Wednesday, February 22, 2012

Two ideas for introducing functions

Here's two ideas for introducing functions to your class - none of them original, but I used them today and was pleased with just how well they worked.

1. Watch the Meat-a-Morphosis video
This amazing video is a winner with students and teachers. Powerful, simple and clear ideas about functions wrapped up in deliciously gruesome humour. 

Before watching the video: We spent a few minutes exploring the key idea of a function as a 'machine' that maps values of input variables (in the domain) to output variables (to a range), looked at the function notation f(x), and tried a few practice examples using f(x) substitutions.

After the video: We discussed the 'function machine' analogy and reviewed some of the fun examples in the video with their corresponding mathematical analogy.

2. Explore an unknown function on the calculator : the ln() function.
This idea comes directly from Mary Barnes' wonderful "Investigating Change" books*. Let the students know they have their very own "function machine" : their calculator. Ask them if they ever wondered what the ln() button does?

Students have been carrying this function machine with them for years.
So what is that ln() button all about?

Let's investigate! I gave each group of students some sheets of butcher paper and pens, and asked the question: "What does this function do to numbers? What is its domain and range?" - then let them at it, encouraging them to write, sketch, draw on the paper to show their thinking.

The results were astounding. As the work progressed, I dropped some hints to different groups to try different types of values and commented loudly (so other groups could hear!) when I saw group making a nice table or beginning to construct a graph. Some groups discovered logarithmic properties - that ln(100) was double ln(10), one group noticed ln(2) + ln(5) = ln(10), while others had discussions about asymptotes or debated with each other if their calculators were doing the right thing - the numbers seemed so odd and error messages kept coming up. For groups running ahead, I sketched y=x on top of their graph and asked them to draw a reflection. They recognised the resulting graph as an exponential one.

After the activity: I fired up GeoGebra on my board, showing how to graph the function (they groaned, having spent a long time doing it by hand :-) ), then we zoomed in and out to explore the interesting parts, referring to conversations and discoveries made by the class.  Then a good discussion on how to determine the domain and range. My not-so-secret agenda is to convince the students the value of  GeoGebra for this course - coming soon to a lesson near you!

I highly recommend this activity. Don't rush it - it will take at least fifteen minutes. Many great opportunities to develop and practice mathematical investigation skills.

* See for a preview. 

Some teaching thoughts:
  • I was surprised how much the activity of exploring ln() on a calculator allowed for differentation through asking different groups different questions. One group finished early, so I gave then the challenge to investigate the hyp() button (hyperbolic trig functions .. hehe!).
  • I don't think it's a problem to explore the ln() function a good six to twelve months before we might otherwise look at it. Not knowing about the function is the whole reason the exercise works.
  • I think it's a mistake to start with the formal definition of a function that distinguishes between a relation and function. This puts to much focus on the idea of one-to-one mapping, before the deeper idea of the mapping aspect of functions. Start with an interim definition of mapping of a domain to a range - the refinement can come next lesson. This is also Mary Barnes' approach.
  • Be ready to explain why we care about functions, as distinct from just working with our usual y = x + 2 expressions. To my thinking, the answer is that functions are themselves distinct mathematical objects - taking us to the next level of abstraction from number -> variable -> function. Equally   importantly, functions are the powerful idea we use for mathematical modelling.

Sunday, February 19, 2012

The best 12,000 rupiah I've spent this year

Earlier this year, while waiting for a train in Yogyakarta, this caught my eye:

A wonderful lesson plan for only 12,000 rupiah!
Click on the image for a larger view.

What really intrigued me was the small Arabic numbers above each Western number - even more interesting, the numbers didn't match:

I knew I had struck gold! 

A few weeks later in my Year 7 class, I split up the calendar pages, and put two sheets on each table group.  The class had recently done a lesson on ancient Egyptian and Roman numbers and heard that we use Indo-Arabic numbers - although at this stage they hadn't seen modern Arabic numbers. "Your mission", I told them, "is to find five interesting things about this calendar- and can you tell me where I bought it?". I let them at it, thinking this would be a ten minute activity. How wrong I was - the activity lasted fifty minutes including a whole class discussion at the end - I was surprised how involved the students were, and how much quality mathematical discussion resulted from their findings and their questions.

At first students focused on the obvious : the Arabic numbering system. Most noticed there was something 'wrong' with the numbering - they didn't match.  I asked a few more questions to groups who hadn't seen the disconnect between the two sets of numbers. One student could read the Arabic text and recognised the names of the Muslim calendar months - however she didn't know why the numbers were so different.

Another student worked out the year was 1433 - not 2012! I hadn't even noticed this little teaching gem.

The year is actually 1433! Or is it 2555 (if we were in Thailand)?

I threw in the fact that if I had bought the calendar in Thailand, the year would be 2555 - which really caused a stir. How could that be? Another girl recognised the Muslim prayer times at the bottom of the calendar -  showing how the times change throughout the month. A student with Chinese background told us the Chinese New Year was different each year and wondered if there was a connection.

By the end of the activity we had explored different number systems, how different cultures have different ideas where to start counting the year from, even how long a year is, depending if you use solar or lunar months. We shared what we knew about the words used for saying numbers in different languages - comparing those which sensibly use a simple power of ten system to those which are more complicated (try 'ninety five' in French).  Indonesia is definitely a great country for learning numbers! One student came up to the board and taught us how to do the stroke sequences to write Arabic numbers so we could improve our writing of them, then we did a few math problems with them.  After the class worked out my 12,000 rupiah purchase was actually only about $1.20, students shared their experiences of using different currencies. (Reminder to self: I must buy some of those $1,000,000,000,000,000 Zimbabwe dollar notes!)  

So many different aspects of mathematics culture embedded in just one calendar. In hindsight, I wish I had bought the entire stock of calendars at that train station. Oh well - a good excuse to return to Indonesia before the next Year 7 class.

Some teacher thoughts:
  • The calendar was cost-effective, ready-to-use resource: tear off the sheets, and each table group has a large sheet with lots of space to write their ideas on.
  • This resource was engaging because it contained conflicting and confusing elements - presenting a challenge for students.
  • This resource should work across a wide range of ages and achievement levels - adjust your questions as needed.
  • This activity is a great way for students in a culturally diverse class to share their different experiences. I was surprised how much my class taught me about the calendar - a great chance to demonstrate to how everyone can learn, even the teacher.
  • At the end of the lesson, you have something to put on the wall!

Sunday, February 5, 2012

When do we get to read Hamlet (for mathematics)?

Our young students explore their cultural and intellectual heritage:

As interesting as we teachers may find middle school and junior school mathematics, our students have been experiencing revolutionary ideas in other subjects for years. The amazing explosion of new, modern radical ideas hasn't hit the mathematics classroom yet - we've only barely touched the beginning of the scientific revolution.

But now in senior high school, for those who chose to follow the path ...

I've just started a calculus course with my Year 11 students - and I really felt it was important for them to gain a sense of the sweep of history and mathematical thought through the ages, and hopefully build a sense of excitement that something amazing is about to be added to their mathematical world. Something that parallels the excitement they hopefully felt when they first encountered modern literature, explored genetics, or wondered if Jackson Pollock's work really was art.

Now in Year 11, our students finally get to open up the great mathematical books of the early modern era - the ideas of Newton, Leibnitz and Euler - that's our "Hamlet"!

Continuing the story ...

Strongly aware I was showing these pictures to a class of  intelligent young women:

Challenge questions:
  • Do your students know who Euclid is? Who Descartes is?  I was quite disturbed to discover my new class of students had absolutely no idea. Is there any other school subject where we would not place knowledge in a cultural and historical context?
  • Can you fit some Cantor,  Godel or Mandelbrot into your teaching? Young people are fascinated, indeed sometimes troubled, by the concepts of truth, reality and infinity - it's part of growing up - but they probably never mention this in mathematics class.
  • Who is Emmy Noether? And why did Einstein refer to her as one of the greatest mathematicians of the century? What is the connection between Emmy Noether and the Large Hadron Collider? We really ought to have a plaque commemorating her in every mathematics and science classroom.

This collection of images and ideas is based on a presentation I prepared for my incoming Year 11 class this week.  At the end of the sequence we celebrated our culture by watching my favourite Sesame Street video (seriously!) Now that's culture for you.