This is a test - 4/30/24

 This is a test.

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)