Tag Archives: problem solving

Recap: NCTM 2017

San Antonio, CA , April 2017
I have summarized each session with some simple (•) bulleted notes, red underline to encapsulate my major take-aways, and occasionally a brief italicized commentary.


Math Task Makeover with Desmos Activity Builder — Michael Fenton (Desmos), Jed Butler (Heritage HS), Bob Lochel (Hatboro-Horsham High School)

  • The Big Take-Away = Use Desmos activities to generate intellectual need to learn the lesson objective.”
  • Generate need for Graph of a Linear Inequalities ….

  • Generate need for Definition of Ellipses …

  • Start with informal investigation, then move to formal language.
  • Teacher facilitation is key.
  • Where to Learn more: learn.desmos.com

I’ve got to starting using the overlay function!


Numberless Word Problems in the Elementary Grades — Brian Bushart & Regina Payne (Round Rock ISD)

  • The Big Take-Away = Have students make sense of word problems prior to computation by removing the numbers.”
  • The origin: Press kids to stop just circling numbers in word problems and applying random operation.
  • Not all day every day. It is a tool for sense making.
  • Focus on the relationship and the operation, formal language, and what the question would be, not the answer.
  • #numberlesswp

This makes sense for secondary grades as well.


Rich Tasks as Landmarks for Students to Use in Navigating Their Mathematical Learning Journey — Peg Cagle (LAUSD)

  • The Big Take-Away = Students’ work on Landmark Tasks throughout the year that should be visible in the classroom so that students can map their learning.”
  • We don’t take advantage enough of narrative in math class.
  • “Imagine shrinking down an entire map to the size of an index card. All the details get lost and the map becomes unreadable. What are the landmarks that will help students navigate the mathematical landscape”.
  • Peg presented the criteria for a Landmark Task …

  • … and presented us with a LandmarkTask …

Tied Up in Knots: In your groups, measure the length, in centimeters, of the piece of rope that you have. Then tie a single overhand knot and remeasure the length. Repeat the process several times. Create a data table, graph and equation relating the number of knots to the the length of the rope.

  • … then she analyzed the task according to the criteria …

  • … and showed how this landmark was made visible in her classroom.

  • The Speech Bubbles were created by the students to make comments on other groups’ work.
  1. This is the second year in a row at this conference that I have seen Peg give a year-long, big picture vision of using tasks in the classroom.
  2. This is also the third presenter who has mentioned some variation the Speech Bubbles. Time to use them in my classroom.
  3. Peg made a statement that has me thinking deeply and that I have quoted several times already: “Students have ample amounts of robust evidence that they are not good in math.” We need to help them overcome that.

Changing Teacher Practices: Transforming Teaching 101 to PD 101 — Audrey Mendivil (San Diego County)

  • The Big Take-Away = Shift from Professional Development to Professional Learning.”
  • 5 Principles of Effective PD
    1. On-Going
    2. Support during implementation
    3. Model new practices
    4. Variety of approaches and active engagement
    5. Specific to discipline/grade level
  • Shift from Professional Development to Professional Learning

  • How to Change:
    1. Small Steps. Stick to only 2-3 short term goals.
    2. Rethink Our Norms:

  • Why PD often FailsHow can we set-up for success?
    1) Top-Down Decisions: How can you include teachers in the decision making process?
    2) Little or no support in transferring ideas to the classroom: What support is available?
    3) Idea that teachers need to be fixed: How are you communicating your why?
    4) Lack of variety in delivery modes: How can you differentiate for teachers?
  • Essential Elements. Audrey took us through a terrific activity for those who create Professional Learning experiences. She gave a sets of cards that were color coded, and asked us to work together to sort them into 4-6 groups, and then name the groups.
    She then shared how she grouped them (which is what the color scheme was for). The idea was to take ALL the things that we want teachers to know and do and rather than create a checklist for them, cluster these concepts into Themes or Essential Elements and have teachers learn that.

  1. This was yet another session at NCTM that focused on Vision and the need to put the WHY in front of teachers.
  2. The re-structuring of the norms resonated with me. I’m still thinking deeply on this one. The norms drive the culture of the meetings, so they offer great leverage.
  3. In her call to keep the list of goals short, Audrey discussed the need to set short-term, intermediate and long-term goals. This falls in line with the concept of “leading and lagging indicators.” Student data may take awhile to improve (lagging) so what are the improvements in teacher moves that we can credit to our PD (leading)?
  4. The objective of the card sort activity gets at the heart of what I see killing most PD in districts … too many short-lived initiatives. Keep the broader concepts in mind. Bigger, slower moving targets are easier to hit. 

The Struggle is Real: Tasks, Academic Status, and Productive Problem Solving — Geoff Krall (New Tech Network

  • The Big Take-Away = Developing a culture of productive struggle requires holistic vigilance on the relationships between Quality Tasks, Effective Facilitation & Academic Safety.

  • Protocols for Problem Solving
    1) Make it visual
    2) Estimate Before Solving
  • Record what students know…
    vs what they are assessed on.
  • Promoting Access:
    Example: Make the smallest (or largest) difference by filling in numbers 1-9 no more than one time each.

I am challenged by Geoff’s two graphs of the linear regression of student growth. My Claims-Based Grading needs a little more work in the area of reflecting cumulative knowledge rather than recent learning.


Logarithmic Earthquake Project: An Algebra 2 Project with Real Applications — Tanisha Fitzgerald-Williams & Beverly Heigre (Notre Dame High School)

  • The Big Take-Away = Have students view videos of earthquake damage and do their own research on Richter Scale, before formal presentation of calculating Magnitude difference with Logarithms.”
  • Step 1: Research

  • Step 2: Calculations

 

 

 

 

  • Step 3: Student Groups make Presentations
  • Note: Tanisha & Beverly also have students offer possible humane response to victims of earthquake presented.
  • There is a google drive folder available that contains materials for this projects: goo.gl/Y197YR

Clothesline: The Master Number Sense Maker — Chris Shore (Me)

  • The Big Take-Away = Number sense and conceptual understanding of current content can be taught simultaneously with Clothesline Math.”
  • I presented the power of the Clothesline to teach Algebra, Geometry and Statistics.
  • clotheslinemath.com
  • #clotheslinemath

There were at least 5 sessions at NCTM Annual in which the Clothesline was a part or the focus of the presentation. 


Fun Sidenote: The ceiling rafters and the carpet print of the convention center had the same Geometric Pattern. I am sure there is lesson to be created out of this.


There are videos of keynotes, ShadowCon and Ignite
at NCTM’s Conference 2017 web page.


The city of San Antonio enhanced an already fantastic trip!

 

 

 

Introductions & Neuron Facts in Algebra 2


neuron vertical
Day 2, Thurs Aug 11, 2016

The Brain Surgeon
Today, we began my regular routine of designating a daily Brain Surgeon. Since this was our first day of the Brain Surgeon, I introduced the routines of the Drum Roll, Reading of the Dual Target, Music Cues, and the Wrinkle Sprinkle. The students seem to embrace the spirit of of it all.

Student Introductions
As with every new school year, I had each student briefly state their name and something interesting about themselves. When they were all done,  I recited all their names. That always impresses a class. Then I told them things about myself. I state that yesterday we started with math, because that is what we are all about here. But since I teach math to them, they are also important and I need to know who they are.

Growth Mindset
Most of our Course Teams across the district agreed to do some kind of growth mindset activity. Here was mine.

I started by summarizing the plethora of lists of fixed vs growth mind set statements with two pictures. I told the students that research in student learning is showing that self-perception of talent as a limit or as a starting point has a tremendous influence on their learning.

Talent Wall

Then I shared that scans of the brain of someone with a fixed mindset versus a growth mindset, shows something very interesting. When faced with a challenge, the fixed mindset brain “goes cold.” It literally shuts down. However, when faced with the same challenge, the growth mindset brain “fires up.” It knows that more is being asked of it, so it kicks into high gear to meet the challenge, rather than duck it.
Brain MindsetsNow it was time to test out where we see ourselves demonstrating  a fixed or growth mindset.

Neuron Facts
I gave the students the worksheet with the Neuron Facts on the front side. I found these on the internet and thought they would make for a good lesson since they highlight the amazing function of our brains. I added the subheadings of Fast, Crowded ,etc. I started with a common practice of mine Notice & Wonder popularized by Annie Fetter (@MFAnnie) of Math Forum.  My Gradual Reel-In process looked something like this:

  1. You Do: Independent response.
  2. Ya’ll Do: Each member of the group shares both their notice and wonder.
  3. We Do: Each group decides on one Notice and one Wonder from those shared. These get shared out by each group as I write them on the board.
  4. I Do: I summarize the major point(s) that I want all students walking out with. Here it was the process of Noticing and Wondering and how we facilitate group discussion in class… And of course how amazing our brains are.

The groups were then tasked with doing one problem together. Homework was to do one more.

Wrinkle Sprinkle
Tying into the concept of the plasticity of the brain, I joke that when we learn we get a new wrinkle on the brain. Each class then concludes with what we learned that day. The brain surgeon leads and records the discussion. The students today stated that they learned…

  • Negative thoughts shut down your brain
  • Speed of the brain cell
  • The amount of oxygen the brain uses

Re-Cap: NCSM 2016

 

Oakland, CA , April 2016NCSM Logo

I have summarized each session with some simple (•) bulleted notes, red underline quotes to encapsulate my major take-aways, and occasionally a brief italicized commentary.


Game-Based Learning: The Hype is Starting to Give Way to Some Surprising Substance  — Keith Devlin (Stanford)

  • Pic Keith_DevlinBig Take-Away = Start with the thinking (which is the more important), then follow with the notation.
  • The “Symbolic Barrier”: Symbols are a terrific way to use mathematics, but a horrible way to learn them.
  • The vast majority of our population is taught symbolic notation, yet most need mathematical thinking.
  • Students using Dragon Box Algebra learn the Algebraic thinking needed for solving equations in 90 minutes. However, this ability did not transfer to paper/symbolic test, therefore, both are needed.
  • We teach students to play music, before we teach them to read it. The same should be true of mathematics.

Personal note: I’ve had Dr. Devlin’s book, Goodbye Descartes, for almost 20 years; after his talk he signed it for me.


Developing Deeper Student Thinking  and Reflection — Patricia Rogers (Gilroy USD)

  • Big Take-Away = Use “structured” student collaboration to enhance student reflection, and thus student thinking.
  • Good collaboration needs to be: Regular, Brief, Prepared, Open-Minded.
  • 3 Teacher Moves (Phil Daro)
    • Student thinking made visible (to other students, not just the teacher)
    • “Everyone Ready” (ALL students individually prepare themselves to share thinking.)
    • “Make an Expert” (of a students who has viable strategy) then have the rest of the class “Turn and Talk” when productive struggle weakens in order to focus on targeted math topic.
  • Classroom Discussions (Chaplin, O-Connor, Anderson)
    • Wait Time
    • Revoice (The teacher rephrases what the student just said.)
    • Restate (Student(s) rephrase what a student just said.) 
    • Add-on (Student(s) extend or challenge another student’s conjecture.)
    • Apply (Students apply their own reasoning to someone else’s reasoning …” just try it on.”)

I’ve seen the two techniques of revoicing & restating demonstrated a great deal lately and have now been challenged to bring these into my class more often.


SFUSD logoThe San Francisco USD Mathematics Teaching Toolkit: Changing the Practice Along with the Content — Glenn Kenyon & Kathy Bradley (SFUSD)

  • Big Take-Away = Established Vision, Beliefs and Goals before building district curriculum

Vision
“All students will make sense of rigorous mathematics in ways that are creative, interactive, and relevant in heterogeneous classrooms.”

Beliefs
1. All students can and should develop a belief that mathematics is sensible, worthwhile, and doable.
2. All students are capable of making sense of mathematics in ways that are creative, interactive, and relevant.
3. All students can and should engage in rigorous mathematics through rich, challenging tasks.
4. Students’ academic success in mathematics must not be predictable on the basis of race, ethnicity, gender, socioeconomic status, language, religion, sexual orientation, cultural affiliation, or special needs.”

3 Goals
1. Help students express, expand and clarify their own thinking. 2. Help students to listen carefully to one another and negotiate meaning.
3. Help students deepen their reasoning.

“The teaching strategies in the SFUSD Math Teaching Toolkit are designed to support an inquiry-based approached to learning mathematics, with an emphasis on classroom discourse. This approach reflects the shifts of pedagogy required to promote the Common Core Standards for Mathematical Practice.”

  • Unit Design Structure to incorporate tasks

SFUSD Unit Design.png

1) Math Talks
(SMP#3. “Math Talks”, instead of Number Talks, so discussion can broaden {e.g. strategies for computing area})

2) Three-Read Protocol
(Model for close reading of complex math text)
First Read (Teacher Read Aloud) = What is the Situation?
Second Read (Choral Read) = What are the Quantities & Units?
Third Read (Individual Read & Think) = What question can be asked?
This only runs 10-12 minutes. Take away the question to create a rich task.

3) Participation (Group) Quiz A technique to give public feedback on group work. Lists ways a student can contribute (“You can help your group if you can…. create a table, draw a diagram, listen to people’s ideas and ask questions, etc) Also publicly list teacher expectations (e.g. How groups … us shared space? ask question? explain thinking? etc)

  • Video Exemplars & PD modules are available on district web site.
  1. SFUSD has a PHENOMENAL math web site chalked full of resources for supporting teachers implement the vision and the curriculum. Check it out!
  2. The description of their Group Quiz speaks to the need to explicitly teach students how to productively collaborate.
  3. This was the first of three sessions that spoke about the importance of vision. It will be the predominant point that I take home with me from this conference.

Beyond Relevance and Real World: Talking with Teachers About Engagement in Mathematics? — Dan MeyerPic Dan M

  • Big Take-Away = ‘Real World’ does not have to be real, just accessible and engaging.
  • 62% of teachers surveyed : Greatest challenge is “unmotivated” students. Interesting that they didn’t say motivating students was the challenge.
  • Question: Why don’t teachers spend more time developing good questions?
    Teacher Response: “Because we don’t have the time.” (True that.)
    Real Issue: “Lack of creativity. Giving the answers does not require creativity.” (True that, also, but ouch!)
  • A stronger option than the typical “engaging images or context” in a textbook: Redefine Real World. A situation is in the process of becoming real to you if you are able to … 

1. Ask a question about it.
2. Guess about it .
3. Argue about it.


High School Coaching Model: Building Bridges Between Coaching and PLC Culture — Kris Cunningham & Jeanette Scott (Phoenix UHSD)

  • Big Take-Away = Roll out PD through PLC teams.
  • New initiatives first unveiled during PLC team meetings.
  • Most powerful change agent was a lesson study. (1st day by 1 teacher, next day by all teachers)
  • Most teachers took 3-4 years to show change; 4 of 5 teachers showed significant change within 5 years.
  • There exists a Common Lesson Plan format for lessons studies and co-planning.
  • Professional Development certificates tied to evaluations. (i.e. Professional Growth affects evaluation outcome.)

The fact that teachers took 3-4 years to show change aligns with Maggie McGatha’s research shared at last year’s NCSM conference


Practicing the Five Practices: Coaching Teachers to Use Student Work in Planning  — Max Ray-Riek (Math Forum)

  • Big Take-Away = Walk teachers through the 5 Practices of Discourse with student work samples.
  • Max shared with us the Teddy Bear’s Banquet pattern problem. He had us determine the Math Goal for the lesson, and then Anticipate the student responses.
  • Max then offered 16 samples of true student responses (Monitor) and then had us Select and Sequence some of the responses for classroom discourse and share why. We were then asked to Connect the responses to the Math Goal.

This is a great training tool that can be brought into any PLC structure.

I also witnessed Max slyly counting on his fingers. This was his way of giving is all wait time on his prompts. 


Smarter Balance – Making Connections: Eliciting to Acting on Evidence —  Judy Hickman (Director of Mathematics, SBAC)

  • Big Take-Away = When the scoring focus is on Reasoning, students can still score full credit with a minor calculation error, if they show understanding.
  • Do NOT put too much emphasis on Interim Assessments. As “snapshots” they will give you good information, but it will be an incomplete assessment.
  • The authors of the exams were shocked that students answered so few questions correctly.

Four Keys to Effective Mathematics Leadership — Mona Toncheff & Bill Barnes  (Activating the Vision )

 

 

 

 

  • Big Take-Away = Vision needs to be created by ALL stakeholders
  • The Four Keys:

1. Establish a Clear Vision for Mathematics Teaching & Learning
2. Support Visionary Professional Learning for Teachers and Teacher Leaders
3. Develop Systems for Activating the Vision
4. Empower the Vision of Family and Community Engagement

This was the second of three sessions that spoke about the importance of vision. This one stressed the need to have all stakeholders (admin, teachers, classified staff, parents and the business community) in on the creation of the vision. Mona & Bill then asked, “If you were ask 10 people on your campus, ‘What is our vision,’ how many answers would you get?”


The Secret to Leading Sustainable Change: Vision, Focus, Feedback, and Action! — Dr. Tim Kanold (Turning Vision into Action )

  • Big Take-Away = Set the Vision, Help people advance the Vision,  Celebrate Evidence that the people are advancing the Vision, and take Action on the feedback towards the Vision. 
  • Sustainable change requires evidence that the change is bigger than their opinions.
  • Is the work you are doing formative? Meaningful feedback must be followed with results in action by the teacher or teacher team.
  • Meaningful Feedback = F.A.S.T. Action: Fair, Accurate, Specific, Timely. Action from your feedback is required.
  • Mary Beth call. Dr. Kanold told a story of when he was Superintendent of Stevenson HSD. He called a secretary at one of the schools, restated that ‘engagement’ was part their district vision, and asked “What does engagement look like in your job.” That’s keeping the vision in front of the people!
  • The Popeye Moment: Change happens when the moment of moral courage vocalizes what Popeye often said, “That’s all I can stands, cuz I can’t stands n’more!

This was the third of three sessions that spoke about the importance of vision. The story of calling the secretary is tattooed on my brain. Dr. Kanold stressed that the vision should be posted visibly during every PLC meeting, and that any unproductive dialogue can be redirected with the simple statement, “How does this conversation advance this vision?”


A Math Coaching Package — Donna Lione, Rebecca Williams & Chris Shore (Me) (Temecula Valley USD )

 

My colleagues and I presented the framework for developing a comprehensive math program. The details of each of the 8 components will be posted as separate posts.

  • Vision
  • Relationships
  • Humility
  • Influence
  • Passion
  • Faith
  • Focus
  • A Plan

Recap: NCTM 2016

San Francisco, CA , April 2016
I have summarized each session with some simple (•) bulleted notes, red underline to encapsulate my major take-aways, and occasionally a brief italicized commentary.


The Status Quo Is Unacceptable: A Common Vision for Improving Collegiate Mathematics Diane Briars, & Linda Braddy, Christine D. Thomas & Dr. Uri Treisman

  • Big Take-Away #1 = College failure rates are 55% higher than for more active forms of instruction.
  • Big Take-Away #2 = The math ed reform movement is now reaching the post-secondary level.
  • Big Take-Away #3 = The change must be institutional.

  • The challenge facing the Math Ed Community (the dismal stats)
    1) Only 50% of students earn A, B or C in college algebra.
    2) Women are twice as likely as men to not continue past Calc 1.
    3) While 20% of all Bachelors Degrees are awarded to Blacks & Hispanics, only 12% of Math Degrees are.
    4) Math is the most significant barrier to degree completion in ALL fields.
  • Innovation does not affect normative practice. Out of 81 different projects (2-3 yrs) connected to a grant or leader, NONE replaced normative practice, because they were based on faculty development, not institutional change.  Dr Treisman, “Institutional change is a bitch.”
  • Historically, school system does change when necessary.

The Learning Mindset Movement and Its Implications for Addressing Opportunity Gaps — Dr. Uri Treisman (The Dana Center)

  • Big Take-Away  = Besides Growth Mindset, there is Belonging Mindset and Purpose Mindset.
  • “I find Algebra beautiful, but will it knock the socks off of a 13 year old. Algebra well taught should leave them barefoot in the park.”
  • “Why do kids give up? Most of the work I do is confusing, cause no one gives me problems in the back of the book.”
  • Growth Mindset = “Can I do this?”
    Belonging Mindset = “Is this where I belong?”
    Purpose Mindset = “Does this connect to who I want to be?
  • Dr. Catherine Good:  Building Bridges to Belonging: Mindsets that Increase Participation, Achievement and Learning
  • Build Belonging through effort & engagement, not talent.
  • Positive Belonging Mindset = Assume they belong.
    Negative Belonging Mindset = Need to be invited in.

Paper Cup + Gust of Wind = Yearlong Rich Task — Peg Cagle

  • Big Take-Away = Revisiting the same task through-out the year emphasizes math as reasoning not simply answer-getting.
  • Peg had us roll a paper cup on its side. She then left us to our own devices to answer several questions, each of which addressed a different mathematical topic throughout the school year.
  • Day 35 Question: How can you convince a skeptic of the shape that the cup traces out as it rolls?
  • Day 70 Question: How can you locate the center of the shape that the cup traces out as it rolls?
  • Day 105 Question: How can you use a cup’s dimensions to determine the area of the shape it traces out as it rolls?
  • “Efficiency is overrated: That is a concern after you learn something.”

Coding to Enrich ALL Math Classes — Jason Slowbe

  • Big Take-Away = Coding in Math class helps teach the Math, not just the coding.
  • Coding can be done on the TI-Calculator
  • Can help students understand the meaning and power of mathematics. For example, Archimedes’ method for approximating the area of a circle.

Rich Problem Solving to Support Today’s Standards — Chris Shore (Teacher Created Materials)

I conducted a product promotion for Teacher Created Materials. The session was on Problem Solving and Linda Gojak’s What’s Your Math Problem Anyway? My presentation focused on the following questions about the teaching of problem solving, each of which I will answer in its own post:

  • What is problem solving?
  • Why teach problem solving?
  • Who should learn problem solving?
  • When should we teach problem solving?
  • How should we teach problem solving?
  • Where do we find resources for teaching problem solving?

A Call to Substance, First Interview with Dr. William Schmidt

Schmidt BookcaseIn March of 1998, during the inaugural year of The Math Projects Journal, we had the unique opportunity to publish our interview with Dr. William H. Schmidt, of the University of Michigan. At the time, Dr. Schmidt was the National Research Coordinator and Executive Director of the U.S. National Center which oversaw participation of the United States in TIMSS. The results of the TIMSS report directly led to the developement of the Common Core 20 years later, which is why Dr. Schmidt is nicknamed “the Godfather of the Common Core.” He is also widely published in both journals and books on mathematics education.

We had the opportunity to interview Dr. Schmidt again about the rollout of the new curriculum. Before we post the current interview, we thought it would be valuable to reprint what Dr. Schmidt had to say in the early years of the research. He emphasized focusing instruction on conceptually understanding and higher order thinking skills, rather than on methodology. This is an important message now more than ever with so many untested techniques and ideologies being promoted widely on the internet. This message heavily influenced the trajectory of MPJ‘s lessons and my own classroom teaching. I hope it does the same for yours.
*************

MPJ: Can you give an example of a model lesson from one of the top achieving countries, either Germany or Japan, which are the focus of the videos?

Dr. Schmidt: If you look around the world, there just isn’t a single way to teach that is dominate among the top achieving countries. Some of them are very didactic, lecture-oriented classes. Some of them are like the kind that you see in the Japanese tapes. If teachers know their mathematics well, they can be just as engaging through a lecture format, as they can teaching as the Japanese do. It is very clear to me that there isn’t one way to do this. Instead, the more analysis that I do, the more I believe that there are some principles involved here that just might go across countries.

MPJ: What is the common thread?

Dr. Schmidt: I think the common thread that makes for the top-achieving countries is pure, honest-to-goodness mathematical substance. If the teachers really know and understand the mathematics, then they bring that to the students, through whatever means they know best. Also, a large part of this idea is to develop this stuff conceptually and not just algorithmically. I think many people misunderstand the Japanese videos. It is not so much the methodology, as it is the mathematics. You watch those lessons and the instructor really understands the mathematics, engaging those students in more ways than we do in this country.

MPJ: So, if a teacher were to do a dog-n-pony show lecture with drill-n-kill practice, and do it well, would it work?

Dr. Schmidt: The dog-n-pony show lecture, yes; the drill-n-kill, no. That’s what I said about there being some principles. I think the common element is a clear understanding of the subject matter and then going through it much more conceptually than algorithmically.

MPJ: Can you give us a model to how to teach math conceptually rather than algorithmically?

Dr. Schmidt: A U.S. lesson typically starts out with the algorithm. For instance, there is the example in the videos of a guy teaching geometry. He says to the kids, “Here are two supplementary angles, one is thirty, how much is the other?” A student says, “a hundred and fifty.” And the teacher says “Good, now why is it that?” And her response is, “Because they are supplementary.”

Instead, conceptually, you could show them that if they measure a straight line, it’s always one hundred and eighty degrees. Then they realize that if they put a line anywhere its going to cut it into two parts. That’s conceptual; you start with understanding why, so if you forget the stupid name, supplementary, and you see a line with an angle you’ll know what the other one is. That’s the difference.

MPJ: How is a strong conceptual understanding of the mathematics developed among teachers?

Dr. Schmidt: It comes from two sources. In some countries, they must major in these fields. The other thing we don’t think about is that they are products of their own systems. For instance, Japanese teachers don’t necessarily take more mathematics at the university level than we do. But look at what they already know before going to the university. They are already ahead.

MPJ: In regard to the things that our readership is looking at, active learning, projects, manipulatives, do you have any models from these other countries, or that you think could be done here?

Dr. Schmidt: You don’t find very much of that anywhere else. They seem to be uniquely American inventions, especially the cooperative learning. We asked teachers how much they use groups, and it’s pretty much nonexistent. We are too much into the methodology in this country, and we miss the substance. We start talking about small groups and manipulatives and it just becomes process. Therefore, the substance behind it gets lost in the shuffle. And for a lot of these ill-prepared teachers, that’s what they grab onto because that’s what they understand.

MPJ: We hear that the US teachers assign more homework and spend more class time dealing with homework than the top achieving countries.

Dr. Schmidt: The dominate activities in the U.S. lessons were reviewing homework and doing seat work. One thing that was startling is that the typical American lesson had only 10 minutes or less of instruction.

MPJ: What role does homework play in other countries?

Dr. Schmidt: It varies a lot. Japan doesn’t give a lot of homework, but the kids study for the next lesson. There’s a difference, of course. Studying is what you do at the university, and homework is what you do in grade school. But Japan is unique. Worldwide, homework and seat work are still the dominate activities. I think if you do that and you do it well, and develop the topics conceptually, it can work.

MPJ: Is this a curriculum issue instead?

Dr. Schmidt: It is the core issue, but just putting that in place by itself wouldn’t work. You have to help teachers teach in ways that engage kids.

MPJ: So, that is something that teachers could start doing today. We could focus on engaging students and developing topics conceptually?

Dr. Schmidt: That is my point. We must start paying much more attention to the subject matter and teach it more conceptually and less algorithmically. And that is why we are in a catch-22. The Japanese teachers grew up in their system seeing math developed conceptually, no matter what they learned at the university level. For our teachers it is a lot more difficult; they have to break out of a mold that they’ve been put into. But I think that is something that teachers can do — Get off the algorithmic side. Don’t just give an equation and when a kid asks why say, “Because that’s the equation.” Try to get them to understand what lies underneath some of this stuff.

MPJ: It seems that, chronologically, you are suggesting a lesson should move from concept to algorithm to application.

Dr. Schmidt: A lot of the lessons that we’ve seen, like in France and such, start out with an application as a motivator. An example is a science one about transformers. They started out by looking at a map of the city and looking how electricity would flow. This got them hooked on the issue, then they hit them with some good hard science about the transformer. That’s very often how it happens: hook them with some kind of application, then take them into it conceptually, let them flounder — that’s where I think what the Japanese do is a good idea — let them talk about some of their ideas, then give them an algorithm, a formula and a few examples. Whereas we typically start with the formula with a few sentences about it, and then have them do worksheets.

MPJ: The report states that American textbooks cover too many topics, yet they typically have only fifteen chapters.

Dr. Schmidt: That is mistaking the notion of what a topic is. The definition of topic has to do with the substance of the mathematics, and when we defined it that way, the measurement across all these topics is not how many chapters are in each book.

MPJ: Can you give us an example of four or five topics?

Dr. Schmidt: Congruence and similarity, three-dimensional geometry, linear equations, and fractions. We actually tested 44 topics and determined how many of these topics were in any given textbook. Our 700 page books address about 35 topics. The Japanese, on the other hand, spend half of the eighth grade year on congruence and similarity alone, and their gain in that year is higher than in any other country. The dilemma I have in telling you what to do is that the teacher shouldn’t decide which five to ten topics should be studied in a year. It only works if somebody coherently lays this thing out as to what needs to be done.

MPJ: Do you have any last things to add?

Dr. Schmidt: People still think that there are general things a teacher should do, like cooperative learning. That’s what people push. We push all the things that have nothing to do with subject matter. I’d like to challenge the notion that there is a single way to do things. If you listen to the ideological left, they say that there is only one way to teach. And the data just do not support that. Among the top achieving countries you cannot find one dominate way of teaching. On the other hand, the ideological right are calling for “the basics.” Yet, the latest analysis shows that the United States, through 8th grade, does average or above average in all the standard arithmetic skills. This is not the place were we are hurting the most. That is all we teach. That is what’s wrong, we never go beyond the basics.

If I wanted to become rich and be an advisor to schools to jack their scores up, I know how to do it. We have certain areas of math that we have the international comparisons on. I can tell you the seven items that we are the weakest on, and if schools just did something in those areas, we’d go up in the international rankings. None of those areas is anything that we would consider the basics.

MPJ: What are those area of weakness?

Dr. Schmidt: Measurement, error analysis, geometric shapes, perimeter, area and volume, congruence, similarity, vectors, geometric transformations, and three-dimensional geometry. These are not the basics.

MPJ: Tomorrow, our readers will not be able to change the textbooks or create national standards. What can a teacher do in the classroom today that will model the type of change that you and the TIMSS report call for?

Dr. Schmidt: That’s a tough question, because most of what I have argued is, based on the data, these really are systemic issues. However, the data also shows that how we teach is as important as what we teach. Teachers should challenge students with more mathematical substance and develop the ideas more conceptually rather than algorithmically.

Get to the Core of The Core

apple coreThe Common Core curriculum can basically be summed up in the following sentence:

Teach your students to THINK and COMMUNICATE their thinking.

Thinking and communicating are the 21st Century skills. Many people believe that the skills of the future involve the competent use of technology. While it is true that using digital tools in school and the work place is the new reality, it is actually the proliferation of technology that makes thinking and communicating imperative in the information age. When all the knowledge of humankind is available at anyone’s fingertips, memorizing information becomes far less important than being able to construct, evaluate and apply it. You can Google information; you cannot Google thinking.

So the core of the Core truly is Thinking & Communicating.

To make my case for this, I would like to pose that the following equation

6 + 4 + 4 + 8 = 22

be adjusted to

6 + 4 + 4 + 8 = 21

Before you start shouting that everything you have read on Facebook about the Common Core is true, let me declare that I am using this equation simply as a teaching device, not a true mathematical statement. You will understand what I mean after I present my evidence.

6 Shifts

Let me start my case that the core of the Core is Thinking & Communicating with the 6 Shifts, which are best represented by the following document found at Engage NY.

6 Shifts

In essence, these shifts are redefining rigor. Old school rigor was defined as sitting quietly taking notes, and completing long homework assignments in isolation. The new school definition of rigor envelops the last 4 shifts on the list: Fluency, Deep Understanding, Applications, and Dual Intensity. The rigor is now placed on the students’ minds instead of on their behinds.

The shifts are also calling for balance. Dual Intensity insists on both procedural fluency AND critical thinking by the students at a high level. It is not about dual mediocrity or about throwing the old out for the new, but a rich coupling of both mechanics and problem solving.

Therefore, I make the case that:
                     6 Shifts = 21st Century Skills,
which are to
                     Think & Communicate.

4 C’s

Another list that is framing much of the Common Core dialoge is the 4 C’s. Resources for this list can be found at Partnership for 21st Century Learning (p21.org).

4 C'sThese C’s redefine school…

The old school definition: A place where young people go to watch old people work.

The new school definition: A place where old people go to teach young people to think.

… and they redefine learning.

The difference of old school vs new school learning can best be contrasted by the following images of the brain.

Brain Chillin   Brain Build

The image on the left shows a passive brain that just hangs out as we stuff it with esoteric trivia. The image on the right shows a brain being built, symbolizing its plasticity. We now know that when the brain learns, its neurons make new connection with each other. In other words, learning literally builds the brain. The 4 C’s  claim that this building involves the capacity of the students’ brains to Critically Think, Communicate, Create and Collaborate.

Therefore, I make the case that:
                     4 C’s = 21st Century Skills
which are to
                     Think & Communicate.

4 Claims

Smarter Balance creates it’s assessments based on 4 Claims. (I teach in California. PARCC has 4 Claims that closely align to those of SBAC.)

SBAC

4 Claims

Notice that Claims #2 & 3 are explicitly stated as Thinking & Communicating, which also overlaps with two of the 4 C’s. Mathematical modeling is #4, which will be discussed later. I want to point out here that Claim #1 reinforces our idea of Dual Intensity from the 6 shifts.

There are two important notes for teachers about this first claim. 1) It says Concepts and Procedures, not just procedures. The students need to know the why not just the how. 2) The Procedures alone account for about 30% of the new state tests, so if we continue to teach as has been traditionally done in America, we will fail to prepare our students for the other 70% of the exam which will assess their conceptual understanding as well as their abilities in problem solving, communicating and modeling.

Therefore, I make the case that:
                     4 Claims = 21st Century Skills
which are to
                     Think & Communicate.

8 Practices

If you open the Common Core Standards for Mathematics, the first two pages of the beastly document contain a detailed description of the Standards of Mathematical Practice. Then at the beginning of each of the grade level sections for the Standards of Content you will find 8 Practices summarized in the grey box shown below.
8 practicesWhat do you notice about the list? Indeed, these habits of mind all involve Thinking & Communicating. While the content standards change with each new grade level, the practice standards do not. With each year of school the students are expected to get better at these 8 Practices. Notice that the first half of the list has already been included in the ones discussed previously: Problem Solving, Communicating Reasoning, Constructing Viable Arguments and Modeling. A case is often made that the other four are embedded in these first four. However one might interpret the list, “Memorize and Regurgitate” is not on there.

Therefore, I make the case that:
                     8 Practices = 21st Century Skills
which are to
                     Think & Communicate.

The Sum of the Numbers

So, as you can now see, the 6 Shifts, the 4 C’s, the 4 Claims and the 8 Practices are all focused on the 21st Century Skills of Thinking & Communicating. Therefore, I can finally, explain my new equation …

Since,

    6 Shifts
    4 C’s
    4 Claims
+  8 Practices
= 21st Century Skills

then 6 + 4 + 4 + 8 = 21!

None of these numbers represents a list of content, because the content changes brought on by the Common Core, while significant, are actually no big deal in the long run. A few years from now we won’t remember all the fuss regarding Statistics and Transformations, but we will all spend the rest of our careers learning how to teach kids to Think & Communicate.

I rest my case.

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.

 

 

Multiple or Best Reps?

I find it interesting that on the day that we post our most recent lesson, 4 x 4, (sample page), Dan Meyer posts the question: Aren’t we doing kids a disservice by emphasizing “multiple representations” rather than the “best representations?”

I understand where Dan is coming from…Why the overkill, when one proper tool solves the problem. I have three quick responses to this.

1) If the goal of the current activity is to apply previously learned skills, then I agree with Mr. Meyer. Students should develop the savvy to choose the most appropriate tool at hand, and implement it properly. When faced with embedding a nail, is there any sense in using both a hammer and a rock?

2) If the goal of the lesson is to build conceptual understanding of the four formal representations of a linear relationship (words, equations, data, graphs), then generating the other three from any given representation develops this insight. How many students can graph a line by plotting the y-intercept and then counting the slope up and over, but have no idea that they just stated the infinite set of points that satisfy the equation?

3) If the goal of the day is to offer a point of access to the students, then the temporary representation will eventually give way to a higher level of abstraction. Look at the banner on Christopher Danielson’s blog. These multi-link cube models can represent the various ways to factor the number 24. Alongside these 3-dimensional arrays, students could be representing factors symbolically, 2 x 2 x 6, 2 x 3 x 4,  3 x 8 etc. In time, the blocks are left behind for a level of abstraction that is far more efficient. Afterall, it is faster to write the factors on paper than it is to build them with the blocks, especially when the students start factoring much larger numbers. So to push back a little bit on the original question: Are we doing kids a disservice by offering training wheels when learning to ride a bike?

The answer to the question of “multiple representations” or “best representation” is, as always, up to the judgement of the teacher at the time.

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