect of special ed. That is, in schools that rely on programs that follow math reform principles, approaches used in traditional math teaching are generally not an option unless a student qualifies as being learning disabled. And if under RtI, a student does well with direct instruction this is taken as evidence that the student does not have a learning disability.

I suspect that the use of RtI is higher in schools that rely on reform-based programs. I would like to see research conducted to see if that is true. From where I and many teachers and parents sit, the effective treatment for many students who are LA, is also the effective preventative measure.

Based on conversations I’ve had with education professors, I believe the educational establishment will likely continue to resist recognizing the merits of traditional math teaching and direct instruction. The following statement from James McLesky (2015), one of the authors of the textbook we used in the special ed class and a professor at University of Florida’s College of Education, is typical of what I’ve been told:

If we provide only (or mostly) skills and drills for students with disabilities, or those who are at risk for having disabilities, this is certainly not sufficient. Students need to also have access to a rich curriculum which motivates them to learn reading, math, or whatever the content may be, in all of its complexity. Thus, a blend of systematic, direct instruction and high quality core instruction in the general education classroom seems to be what most students need and benefit from. 

Statements such as these imply that students who respond to a diet of more direct instruction constitute a group who may simply learn better on a superficial level. I fear that RtI will evolve to incorporate some of the pedagogical features of reform math that has resulted in the use of RtI in the first place.

I am hoping that the publication of Sweller’s latest paper and the reaction to it that I’ve seen so far, will result in an increasing recognition of the benefits of direct instruction specifically and traditional instruction generally, as well as the harm that can result from inquiry-based approaches. Unless and until this happens, the group-think of the well-intentioned educational establishment will prevail. Parents and professionals who benefitted from traditional teaching techniques and environments will remain on the outside — and the methods that can do the most good will continue to hide in plain sight.

Reference:

McLesky, James. (2015) Private email to Barry Garelick. November.

Rosenberg, Michael; D. L. Westling, J. McLesky (2008). Special Education for Today’s Teachers: An Introduction. Pearson. New York.

Doing it wrong, Dept.

Talking about teaching math opens one up to choruses of “You’re doing it all wrong” among those who have been indoctrinated into the various catechisms of math education. One of those is “Never tell a student they made a mistake”. I guess this is because it affects their confidence and self-esteem and therefore is anti-growth-mindset. (On the other hand, we have Jo Boaler telling us that students should be encouraged to make mistakes because it makes their brains grow.)

I have no problem telling a student they made a mistake, though I do it by saying “That’s not what I got. Anyone else get an answer?” When many students make a mistake I capitalize on this and say “OK, so far we have …. ” (I rattle off the various answers), and then many hands go up among those who want to be part of what is now perceived as a fun game. But if only one person makes the mistake, I’ll try to see if they know what they did wrong. Sometimes the student knows; other times, I might know and I’ll give my opinion. And still other times we don’t know, but I’ll give another similar problem and the student who made the mistake usually will try again. At least that’s been my experience. But to make the math ed progressives reading this entry feel better, I’m no doubt doing something wrong.

For eighth graders, it’s a little trickier, because they are very self-aware at this stage of their lives and can be very guarded. Some teachers use mini-whiteboards on which students write the answer and hold up the boards for the teacher to then say “Right, right, nope–try again, …” etc. I do a variation of this. I don’t use mini-whiteboards. Instead, I’ll tell them to do the problem in their notebooks, and then I go around. If someone has the wrong answer and they write out their steps, I can point out the mistake, and they can then re-do it. For those who get it right, I’ll tell them so. If the person who got it wrong initially then gets the right answer, I call on that person to tell the answer to the class. In this way, the person is not singled out for making a mistake, and they feel confident in giving the answer to the class, knowing it’s correct and not fearing the teacher saying that it’s wrong in front of their peers.

But when time pressure is an issue, you sometimes have no choice but to tell someone they are wrong. I make note of those who are not getting it, and during the time that I allot for students to start on their homework (a term which has now morphed into “practice problems”–I guess “homework” is too risky a word in view of self-esteem and growth mindset fantasies) I spend the most time working with them.

For those students in eighth grade who really should not be in such class but who are placed there because of parents’ insistence, there are a number of options I exercise. I may recommend to the parents that they hire a tutor. Another alternative (which may occur even if the student has a tutor) is to recommend that the student repeat algebra in 9th grade. Some go along with this, but others do not.

If these ideas are offensive to some of you, please realize that I wear my shirts tucked in, avoid Apple products, and use a point-and-shoot digital camera rather than take pictures with my cell phone. I am semi-anachronistic and am determined to stay that way. It’s only a matter of time before my out-of-date habits become the latest fad. By that time, I’ll likely be dead, in which case they’ll probably name a brick-and-mortar bookstore after me.

The Math Wars Continue, Dept.

I always get a kick (as well as a wave of nausea) when I hear arguments about how math should be taught referred to as “math wars stuff”. Such criticism implies that the we are long past the math wars and that they were just trivial spats that signified nothing. In a communication I once had with Jay Mathews–who for many years has spewed his arrogant views of education in a column he writes for the Washington Post–he said that the math wars were two groups of smart people calling each other names.

I won’t comment on the word “smart” here, other than to say it’s overused to the extent that it means nothing, and has become a code word for edu-pundits who compliment each other by saying so and so “wrote a smart and thoughtful post” about whatever.

Well, I had the opportunity to write a “smart and thoughtful post” on math education, courtesy of Rick Hess who invited me to do so. It was published first at Ed Week, then at AEI, and finally at Education Next’s blog. While it has proved a popular piece, there was a recent take-down of it, also published at Education Next’s blog. The author works for TERC which publishes Investigations in Number, Data and Space, which in my opinion and others whose opinions I respect, is one of the worst of the NSF-sponsored atrocities.

I was about to defend my stance, when to my great relief, Sanjoy Mahajan, a research associate in mathematics at MIT did the heavy lifting for me on Twitter, reproduced below:

  1. There’s so much to say about that clever nonsense. There’s the straw man of “practicing procedures alone” bringing understanding. But Mighton’s new book _All Things Being Equal_, pp. 98-102, has a great treatment of the long-division procedure with understanding.
  2. There’s the sneaking in of “when division applies in solving real-world problems” onto (into?) the list of concepts underlying the long-division ALGORITHM. Sure, it belongs on the list underlying division — but not underlying the algorithm.
  3. After these concepts, mostly valid, comes a call to develop a “multi-faceted view of division.” But I want students to understand not all ways that one could divide but rather the long-division algorithm.
  4. And the method offered will not help: the “rich task” of justifying the algorithm for “any two RATIONAL NUMBERS.”  That choice is either sloppy or insane. I have never used the long-division algorithm for arbitrary fractions, only decimal numbers.
  5. About the incessant calls for “authentic mathematics.” It’s rich coming from educators whose favorite incantation for stopping any rigorous teaching, e.g. long multiplication, is “development [un]readiness.”
  6. Proving this conjecture is not at all authentic mathematics. The main effort of mathematicians, not evident from the format of journal articles, is making the conjectures.
  7. Finally: Even if it were authentic, where’s the argument to show that acting with the outer forms, but without the inner knowledge, of mathematicians makes you understand math like mathematicians? It’s cargo-cult thinking.

I hope you enjoyed this foray into the supposedly defunct math wars.

Word problem, Dept.

The following word problem appeared in a traditional algebra textbook around. I will identify the book later. Right now I want to know your opinion of the problem, good or bad. If you dislike it, please specify why. Same if you like it.

Thanks.

Two boys are camped at a spot where a river enters a big lake.  One boy is injured so severely that every minute counts.  His companion can use an outboard motorboat to get a doctor by going 3 miles down the lake and back, or by going 3 miles up the river and back.  Show that even though the boy does not know the speed of either the boat or the current, he should choose the lake.

I must be doing something wrong, Dept.

Reading comments about teaching on social media is similar to reading about various maladies and diseases in a medical book. You come away thinking you have every one of the illnesses. Similarly, the pithy do’s and don’t’s from various authorities, distilled to one sentence leave me (and others) with the feeling of “Guilty as charged; I must change how I do things.”

Recently, a well known blogger and author quoted another well known figure among the edu-literati Katherine Birbalsingh who is headmistress of the Michaela School in London. The school is well known for getting good results for its students, many of whom come from low-income families and who would otherwise do poorly in other schools.

So of course I took it to heart when I saw her quote:

Always judge yourself by the bottom 5 kids in the class, not the top 5

This is the type of quote which, if you question it, makes you look like a jerk. So I’ll go out on a limb here and take a chance. Many people think I’m a jerk anyway, so I have little to lose.

Snippets like this don’t tell the whole picture/ I’ve taught classes with math deficits and severe immaturity levels. My current Math 7 class is one. Now among my bottom five, one boy took off Friday and Monday (with no advance notice to the school) with two other boys in the class, with his father to go to some cabin. Neither he nor his friends were up on what we were doing in class, despite a review when they came back. They did poorly on the quiz I gave that week.

I’m a firm believer in the parable of the lost sheep–do you tend the 99 who are obeying or go after the one who goes astray? I try my best, and sometimes that’s all you can do. On the other hand, a girl with significant deficits, has done quite well in my class. Another girl who is doing poorly because she cannot remember procedures (not even plays in basketball) continues to do poorly. Her mother refuses to have her tested for fear of her being segregated and put in lower level classes. (I sympathize; even if she gets special ed assistance, there is little to no chance that anyone with expertise in learning disabilities will provide help, other than to provide more time on tests, and other things that don’t address the underlying problem).

As my “parole officer” Diane said (in last chapter of my latest book), when I remarked that you can lead a horse to water: “You dragged them, kicking and screaming.” And sometimes that’s the real truth despite what the glorified experts on social media say.

Regarding the pithy quote from Ms Birbalsingh; to use the parlance of Twitter: “It lacks nuance.”

Formative Assessments Dept.

(A preview/snippet from my book “Out on Good Behavior: Teaching math while looking over your shoulder”)

Some say summative assessments can be used formatively, by using the results to guide approaches in subsequent courses. The overlapping nature of how these definitions have evolved give me much cover in my quest to appear aligned with the edu-party line. During my first year at Cypress, I allowed my classes to use notes for quizzes, but not tests. I felt that this would reinforce the idea of the value of notes. The problem was that some students’ organizational skills were lacking—resulting in this typical conversation:

Student: How do you do this problem?

Me: Look in your notes.

Student: I can’t find it.

Me: (Drawing a diagram on a mini-white board.) How would you find the time each of the cars are driving?

Student: I don’t know.

Me: (Writing “Distance = Rate x Time” underneath the diagram)

Student: Oh!

I knew that there was a potential that such approach could quickly blossom into grade inflation and an artificial sense of achievement. So I justified my giving them help by telling myself “Well, I guess this is a formative assessment and I’m using the results to guide future instruction.” But I knew there were limits.