This is a test - 4/30/24

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1/28/2023

New Blog!

I'm now continuing to blog at my new site http://dmcpress.org

I hope to see you there.

Ihor Charischak

What is a Coherent Pedagogical Framework?

In a recent blog, Henri Picciotto (following a successful workshop series that he led) shared a participant’s comment. Henri writes:

“One of the participants in my Making Sense in Algebra 2 workshop had an interesting criticism. That anonymous participant pointed out that I presented no coherent pedagogical framework for the activities I shared. Good point! I did not present a coherent [pedagogical] framework because, well, I do not have one to present.”

I was puzzled. Which coherent pedagogical frameworks was the participant referring to? Webster states that a framework is a basic structure underlying a system, concept, or text. For math education that structure is a curriculum. Pedagogical refers to the myriad of approaches that a teacher can take in presenting a curriculum to students. And a coherent pedagogical framework would be a pedagogical framework that made sense. So conjuring up the meaning of those three words together Henri continues with why he doesn’t have one to present.

“During my four-plus decades in the classroom, I've seen many math edu-fads come and go: new math, individualization, manipulatives, problem-solving, group work, constructivism, constructionism (yes, that's a thing), portfolios, complex instruction, differentiation, interdisciplinary-ism, backward design, coding, rubrics, problem-based instruction, technology, Khan Academy, standards-based grading, making, three acts, flipping, inquiry learning, notice-wonder, growth mindset... not to mention various generations of standards.”

So instead of following some fad-like frameworks, Henri says:

“We need to be eclectic, and select "what appears to be best in various doctrines, methods, or styles." Instead of rejecting the fads wholesale, we need to consider each one as it comes along, as all (or almost all) have some validity. Instead of shutting our classroom door and continuing business as usual, we should keep it wide open.  Without becoming a dogmatic across-the-board adopter of each pedagogical scheme, we need to learn what we can from it, and incorporate that bit into our repertoire. This is how we get the sort of flexibility that makes for good teaching. If we do that, our lessons will not fit a standard mold. Quite the opposite: they will depend on the myriad variables that make teaching such a complex endeavor.”

I too like Henri have spent more than 4 decades working in math education. I’ve also worked with many of the edu-fads he mentions. In my private school teaching days I eclectically developed my own curriculum which included lessons borrowed liberally from Harold Jacobs’ “Mathematics: A Human Endeavor.” In fact, Harold’s work helped me to develop a coherent pedagogical framework - a classroom strategy model - that served extremely well in my modeling how to teach coherent lessons to the teachers I worked with. My model went something like this.

Each lesson (approximately 45 minutes) had three parts.

The first part I called: Set the Stage. This part would motivate the activity that followed. (I never wrote objectives on the board.)

The second part was: Do the activity. Students would usually work in groups. They would discuss and record their findings on a handout I would give them. (See 5th grade example.)

Finally (and maybe most importantly) was Debrief. What did we learn today? This is where the objective is revealed or left open for further noticing, wondering and even debating.

This was the model I used with teachers who were teaching math in conventional ways. For teachers who were interested in exploring more innovatively, I also modeled a collaborative project approach - usually referred to as Project Based Learning (PBL) which was an edu-fad back in the 1920s, but recently is undergoing a revival according to the Buck Institute. What I like about PBL is that it takes into consideration student interest. My example of PBL is the Noon Day Project which is a recreation of the measurement of the earth done by Eratosthenes in 200 BC. See my blog entry about it here.

Factor Game (Updated)

Factor Game - Illuminations

During the 1980s I was creating math "apps" that I called Microworlds because they were written in the Microworlds programming environment. Today some of these microworlds still exist using a different platform. In this blog I am featuring the Factor Game which you may be familiar with because of its availability in NCTM's Illuminations library. NCTM acknowledges Connected Math as the source for this game. I used the game in a framework similar to the lessons that Connected Math uses. They do a "3 Act" sequence popularized by Dan Meyer only they call it the "Launch-Explore-Summarize" sequence. In my teaching days I used the “Set the Stage-Do the activity-Debrief" strategy. Of course, Madeline Hunter set the stage for all of us "3 Act" fans with her very dated  "anticipatory set-instruction-independent practice" strategy - not anything I would recommend today. Here’s my Factor Game’s three part strategy.

Factor Game - The Launch (Setting the Stage)

Figure 1 

How to play the game 
This is  a large group activity. Split the class into two groups and assign a captain to each group. On the blackboard or white board tape sixteen 3 by 5 cards (or use post-its) numbered from 1 to 16. (See Figure 1.)

Figure 2 - Game board

The teams compete for the highest score by picking numbers from the game board (Figure 2).  For example, let’s say Player A chooses 15. That means that the 15 card gets moved to team A’s column and team A has 15 points. Meanwhile Team B is entitled to receive the factors of 15 (1, 3, and 5) for a total of 9 points. (See figure 3.)

Figure 3

Whether Team B gets the points or not depends on them knowing that they are entitled to getting those cards. The students have to tell you (the teacher) what to do. Here’s an example. Let’s say Team A goes first and after some discussion they decide to choose 15. The team captain will then announces their choice. You then move the 15 card to Team A's total. You then ask Team B if there are any cards on the board that they are entitled to. The team B captain would direct the teacher to move the 1, 3 and 5 to Team B's hopper for a total of 9 points. (See Figure 3.) If team B doesn’t know or makes a mistake it is the obligation of the other team to catch it. This keeps the students attentive and engaged. If some errors are not picked up by the students, the teacher should make sure they are aware of it. One problem might be that team B chooses a number that is not a factor of 15.  Team B would then lose their turn.  Play continues until all the remaining cards do not have a factor on the board. The game ends at that point. Team with the highest score wins.

Important note: Play one game as a practice learning game. In this way the students discover the rules for the game on there own. And that makes it more exciting for them.

Here’s a quick sample game (figure 4).

Figure 4

Though Team A went first they made a bad choice because they gave up 9 points. A better first choice would have been to take the largest prime number which was 13. Team B would have received only 1 point for a 12 point advantage. That's a very large disadvantage to overcome in a game consisting of only 16 numbers.

Here's another explanation of playing the factor game with 30 numbers on the board.

Explore (Do the activity)

Once the students get the hang of playing the 16 game with cards or post-its on the board, have the students open the Factor Game on their computers. Set the board to show numbers from 1 to 16. Have the students play several games against the computer. The challenge for the 16 game is to figure out if going first is always an advantage. In other words can they always beat the computer in the 16 game if they go first? Once they figure that out, have them play the 25 game. Does the team going first still have the advantage? Try the 30 game and see if going first continues to be a winning pattern or not.

Summarize (Debrief)

Question for students: What did we (including the teacher) learn from playing the Factor Game? Did you find that some numbers are better than others to pick for the first move?

Followup activity: Make a table of all possible first moves (from 1 to 30).

Figure 5

What is the best first move in a game of 30? Explain.

Extensions (for student projects):

The Factor Game applet was adapted with permission and guidance from "Prime Time: Factors and Multiples," Connected Mathematics Project, G. Lappan, J. Fey, W. Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1996), pp. 1‐16. However the idea for the Factor Game was predated by Dr. Factor which originally appeared as one part of a four part program called Playing to Learn published by HRM and Taxman circa late 70s early 80s. David Bau writes about Taxman in his 2008 blogpost:

The Taxman game is (apparently) an old programming exercise. But it is also a good game for practicing factors [and problem solving]. […] Here is a gadget that applies the rules of the game for you. The board defaults to 100 numbers but you can start with 20 by changing the number next to Restart button. Can you beat the Taxman?

The Taxman metaphor is a good one because the factors can be thought of as the currency to be paid to the taxman. If no factors remain for the numbers that are left on the board, the taxman (greedily) gets the rest of the numbers and the game is over. It’s challenging to beat the Taxman, but if you keep trying there is a sequence to beat him in the 20 game. Try it for other numbers as well.

David Bau continues:

It is worth playing without reading anything else - it is not too hard to find a heuristic that beats the taxman. The game was written up in an article by Robert Moniot in the Feb 2007 MAA Horizons - an optimal strategy is not known. I've gotten up to 121 points on the 20-size board; I am pretty sure this is not optimal. Can you beat the board with say 30 squares? What is the best score you can get?

Robert Moniot shared a little history about the Taxman game:

After the Math Horizons paper appeared, I learned that the game (Taxman) was invented by Diane Resek of San Francisco State University. She writes: “I came up with the game when I was working at the Lawrence Hall of Science in Berkeley from about 1969 to 1972. I was coordinating a grant Leon Henkin (UC Berkeley) and Robert Davis (I think he was at U of Illinois at that time) had from NSF to work with K-6 teachers in the Berkeley Unified School District. One of the things I tried to do was to come up with interesting ways for kids to practice their skills or their facts which would involve them in some thinking and not be so boring. The Taxman was one game I came up with for multiplication facts. It was named for the Beatle's song -"Taxman". At the same time other people were working with kids on teletype machines. They taught them Basic and had games on it for them to play. When I came up with a game or an activity, they would turn it into a program. (slightly edited). (Source)

Look at me. I’m a teacher!

November, 1967

My first day in front of a room filled with kids scared the hell out of me, but I kept my confident veneer. As a first year teacher, I was assigned 5 classes – 3 Algebra 1 and and 2 General Math. When I asked for advice about teaching the general math classes I was told to give them busy work. The textbook we used was at least 10 years old, and there wasn’t much in it that my students could relate to. Sometimes I pretended to be doing something important at my desk so I would let them chat away the “study” period that I had given them.  Fortuntely, it never came back to haunt me, although I did feel guilty about it. What was the matter with me?  I wasn’t sure which was more boring  - teaching them or watching them learn. A couple of them reminded me of the girls who used to sit in the back of the room with the sweathogs in the Welcome Back, Kotter TV show.  Why was I so irresponsible? Because I really didn’t know what to do with them.  So I chickened out and hid behind busy work pretending it was important.

The Algebra classes, on the other hand, were fun for me. The students were motivated. About 35 years later I heard from a couple of them thanking me for making it interesting for them. (Thank you, Internet.) I appreciated it.  Better late than never.  I still plan to visit one of the students in California. OMG, he’s in his 50s.

The Stock Market Game

So three/fifths of my day was fine, but the other two/fifths of general math classes was boring for both my students and me.  I really didn’t know it, but we had an unspoken compromise, a kind of truce where we agreed that if they didn’t act out and looked busy, I wouldn’t bother them or try to make them think. They had collectively given up on math long before they even got to me.  I knew I wouldn’t be able to do this forever. I could stand it for just so long.  Something had to give.

Then I got a break. On one of my caffeine energized mornings in the spring of 1968, I  read about how the stock market was doing well, gaining momentum.  It seemed that any stocks in those early 60s carrying the name “tronics” became “highflyers.”  The electronics sector was responsible for a 2 year bull market after President Johnson’s state of the Union address on January 10, 1967.  Despite my math background the electronics field was virgin territory for me, as was the stock market  It wouldn’t be until 1970 that I’d  spend $70 on my first Bowmar “brain” calculator.  I was intrigued by how the mainframe (whatever that was) seemed to be rocket propelling the stock market.

I was still thinking about the stock market when I discovered a Stock Market game in a department store. I thought it would interesting to try it with my general math classes. At this point, I had no real hope for anything. Nothing to lose.  I just wanted another way to get through the day.

Okay – despite how this sounds – like something scripted in a movie – It really did happen.   I bought the board game and we played it. It was a hit!  Even the principal stopped by my room to see what all the ruckus was about. He just stood there dumbfounded not sure what to make of it. The sweathogs were having fun doing something educational.  Kids engaged in buying and selling stocks. Now, fast forward 50 years and I’m playing the same game with a group of distracted 6^th graders. But I’m getting ahead of myself. I’ll return to Stocks and Bonds in a later chapter.

We had fun and my students were doing math willingly because they wanted to not only win the game but doing math made sense to them. I had stumbled upon something so pedagogically important that I would never forget it: sustained willing engagement.*

* This is an excerpt from my book The Wannado Curriculum A Math Teacher's Journey to the Dynamic Math 2.0 Classroom available from Amazon.

Achieving Mediocrity and what to do about it

This comic reminded me of my days teaching math at Brooklyn Friends School (BFS) in the late 1970s when I offered every student an opportunity to get an "A" if they were willing to do what was necessary to achieve it. I was always surprised by how few of my "C" students would take me up on it. Today I know why. It's because I was "selling" a product that those students didn't really care about but were forced to "buy." For Curtis a "C" is good enough for a subject that he feels he is terrible at and doesn't think is worth the effort. Kids want to achieve success, but they are not willing to work hard at something they think they can't be sucessful at. 

Math becomes more engaging for students if it is embedded and integral to the to the goal of the activity but not the main focus. For example, games or challenges can motivate students to learn basic skills where without the game context learning the skills would be a bore.

arcademics.com

Since math is an arena where context matters, solving contrived math problems is not engaging enough for most students. So we as teachers do the best we can to make traditional problem solving as interesting as possible.

Let's say our goal is have students know something about graphing linear equations. In the flipped classroom model, the teacher could assign students  Salman Khan's presentation of the skill (albeit rather sloppily) on video. Then in class the teacher could have a handout ready with a bunch of linear equations on one side of the paper and corresponding coordinate axes next to it. (I downloaded this piece from workshopworks.com) 

The solutions are provided so students can check them and then move on to the next lesson. Very high tech, Right? Yes, but overall boring and pointless. Most students don't really care enough to even ask "What's the point?" because they know it will be on the next test/quiz.

An alternative method is to use the computer program Green Globs. Here drawing lines has a purpose. They blow up globs! Students are presented with an array of 13 globs randomly distributed on a coordinate axis. The goal is to get the highest score by exploding all the globs using algebraic functions in this example we can stick to just linear equations which produce straight lines.

Below is a game in progress. The first shot y = -2 hit two globs that have y values equal to -2. And since the scoring doubles for each additional glob I hit with one shot, I get 3 points (1 for the first and 2 for the second glob.)

 I can do the same with x = - 4 and get 2 more globs for a score total of 6. To get more points I would shoot a line that slopes downward from left to right. For example, y=-x+5 gets me 3 globs which is 7 (1+2+4) points.

Upon reflection I can see that I could probably get 4 globs instead of 3 by tweaking the previous equation. If I make the slope -.8 the line hits all 4 globs which increases my score from 7 to 15 points since the 4th glob was worth 8 points for a total of 21 points. (Watch this explanation in more detail here.)

(To give students some intuition about slopes and Y intercepts they could watch Salman's demonstration of developing intution in getting lines to go through points.)

Notice that the focus is on blowing up globs, not on the math. However the math is needed for the student to succeed. And the better he or she knows the math the higher the score. And students really do "wanna*" get a higher score.

We need new curriculum that will make the math intrinsically interesting for students where the focus is outside the math. I call these wannado activities because the students are motivated intrinsically. They love the game and they want to learn the math to be successful which is within their reach. 

Many wannado activities are already out there. You can start to move your curriculum in that direction by trying out existing activities. I will be sharing examples in future blog entries.

Wannado Activities
There are 3 steps:
1. Set the stage for the learning of the math. Announce a game or challenge or puzzle or whatever else is instrinsically interesting to your class. It will vary from class to class. But from my experience there are strategies that are universally motivating to students.
2. Do the activity. Here students shoot globs. The teacher helps them to improve their skills by practicing certain strategies that can get them better results.
3. Debrief the activity. Ask the students to demonstrate what they have learned. 

Here is a detailed account on introducing Green Globs.

Currently Green Globs is available from David Kibbey at greenglobs.net

*I began to distinguish between "want to do" and "wanna do" after a student who was told he was not allowed to use the computer said "But I really, REALLY wanna use it." (circa 1981)

The Motivation Equation

The heart and soul of student learning is intrinsic motivation. And no one does a better job of describing this phenomenon than Kathleen Cushman in her refreshingly short and web-based book "The Motivation Equation" which is available to read online or download for free.

The table of contents gives you a nice snapshot of what the book contains. Here are the chapter titles:

Contents
Preface. In which we meet Ned Cephalus, his teachers, and their learning scientist friends
Introducing the Motivation Equation. In which we consider what learners value and their expectations of success
Chapter 1. Make sure we’re okay. In which teachers make it safe to risk a try
Chapter 2. See that it matters. In which students discover a reason to care
Chapter 3. Keep it active! In which fun, play, and surprise create a culture of curiosity
Chapter 4. Get us to stretch. In which students see in different ways and reach beyond their grasp
Chapter 5. Act like a coach. In which teachers guide practice and reinforce new skills
Chapter 6. Ask us to use it. In which students explain, teach, present, and perform what they learn
Chapter 7. Give us time to reflect. In which students think back on their learning and growth
Chapter 8. Have us make plans. In which students figure out where to go next
Appendix 1. Teachers and their lessons. In which teachers use a protocol to study learner motivation
Appendix 2. Resources. In which we offer practical resources for teachers

If you only have time for one chapter read Introducing the Motivation Equation (linked above) to experience what this is all about.

Though I loved the book, it is not easy for a teacher to implement given the constraints of a typical classroom. But its definitely a worthwhile goal on the road to the Wannado curriculum*.

*In my book, I define “wannado” as an excited form of “want to do.” For example, I wanted to do my math homework, fearing the consequences of not doing it, whereas I always would wannado (play) baseball in whatever form it appeared. The same for having to do something. “Haftado” is an extreme, distasteful form of “have to do.”

“Instead of making kids learn math, let’s make math kids will learn.”

-Seymour Papert

Conclusions from the Wannado (Math) Curriculum (a recently published book written by Ihor Charischak)

It’s entirely possible to fall in love with mathematics if the context is
right, like the “perfect storm,” where all the elements come together.
Beyond the day-to-day usefulness of math, mathematics can be
dynamic, fascinating, and empowering. Intrinsic motivation should drive learning. The math curriculum should be open-ended and allow for student and teacher creative flairs. There is a place for teachers and students to be partners in their learning enterprise, so that creating stories can bring a new life to what students would otherwise say is boring.

Weaving other subjects into the teaching of math is an incredibly powerful way to engage student imagination and help them to see math’s relevance to the real world. It’s fine to be able to solve an algebraic equation, but if students have no idea what it’s used for […], then what’s the point? Without a context, it’s just “mental gymnastics.” You don’t have to go far to see the math in history, science, music, and art. The list is endless.

Currently, story-based learning adventures are not part of most math curriculums. The focus remains on the “haftado” curriculum (e.g., passing through all the gates on the "royal road to calculus," where rewards are mostly extrinsic). However, one can invent—or better yet, reinvent—mathematics. The shift from a “haftado” to a “wannado” curriculum does not need to deprive students of the basic skills they need to be successful. Rather, it provides the perfect context for understanding the relevancy of those skills and the motivation to learn them. If kids “wannado” the projects, they will learn whatever hard stuff they encounter in order to accomplish their project’s goals … just as they do when they play video games. We, as a math community, need to develop alternative routes for students with unique needs and skills. Technology opens the door for a whole host of alternatives. To keep the focus on math, it is imperative that technology be integral to the curriculum, rather than integrated. This is a subtle but important distinction because technology-based microworlds empower students to focus on getting to know powerful mathematical ideas seamlessly. I firmly believe that technology can transform teaching and learning environments and help students achieve beyond what is possible without it.

Excerpt from "The Wannado Curriculum A Math Teacher's Journey to the Math 2.0 Classroom" by Ihor Charischak (2015)

MTBoS and other NCTM conference adventures

"We are math teachers who share what we've learned, cause we don't want our classes to suck the energy from students. Professional development among friends, not just colleagues. Fun! Immediately useful! Interesting!" So starts the description of MTBoS a 

The chart

refreshing new movement in the NCTM world. Armed with a table in the exhibit hall at the annual NCTM meeting in Boston this fledgling group of young social network activist teachers are slowly yet exponentially changing the face of math education. At least that's how it appeared to me every time I passed the booth and could barely squeeze in to say hello to the latest facilitator (of which there were many) at the booth. Led by Tina Cardone's enthusiasm the MTBoS booth was the best place to visit. What did they have to offer? Lot's of free stuff that members created and shared passionately with visitors. "Do you tweet? Do you blog?" If no was the answer then newbie visitors were given a 5 minute overview of the advantages of these socially viable venues. I'm sure many "joined" the movement and signed their names on the chart with their new twitter handles.

At the MTBoS booth

Tonight MTBoS will be doing a webinar having participants share their experiences at the conference. Click here for details.

I hope to "see" you there!

More NCTM conference adventures in my next blog entry.

The Wannado* Curriculum: A Math Teacher's Journey to the Dynamic Math 2.0 Classroom

Ihor Charischak

In his new book Ihor Charischak describes the dynamic classroom as a place where the interaction between teacher and students produces engagement and learning. Success depends on what the teacher does, how it fits in with the needs of the students, and the quality and utilization of resources. In The Wannado* Curriculum, Charischak tells how he turned his vision of a dynamic Math 2.0 classroom into a reality. Part memoir, part teaching tool, the Wannado Curriculum offers insight into helping teachers establish a context for creating their own "wannado" curriculum, a project-based approach where the context makes the content interesting to students. The book:

• describes how growing up as an immigrant in America impacted his learning
• tells how he discovered the secret to working with unmotivated students 
• Explores the idea that alternative ways of teaching and learning are the keys to powerful, dynamic teaching and learning that motivates students 
• discusses his experiences in a private, child-centered school, where he used computers to practice the teaching and learning he was excited about 
• relays how the real-life game of craps inspired a reluctant student to ask questions about the mathematical intricacies of the game 
• brings to life his experiences with computers in teaching math 
• details his vision of the dynamic math classroom 
• introduces Math 2.0, a powerful environment that uses mathematics software and collaborative Web 2.0 tools in a dynamic classroom setting 

The Wannado Curriculum presents glimpses of what twenty-first century math teaching and learning could look like if a student-driven and teacher-supported method was universally embraced.

More information about the Wannado Curriculum is available here.

*Wannado is the heightened version of “want to do” It’s what kids (and adults?) say when they really, really want to do something.

The Wannado* Curriculum - A Math Teacher's Journey to the Dynamic Math 2.0 Classroom - now available!

Now Available at Amazon

“The Wannado* Curriculum: A Math Teacher's Journey to the Dynamic Math 2.0 Classroom” presents glimpses of what 21^st century math teaching and learning could look like if, we truly embrace a student driven, teacher supported, national approach. The flawed NCLB and recent “Race to the Top” reforms are doomed to failure because they depend on a model that stubbornly will not scale, namely having cadres of well-prepared teachers who are experts in their field in every school. False hope continues to support these misguided efforts because of success stories from smaller, more homogenous countries like Finland and cities like Shanghai. Our more diverse, and heterogeneous approach does not lend itself easily to copying their successful(?) platforms. We need our own style that fits our needs better.

In this book, I plan to address this issue head on and explain how we could have a “tipping point” [1] where math achievement dramatically improves without having to resort to “super teachers” in the classroom to save us. Yes, we need good teachers, but to achieve progress in student math learning we need to follow a variety of paths, not just the one set over 100 years ago by the Committee of Ten [2] that outlined the current order of math topics in play today. What I propose is not a new idea. John Dewey and other progressive pioneers (mostly ignored by the mainstream decision makers in proposing solutions) offered it and practiced it successfully in their pockets of influence and notably in places where research studies acknowledged their success. Unfortunately, most reform efforts just tinker around the edges and don’t get at the heart of the problem: Most students find formal math learning boring and unrelated to their everyday life.

Even the top kids who are successful have little choice in how they study mathematics because the path has been set in stone, a path I call the “Royal Road to Calculus” which has its origins in the aforementioned Committee of Ten report.

The question, then, is what can educators, parents, and mentors do to offer alternatives to the current “one size fits all” path so that students will want to go to school not just because they need the necessary credentials and grades, but because they see the content of what they are learning as significant stepping stones to their dreams and ambitions and because they have adults around them who support their visions and provide the opportunities for them to explore the paths that interest them. Everyday math [3] is important but it can be learned in much more creative and empowering ways that enable students to see the value of math in their lives. This book will share examples of how this can be done.
Now available at http://amzn.to/1E4b1RV

*Wannado is the heightened version of “want to do” It’s what kids (and adults?) say when they really, really want to do something.

---

[1] Malcolm Gladwell – Tipping Point

[2] Committee of Ten 1893 report

[3] Keith Devlin – Mathematics Education for a New Era. Chapter 3. P. 23

Three Wannado Activities

In the preview to my upcoming book (previous blog) I mentioned that one of the challenges in changing the culture of schools is the difficulty in upgrading the level of teaching abilities. Unfortunately, that is a bit like waiting for superman (or woman) and it's not going to happen anytime soon.  Michael Fullan writes that unless the teachers are motivated to continue to learn and improve their craft not much will change.

"The key to system-wide success is to situate the energy of educators and students as the central driving force. This means aligning the goals of reform and the intrinsic motivation of participants. Intrinsic energy derives from doing something well that is important to you and to those with whom you are working. Thus policies and strategies must generate the very conditions that make intrinsic motivation flourish."*

Motivating teachers to improve their teaching is more likely to happen if the activities they do are intrinsically interesting to students.

An example is “13x7=28” starring Abbott and Costello. (See lesson - student and teacher pages.)

Another example of this is the famous jinx puzzle that I wrote about here.

A third example is to script a sequel to the video Weird Number which I describe in detail here.

In each activity the teacher presents an engaging scenario followed by a hands-on activity or discussion. This leads to some surprise twists or conclusions which are discussed and debriefed at the end of the activity.

*https://michaelfullan.ca/wp-content/uploads/2016/06/13396088160.pdf

Wannado Wannabes

"It is time for a change. We do not have to accept that the school we have always had is what we have to have now. Times have changed. Now everyone goes to school and now we have computers and the internet. The possibilities are endless. The economics of school can be quite different than what they are now. We can let kids learn what they want to in the way that works best for them. We will have happier and better functioning society because of it."

-Roger Schank

When I think back to my high school days what I remember most was playing baseball and basketball on my high school teams. Those were my favorite subjects. They served me well. I continued to play those games until my early 50s when my knees couldn't take the pounding anymore. The other subject that served me well was math. Without that I wouldn't have had the career I had being a math teacher. But math was not something that I would go out of my way to do. I never was interested enough to explore the wonderful math books in the library. If it wasn't for my math crisis in my sophomore year in college when I almost quit majoring in math, I would have never discovered the books in 510-599 section of the library which got me to see that math could be interesting and even empowering. I'm sorry to this day that the traditional math curriculum that I followed didn't allow for excursions to those books that might have fostered a love for math that I really didn't have despite being a straight A student in math.

Roger is right. Schools should allow students to study things that interest them; to follow what I call a wannado curriculum. I'm sure we would have a lot of wannado wannabes in schools everywhere.

I'm writing about my path to the Wannado Curriculum in my forthcoming book titled "The Wannado Curriculum - A Math Teacher's journey to the Dynamic Math 2.0 Classroom"

More from the Wannado Curriculum (Book in Progress)

Click to see Youtube video

The Weird Number

Back in 1970 I saw an animated film that changed my perspective on teaching fractions. It was called The Weird Number. (1) Here’s a piece of narration from the movie:

“My story concerns a strange event that took place in a little town nestled in the mountains.  A little town inhabited only by natural [counting] numbers, but whenever the townspeople gathered together rumors were exchanged; rumors that other numbers lived in the dark woods beyond the mountains but no one could imagine a number that wasn’t a natural number so no one believed the rumor.”  

So starts the story of the Weird Number a delightful excursion into a fantasy world of a town inhabited by natural numbers. Two citizens of this town 9 the baker and 736 the sheriff play key roles in the development of the story.

The narrator continues:

“Now one thing that never happens in this town is robberies.  This is because the thief is easily identified since the number of items stolen is always the same as the number of the thief.  For example, if 4 stole something, he would steal 4 of that particular item so he would be easily identified.  Therefore, there were never any robberies. But one day there was a robbery.  9 who was the baker rushed over to the sheriffs office to tell him about the robbery. ‘What was stolen?’ the sheriff asked 9.  “Just a little piece of bread.” replied 9.  ‘One piece of bread?’ said 763.  ‘I can’t believe it.  One is the mayor.  He would never steal.’

‘No, no,’ said 9, ‘not one piece of bread, a little piece of bread.’

‘Not one piece of bread, but a piece,’ said 763.  “What kind of nonsense is that?’”

Actually this makes a lot of sense once you understand who the thief was.  It turned out to be 2/3 a number totally unknown to the residents of this town. The sheriff organized a posse to catch the thief, but 2/3 was a clever escape artist because he was a master of disguise.  When the posse discovered his location in a barn, he stepped out wearing his 4/6 disguise and told his pursuers that he had no idea where 2/3 was. In the meantime the posse learns from 4/6 a trick that all whole numbers are capable of since they are also masters of disguise. For example, One, the sheriff could become 2/2, 3/3, 4/4 etc. Five could become 15/3 and so on.

Soon after 4/6 left them, 763 realized that 2/3 and 4/6 were one and the same number.  So he continued to pursue him.  But 2/3 was always clever enough to take on a new form (in this case 18/27) to outwit the sheriff.

After the whole numbers realized their ability to transform into other forms numbers in fractional form became common site on the town square.  Even 2/3 was not afraid to hangout in town, albeit in a different form. (2)

The movie ends with this cliffhanger that inspires a wannado followup.

The fractions and whole numbers are sitting around at tables in a pub pleased as punch that they were all members of the same number family when they learned of rumors that were other kinds of numbers living in the dark woods beyond the mountains.  Numbers that could not be written as a natural number on top and natural number on the bottom.  But no one paid any attention to that. The movie ends with a flash of lightening in the window followed by “The End” splashed on the screen.

After I saw this movie for the first time I was hoping for a sequel, but it never materialized as far as I know. So who or what were these mysterious non-rational (irrational) numbers?

Here’s a suggested activity/project for your class: Have your students create a video sequel about this mysterious number. 

What kind of story line could you have?  Here’s a suggestion. Start off with this:

“It’s a stormy day on the sea off the coast of Greece.  The year is around 520 BC.  A man, fighting for his life, is heaved over the side of a boat and plummets into the open sea to die.  His crime?  Stealing the crown jewels?  Murdering the King?  Nope. He was telling the world a mathematical secret. The secret of the dangerous ratio. This was the fate of Hippasus, a follower of Pythagoras who was forced to walk the plank and drown as a punishment for this crime. it’s difficult to imagine what a stir it created when it was first proposed! The Pythagoreans just couldn’t imagine that there was no ratio that equaled the length of the diagonal of a 1 unit square. The value was the square root of 2 – an irrational number. So they wanted to keep it a secret. Thus Hippasus who knew otherwise was doomed to his fate. (3)

________________

1.    Xerox, 1970. The Weird Number. Video. 

2.    A professor of math education used the Weird Number video as a motivator for a lesson development assignment for his students. Here’s what they came up with. http://edu320.blogspot.com/2006/09/weird-number.html

3.    Read more about the Hippasus story in Brian Clegg’s “A Dangerous Ratio” http://nrich.maths.org/2671

My First Days of Teaching - 1967

My first day in front of a room filled with kids scared the hell out of me, but I kept my confident veneer. As a first year teacher, I was assigned 5 classes – three Algebra I and and two General Math. When I asked for advice about teaching the general math classes I was told to give them busy work. The textbook we used was at least 10 years old, and there wasn’t much in it that my students could relate to. Sometimes I pretended to be doing something important at my desk so I would let them chat away the “study” period that I had given them.  Fortunately, it never came back to haunt me, although I did feel guilty about it. What was the matter with me?  I wasn’t sure which was more boring  - teaching them or watching them learn. A couple of them reminded me of the girls who used to sit in the back of the room with the sweathogs in the Welcome Back, Kotter TV show.  Why was I so irresponsible? Because I really didn’t know what to do with them.  So I chickened out and hid behind busy work pretending it was important.

Another memorable moment with that class was having my life threatened by one of the students. I had to throw him out of class for talking back to me. No curse words were uttered, but his intense defiance not only scared me, but also put a serious dent in my façade of being in charge.

The algebra classes, on the other hand, were a joy in comparison. The students were motivated. About 35 years later I heard from a couple of them thanking me for making it interesting for them. (Thank you, Internet.) I appreciated it.  Better late than never.  I still plan to visit one of the students in California. OMG, he’s in his 50s.

The Stock Market Game
So three/fifths of my day was fine, but the other two/fifths of general math classes was boring for both my students and me.  I really didn’t know it, but we had an unspoken compromise, a kind of truce where we agreed that if they didn’t act out and looked busy, I wouldn’t bother them or try to make them think. They had collectively given up on math long before they even got to me.  I knew I wouldn’t be able to do this forever. I could stand it for just so long.  Something had to give.

Then I got a break. On one of my caffeine energized mornings in the spring of 1968, I  read about how well the stock market was doing.  It seemed that any stocks in those early 1960s carrying the name “tronics” became “highflyers.”  The electronics sector was responsible for a two year bull market after President Johnson’s State of the Union address on January 10, 1967.  Despite my math background the electronics field held little interest for me nor did I have much interest in the stock market. It wouldn’t be until 1970 that I’d spend $70 on my first calculator.  But I was intrigued by how the mainframe (whatever that was) seemed to be rocket propelling the stock market.

I was still thinking about the stock market when I discovered a stock market game in a department store. I thought it would interesting to try it with my general math classes. At this point, I had nothing to lose.  I just wanted another way to get through the day with those general math students.

Okay – despite how this sounds – like something scripted in a movie – It really did happen.   I bought the board game and we played it. It was a hit!  Even the principal stopped by my room to see what all the ruckus was about. He just stood there dumbfounded not sure what to make of it. My “sweathogs” were having fun doing something educational.  Kids engaged in buying and selling stocks. Now, fast forward 50 years and I’m playing the same game with a group of distracted 6th graders. But I’m getting ahead of myself. I’ll return to Stocks and Bonds in Chapter 12.

We had fun and my students were doing math willingly because they wanted to not only to win the game but doing math made sense to them. I had stumbled upon something so pedagogically important that it would never forget it: sustained willing engagement.

Excerpt - The Wannado Curriculum - in press (due date: Nov. 2014)

Why can't math textbooks be more engaging reading for students or is that an oxymoron?

Curriculum of prestigious private K-8 school

Aritstotle Academy, a chartered school in Utah, uses the wonderful book byJoy Hakim “A history of US” for history and social studies and Delta modules for science. But in math students are forced to do Glencoe Math which reads like a recipe for Minestrone soup. I’m surprised that Aristotle Academy doesn’t use at least one of Joy Hakim’s science books. (See http://www.joyhakim.com.) For math Ivar Ekeland's The Cat in Numberland is not only interesting to kids, but contains some real math worth considering. But then something would have to give from Glencoe Math's maze of topics which of course are way more important than a silly book about cats. (Really?)

 Since textbooks are written by committee and need to cover all the bases for all the stakeholders involved, adopting a creatively written textbook is a long shot at best. Finding one is not easy either. Julie Brennan over at http://livingmath.net has many good ones to recommend. Her audience is mostly homeschoolers, but good alternative schools will benefit their students by exploring her list.

Think Math - Wonderful video that begs an important question

http://youtu.be/oD8-8uoS5nA

Watch this video. It's truly wonderful. Yes, we need to get kids to think about math. But what math? Rick's example showing the dynamics of generating a sine curve is great, but will all kids appreciate it and do we need to have every student appreciate math at the level that I and other math teachers appreciate it? We always get stuck in thinking as a teacher and what we think is wonderful but not always what the student wants to do. The "Wanna Do" curriculum is what I endorse. Students get to explore math in a way that makes sense to them. Sine curves evolving from a point spinning on circles is wonderful fodder for Youtube so that like minded students and mostly teachers appreciate. I would have chosen a different example that would resonate with more students. and adults. (To be continued.)

My Fraction Darts Story

This entry first appeared as an online article in 2005 inspired by the Math Forum's Toolfest 2005 event. It's been updated to be a blog entry in Scenes from a Dynamic Math Classroom.

I spend a lot of time in schools helping math teachers and students use various software programs. Some of these programs I especially like because they usually reveal something interesting about the learners and their approach to solving problems. One example is Fraction Darts (1) which is a microworld designed to help students with comparing fractions. The context is an engaging darts-like game where the object is to pop a balloon located on a number line between 0 and 1 by "throwing" a number in fractional form. Where the dart lands in relation to the balloon offers a clue as to the selection of the next "fraction to throw."

In one of my staff development sessions I had two veteran 6th grade teachers play a few rounds of Fraction Darts. Given their many years of teaching 6th grade math I assumed that they would like the program and find it easy to use. I was right about them liking the program, but I was surprised that they found the activity also very challenging!

Here's a game of Darts in progress. Each teacher has taken a turn throwing a dart. The first teacher tried 3/4. Noticing that 3/4 was too large, the second teacher chose 5/8 which was too small. Since these teachers had a lot of years under their belts teaching fractions, I just assumed they would take the (obvious?) strategy of finding a common denominator to choose their next dart. For example if they chose 8 for a common denominator then they would need a number between 5/8 and 6/8. With fractional notation this is difficult to "intuit" since it is not obvious what is in between 5/8 & 6/8. (Fraction Darts won't allow 5.5/8.) Of course 16 is a better choice for the common denominator because you get 10/16 and 12/16 then the number in between, 11/16, is easy to determine. However these teachers took a different route. Here's the dialogue that followed (as best as I can remember it.) I'll call the teachers Alice and John.

Alice: So I need to throw something bigger than 5/8 but smaller that 3/4. Hmm.. Let me try making the denominator [in 5/8] smaller. Say 5/7?
John: 5/7 made it bigger, but by too much.

Alice: Woops. I'll try 5/9.
John: That made it too small - even smaller than 5/8.

Alice: We now need something smaller than 5/7 and bigger than 5/8.

Alice: Smaller than 5/7? Then it must also be smaller than 10/14. Right? So 10/15 should be smaller right?

John: Let's try it. (The balloon pops.) Bingo!
Alice: This is cool!
Alice and John continued playing several more rounds.  So what skills did they need to know to eventually pop the balloon?
1. You can make a fraction smaller if you leave the numerator alone and increase the denominator. (For example, 4/6 is smaller than 4/5.)
2. You can make a fraction larger if you leave the numerator alone and decrease the denominator. (Revisiting the previous example, 4/5 is bigger than 4/6.)
At first glance this appears counter intuitive. But it works because in 4/6 you are dividing your unit into more pieces than 4/5 so each piece of 4/6 will be smaller than 4/5.

11/16 is what I expected the teachers to come up with but they surprised me by choosing a trial and error way to do it.

My teachers were both pleased as punch doing this activity and couldn't wait to try it with their kids. On my next visit to their classrooms, I wanted to give their 6th grade students a pretest before starting  to play with Darts. Here is what I gave them.

There were 17 replies. Here are some of them.

• 4/8, since 3/4 was too big and 5/8 was too small, 4/8 would be in the middle.
• 11/16 because its a pattern.
• 4/6 because 3/4 is too big.
• 4/8 because that's what I think.
• The answer is one half because 3/4 and 5/8 are to big and too small you have to chose the one in the middle which is 1/2.
• 11/16 because 11/16 is in the middle of 3/4 and 5/8. I think.
• I would throw 4/6 because 3, 4, and 5 go after each other and 4, 6 and 8 are all even numbers. Also, its going like a pattern.

5 out of the 6 students who chose 11/16 (a fraction that would pop the balloon) did not explain why!

Another successful choice was 4/6. They chose it (I think) because 4 was in between 3 and 5 in the numerators and 6 was in the middle between 4 and 8 in the denominators. Unbeknowst to them and me at that time was that if you take the average of the numerators and the average of the denominators and make a new fraction out of  it, that number will always fall midway between the original fractions. (See my proof.)

As time for my inservice was coming to a close the teachers were investigating another question: What happens to the fraction if you add 1 to (or subtract 1 from) both numerator and denominator? Subtraction makes the new fraction smaller and addition makes it larger. Another interesting strategy to use for playing darts. (2)

1. The idea behind Fraction Darts is an old one. Darts was created in 1973 with support from NSF in the public domain. The authors were Sharon Dugdale and David Kibbey. Many versions of the program have appeared since. The version I'm using was written in Flash in 2005 by Jason Sayres for CIESE. 

2. Inventing new strategies is what Math 2.0 is all about; It's the human synergy created in learning environments when dynamic tools are used for exploration and discovery.