Saturday, December 27, 2014

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.

Wednesday, November 12, 2014

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."

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"

Tuesday, October 14, 2014

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.
3.    Read more about the Hippasus story in Brian Clegg’s “A Dangerous Ratio”

Tuesday, August 19, 2014

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 fa├žade 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)

Sunday, July 27, 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 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 has many good ones to recommend. Her audience is mostly homeschoolers, but good alternative schools will benefit their students by exploring her list.

Monday, January 27, 2014

Think Math - Wonderful video that begs an important question
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.)

Thursday, January 9, 2014

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.