Tag Archives: textbooks

Interview: Dan Meyer on Using a Ladder to Clear a High Bar

Pic MeyerDan Meyer is as close to a celebrity that a math educator can get. We all owe him a debt of gratitude for making math nerds look cool. He deserves his rock star status because he is an amazing presenter, a thought-provoking writer, and an ingenious creator of math tasks.

Behind all the hype, though, is some serious substance. Dan’s ideas are akin to the spirit of MPJ’s lessons in that they seek to engage students in meaningful mathematics, and aspire to teach them high-level cognitive skills. Dan’s methods, though, have a unique twist that challenges teachers’ thinking. I know he has given me a great deal to think on. I hope he does here for you as well.

MPJ
From what we read on your blog, you are about two things: 3-Act Lessons and the Ladder of Abstraction. Let’s start with 3-Act Lessons. Give us the gist of what they are and why they are an effective tool in teaching math.

Dan Meyer
We make huge promises to kids about the power of math in their world. But then we give them these problems that don’t do justice to that power or to the world they live in. Three-act math uses photos and videos to show students a more faithful reproduction of their world and a more faithful reproduction of the practices of applied mathematicians.

MPJ
MPJ has preached for over a decade the need to have students create their own mathematical models (abstract generalizations). Describe your Ladder of Abstraction and how it applies to teaching mathematics.

Dan Meyer
The process of abstraction is extremely powerful and also not something I understood intuitively until I was a long way out of my secondary math education. Basically, whenever we turn the world into a table or an equation or a graph, we LOSE something. People don’t run at a constant rate. The Earth isn’t a perfect sphere. But we abstract a runner into a linear equation and the Earth into a sphere because those abstractions are perfect ENOUGH to help us answer questions. That’s an important part of modeling. Asking, “Is this model perfect enough?”

MPJ
With so many teachers reliant on teaching from the textbook, do you have any ideas on how to get these practices used more regularly in classrooms?

Dan Meyer
I tell teachers what I tell myself: whatever you’re going to teach today, whether it’s pure math or applied math, make sure students have some NEED for it. A better need than “I don’t want to fail this class.” And I offer them techniques for provoking that need. I also offer teachers a homework assignment, an exercise like push ups, to get better at provoking that kind of need: take a photo or a short video and ask people what questions they have about it, if any. If they don’t have any questions, retake the photo or video in a way that provokes more questions. That homework assignment has been incredibly helpful in my own growth.

MPJ
How well do your theories mesh with what is coming down the pike as the Common Core?

Dan Meyer
The modeling practice of the CCSS gets focused treatment in high school. I encourage all of your readers to study high school modeling (it’s only two pages) and ask themselves, “Are the ‘real-world’ problems I assign preparing students to clear this high bar?” Then Google “three-act math” and see if my work can help.

MPJ
What do you intend to prove with your PhD research?

Dan Meyer
I’d like to understand how any or all of this translates to online education.

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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.

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