Reversing the Meat-a-Morphosis machine : inverse functions

A short followup to the post Two Ideas for Introducing Functions.

Yesterday I discovered another payoff from the hilariously gruesome Meat-a-Morphosis video: a powerful and memorable analogy for Inverse Functions. Imagine if we ran the machine backwards - if we put chicken nuggets in and the original chickens came out? My students thought this was hilarious.

Inverse Functions: What if you put in nuggets and out came chickens?
Image adapted from original at Meat-a-Morphosis ( Patty Hill &  Michael Word)

Even more fun: for students still coming to terms with f(x) notation, faced with this:

doesn't this convey the same idea even more clearly?

Put a chicken into the nuggetiser, then into the de-nuggetiser and you get back your chicken
(OK , maybe with  few ruffled feathers and a lot of squawking!)

Who would have thought there was so much laughter in a lesson on functions? Thanks to the Meat-a-Morphosis team at Kealing Middle School, Austin Texas for a wondeful teaching resource.

Who's afraid of the error monster?

Maybe it's built into our very survival instincts : if something is wrong, it's uncomfortable - so run away and hide from it. We see it in class every day, every hour - students (and teachers!) running away from errors. No-one wants to be wrong, or even worse, be seen to be wrong. And yet, when it comes to learning, "errors" are valuable tools. I see it as one my most important roles as a teacher to convince students not to be afraid of errors - on the contrary, to look for them, appreciate them and share them.

When I talk about errors with my students I introduce them to the Error Monster:

Image from
http://conservationbytes.com/2009/10/21/sleuthing-the-chinese-green-slime-monster/ 
(by CJA Bradshaw?)

We discuss ways in which this scary monster is in fact a good friend - how every error we make, or another person makes, is a valuable gift to our learning.  I encourage students to face their monster head on - whenever they do an assessment and see their mistakes, to run towards their monster and embrace it:

Adapted from CJA Bradshaw's slime monster.

Now our monster becomes an object of fun and affection - helping overcome embarrassment and disappointment at making mistakes - and allowing us to instead focus on resolving those errors.

Teacher notes:

• I like to use the analogy of a blind person using a walking cane. How could they see where to go, if they didn't make "mistakes"? It's only by having the cane bump into things the person can see where to go. Errors help guide us on our learning path.
• You have to walk the walk : be happy - and not embarrassed - to face your own errors in class. I highlight to my students my specific weaknesses when doing algebra : I know (and they know) I make silly errors with signs and expansions - so I laugh at my monster and then keep a careful eye out for him. I make a show in front of the class of checking for my common errors. Hopefully over time I will get better at these!

A visit to the Function Zoo

Do you remember your early encounters with the animal kingdom? So many wonderful different animals - it may even have been a bit overwhelming at first. But very quickly we learnt to group the animals into a scheme that made sense to us. In mathematics we have a similar extravaganza of different 'animals', which can be overwhelming for students to make sense of. Enter the idea of The Function Zoo - first introduced to me by Mary Barnes in her amazing Investigating Change books.

Here is how I worked the idea into a Year 11 class, several lessons into the Functions topic:

A look at the different species of animals ....

... and how we might organise them.

The challenge:

Students worked in groups of four, using large sheets of butcher paper to sketch their ideas. There were at least two laptops per group and the students had just enough GeoGebra skills to be able to turn algebraic expressions into graphs. 

The results were incredible: great conversations between students about functions. With GeoGebra on hand, I was able to encourage students to explore their questions, rather than give them answers, and even ask them more questions if they were ready for it.

Twenty minutes later I quietly threw this slide on the screen but otherwise said nothing:

The groups noticed it soon enough - and went wild. Seeing a few more functions they knew but had forgotten gave them new energy to keep going. Others asked each other questions, trying to work out the graphs they didn't recognise.  Most recognised the last graph from our "explore your calculator" game. We then debriefed as a class, and explored why the idea of the Function Zoo is helpful and interesting. Apart from the obvious benefit of being able to organise our thinking, the real benefit comes in being able to make connections - as I suggested in these slides:

As often happens in student exploration activities, the class produced something unexpected, a gift from them to extend the lesson idea.  One group drew the absolute value of a quadratic function - a blend of two of our function families. We decided this new function was like the cross-species breeding you sometimes see on display at the zoo : the lion bred with a tiger to make a liger.

Absolute value of a quadratic function : a "liger" in our function zoo.
Liger drawing: St Hilare (1772-188). Function by GeoGebra.

A fun and powerful idea - allowing students to see that even quite unusual functions can be seen as blend of function attributes they already know how to work with.

Download lesson slides & annotations: (Google Drive)  PDF  PowerPoint

Teaching Notes:

• A graphing tool makes a huge difference to the success of this lesson. Without it, students would spend a very long time plotting to explore their ideas. There is time for careful plotting later - this lesson is about seeing the bigger picture.
• I found the group structure allowed for a high degree of differentiation - I could customise leading questions for each group, depending where they were up to on the functions journey.
• I can't stress enough the value of developing students' GeoGebra skills (or other computer graphing application) when doing mathematics at this level. I sneak some GeoGebra learning into every lesson - even if it's just the class watching me do a quick check of an equation or a graph. Show them one small GeoGebra idea per lesson and by the end of term they will know the product well - especially if they are using GeoGebra at home as part of their study.
• Why am I such a GeoGebra fanboy? Most importantly because all my students can download a copy to use home. GeoGebra is free and runs on Windows and Macintosh and it doesn't need an internet connection to run.

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 http://books.google.com.au/books/about/Investigating_change.html?id=BjOJBR54jkIC 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.

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!

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.

SBG - A view from a rookie, one year later

It's almost a year now since I started using Standards Based Grading ideas in my teaching. Actually it's almost a year since I started teaching! So how did SBG work out for this rookie teacher?

Happy Birthday : 1 year using Standards Based Grading
Photo: Theresa Thomas (Creative Commons)
http://www.flickr.com/photos/theresasthompson/2311733808/

SBG in your first year of teaching

Hmm .. it's a bit ambitious. There are so many things to come to terms with as a new teacher, adding SBG to the mix might be pushing it - especially if you are the first teacher in your faculty to try it out. I took the plunge - probably a little foolishly - SBG seemed just so right for so many reasons. I did at least have the sense to try a full implementation with only one year group.  

Preparing SBG resources (quizzes, grading sheets) and the extra marking will take more time, something in very short supply in your first year,  but it will help you develop some critical teaching skills. Creating and using a list of standards will ensure you know the content, help you focus on what is important to teach, and provide clear direction for your lesson planning. Using quizzes on a regular basis will give you continuous feedback how your teaching efforts are translating into actual, measured student learning.  It's your first year - and you most definitely need the feedback - the sooner the better to help develop good habits. As a new teacher, you are likely still learning the basics of providing and receiving student feedback, so SBG will provide you a scaffold for these essential teaching skills. If you aren't using SBG, you will likely not find out until end-of-topic tests - which is too late to find out you were teaching it the wrong way.

Would I recommend SBG for new teachers in their first year? Cautiously - and only with support from your head teacher. Try it with only one class, carefully choosing a class that will not be overly testing your still nascent classroom management skills or content knowledge. If you don't have head teacher support or a class you feel confident with, then perhaps consider what I'm calling SBG-Lite for the first year. 

SBG-Lite

While faculty and legislative constraints prevented me using a full SBG implementation with all but two of my classes, I found that even in classes where I was not allowed to use SBG for end of semester grades, I used many SBG tools while teaching each topic. Even if you can't use SBG data to determine official school grades, the information gained about a student's learning aligned to your standards provides rich data to include in written comments in school reports and in conversations with students and their parents.

So what does SBG-Lite look like? You still have standards, you still have quizzes, you still receive and give regular feedback on student mastery of the standards. The only real difference is it doesn't directly translate into end of semester grades. It's up to you to convince students their efforts to achieve mastery of standards will ultimately translate into better grades and that tracking standards and repeat attempts on quizzes are worth the effort. It's a harder sell - but you believe it - don't you? While SBG-Lite most likely doesn't help improve motivation and engagement to the same degree as a full SBG implementation (because it lacks the element of direct student control over their marks), it remains a helpful addition to the learning environment.

Of course I would like to be able to use a full SBG implementation with more of my classes, but sometimes you just have to adapt with your current system - especially when you are the new kid on the block. Pushing too hard will just trigger a response to shut down SBG in those classes where you can use it.

SBG and the challenge of "Working Mathematically"

A serious problem in most mathematics assessment practices, and consequently in most mathematics teaching practices, is an overemphasis on procedural skills development, often at the cost of understanding, reasoning and problem solving. There is a risk that using SBG amplifies this focus on skills fluency - reducing learning mathematics to a list of skills to be evaluated. But it doesn't have to be that way - it depends on the standards you use. My solution was to label my standards with the "Working Mathematically" proficiencies (Understanding, Skills, Problem Solving and Reasoning). This helped me see the balance (or lack of it) in my learning outcomes as well as offering guidance as to the different types of  learning strategies which help students master the different types of standards.

SBG: What worked ...

Can I categorically say SBG improved the results of my class? No. But I can say the class did well in their final results - most students mastered most of the standards, as shown in end of year tests, and I do know we had a very positive class environment - so I stand by a 'did no harm' claim while suggesting SBG may have led to better outcomes.

Students responded well to SBG - in their feedback they indicated they appreciated having more control over their grades. Many students and even some parents remarked at the end of the year how student attitudes to mathematics had changed from one of fear and anxiety to one of feeling able to do the work - and even enjoy it! SBG definitely helped me develop my teaching skills, providing an anchor in the maelstrom that is the the first year of teaching.

.. and what didn't work

Teaching for my first year was challenging - and that's an understatement. As the year progressed I found it harder to be consistent in my SBG implementation. In the rush to complete my teaching program, SBG sometimes fell by the wayside. I'm hoping with more experience this won't happen in 2012. 

In the middle of the year, I tried using SBG with my youngest class, fresh from primary (elementary) school  but I couldn't get the class in a 'learning about learning' space and I had to give up. I'm not sure if this was because SBG is too demanding for students of this age group, or if I need more experience working with younger students - possibly the latter. I'll find out this year because I'm trying SBG again with the same year level.

One of the central ideas of SBG - being able to reassess individual standards - started to slip - most likely because I didn't promote the link between quizzes and the end of year mark sufficiently - so as the year progressed, students didn't retake quizzes. One solution would have been to use a web-based tool like ActiveGrade - unfortunately several factors conspired against this - not the least an inflexible attitude from my school district network administrators who deemed the application unsafe for schools (go figure).

Three ideas for 2012:

• SBG for the struggling, disengaged class: Like most mathematics faculties, we use so-called 'ability streaming' to allocate students to classes. As a consequence, we end up with entire classrooms of students with a well-established pattern of low mathematics achievement and high levels of disengagement, factors which typically get worse as the students progress through the school system.   Previously I was concerned SBG would be too hard to manage for this type of class, but now I realise we have nothing to lose by trying something different. I'm going to try SBG-Lite with one of these classes (unfortunately full SBG isn't an option due to faculty grading policy for this year level).
• Provide better visibility of student standards achievement: At the close of each topic, I will provide each student and their parents with a view of their grade book, making it clear to them which quizzes they might wish to retake in order to improve their topic result.
• Leverage SBG through student summary books. I have designed workbooks made of one or two standards per page with lots of blank space between each standards where students can write their own summaries of what the standard means to them.

The verdict after the first year? SBG was a valuable and helpful tool - it's a core part of my teaching practice. I'm ramping it up for Year Two!