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.

Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

Sue bookcoverPlaying With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is being published by fellow math blogger, Sue VanHattum of Math Mama Writes. In her book she “brings together the stories of over thirty authors who share their math enthusiasm with their communities, families, or students. After every chapter is a puzzle, game, or activity to get you and your kids playing with math too.”

Sue was kind enough to include an article I wrote, Textbook Free, as one of the chapters. I am honored to be included in a body of work that is best described by one of the authors as …

a collection of love stories because the authors, including yours truly, want to share something we’re pretty crazy about.  — Fawn Nguyen

In order to raise money for printing costs, Playing With Math has started a crowd-funding campaign. Contributions of any size are welcome, but $25 gets you a copy of the book. If you are interested in supporting the cause, please visit incite.org.

Sue VanHattum has assembled a marvelously useful and inspiring book. It is filled with stories by people who don’t just love math, they share that love with others through innovative math activities. Playing With Math is perfect for anyone eager to make math absorbing, entertaining, and fun. — Laura Grace Weldon, author of Free Range Learning

Let’s all help Sue make this terrific resource happen!

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!

SMP Posters by MPJ

SMP Posters Pic 2_Page_8I created my own posters for the Common Core Standards of Mathematical Practices. I combined the best from what I found from others and added my own structure. Necessity dictated my doing this for two reasons: 1) I wanted to respect others’ copyrights, and 2) I couldn’t find any that were appealing to secondary students.

With that said, I offer MPJ’s SMP Posters for use in the classroom. Each poster here has the following features:

The summary of the Practice straight from the Common Core documents, as listed in that famous grey box

SMP Posters Pic 1

The verbage of the Practice written in kid-friendly, first person language

SMP Posters Pic 2

A single word that embodies the particular practice

SMP Posters Pic 3

A diagram that displays an application of the practice, using Algebra as an example so as to span both middle and high school

SMP Posters Pic 4

A group of words that relate

SMP Posters Pic 5

A list of questions that pertain

SMP Posters Pic 6

A clip art image of a high school student to drive home the point that the practices are for them and not the teacher

SMP Posters Pic 7

An instructive statement that includes the word “Think”

SMP Posters Pic 8

A special shout out goes to the Jordan School District’s SMP posters for elementary schools which were the initial inspiration for this set. Other sources include: Eastern Bristol High School and Carroll County.

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.

Common Core and The Land of Oz

Oz FourThe Common Core is a noble cause. Who would argue that teaching kids to think and communicate their thinking is anything but a virtuous goal? It’s like the Emerald City in the Land of Oz, and standing between us and that bright shining city is a Wicked Witch and a bunch of Flying Monkeys. We know how the movie ends, though; we will melt that witch and make it down the Yellow Brick Road.

I made this comparison for a news reporter after my keynote address at the Idaho State Math Conference last fall. My analogy made NPG News at the same time that my math coaching colleagues and I back at Temecula Valley Unified were developing a four-year plan for professional development and student support in our district. So we wove the Wizard of Oz theme into our plan.

It turned out to be more than a catchy metaphor. The theme is actually quite symbolic to the trials and potentials of rolling out the common core.

4 Year PlanLet’s begin with the Emerald City. The Common Core claims to teach students 21st Century skills. In our district, we have summed up those skills as the ability to “Think and Communicate.” This, then, is our noble cause, our shining city.

Along the Yellow Brick Road is the infamous Wicked Witch and her Flying Monkeys. Our number one issue for teachers in Year 1 of the roll out was the lack of resources, and therefore, the demand upon them to find and create their own curricula. We did not anticipate this phenomenon, but it quickly consumed our role as math coaches. Our first year will end (hopefully), with Units, Pacing Guides & Model Lessons in place, and with an infrastructure to share them among the 130 secondary teachers in our district. Since this is by far the biggest obstacle facing us, and the ugliest work to overcome, establishing the content, scope and sequence gets the tag as the Wicked Witch. In Year 2 (the first of the Flying Monkeys) our primary purpose is to change our method of first instruction. The Common Core is calling for radical shifts in how we teach as well as what we teach, so that will be the focus of Year 2. Year 3 then focuses on what to do for those students who don’t get it (Tier 2 intervention). Finally, while we continue with the work that we laid out in the first three years, Year 4 will emphasize enrichment for students who easily learn the material and on implementing student use of technology.

Reflection FrameWhile many of the obstacles listed above deal with the work of us math coaches, the work of the teachers is personified by the four main characters of Oz: Dorothy, Tin Man, Cowardly Lion and Scarecrow. Their training is structured around the four Essential Questions of a PLC (Professional Learning Community). Dorothy must go first, because she was all about direction (“There’s no place like home.”)  So she asks the question, “What do we want the students to know and be able to do?” The Common Core has defined this question very clearly for us, particularly when it comes to the Mathematical Practices. We summed up these practices on a Reflection Fame that we use to debrief with teachers after our elbow coaching sessions. Year 2 calls upon the Tin Man, because it takes a heart to care for those students who don’t get it, especially in secondary schools. We are now commissioned to deliver a “guaranteed and viable curriculum to ALL students.” Year 2 will focus then on Tier 1 interventions … reaching and teaching ‘those kids’ … within the classroom. In order to do this we must have formative assessment and data collection protocols in place to be able answer the question “How do we know if they know it?” The Lion personifies Year 3, because it will take Courage to deliver Tier 2 intervention in response to “What do we do when they don’t know it?” Then, to answer the question “What do we do when they do know it?,” the Scarecrow and his brain will be employed in Year 4, when all the mighty work of the first three years is in place, and we can focus on the needs of the advanced students and on teaching all students to Think with and Communicate through technology.

Finally, and most importantly, we turn our attention to the students results. These are personified by who else, but the Munchkins. We plan to establish Student Mile Markers. These will be Performance Task benchmarks that will be given each year with the Final Exams (but not necessarily counted in a grade) to be used as a gauge to our collective progress (that of students, teachers, coaches and administrators) down the Yellow Brick Road.

The Wizard of Oz gives us a nice frame to dialogue within, but it also offers an important lesson for all teachers. The Wizard gave Dorothy and her friends absolutely nothing, other than the realization that they already had inside each of them that which they had been seeking all along. As do we. Brains, Courage, a Heart, and a Direction Home.

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.

Making a Lasting Impression… Without Even Knowing It

Footsteps in the Snad

GUEST POST: Today’s article is written by Greg Rhodes, the co-founder, creative director, and overall tech guru for MPJ. Usually, he stays behind the scenes, but a recent email made such an impact on him that he just had to share it with our readers.

I’ve been out of the classroom for a long time now, over fifteen years. But prior to my career transition, I was a math teacher like many of you. During those years, I did my best day in and day out to help students think logically and solve problems creatively… and maybe even have some fun in the process. But did I ever think that any of my hands-on lessons or outdoor projects made any lasting impression on my students? Not for a second.

But all of that changed one day when I received an email from Andy, a former student of mine. It left me absolutely speechless.

Hi Mr. Rhodes!  This email is likely to be out of left field since I haven't seen you in about fifteen years (assuming this is the correct "you"), so I apologize for potentially appearing to be an internet stalker.

Background:  I took geometry with you at Trabuco Hills High School in 1996/97 as a freshman and then struggled with algebra 2 honors the following year.  I am currently (after a rather circuitous journey) in a single subject credential program for chemistry at a local state college.

I have found myself bringing up your geometry class over and over again in class discussions of late, and reflecting somewhat extensively upon that time in my life.  Now, as I write the TPA 2 that is due this Monday, I just wanted to take a moment to tell you that it was a good class, and that it made a lasting impression.

So, thank you.

Wow! Fifteen years later and he’s still thinking about my little geometry class… and even discussing it with his classmates. In my wildest dreams, I never could have imagined that my teaching would leave such a lasting impression on any of my students.

So, here’s a word of encouragement as you prepare to go back into the classrooms (or as already there): Keep pursuing great teaching. Keep asking yourself how to make your lesson better, how to help your students understand deeper.

You are making a difference in the lives of your students. Never forget that. Some will thank you with a card or an coffee mug on Teacher Appreciation Day, and others may never say a thing. But don’t let that stop you from being the teacher they remember with fondness fifteen years from now.

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.

Innovative math lessons you can use in your classroom today

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