Getting the most out of graphing software

"GeoGebra is your friend!" - my students must have heard me say it a hundred times.  If a student asks me about a homework question, they know my immediate response : "Did you check what it looked like in GeoGebra?". If they haven't, then I will usually ask them to sit with me while we explore it together using the software.

Some teachers worry using mathematics software will weaken student's skills, but here's a mantra I recite in class which I believe not only develops mathematical skills but also stimulates deeper learning:

I believe the essential ingredient in using graphing software to answer questions is to stop and think before using the software and then predict what you expect the software to display. If you are fortunate, you'll find the software doesn't match your prediction. I say fortunate because you have discovered a misconception, an error - or in some cases, managed to confuse the software. Prediction and the subsequent reveal of an incorrect prediction is a powerful learning tool.  With a positive attitude to the error monster this revelation will stimulate questions and further exploration.

Another key learning idea I advocate is to take a few extra minutes once you have your answer to extend the problem with some "what if?" questions: "What if I changed that positive x to a negative x? What if that was to the power 3, not power 2? What if that parameter was 4 not 5? Can I reflect that curve?" Here the power of the software comes to the fore: we can ask many questions and rapidly get answers - something not possible in reasonable time without the software. Of course students won't have the time to do this for every question, but even just doing this once in a study session is rewarding.

One more powerful pedagogical factor is at work when students use a graphing tool to help with their homework: they are forced to translate their problem into a representation suitable for the tool. For example, an algebraic equation has to be split into two (or more) graphs and intersections found. This serves to build and reinforce understanding of the links between the different forms of mathematical representation. Often a student needs break down the problem into steps, introducing parameters and intermediate results or constructions, providing 'hooks' they can use to explore how the problem changes as parameters are changed. 

A topic I recently taught was based totally on drawing graphs by hand - and students have to be able to do this in an exam situation, without software.  For a course like this, I think the graphing software is an even more valuable learning tool. Why check your answers in the back of the book when you can do this:

This approach means students are still learning to work by hand - and maximising the benefits of having software during the learning of the topic - without becoming dependent on it - a bad thing at exam time!

So to my way of thinking, there's no question dynamic geometry software is a powerful learning tool: when coupled with a mindset that thinks and predicts prior to using the software, and then extends a problem through questioning and exploration with the software - it's like having a personal tutor. GeoGebra is indeed your friend!

Practicalities: There's lots of good quality dynamic geometry and algebra software available to students: I'm a big GeoGebra fan, and I also like the Desmos tool. I'm beginning to really appreciate AutoGraph - but sadly the cost factor rules it out for most of my students.  For intensive algebraic work, I point my students at WolframAlpha - especially the WolframAlpha iPad app which is great value.

Exploring inequality : an entry point to calculus

"Have you ever noticed .... ", I said to my senior maths class, as I walked in bearing a huge and very obvious glass bowl containing about 40 packets of Smarties, ".. how some people seem to have so much more than other people?"

"Taking it Back - Occupy Oakland" by Glenn Halog
http://www.flickr.com/photos/ghalog/6271929376/in/photostream/ CC-BY-NC-2.0

I then proceeded to "share" out the Smarties: first I gave 20 of the 40 packets to one student - making a huge pile on her desk. Her eyes popped out - while the other students looked with disbelief and some concern for their own anticipated share. I gave a wicked grin and 10 packets to the student next to her. To the rest of the class I handed out 2 or 1 packets - apart from a few students at the end of line who received nothing. Oh the looks they gave me!

And so we started a lesson exploring the question of how we could measure income distribution - a hook (although the class didn't know it yet) - to introduce our next calculus topic: integration.  Here are some notes on my first attempts at a lesson design using an idea from economics as a motivation why we might want to find the area between two curves.  But first a big thank-you to mathematics teacher Alastair Lupton who showed me how to bring the Gini Coefficient into the classroom and encouraged me to try it out in my classroom.

So here's the sequence I tried this year.

Step 1: Build interest in the problem. With strict instructions not to eat or worse yet - share - their Smarties, we looked at a short OECD video about the rising inequality in income distribution:

Depending on the time available, you might want to explore some other video material, perhaps some recent news footage of the Occupy movement protests, or look at some studies of global income distribution.

Step 2:  Thinking how to organise the data: I lined up the students, holding their very unequal distribution of Smarties. We ordered the line by 'income' and partitioned into 5 groups - helping the students see the organisation of the data into quintiles.  We returned to our desks and looked at some local and international data on income distribution, also organised into quintiles. Here is some recent Australian data:

Click on the image for a larger view.
Source: Australia Bureau of Statistics 6503.0Household Expenditure Survey and
Survey of Income and Housing User Guide 2009-10

Step 3: Ask the question: "How could we measure inequality?" This isn't easy or obvious. Give the class some time to explore ideas. Then it's time to look at how economists do it...

Step 4: Develop the idea of  graphing cummulative quintiles.  After trying some different ways to plot our quintiles, I showed the students how the economists do it: reorganising the data into cummulative quintiles. This allows us to make normalised curves which work for all situations, regardless of the size of the total income pool. We drew our first Lorenz Curves:

The Lorenz Curve is used to calculate the Gini Coefficient. The area A is the difference from total equality.
The larger the area A as a proportion of the total area A+B, the greater the inequality.
Source: Wikipedia Lorenz Curve Image by Reidpath,

To help explore the idea, we discussed what the Lorenz Curve would look like if one person had all the Smarties, and if all the Smarties were shared equally.  We also considered if the curve would ever go above the "Line of Equality" (it won't!).  We selected different data sets (see references below) and plotted them.  Here is the 1993 World Bank data for Nigeria plotted in GeoGebra, with a polynomial fitted to the curve:

By modelling the curve with a polynomial, we can use integration
to calculate the area under the curve and hence the area between the curves.
Data is entered into the GeoGebra Spreadsheet window, then plotted and
a function calculated to fit the data using FitPoly[].

Step 5: Ask the question again: how could we measure the inequality?  After looking at a few different data sets, students will quickly come to the conclusion that measuring the area between the line of equality and the Lorenz Curve will give us a nice single number. And now you have them hooked: here's a very interesting and practical reason we might want to be able to calculate the area between two curves.

Step 6: Declare a communist revolution.  I then ordered a redistribution of the Smarties so everyone was equal.  This was actually quite funny because several of my diet conscious students insisted they did not want any Smarties. Tongue-in-cheek I told them this was not an option - it was a revolution and everyone had to be equal whether they wanted it or not!  A nice opportunity to open up the discussion to different views about income distribution.  I gave my students a selection of recent articles from The Economist which seemed to provide a good balanced discussion on the topic.

Step 7: Begin the mathematical discussion on ways to calculate the area between the two curves. Your students will have many useful ideas! Try them out with the tools available. And now you're ready to start a calculus based exploration: What is the area under a curve? 

Where could you go with this lesson idea?

• Get students to make up a small poster using their data and stick them up on the wall. Then as you move through the Integration topic, you can refer to them in the context of each new idea.
• Once students know how to integrate, get them to model their curves as a polynomial - I like to use the GeoGebra FitPoly[]function - and then do calculate the integral, comparing their result to given Gini Coefficient for the data set.
• The student data makes for a great application of the Trapezoidal Rule : they can calculate the area without knowing the equation of the curve.  A good example of why you might want to use the numerical approaches to calculating integrals.
• Challenge activity: calculate the area under the curve using Simpson's Rule. If you only have the standard Simpson's Rule, you can't do it because there are an even number of data points! But there is more than one Simpson's Rule - challenge your students use the internet to find one that will work for this data. [Hint: Simpson's 3/8 rule will work].
• Apply the concept of the Lorenz Curve to another field of study. An interesting application is to social networks - some people contribute significantly more than others, while others 'lurk' in silence. I use edmodo with my class and there is a high degree of inequality in the number of postings per student - counting postings per students could make for an interesting Lorenz Curve.

Thinking beyond the mathematics:

• Talk to the economics teachers at your school. I discovered mine do teach the Gini Coefficient, but they don't go into how it is calculated.  I think it could be a very powerful lesson to develop a  sequence of combined economics/calculus lessons with an economics teacher at your school. The more I explored the subject, the more interesting I found it. Options to consider include: the effects of taxation policy on the Lorenz Curve; the differences in the Gini Coefficient between different types of economies; differences within one country over a time series; challenges to the validity of the measure; economic and social arguments on the topic of income distribution.  All highly suitable for deeper mathematical and social science exploration.
• Take some time out to look at the Gap Minder website which options to view the data through the Gini Coefficient.

Resources

• The Wipedia pages on the Lorenz Curve and the Gini Coefficient are a good starting place, with good entry points to more nuanced discussion on the use of the Gini Coefficient.
• Some excellent Australian data and good explanatory notes in the ABS publication 6503.0 Household Expenditure Survey and  Survey of Income and Housing User Guide 2009-10. See Table A1 in Appendix 1, and a high quality discussion on the Gini Coefficient in Appendix 3.
• International data from World Bank: World Development Report  2000/2001: Attacking Poverty - Table 5 Distribution of Income or Consumption 
• The October 23, 2012 issue of The Economist contains some excellent articles on the challenges of income inequality seen from a pro-Free Market view. I found these particularly interesting given one could hardly call The Economist left wing!

Some teaching reflections:

• The students really loved the lesson - they were engaged and it was interesting.
• I planned carefully for my 'inequitable Smarties distribution'. Our class was well established and we knew each other well enough that my students would know I was up to something and trust me when I played this game. I also made sure the students who didn't receive Smarties were the most resilient, confident students.
• I did however make the mistake of trying to do this opening lesson in a single 50 minute period - it wasn't enough time and I rushed it, making it less student centred than I had hoped. This lesson needs a double period to do it justice. 
• Is it worth taking the time out from a busy course to do this activity? I think so. Once I realised I could leverage this work into my teaching of the Trapezoidal Rule, Simpson's Rule, the area between two curves and also do some polynomial modelling, I saw it was a lesson that  just "keeps on giving".
• Coming from a physics background, it was wonderful to find an interesting and practical application of calculus to a completely different field. Many of my students are planning a career in business and are interesting in economics - here was something to show them the calculus applied to money as much as to speeding particles!

This is part 2 of a sequence of posts on teaching integration. 
Part 1: Slicing and Dicing.  Part 3: Integration in the world around us

Polynomial stories

What's not to like to about polynomials? They look amazing - and they are just great fun to play with - especially if you have dynamic graphing software to explore their shapes. Here are a few teaching ideas I developed over the last few weeks.

First and foremost we need a character : meet Polly the Amazonian parrot. There's a reason she is from the Amazon... you'll see soon.

Amazona agilis by Jacques Barraband(1767-1809)
http://commons.wikimedia.org/wiki/File:Amazona_agilis_-_Barraband.jpg

Polly featured throughout my lessons - my favourite was the zero polynomial $P(x) = 0$

Reminds me of a Monty Python sketch ....

and later when we looked at taking the second derivative, then the third, fourth, and fifth derivatives - the disappearing Polly:

The Disappearing Polynomial

I'm a big believer in having 'characters' to help teach mathematics - I think they act as 'mental anchor points' to help link related concepts, and then make high level linkages across topics more visible - as in the example above of the disappearing polynomial.

Then we need to explore the properties of polynomials. If we have a quartic polynomial, how does its graph change if we change the roots? I got a good response to my students with this homework exploration tool - I used it for a 'flipped' lesson:

Locus and Parabola Make a Parabola GHT0301

Lesson 02 FLIPPED Graphs of Polynomials

Early on in the presentation of polynomials, I think it's a good idea to show some of the interesting applications:

Using polynomials for modelling. I used this example of a photograph of the Amazon river, loaded into GeoGebra, fitted to a polynomial using the FitPoly function.  Why would we want to do this? I suggested in this case, having an equation for the Amazon River could help us model water flow - perhaps helped by working out the gradient function:

Amazon River – photo from NASA. 
Curve fitting using GeoGebra FitPoly[] function.   
http://commons.wikimedia.org/wiki/File:Amazon_57.53278W_2.71207S.jpg

Using polynomials to approximate other functions: A good time I think to introduce the Taylor Series:

No need to go into deep explanations - just show what is possible with polynomials. I returned to this idea in the next topic when showing higher derivatives, and have another visit planned when we do complex numbers to help demonstrate the famous Euler Formula.

And finally, after demonstrating the closure of polynomial operations for addition, subtraction and multiplication, students may find it interesting to learn about the role polynomials play in many encryption systems.

I looked hard for an online paper suitable for advanced high school students - the best I have found so far is Christopher Cooper's notes for his "Languages and Machines" course at Macquarie University.

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More GeoGebra HowTo Sheets

Toys and tools for exploring the Parabola

Following on from three ideas to introduce locus, here are three ideas I used to help make the locus of the parabola come alive for my students. Regular readers of this blog will know how much I believe in the benefits of hands on exploration of mathematical objects - and these are very hands on!

The three ideas are:

• Use a MIRA mirror to construct a parabola. My senior students loved this activity- a chance to revert to back to childhood, while still being challenging.
• Use GeoGebra to construct a parabola given any arbitrary focus point and directrix. Try this with non-standard orientations.
• Be entranced by a wonderful 3D optical illusion toy that exploits the properties of the parabola.

1. Using a MIRA mirror (MIRA math tool) to construct a parabola
I'm extremely fortunate to have a box of these in my faculty storeroom:

Source:  http://www.enasco.com/product/TB14953T 

While they look like tools for the junior math room (and they are wonderful to use in this context!), there's no reason our senior students should be locked out from using them! Here is a worksheet that give instructions on constructing a parabola with the MIRA mirror.  It's a really fun activity - a chance for senior students to play a little - and a great opportunity to ask the "why" question - reinforcing the idea of locus and the locus definition of a parabola.

Locus and Parabola MIRA Parabola GHT0501

2. Constructing a parabola using GeoGebra.
Why do I need special help to construct a parabola in GeoGebra you may say? Of course GeoGebra can construct a parabola with zero effort. But this guide explores how to construct a parabola using the locus approach.

Can you find the parabola given any arbitrary focus
point F and directrix AB?

Locus and Parabola Make a Parabola GHT0301

3. Discover something special about the parabola
A terrific toy worthy of being in your mathematics (and science) classroom is the Optigone Mirage®.

The Optigone  Mirage® is a pair of twin parabolic mirrors, arranged to project
a 3D image floating above the top of the kit. In this image from a paper by Christian Ucke,
the pig is actually inside the mirrors.

As always, encourage students to play with the toy (not that they will need encouragement - my students had their smartphones out takings photographs within seconds!), then ask the Why? question. Even though I purchased* one of these for my senior mathematics class, students across all my classes were entranced by it - and it gave me great pleasure to say to the juniors "you will learn how it works in your maths and science classes in a few years".

* Sadly I could not find a convenient way to purchase one of these in Australia - so I ended up buying a  clone from Australian Geographic.

How do you do that in GeoGebra?

Looking back on my own high school mathematics education, I realise I never really knew what a parallelogram was. I never knew how it 'worked', how its angles and diagonals operated, how they changed when the slope of the parallel lines was changed. The rhombus? All I could really say - if I remembered it at all - was it was a kind of squashed up square. If only I had  been given a dynamic geometry tool to play with! As a teacher now, I strive to have my students actually touch mathematical objects - to move them, push them, pull them, to watch what happens. I'm convinced that if students do that, so long as they are reflecting on what is happening to the objects (and why), they will remember them for life.  And the ideal tool for hands-on interaction: GeoGebra.  Free software, runs on Windows, Apple and Linux (anything that runs Java), backed by a community of hundreds of thousands of teachers using and sharing GeoGebra resources.

A resource I haven't found yet though is a set of simple, one page instructions I can give to students showing how to construct a certain mathematical objects in GeoGebra, so I have begun building some.

Here's the first installment:

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For Junior and Middle School

How to construct a parallelogram.

How to construct a rhombus. This one appears simple but can be confusing - practice it first before giving to students.

For Senior School

How to construct a parabola using a focus point and a directrix.
How to build a polynomial factor exploration tool.

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I have put these links on a new GeoGebra HowTo page in this blog. These files are also available at the Maths Faculty sharing repository.

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.

Getting personal with rates of change

The key to mastering calculus seems to be gaining a good understanding of rates of change, how this relates to the idea of a function and then seeing how we can use the tools of algebra and geometry to develop the gradient function. Now as exciting as it is for some of us to play with a quadratic or a cubic function, I recently discovered, quite by chance, a very personal and highly engaging way to explore rates of change of a function.

This data is all about ME!

I was exploring the introductory concepts with a Paul*, a teenager just starting on the calculus road, when he made the connection that he was experiencing a very dynamic change process: his height had started shooting up in the last few years and very soon he expected to be nearly 2m tall.  He knew he was experiencing a "growth spurt", growing at a faster rate than when he was younger. Just as I was wondering how we could use this connection in our exploration, Paul told me his parents had been marking his height on the kitchen wall for the last 10 years. Wow! This was exciting - some real data we could plot and explore.  Paul measured the markings off the kitchen wall and made a table showing his height at different ages. And here's what we were able to do with that data in GeoGebra:

In the process of this exploration we uncovered many ideas about slope, functions and the use of modelling, each time applying them in a context Paul had a profound personal interest in - we even named our  polynomial the "Paul function" in his honour. I'm certain he will never forget the idea of rates of change, the gradient function or the power and fun of modelling a function based on data points.

Do your students have a wall somewhere with their heights measured over the last 15 years? If so, I highly recommend working this into your calculus activities.

I'm a deeply indebted to the work of Mary Barnes, in particular her 1999 series "Investigating Change", on teaching and learning calculus, still available at Curriculum Press. Google Books previews are available. Thanks also to a new Google+ friend and teacher Steve Phelps for showing me how to use the GeoGebra FitPoly function. Using GeoGebra to build a gradient trace function comes from the original GeoGebra documentation by Marcus Hohernwater, however I have found in practice it can take students some time to understand what is going on. Priscilla Allan's YouTube demo shows a good way to use colour to make it clearer and I have extended her idea to actually show the trace point moving along the x-axis, prior to adding in the gradient value. 


And special thanks to Paul (*not his real name of course) and his parents for allowing me to share this story of our ongoing exploration of mathematics.

Why is maths different when it comes to laptops?

Continuing the series on 1:1 laptops in the mathematics classroom. This post may be a little uncomfortable for all of us, but the factors considered come up again and again for all mathematics teachers - even those of us (like myself) who have drunk the Cool-Aid and are eagerly looking for ways to enrich our teaching through use of technology. In later posts we shall consider the wonderful and amazing things mathematics teachers can  do with laptops - but first we need consider some of the barriers.

In the previous post, we looked at what for many people is an unexpected finding: mathematics teachers have their students use laptops much less than teachers in other subjects. Some reports put this figure at 50% less than other subjects.  When all other factors are taken into account - access to technology, training, confidence, skills - we still find a reluctance to use the 1:1 laptops in the mathematics classroom. And so we ask: Why do maths teachers make these decisions? Is there something different about mathematics?

My research has led me to conclude there is indeed something different when it comes to mathematics.

Mathematics teacher beliefs + mathematics teacher practices :
a powerful combination which often acts as a barrier to using technology.

A core set of beliefs about mathematics and mathematics teaching in conjunction with some strongly entrenched mathematics teaching practices act together as a powerful barrier to widespread use of the laptops in mathematics classrooms.  I see the three key themes at work:

• "Maths is something you do on paper"
• "Laptops aren't suitable for low achieving students"
• "The teacher leads, the student follows"

I make no explicit comment on the validity or otherwise of these commonly held mathematics teacher beliefs and teaching practices - but there is no getting around their effect on 1:1 laptop programs.

"Maths is something you do on paper"

When you ask mathematics teachers what 'real' maths is, and how you 'really' learn it - pen and paper, and I really mean pen and paper the physical media - eventually emerge as a key requirement. Software may be fine to demonstrate and maybe explore mathematics (for some mathematics teachers)- but it's not properly learnt until it's done on paper.

Now consider the very strongly established practice of managing student learning by working in and monitoring output in the student exercise book. Learning outcomes aren't tangible - can't be verified until they are seen in the exercise book.  Entire sequences of classroom practice, homework, outcomes tracking are based on physical movement in and around the student exercise book. You won't find this combination of beliefs and practice in most other subjects. No-one would say you don't understand science, or history, unless you do it on paper. And other subjects are much more receptive to accepting digital learning artifacts as evidence of learning.

Unfortunately there is also a technical hurdle : unlike other subjects, writing in the language of our subject with a computer is hard. It's actually very awkward to write a continual flow of mathematical ideas with standard or even specialised software. 

Try writing this   without taking a software detour. Now do twenty lines of it. For now at least, the technology gets in the way of expressing the ideas. While there is powerful and non-intrusive software such as GeoGebra for exploring and demonstrating some parts of mathematics, actually writing long sequences of mathematical language is hard work on the computer.

So: combine the belief that real mathematics is done on paper with a key teaching practice based on writing in exercise books and there isn't much space left for using laptops beyond peripheral extension activities.

"Laptops are not suitable for low achieving students"

We have a real problem in secondary school mathematics: many students are not achieving the learning outcomes. It's no surprise these students don't enjoy maths and are looking for other ways to occupy their time and energy during maths class. Now ask mathematics teachers if using laptops might help make the classroom more engaging, or possibly even provide new ways to help these students with learning mathematics. The answer is a pretty resounding 'no' - there is a widely held belief that laptops are not suitable for low achieving students. Two lines of reasoning are offered: the low achieving students are actually incapable of using the software; and the low achieving students are using the laptops to escape from mathematics and instead engage in off-task behaviour - watching videos, listening to music, playing games. Not like the high achieving students who want to use their laptops for maths.

Some mathematics teachers strongly believe it is in their lower achieving students' best interests to turn off the laptops. These students need to do more maths, and allowing them to use the laptops, which provides more distraction, is actually harming them - teachers motivated by care and compassion for their students make the decision to block use of the laptops. Personally - I don't agree with this approach - indeed I believe the laptops offer us possibilities to re-engage students with mathematics - but this reaction is understandable and consistent with those teachers' beliefs.

Now consider the strongly entrenched teaching practice of  ability streaming, used in mathematics faculties across the country almost without exception, and to a degree not seen in any other school subject. We put the highest achieving students in one class, and then progressively lower achieving students into progressively "lower" class groups, creating entire classes of disengaged, low achieving students.

Combine the belief that low achieving students can't or won't use the laptops for learning with the practice of ability streaming, and we have effectively created entire classrooms where the laptops just will not be used. And indeed this seems to be the case.  Chances are when it comes to secondary mathematics, you will see the laptops being used almost exclusively in the top achieving classes.

"Teacher leads, student follows"

And finally, we consider the strong prevalence in secondary mathematics education of the idea that the teacher should show-and-tell, and that students should follow-and-practice. While it would be an overstatement to say this is always the case, it is the prevalent belief among maths teachers.  A student armed with a laptop can be disruptive to 'teacher leads, students follows' - and although the presence of the laptops doesn't automatically guarantee a change in pedagogy, the benefits of the laptops seem to me to be diminished if they are merely used to automate lead-and-follow practices.  This combo of belief+practice isn't unique to mathematics teaching by any means, but I do think we are more likely to follow traditional teaching and learning approaches than other subjects.

In conclusion ...

So by considering these three powerful belief+practice combinations, which are to a large degree unique to secondary mathematics education, we can begin to see just why laptops are used at up to 50% less than in other subjects. I find that even in my own practice, keen as I am on using technology with my students, I'm often falling into these memes: I do worry about not doing the maths on paper - "is it real maths?", I do worry about exercise books, and I do find myself dismissing using the laptops with my lower achieving students. And I catch myself 'holding the mouse' often.

Am I optimistic about using technology in the mathematics classroom? Absolutely. But I also recognise there are powerful beliefs and practices in our subject domain - and these contribute to making using the laptops harder in mathematics teaching and learning.

This post is high level summary of research I conducted during 2010. The study examined the use of the laptops in mathematics classrooms at five schools, looking at usage levels, how the laptops are used and the role of teacher skill, confidence, knowledge and beliefs factors. The work builds on a body of knowledge as found in nearly 100 published papers on technology in education, mathematics teaching using technology, and mathematics teaching beliefs and practices. An academic paper is currently in preparation.

Powerful learning with GeoGebra - but who is holding the mouse?

This is the second of two short reflections on using (or not using) laptops in my maths teaching.

In the last few weeks I have had the pleasure of tutoring a bright young student, let's call him Paul*, who needed just a little extra help with some of the more advanced parts of the Year 10 math syllabus. Heading out to my first session with Paul, I instinctively brought my trusty Asus EeePC ready to fire up GeoGebra. I say instinctively because I just can't imagine doing any sort of advanced mathematics without GeoGebra around - even just for my own purposes. GeoGebra is just part of my kit.

And sure enough, within minutes, Paul and I were exploring the ideas from his course textbook using GeoGebra - what does the parabola look like when when we change parameters? Concepts that Paul had missed previously came to him faster than I could demonstrate them - he saw what was happening in the dynamic graphs even before we discussed them.  But then something even better happened - Paul started asking lots of 'what if' questions - what if we did this to the function, what about that? GeoGebra let us freely and rapidly explore his questions. Later as part of practicing the learning, we played what I call "graph racing games": I wrote down a function like   and gave Paul a few minutes headstart to graph it by hand on paper, and try to get it done (correctly) before I could graph it using GeoGebra. It didn't take long before Paul could do every variation without error.

We moved on to trigonometry, and I asked Paul if he had seen the graphs for sin(x) and cos(x) - he had and we drew them in GeoGebra to be remind ourselves. And then he asked - unprompted - what about tan(x)? He hadn't seen it yet. I was about to show the graph, when a voice from a past teacher of mine said 'Stop!'. I asked Paul what he expected. "Sort of like sin(x), just bigger - maybe it goes from 4 to -4".  We had a look at the graph - Paul nearly fell of his chair:

Even more fun to be had when we moved from solving equations like   to slightly more tricky ones like this

Without GeoGebra we would have just done the problems as an algebraic manipulation - but we did have GeoGebra - so no reason not to take a peek ahead of what is coming in Year 11. Certainly got a reaction ....

This is not what Paul expected to see when solving
an equation rearranged to be a quadratic!

After each surprise, we took time to explore the reason why the graph looked so odd - Paul hadn't seen asymptotes before - and uncovered all sorts of number properties along the way.  I was really thrilled to see how powerful teaching and learning can be when there is a laptop with GeoGebra right there in front of the student.  But there was a problem .....

Who is holding the mouse?

It took me two weeks to realise - but all this time, I was fiercely keeping hold of the mouse. I was driving the application. My own love of the software, the power of the exploration and the idea that I was the teacher was leading me to be the driver of the application.  Perhaps I also wanted to stay focused on the math, and didn't want Paul to get distracted using the software. Finally I let go of the mouse - and let Paul do all the driving. He worked out how to do the basics in a few minutes (seriously!) - and before long, he was investigating his own questions - and even decided to name a theorem after himself.

Paul worked out these two triangles are similar. Knowing this
fact made an entire set of exercises in his textbook trivial.

Seeing how powerful using GeoGebra was, and knowing it's a free application, I suggested Paul install the GeoGebra java app, or even just the WebStart link on his laptop. To my surprise I discovered his school laptop doesn't have Java installed (to stop students playing games), that there is no math software on his laptop, and that in fact, his maths teacher doesn't even allow laptops to be used in their class.

Paul's school managed laptop has no maths software,
and the laptops aren't allowed in maths class.

Fortunately Paul can use GeoGebra at home!

Some take home thoughts:

• Who is holding the mouse? Are you really prepared to let go of the control and allow the student to take the exploration where they want it to go? Will you let them make the mistakes and wrong turns using the software that are so essential to learning?
• When using graphing software - never just show the graph. Stop! Ask first for a prediction. Then see what the graph looks like. And then ask that all important question: Why does it look like that?
• A teacher's personal use of a maths tool makes it much more likely they will use such software with their students.
• Using maths software on the laptops is best when the technology is (almost) transparent - you want the conversation to be on the math, not on the tool. Until the teacher and student have this fluency, using the laptops for math is hard work.
• Just because a school and a student has a laptop - don't assume there is math software on it, and don't be surprised to find the math teacher actively prevents use of the official school laptop in their class.

In the next post in this series, we shall look at the current research on one-to-one laptops and their use in mathematics classrooms. 

* Paul is a pseudonym. My thanks to Paul and his parents for allowing me to share this story.