Tag Archives: methodology

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.

Advertisements

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?

Launching My 180Blog

ChessboardI have discovered several teachers who are posting 180Blogs… Blogs for a 180-day school year. These teachers are basically posting a public diary of their daily math lessons. These blogs serve as terrific professional development for both writers and readers.

In many of the trainings I have conducted, teachers continually want to know how more innovative, activity-based lessons, like those that we publish here at MPJ, fit into the grand scheme of the school year. I thought launching my own 180blog would help answer that question. Today, I beginning sharing it with world: Chris Shore’s 180Blog

The biggest lesson that I learned from the experience so far is that a 180Blog would be awfully dull if all I did everyday was go over homework, share notes on an overhead, and assign odd problems from the textbook. For good or for bad, this blog shows nontraditional methods of teaching, in a very traditional school environment. It shows how the unconventional dovetails with the conventional, the fresh with the mundane.

A few notes about Chris Shore’s 180Blog:

  • This blog will unofficially be 94Blog, in that I didn’t start it until this second semester after winter break.
  • It took me awhile to work out the technical bugs of hosting it on my school’s site, so I will be launching the first three full weeks simultaneously, and continue from there.
  • If I fall behind a day or two, please forgive me. I will make it up on the weekends.
  • I plan on being very transparent with the Good, the Bad and the Ugly. You can already see elements of this coming out the gate.
  • While I am using my classroom site to host my daily log, I will use the MPJ blog here to expand upon various tools, lessons and methods that I display, as well as ruminations on student understanding and misconceptions.

I would appreciate your feedback on this effort. I hope it is as helpful for you as it has been fun for me.

Other 180Blogs:
Fawn Nguyen (Special thanks for your guidance on mine)
Bowman Dickson
Justin Lanier
Dan Anderson
Sadie Estrella

“Too Bad We Can’t Find Out Which Way Works Best.”

“It’s too bad someone can’t do a study to see which way (direct instruction or hands-on learning) works best.”
This comment came from a colleague in a discussion on how well my remedial Algebra class did with the assessment on a particular non-traditional lesson. Before I share my thoughts on this, let me offer some background on the class and some data.

The demographics of the class is quite challenging: 4 SDC (special day/workshop), 5 ELL (english language learners), 7 IS (special ed), 5 Academy (highly at-risk), out of a total of 30. While the others don’t have an acronym after their name, they still have a history of struggling in school. And I absolutely love teaching this group! I share all this first, because the data I am about to show will be all that much more impressive.

I began our previous unit on Solving Equations with a pre-assessment of the 6 types of equations that they were going to learn to solve:

1) x + 4 = 31
2) 4x = 28
3) 7x + 5 = 26
4) 10x + 2 – 4x = 44
5) 11x – 4 = 3x + 12
6) 9(x – 2) = 45

Then I spent a week of direct instruction (D.I.) and 2 days with the simplifying and linear equations components of the Truffles lesson, after which I assessed them for a second time on the same 6 equations. I then led the students through the Hippity-Hoppity lesson for 4 days, and assessed them again on the same 6 equations. The progression of results is shown below. (There are only 22 students shown due to the shuffling of students classes at the beginning of the year.)

 # correct Pre-
Assess
After
Truffles & D.I.
After
Hippity-Hoppity
6s 4 13 19
5s 4 7 3
4s 3 1 0
3s 3 1 0
2s 3 0 0
1s 2 0 0
0s 3 0 0

My class went from only 36% getting 5 or 6 correct, to 90% after the Direct Instruction and Truffles lessons, to 100% after the Hippity-Hoppity lesson. It is worth noting that the number of students correctly solving all 6 equations rose from 59% to 86% after the last “active-learning” activity.

My friend made his “too bad we can’t find out which way is best” comment having only heard about the last lesson, not knowing the work I put into the unit throughout. That work demonstrates a variety of strategies that might be classified as direct instruction or hands-on learning. He didn’t realize that I had implemented several ways, not just one.

I utilized several strategies because there is a great deal of research that shows that the best way is a balanced approached. However, that balanced approach is not between methodologies (direct instruction vs discovery/hands-on/active learning); it is between skill acquisition and critical thinking. I chose to use lectures and guided practice to impart skills, the Truffles lesson to instill understanding of a variable, and the Hopping lesson to offer an application of the topic. These last two lessons also required my students to practice other skills that they lack: reading and writing in a mathematical context, and following multi-step directions.

So we should tell all of our colleagues that we do indeed know a best way. It is not my way, your way, or their way, but a balanced way.

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!

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

Question:

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

Answer:

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

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

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

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

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

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


Chris Shore, Editor