Tag Archives: pedagogy

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.

Dr. Jon Star Speaks HOT Heresy

Pic StarDr. Jon Star, of Harvard University, gave a mathematically blasphemous speech at the 21st Century Mathematics Conference in Stockholm, Sweden last year. The presentation was titled, Neuroscience and Cognitive Psychology of Mathematics. His heretical statement was that mathematics does not teach higher order thinking skills; only the teaching of problem solving actually teaches problem solving. The Math Projects Journal has always preached the teaching of mathematical substance, or what is now commonly known as higher order thinking skills (HOTS), so we reached out to Dr. Star regarding his research.

The belief that just by learning math one gets critical thinking skills is also not well-supported by evidence.

MPJ
You must know that your claim stating that math does not inherently teach critical thinking is very unnerving to the math education community.

Dr. Star
Just to be clear about my goals in the Stockholm talk, I was trying to argue the following:

First, the belief that math plays some sort of special and relatively unique role (as a discipline) in promoting what you refer to as HOTS (Higher Order Thinking Skills) is not well-supported by evidence.

Second and related, the belief that just by learning/understanding math, one gets critical thinking skills as well (e.g., two for the price of one, without explicit or even implicit attention to developing HOTS), is also not well-supported by evidence. Certainly in some instances this does happen, but it does not appear to happen in any widespread way for ‘typical’ students.

And third, given that we do want students to develop HOTS, rather than expecting/hoping that these just emerge as a natural by-product of learning/understanding math, it is essential that we think about how to explicitly promote critical thinking and problem-solving in what we teach and how we teach math. With respect to this last point, arguably generations of math curriculum and pedagogy reformers have sought this same goal – teaching math such that higher order thinking skills develop. But evidence and intuition suggests that this is very hard to do. But certainly we should continue trying…

MPJ
Is it math, per se, that does not impart the HOTS, or is it the way we teach math that is inept in imparting these skills?

Dr. Star
I would say that both content and pedagogy are important, but it seems that pedagogy plays an especially important role. If we want students to be able to transfer knowledge to domains outside of math class – apply reasoning skills that worked in math class to other kinds of problems – it seems necessary to teach with such transfer goals in mind. There are many different (at times competing) pedagogical visions for how to teach math such that this kind of transfer is possible. Some feel that the best approach is to engage students in certain kinds of reasoning and communication that are believed to facilitate application of knowledge to novel situations, and others feel that a certain amount of practice in applying concepts and skills is necessary for future transfer. I can see potential merit in both of these approaches, although empirically there isn’t a lot of good evidence to point us in the right direction.

I would say that both content and pedagogy are important, but it seems that pedagogy plays an especially important role.

MPJ
The 8 Common Core Standards of Practice imply that habits of mind can be taught. In your view, do these practices have value?

Dr. Star
I think that the Common Core practice standards are admirable goals. However, as noted above, I think we are still struggling to determine the best ways to achieve these goals pedagogically.

MPJ
Anecdotally, educated people think, communicate and behave differently than uneducated people. I believe research bears this out as well. Is this then simply a non-associated correlation (people who already have educated traits get an education), or does a quality education truly transform an individual?

Dr. Star
Certainly some people do develop problem solving skills merely by learning math. Some of these people developed (or would have developed) both math understanding and HOTS even if they didn’t have a classroom or a teacher – they could have done so by themselves on a desert island, so to speak. Most people, though, definitely need math training to learn math content, and they need explicit instruction in critical thinking to develop higher order skills as well.

Most people, though, definitely need math training to learn math content, and they need explicit instruction in critical thinking to develop higher order skills as well.

MPJ
What advice do you have then for classroom teachers in the quest for teaching higher order thinking skills?

Dr. Star
Try to identify the places in your lessons where you hope students are developing higher order thinking skills, and consider ways that you can be more deliberate and explicit in your pursuit of and assessment of these goals. For example, ask your students about any broader connections they are making from the mathematical content of the lesson. Give students opportunities to apply what they have learned in a lesson to other mathematical and non-mathematical topics. Let students know what you mean by phrases such as “critical thinking”, “problem-solving”, and “logical thinking”; give students examples of what these practices look like as well as tasks that allow them to develop and experience these important competencies.

Let students know what you mean by phrases such as “critical thinking”, “problem-solving”, and “logical thinking”; give students examples of what these practices look like as well as tasks that allow them to develop and experience these important competencies.

**** Dr. Star may be reached at jon_star@harvard.edu
****For more of Jon Star’s thoughts on Math Education, see this Scholastic video on YouTube.

 

 

Enduring Cosmic Power

Today, I received one of the greatest compliments from a former student in the following post:

Jorge Post Border

This one-time middle schooler, enrolled in my high school Geometry class seven years ago, is referring to my consistent overt effort to have students understand and appreciate the true potential of math. Ultimate Cosmic Power in an Itty Bitty Thinking Space goes beyond the cognitive easter egg hunt that the “answer getting” routine too often reduces math to. I don’t know how to respond to knowing that most students think it is stupid or crazy at the time, but Jorge’s words of enduring impact have me smiling today.

Hey Georgie,
I remember you as a bright, happy young person. Though I am pleasantly surprised that my teaching has lasted with you, I am not surprised that you have chosen a mathematical career path. Build a new world, young engineer!

Graph of the Week (New Site)

Kelly teensI am getting the word out on this awesome site: Turner’s Graph of the Week. My friend Kelly Turner did a presentation at the Great San Diego Math Conference last spring and I loved her idea of having students analyze graphs from magazines and newspapers. These are mostly one-quadrant graphs with a natural context. This ties in directly to the Common Core’s call for applications and for reading non-fictional text. I was so impressed that I encouraged her to go public with the idea. I am serving as her megaphone.

Kelly does this activity once a week with her students, thus the name. The site offers several features:

    • Graphs. You don’t have to find your own. Kelly has already posted 12, and will post more on the GOWS page of the site as the school year progresses.
    • Submissions. If you like the activity and have graphs of your own that would serve others, email them to turner_k@auhsd.us. Kelly will screen the submissions and build the online collection.
    • Templates.  There is a generic worksheet template with writing prompts to guide students in reading, interpreting and analyzing the graphs.
    • Samples. On the home page Kelly will offer the graph that she is currently using for the week. Directly below that will be a student sample of the previous week’s graph.

Try it. If you like it, share with your colleagues. The same graph can be used at multiple course levels, with the level of questioning being adjusted to the level of students.

Kelly thanks for all the work on this. You have made teachers’ work easier and students’ education better.

TMC13 Session Recaps

TMC DrexelIn my last post, I summarized the overall experience of Twitter Math Camp 2013 at Drexel University. Following is my recap of the sessions that I attended. This conference was unique in that I learned something significant in each session.

Geometry Break-Out #1, Megan Hayes-Golding @mgolding, GA & Tina Cardone @crstn85, MA

After the opening greeting, the first morning session was a choice of break-outs according to course (Algebra 1, Geometry, Stats etc). These were intended to be open-ended discussion/work sessions. In the Geometry session, there was an overwhelming need by the group to wrap their heads around the Common Core Geometry Standards. Megan & Tina wisely went with the flow, and had us jigsaw the standards in pairs and share out. It was enormously helpful for everyone. I was already very familiar with the standards, but I still learned something about the CC standards on constructions. Specifically, the standards not only call for the four basic constructions plus those involving parallel and perpendicular lines, but the students are expected to construct a square, equilateral triangle, and hexagon as well. This was time well spent, with the bonus of getting to know Edmund Harriss @Gelada, Jessica @algebrainiac1 and StephReilly @reilly1041.

Edmund ArtThrough out the weekend, I had extended conversations with Edmund from which I learned a great deal. Mostly because Edmund is a math professor and as he spoke of his work with the mathematics of tiling patterns, I felt my IQ rise just by listening to him. Much of our discussions centered around the American education, though. Edmund had an interesting perspective, because while he teaches at the University of Arkansas and also leads special math programs for gifted children, Edmund is British. From that experience, he had a great deal to share about “how to run standards based education correctly.” I hope he blogs about that soon.

“I Notice & I Wonder,” Max Ray @maxmathforum, PA

Max Ray is the “Professional Collaboration Facilitator” at the Math Forum at Drexel. In essence, he teaches teachers how to teach problem-solving. I had heard before of starting lessons with “What do you notice? What do you wonder?” This phrase, which was originated by Annie Fetter @MFAnnie, is intended to initiate student thinking on a rich and robust task. That seemed pretty simple, so I wasn’t anticipating much new learning here … Boy, was I wrong! Max started with a picture of 3 glasses and the phrase “What do you notice? What do you wonder?”

TMC glasses      TMC graphs

We were asked to ponder for a moment, then share our thoughts with our neighbors. (Think-Pair-Share).  “I notice they have different shapes. I wonder if they have the same volume. What kind of drinks go in each one?” Then he posted the picture of 4 graphs, and again posed the same questions: “What do you notice? What do you wonder?” The ensuing discussion resulted in everything from “I notice the graphs are different colors” to “I wonder if the graphs correlate with the filling of the glasses.” The thing that I noticed about this whole activity is that Max let us mull this over without offering a single number or formula. Nor did he offer a single answer to any of our wonderings. Two pictures and two questions occupied us for 15 minutes. In the era of rushing through content it was wonderful to be reminded that mathematics starts with an observation and a question. Speaking of questions, my group wondered what glass shape would correlate to the fourth graph… while Max stood at the front of room silently smiling.

“Practicing the 5 Practices,” Christopher Danielson @Trianglemancsd, MN

Christopher Danielson is a professor of mathematics at Normandale Community College and also teaches methods courses for elementary school teachers. He shared the research published in Five Practices for Orchestrating Productive Mathematics Discussions. In summary, the 5 Practices are:

5 Practices PicAnticipating, during planning, student responses to the lesson prompt
Monitoring students repsonses during the lesson activity
Selecting which student responses are to be discussed publicly
Sequencing those student responses chosen
Connecting the responses to each other and to the mathematical ideas

Chris emphasized that the first and last of these are the two most troublesome for teachers. Chris modeled all these principles by conducting a math lesson on fractions. He knew what the issues would be with the context. He called us specifically by name to present our responses in an order that allowed the discussion to develop from simple ideas to more complex. I was particularly impressed on how he asked us to compare and contrast the various strategies. This is where I personally saw that I needed to bolster my own efforts on connecting ideas in own my class discussions. I walked away with the understanding that while any class discussion is better than none, there truly is an art form to doing class discussion right.

“5 Ways to Boost Engagement,” John Berray, @johnberray, CA

I have to say that the number one way to boost engagement is to teach like John Berray. The joy that he has for the material and for his students was just bursting out of him. With that said, John had 5 other ideas on increasing engagement:

1) Turn the Mundane on its ear
2) Jump on the timely
3) Bring in the outside world
4) Unlikely objects arouse wonder
5) Spill some paint

Translation: 1) Make it fun, 2)Tie math to current events, 3) Use the internet, particularly video, 4) Be goofy, 5) Connect the material to kid’s lives.

The highlight of the session was John showing how to make a textbook problem more exciting (a textbook makeover). The sample problem asked how many ways are there to take a 10-question true-false test (assuming all 10 question are answered). John asked us, “Who wants a shot at the glory?” and offered $5 to anyone who can match his answer key exactly. We were all prompted to number our papers #1-10 and choose T or F randomly for each. Once we all had our answers to this hypothetical 10-question True-False quiz, we were all asked to stand up. He began to display 10 questions, one at a time, about the participants at the conference. This offered humor and another level of engagement, as we were all trying to guess correctly, even though we had predetermined answers. After the first answer was revealed, all those who answered wrong on the paper had to sit down. We were asked to notice how many were still standing. This routine continued as we went through the entire list. Nobody won. The obvious question is, “How many people would we have to do this with in order to expect a winner?” He had just turned the mundane on its ear.

Geometry Break-Out #2

Our group reconvened with a few new people joining in. It was especially nice to See Peg Cagle @pegcagle after so many years. While the first day was a working session, this day was all about discussion. The group really wanted to talk about how to teach all the standards we listed in the previous sessions, while instilling the CCSS Practices. Teachers shared their various ideas, experiences and techniques. There was also a question on grading practices that revealed the dark side of the MathTwitterBlogosphere … We can be a very opinionated bunch. The hot topic for us was standards based grading. This turned out to be a benefit to the new teachers in the room or to old teachers with open minds, because quite a variety of ideas and positions were shared. It was an engrossing conversation, because no matter the positions taken, they were all shared with a passion for teaching students rich mathematics. The end of session came way to soon.

“Still Keeping it Real,” Karim Kai Ani & Team Mathalicious, @Mathalicious, VA

Mathalicious offers engaging, innovative math lessons with a focus on “real-world” applications. Karim @karimkai led us through two Mathalicious lessons that were solidly based in mathematics and loads of fun. The first, Datelines, tied the age of potential dates to systems of inequalities. The age gap on a date becomes less of an issue as people get older. For example, a 24-year old dating a 20-year old is less awkward than the 20-year old dating a 16-year old. This is an engaging topic for teenagers that Mathalicious sets to a graph and poses critical questions according to a given rule on dating ages. Like I said … solid. The second lesson, Prisn, used Venn diagrams to analyze the probability of being wrongfully flagged by the governments PRISM program for mining data. This lesson was about as relevant as any can get. It allowed for rich non-partisan conversation on how much error the public will accept. As I told Karim, these lessons are sexy, but have a lot of substance. At the conclusion, he generously gave the TMC participants a free trial subscription to Mathalicious. I intend on checking out more of their work.

“Getting Students to Think Mathematically in Cooperative Groups,” Lani horn, @tchmathculture, TN

Ilani BookThis one was very special for me, because Dr. Ilana Horn was such an influence on the teacher collaboration model that we have implemented at my high school for the last 9 years. Back in 2004, I was about to be the Math Department Chair for a new high school and was speaking with Jo Boaler about collab models for teachers. She told me that the person to contact was Lani Horn at the University of Washington (She is now at Vanderbilt in Tennessee). A week later, I happened to be vacationing in Seattle, and Lani was kind enough to give up time to a stranger and talk about her doctoral research. She was gracious as well as knowledgeable.

So I was excited to see her again and share how her information helped lead my crew back home to be one of the highest performing schools in the county. She was pleased to hear the news. Her session this time was on student rather than teacher collaboration. The specific model she shared is known as Complex Instruction (CI), in which students are grouped heterogeneously, with intentional methods to have all students participate. The focus of Lani’s session was on how academic status affects student engagement during group work. She was very intentional in telling us that participation is hindered by this perceived status about smartness, which is too often defined in math class as “quick and accurate.” To help make it safe for everyone to participate, the teacher needs to redefine smartness by acknowledging and rewarding “good questions, making connections, representing ideas clearly, explaining logically, or extending an idea.” Lani shared a video of a group of students working on a math problem, and asked us our thoughts regarding each students level of participation. She also asked us to analyze the teachers interaction and prompted us for alternative responses. This analysis of the work done by each student debunked the conventional wisdom that non-participatory children are lazy, stupid or shy. I had learned as much from Lani Horn on this day as I did in our first encounter.

Due to another engagement, I had to fly home early from the conference so I did not get a chance to attend the last session on Friday or any on Saturday. I heard I missed some great stuff,  which I don’t doubt.

Rich and Robust

Coffee beansI recently had the pleasure of learning from Tim Kanold of Stevenson High School fame. I heard him speak on several occasions last fall, and he kept saying that we need to involve students in “rich and robust tasks.” He was addressing the Common Core‘s call to the Standards of Practice. These practices can be summarized by saying that the Common Core is demanding students to think and to communicate their thinking. This can’t get done by taking notes from an overhead and doing the odd problems in the textbook. It gets done by purposefully deciding that students are going to solve rich and robust problems rather than simply watch their teacher complete examples of algorithms.

There is still a time and place for direct instruction and guided practice; but that should not be the complete experience for students, which is what we unfortunately find in the vast majority of American classroom instruction. For quite sometime, MPJ has been producing what we hope to be rich and robust tasks. Due to the growth of the internet, the availability of such rich and robust tasks has expanded tremendously. There are many exotic islands of innovation among the seas of tradition, but the blogosphere has made these islands less remote. Below I have listed a few, alongside my paraphrasing of the some of the Common Core Practices. This is not a comprehensive list by any means, however, I encourage you to take a few minutes and peruse these lessons in order to get a quick taste of what I think Kanold means by Rich and Robust.

Listed here are some additional sites that offer rich and robust tasks. {Note: I will be happy to update this list with any reader-submitted links, subject to review.}

The activities listed above obviously are not your typical math lessons. For good or for bad, the mathematical frontier created by the experiences highlighted here would make for a far different academic education than the gauntlet of lectures that most of us remember from school.

Now, I am going to assume that while the thought of introducing these large-scale examples into one’s repertoire is exhilarating for many, it may be terrifying for some. Let me ease those hearts by saying that rich and robust can be done on a much smaller scale. For example, we could simply ask the students: “Is x times x equal to two x or x squared. In other words, which of the following statements do you think is always true, if either: x·x = 2x  or x·x = x2?”

The CC Practices call for students to construct viable arguments and critique the reasoning of others. If your students stare back at you in silence with this question, then you will know why the Common Core Practices are so needed. If you answer the question for them, then they will watch you participate in a rich and robust activity, while they again participate in mundane note taking. For those that believe that this prompt is too elementary for any course above Algebra, let me assure you that it is not. I posed this very question to my International Baccalaureate students. A handful chose incorrectly, while several “could not remember.” When I asked the rest of the class, which was comprised of some of the brightest seniors on campus, no one could justify their correct answer. The best I got was that they “remember someone teaching us that once.” A simple question turned out to be far more rich and robust than it should have been, but it was a worthwhile day. {Try this one and get back to me.}

I must say here that I am grateful for my math education; it was far better than not having one at all. However, admittedly it was not rich and robust. The question is: Will we make it so for our students? It will take a conscious decision on our part to give our students a different educational experience than most of us had. So ask yourselves: When was the last time that you immersed your students in a rich and robust task? When is the next one planned? Has the time between those two dates been far too long? Are we up to the rich and robust task of offering rich and robust tasks?

The Elusive Relationship of x & y

“Ali has $10 dollars and spends $5 dollars every 2 days.” That was the simple scenario offered for the linear relationship that my students were working with at the time. I was having them write equations from various contexts in order to emphasize the concepts of slope and y-intercept. Boy, did I unravel a serious issue.

Typically, what is asked when dealing with slope-intercept form of lines is somemthing like: “What is the slope and y-intercept of the following equation:  y = -(5/2)x + 10?” Then those values are to be used to graph the equation. The hope commonly is that the students have been paying enough attention to know that the “slope is the number in front of the x.” The frustration commonly is that students still can’t identify the slope from the equation. From a context, the slope is even more difficult to identify, like with the Ali Scenario above. What I found out through a very unique question is why it is so difficult for them.

In the previous contexts that I had offered them, the students would see only two numbers. “Sally has 1 friend and makes 2 new friends every week,” as an example. I was asking them to identify the constant and the rate of change in examples like this. That was a struggle, but they eventually started to get it. When they ran across the Ali Scenaro, I was expecting the three numbers that were representing only two values to cause issues. I was correct. But it was their response to my next question that opened my eyes: “What two quantities are related here?”

I got blank stares. So I asked them to write down the two words that represent the quantities being related in the given context. When I looked at their answers, I was shocked to find the two words that over half the class had written down: spend & has!

Really?! I wrote those two words on the board, and asked them, “Does it make sense that the number of spends that I have determines the number of ‘hasses’ that I have?” I noticed everyone’s attention in class was riveted on the two words on the board. They knew what I was saying didn’t make sense, but they were struggling to reconcile the issue, so I asked another question. “What determines what in this scenario?’ A student finally offered, “days determines dollars.” Yes!

We had to wrap up the day, since I had spent so much of the period mining this one context for mathematical understanding. So the next day, I started the lesson by posting the same Ali Scenario and the two sets of words, “spends & has” and “days & dollars.” We revisited the idea that even though the sentence claims that “Ali spends and Ali has,” we are looking for words that represent numbers. Then we can use those numbers and a few symbols (variables, operations and equal sign), to write an equation (an abstract generalization) to represent the relationship. That equation (like the scenario and eventually the graph) represents an infinite number of combinations of “days & dollars.”

The students now easily identified 10 as the constant and wrote it properly as the y-intercept of the equation. The trouble was in dealing with a slope that was represented by a negative fraction, which was the issue that I originally intended the scenario to pose. However, my adventure into the minds of my Algebra students helped remind me that we too often have them leap from the concrete to the abstract, or that we skip the concrete altogether. How many of the millions of Algebra students in this country are graphing linear equations, yet have no idea that x and y actually represent a quantitative relationship? Focusing on getting the answer correct (“Graph this by starting at this number and counting up and over this number.”) often times bypasses the ultimate goal: mathematical understanding.

First Day Challenge (cont’d) – The 4 E’s

Hopefully, my challenge pressed you to create a new and unique First Day routine. As I shared in my previous post, I was inspired by the story of the martial art student who claimed that for each good teacher he had in China, all the principles that he would master with them were taught in the very first lesson. So I came up with The 4 E’s of the Math Mission. These are basically my answer to the question “Why should we learn this?”

  1. Empower you with fundamental math skills (Skills)
  2. Enhance your critical thinking and problem solving (Reasoning)
  3. Expose you to mathematical ideas and their related fields (Applications)
  4. Enrich your life with a greater appreciation for the power and beauty of mathematics and a deeper understanding of its embedded role in the nature of the universe. (Nature’s Conspiracy)

At the beginning of my first day with the students, I have them take a brief four-question quiz. Each question is headed by and supports one of the 4 E’s above. I have the students take the quiz with the explanation that it is like a movie trailer. Rather than a test on something I hoped they learned last year, the quiz is a  preview of coming attractions. Therefore, I don’t expect them to be able to answer all four questions; however, I do expect them to give an intelligent response. I preach to them that we cannot always give a complete response or even an accurate response, but we can always, always always give an intelligent response. Showing numbers, equations or diagrams is intelligent; leaving the paper blank is not.

The students work independently for a few minutes while I do the normal housekeeping duties, like checking their course schedules and adding students to the roster. Then I have them share ideas with each other. If they like what they see on someone else’s paper, they are free to write it on their own as long as the person explains it to them. “We share; we don’t copy.”

Then comes the time for a whole class discussion. I use this opportunity to inform the students that this process of independent thought, group sharing and class discourse  (think, pair, share) is a normal routine in the class. I point out that the 4 E’s are displayed on the front wall over the board, because this is what I come to work to do each day (my goal). I am not there to get them a good grade (their goal). Of course, if I achieve my goal, they will achieve theirs. So I ask  for volunteers on each of the questions, stressing the appropriate points for each of the 4 E’s :

Empower: There are certain skills that you will learn here, that you can use in everyday life, like calculating distance-rate-time to a long drive or the amount of mix need for the church’s pancake breakfast using the ratio on the side of the box.
Enhance: Just like weight lifting makes you stronger for the rest of your life, some problems we do, not because they have a practical application, but because they actually rewire our brains and make us smarter.
Expose: Part of school’s purpose is to show you what is possible. How do you know that you don’t want to be a rocket scientists until we actually launch one and have you calculate its max height with nothing more than a stop watch and a calculator?
Enrich: There is an age-old philosophical question about whether math is created or discovered by humans. I say both, but this E takes a definite side of the debate. Our daily task in the class is to unveil another “secret of the universe.” For example, the fact that ratio of the circumference of a circle to its diameter (“how far around divided by how far across”) is always 3.14 something. This is true for all circles in the universe, “Almost like there is a conspiracy.”

The second half of the period is committed to getting to know the students. We start with the first student who stands up, states their name and something interesting about themselves. The statement of interest is really just a stall tactic so that I can review the names that have already been mentioned. After every student has done this, I go back and state all their names. And finish the intros with some brief information about myself.

I close the class by telling them that the order of things today was intentional and symbolic. That the class will be about math first and foremost (not points and homework), and of course, it is each one of them to whom I must teach math, so they are important as well. I reiterate some of the points of the class culture like giving an intelligent response, think-pair-share, and “share, don’t copy.” I hand them the grading policy as they leave. We will discuss each of its points when the time comes. Hopefully, I communicated everything in a dynamic manner so the students are anticipating a fun year full of learning.

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