Category Archives: The Paradigm

Kicking the Textbook Habit

Textbook FreeI have had several inquiries about an article I wrote many years ago titled, Textbook Free: Kicking the Habit. I am not surprised, because, in these days of Common Core roll-outs with few valid materials, teachers are having to create and find their own curricula. While the article is over a dozen years old, it could not be more timely, so I thought I would make it available again. I hope this helps encourage teachers that using textbooks as a resource instead of as scripture in the era of the New Curriculum can be easy and fun.

Textbook Free: Kicking the Habit

Originally printed in The Math Projects Journal in May 2001:

I kicked the habit! I am no longer a textbook junkie. I no longer rely on my daily fix of some publisher’s bloated curriculum. I am free of my addiction without the help of an arm patch, rehabilitation clinic or twelve-step program. I quit cold turkey. Here’s how.

At my school, the students are issued a math book that they leave at home and each teacher is issued a class set. I usually keep one underneath each desk. This year, however, the librarian informed me on the first day of school that we were out of Geometry textbooks. Our student population had grown so large that our library ran short. In fact, for two to three weeks many of my students would not have a book at home either. There was talk of teachers sharing class sets and photocopying pages for students. I decided to try a different strategy. I took this as a professional challenge to see how long I could teach without a textbook. I knew whatever happened would be a growing experience for me as well as my students.

Well, by no fault of the school library, two to three weeks stretched to seven. By that time, I was well into my “textbook free” strategy, so I just kept the ball rolling…for the rest of the year. I used only 12 assignments from the textbook in those 180 days. Here is how that unique experience of being textbook free has changed my teaching, forever.

Firstly, I am now much more focused on standards. Rather than leafing through the textbook, I looked at my state and district standards, and established my curriculum from those. After all, shouldn’t they be determining what we teach? From there, I grouped the topics into units, and then scheduled individual lessons. This process naturally pared down the number of topics that I taught and allowed me to allocate a full week of instruction to each concept, rather than one day to each section of the textbook.

The second big change that has occurred is the structure of my lessons. Everything from my homework to my instruction has radically changed. My typical textbook free lesson was comprised of three to six problems of various difficulty. Oftentimes, I began a lesson with one to three review problems from previously learned material which applied to the current lesson. This is similar to a traditional warm-up with the exceptions that the problems are very relevant to the new lesson, and not simply arbitrary review.

Sometimes, I began with THE big problem from the previous night’s assignment, and solicited student responses. It is not hard to see that my old practice of dedicating 20 minutes of class time to questions on how to complete the previous homework disappeared. The intent of the class slowly evolved from getting the answers correct to understanding the mathematical principles behind the question.

These introductory problems served as a terrific assessment tool, also. Previously, it was difficult to know how well the students were doing when only a handful of them were asking questions from a truck-load of exercises. However, when the whole class was engaged on the same few problems, it was easy to walk the room and evaluate their performance and understanding.

The introductory questions naturally lead to the main problem or small set of problems that would drive the lesson. The students were engaged in an investigation, project or activity relating to the concept. Each day my students came to class to solve problems, rather than take notes — a huge change from all the previous “textbook years.” This process of problem-solving and investigation consumed the full class period. Gone were the days of having the students start homework in class. I taught the entire class period.

The homework assignments were only one to three problems long and were typically extensions of the day’s topic, not just practice exercises. I had learned from the international comparisons that America is one of the few countries that pushes the drill-n-kill regime and yet we are at the bottom of the performance pile. So I tried to limit both the number and size of my assignments, and to make them more challenging and contextual.

By doing that, I firmly settled the argument regarding the quantity and frequency of homework that students need to be successful. For the skeptics that are still reluctant to abandon their practice of assigning 30 homework problems a night, I have some strong evidence. My class averages led the district on the district final. With this in mind, I can at least make a case that this new homework philosophy is not hurting my students in anyway.

Another significant change was my lesson planning. Rather than writing examples of how to complete an algorithm or creating cute acronyms to remember esoteric rules, I actually wrote lesson plans. I started planning each lesson by asking: “What do I want the students to know? What is their common misconception of the topic? How can I best get them to understand the topic? How can I challenge them within the context of the topic?” I would then try to create a story/context/scenario and a small set of problems that would best develop understanding of that topic. It was so much fun. This change in my approach to lesson planning was actually a reflection of my new attitude towards teaching. My job description truly shifted from covering material to uncovering knowledge.

Focused, standards-based curriculum; in-depth, problem-solving instruction; short, conceptually-based homework assignments. This experience was so exhilarating that I am now a junkie all over again. I traded my old addiction to the textbook, for a new one — creative lesson planning. This is one habit, though, that I never intend to kick.

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The 4-1/2 Principles of Quality Math Instruction

Imagine a small box with an American flag painted on each side. This represents the box in which we American teachers think about education … how we view the role of the teacher, the nature of the learner and the purpose of school. I say this, because ongoing international research studies show that teaching is a cultural phenomenon. We do not teach the way we were trained to teach; we teach the way we were taught. While there are differences among us American teachers, there are glaring commonalities that we uniquely share. The same can be said for our counterparts abroad. We could make a similar box and cover it with French flags and have a conversation about how French teachers think about education. We could do the same for the Japanese or any country for that matter. The issue is that while our education system has a few great things to share with world, for the most part, countries which outperform us in academic math tests do so because their box is far superior to ours.

That lesson can be found in the Trends in International Mathematics and Science Study (TIMSS). In March of 1998, The Math Projects Journal was granted an interview with Dr. William Schmidt, the American Coordinator of TIMSS. He claims that among the top-performing countries in mathematics (no, the United States is not one of them) there is no common methodology, but there are common principles of instruction that all the top-performing countries share: teaching to conceptual understanding and teaching with mathematical substance. In his writings and public presentations he stresses two additional components: standards and accountability.

Therefore, I have consolidated these findings into what I have dubbed “The Four and a Half Principles of Quality Math Instruction,” or in the vernacular of the digital age, “Q.M.I. version 4.5.” The first four principles come from the research shown in the international comparisons of the TIMSS report:

1) STANDARDS: Focus on a the limited number of topics that your students need to know; don’t just cover the textbook.
2) CONCEPTS: Teach students to understand what they are doing, not just to mimic what you are doing.
3) SUBSTANCE: Intellectually challenge students; raise your level of questioning.
4) ACCOUNTABILITY: Hold students to knowledge and performance expectations that go beyond grades and unit credit.

The fifth principle comes from professional experience and opinion rather than research, and therefore, its emphasis is demoted to a half-principle.

½) RAPPORT: No philosophy, technique, methodology, instructional material or textbook can replace the student-teacher relationship. You must reach ‘em before you teach ‘em.

Since I don’t have a PhD after my name, I found backup from someone who does. I was reading a book titled Six Easy Pieces by Dr. Richard Feynman. It caught my eye because the subtitle of the book was, “Physics Taught by its Most Brilliant Teacher.”  The preface of Six Easy Pieces is full of insights into the teaching philosophies and methods of one of the finest teachers of arguably the most difficult subject in contemporary academia. Here are some quotes by Dr. Feynman regarding teaching:

Dr. Feynman on Standards:

First figure out why you want the students to learn the subject and what you want them to know, and the method will result more or less by common sense.

Dr. Feynman on Concepts:

I wanted to take care of the fellow who cannot be expected to learn most of the material in the lecture at all. I wanted there to be at least a central core or backbone of material which he could get…the central and most direct features.

Dr. Feynman on Rapport:

The best teaching can be done only when there is a direct individual relationship between a student and a good teacher — a situation in which the student discusses the ideas, thinks about the things, and talks about the things. It’s impossible to learn very much by simply sitting in a lecture, or even by simply doing problems that are assigned.

So where is “the use of math projects” in the list? Math projects are not on the list, because in-and-of themselves they are not critical to quality math instruction. Projects are effective tools of instruction only when they embody these four and a half basic principles of teaching discussed herein — in particular, teaching to conceptual understanding and with mathematical substance. To gain further verification of the potential effectiveness of math projects, though, I once again call upon Dr. Feynman.

Dr. Feynman on Projects:

I think one way we could help the students more would be by putting more hard work into developing a set of problems which would elucidate some of the ideas in the lectures. Problems give a good opportunity to fill out the material of the lectures and make more realistic, more complete, and more settled in the mind the ideas that have been exposed.

Thank you, Dr. Feynman, for the encouragement to keep creating and implementing quality math lessons and problems, and to persist in developing good student-teacher relationships. Thank you, Dr. Schmidt, for revealing to us the value in teaching to conceptual understanding and with mathematical substance, and for pressing us to see the need for standards and accountability. May we all keep in mind the most important lesson offered by these studies: The greatest determining factor in the quality of the education that a student receives is the decisions that a teacher makes on a daily basis!

Ultimate Cosmic Power in an Itty-Bitty Thinking Space

“Give me any combination of two numbers that have a sum of seven,” I said to my students. One person offered, “Two, five,” which I wrote on the board as (2, 5). I asked for a few more and got (5, 2), (1, 6) and (0, 7).

“Good,” I praised, “now give me ALL the combinations of two numbers that have a sum of seven.” They chuckled. “I want them all, and I want you to write them down.” The students were hesitant, because they knew there are an infinite number of pairs that have a sum of seven. So I challenged one of them to a race. “You write them down on your paper, I’ll write them on the board. Nobody goes to lunch, until one of us is done. Ready, Go!” I scribbled on the board x + y = 7. “Done!”

They didn’t buy it. “I have just written ALL the combinations of two numbers that have a sum of seven. Since you don’t believe me, I’ll do it a different way. In fact, I’ll take you all on. All of you write down combinations of numbers, that way you get done in one-thirtieth of the time, and I’ll still woop ya. Ready, Go!” I quickly sketched the graph of x + y = 7. “Done!”

This goofy little exercise was intended to impart the idea that mathematics gives us the ability to represent an infinite number of elements in a finite time with little effort. I spread my arms wide in front of the class and exclaimed “Ultimate cosmic power…” then brought my hands to rest on a student’s head and continued, “…in an itty-bitty thinking space.” (A play on the Genie’s words from the Disney movie Aladdin.) No offense to the student, but our brains are not very big. Yet, we were able to take all the pairs of numbers whose sum is seven, shove them all in our heads at once and think about them all at the same time. The ability to then communicate them to the world outside of our heads using equations and graphs is what makes mathematics the Ultimate Cosmic Power.

However, most people don’t share in our awe of this power. I believe that is because we never initiate them into our mathematical club. We keep students on the other side of the room while they watch us speak club code and give the secret club handshake, but we never let them in. I have proof of this …

Thoughts on Math by the Uninitiated

From an Algebra 2 student who was just kidding, but his joke reflects how many people perceive the purpose of math:

 Algebra would be a lot easier if they just told you what x was.
– Scott, Class of ’94

For a moment, I thought this next one was kind of cute when a student had just simplified 3x + 2x to equal 5x:

Only in math do you put two things together and get a smaller thing.
– Neal, Class of ’99

Then I realized … In math we don’t combine to make a smaller thing; we combine to make an equal thing!

Then there was the English Teacher who stopped me in the hallway one day, visibly irritated, and poking me in the chest:

You math teachers aren’t very good. My whole life you have been asking people to find x. Why can’t you find it yourselves?

I don’t know what set her off on that day, but I do know one thing about the three people who made the statements above. They all see the sole purpose of math as a cognitive Easter egg hunt. They close their eyes, while the teacher hides the variable. Ready, set, go. Praises and smiles when the basket is full.

They have no appreciation for the true beauty and power of Algebra, because we never initiated them into the club. So how do we teach them the secret handshake?

The Initiation Rites

  • Context
  • Multiple Representations
  • Complexity
  • Abstract Generalizations

Context

Too often we jump straight to the naked math problem. For example,  we ask students to graph y = 2x + 1 or evaluate 2 – 5, without offering any kind of context. Context gives the students something to cognitively hold onto while they are grappling with the math concepts. Take for instance, the teaching of negative integers. In the Wallflowers lesson, students are asked to mathematically represent scenarios that they can relate to (a high school dance). While these scenarios are a bit contrived (girls are positive and boys are negative), students can “see” how a balance of positives and negatives equals zero, taking away negatives leaves behind positives, etc. From here it is useful to go to other contexts that are more applicable. The Postman Always Ring Twice relates positive values to checks and negative numbers to bills, showing how truly “subtracting a negative is the same as adding a positive.” By presenting the context first and then asking the students to represent it symbolically, we give them a framework in which to think when eventually presented with the naked math problem.

Multiple Representations

Thoughts on Math by One of the Initiated

I once met a Calculus teacher in Massachusetts who was originally from India. She was very distraught about teaching in America, She said that she kept getting complaints from both students and parents about her teaching style. She said that all she was doing was teaching the way they do in India. When I asked her to characterize the style for me, she said that in India there is a saying:

If you know how to do one problem inside and out, you can do a hundred just like it.
– Seheti, Math Teacher from India

I could easily see the contrast. In India they teach students a hundred ways to do one problem. In America we teach one way to do a hundred problems.

To further make Seheti’s point, I had a conversation with my daughter’s third grade teacher on this very point. She asked me:

Thoughts on Math by the Uninitiated

How would you do this problem?
              3,165
             -2,987

Of course, she gave me this example because it requires multiple borrowing in a traditional algorithm. I told her, however, that I don’t see it as a subtraction problem; I see it as an addition problem. I have 165 above 3,000 and 13 below 3, 000, therefore, 165 plus 10 plus 3 was 178 … with no borrowing. Her response was, “But why would you go through all that trouble?” I chuckled at the unintentional irony, placed the pencil on the paper and challenged, ” Do your method without picking up the pencil.”

These examples show the strength in teaching students Multiple Representations of a problem, and the weakness of teaching only one method. In the Candy Bars lesson, multi-link cubes are used to demonstrate why we need a common denominator when adding fractions. Fractions, after all, are merely relations to the whole. Operations on fractions can only be done then on the same size whole!

When students are first asked to show one-half of a candy bar with the cubes they put together two cubes of different colors. When asked to “build” two-thirds of a candy bar with the cubes, the construct a stick of three cubes, one being of a different color. When prompted to “show” what fraction of the candy bar they have if they are given portions, they naturally connect the two to make one, which shows two-fifths, which is exactly how they incorrectly complete the algorithm for adding fractions: they add both numerators and denominators.

When corrected and told that the bars must be the same size and still show one-half and two-thirds, the students independently build sticks of 6 cubes each (three of one are colored, two of the other are colored). When then pressed to now tell us how much “of the same size candy bar they now have, they combine the colored cubes, but keep the stick the same 6-cube length and present … five-sixths, which is correct. Publically generating an algorithm that now represents what we do is easy, and the cubes offer a model for students to fall back on in the event that they forget the procedure.

Complexity

Too often we think that our job is to always make math simple for our students. This initiative then leads us to break problems down to their tiny, separate parts, and we never ask the students to put it all together. This point was made by Tad Watanabe in NCTM’s Mathematics Teacher (Vol 93, No 1, p 31). Mr. Watanabe showed a high school entrance exam from Japan. It had only 7 problems. One of them is displayed on the right. (Click to enlarge the image.)

Take some time to do the above problem. It puts to shame what we expect from our 8th graders: “There are 10 marbles in the box. 3 are red. What is the probability of drawing a red marble?” The Japanese expect there students to do complex problems, therefore, they teach them to do complex problems. We Americans often feel that we have failed if we pose students with difficult problems. One would have to look long and far to find an Algebra final exam in the United States with the level of complexity of the Japanese example above. It is not that Japanese students are more capable than American students and therefore, they can do these kinds of problems. The Japanese students are more capable because they are regularly asked to do these kind of problems!

To further the point on Complexity, I would like to share the story of the American math teacher who visited schools in Bulgaria. When asked to contrast math education between Europe and the U.S., he said he could do so by showing a typical question from both countries:

Typical Geometry Question in Bulgaria: Draw a triangle. Draw a semicircle on each side. Within each semicircle , inscribe the rectangle of the greatest area. Draw the lines that pass through the centers perpendicular to the side of the triangle. Prove that these lines are concurrent.

Typical Geometry Question in America: Draw a triangle.

Abstract Generalizations

Finally, the ultimate in Ultimate Cosmic Power: Abstract Generalizations. In other words, students are asked to model their world mathematically. As when I ask my daughter and her friend when they were in third grade, “Your class has 20 students. How many are boys and how many are girls ? How many boys and girls might be in another class of 20 students?” We went through several scenarios, and then I asked, “If we allow b to represent the number of boys, and g to represent the number of girls in a class of 20 students, what would you say about the the number of boys, girls and the total students?” They said “b plus a equals 20.” I then showed how to write b + g = 20, and they agreed. There were several abstract representations going on with the girls. Words and an equation, and a brief encounter with data.

The Rising Water lesson does the same thing, but more formally. It first poses a context in words (one representation): A swimming pool contains water 10 cm deep. The water is rising 3 cm per minute. The students are to then generate a table of values, an equation and a graph for this scenario (3 more representations). The objective of the lesson is to teach students that all four representations describe the same relationship between two quantities (time and water depth). The students are then asked to generate their own scenarios, with the four representations. The more that students are asked to create their own models, the better capable they will be when they are presented with one.

So our Initiation Rites into our math club are these four components of Ultimate Cosmic Power (Context, Multiple Representations, Complexity and Abstract Generalizations). We will go back to these constantly in our discussions, as well as to the Four and a Half Principles of Quality Math Instruction posted previously. To show that there is hope in teaching in this manner, I share with you a statement from a former student at the end of the year.

Thoughts on Math by One of the Initiated

Poetry is the language of love. Math is the language of everything else.
– John, Class of ’99

Q&A: Lessons and Homework in a Activities-Based Classroom

Question:

I really have the desire to turn my classroom into a hands-on, activities-based classroom, but am still struggling with how to cover it all. I guess I am still not sure how to “get away” from the “drill-and-practice” method. I was curious as to how much homework you assign (on average) and whether or not it was from a textbook or more activity-type problems. I was wondering if you would be able to give me a sort of play-by-play of some of the topics you teach. For example, Graphing Linear Equations. What do you do from day to day and do you ever use the textbook and the traditional problems therein?
Rachel Rosales (Owensboro, KY)

Answer:

In Textbook Free: Kicking the Habit, I give a very detailed synopsis of a textbook-free lesson. My math department at Great Oak High School is known for not following the textbook. We still have students read and do problem sets from the book, but our lessons are not bound to it’s pages. I have been teaching like this for over ten years now, and it has been a wonderfully liberating experience for me, and a productive experience for my students. In short, I can share with you the following.

First off, change your focus. American teachers believe that a day in the classroom revolves around the homework assignment rather than around the lesson. Plan a lesson by asking, “What is the main concept that the students need to understand?” This is different from the tradition of asking, “What do they need to know to get through this problem set tonight?” Then find/create a few problems that address the misunderstanding that students have of that concept. The purpose of the lesson should be to get students to understand what they are doing and not just mimic what you are doing.

Regarding assignments though, I consider “homework” assignments different than “project” assignments. Projects are done about once a week in class, for which the students may go home needing to complete or expand upon. Homework is assigned about 2-3 times a week (never on the weekends), and is usually a handful of questions that expand upon the ideas addressed in class. I usually save the practice/drill problems for the warm-ups the following day (these are also few in number). In my lower level classes, I never assign homework. My school district’s final exam results show that my students are not hindered by sparing them the grief of the drill and kill routine. In fact, my school is our district leader in high school state math scores, and we give the least amount of homework.

In regards to Graphing Linear Equations: I want my students to understand the idea of rate (slope), initial value (y-intercept), what a linear relationship is and how to interept the graph of a linear equation. An example of how I might teach this concept is described below: I usually start with discussing the idea of rate. The lesson centers around the Tic-Tac question:

The following equation represents the relationship of the number of Tic-Tacs eaten, t, to the number of kisses received, k, on a date: k = 2t + 1.

The students are required to describe in writing what the equation says about the scenario. (e.g. “You receive two kisses for every one Tic-Tac you eat, plus you get a guaranteed kiss on the door step at the end of the date.”) They are also required to show a table of values and graph of the scenario. Their homework is to take a new table of values and write an equation and graph relating to the table. Then I take them through Jennifer Sawyer’s “Rising Water” project which has the students generate a variety of linear situations and then compare their equations and graphs to reveal the role of the slope and y-intercept. I follow this up with the “Speeding Ticket Investigation” to reinforce graphing. At this point, we take notes. I usually provide a graphing organizer and take a few days to formalize procedures like writing an equation given slope and a point, given two points, writing equations from graphs, etc. I use the lapboards quite a bit here. I finish with the “Jogging Hare,” by going through the lesson as a class with one set of numbers, then the next day giving the same problem with a new set of values as an assessment. (They may use their notes from the previous run through).  If you teach line of best fit, then I would go one step further with “Cool Shoes.” This usually takes about three weeks, and then I generally spend a couple of days reviewing before the test. (4 weeks total).


Chris Shore, Editor