How to get your math class SCREAMING

I know students are not supposed to be using mobile phones in class for private communications*, but I couldn't help but smile when one of my students showed me a text message she had just received from a friend in the class next door : "What are you guys doing in there? We can hear you screaming!"  

If you don't know who these five boys are,
 then you're definitely not teaching at a girls school.

So a little context first (context is everything!): This class doesn't really like maths that much - they tolerate it - I try my best to make it relevant and pleasant, trying to raise their confidence and skill levels. We've been studying a fairly dry topic for the last few weeks - they've done reasonably well in the topic test but need more practice. Looking for something engaging to make the second half of a long double period interesting, I turned to Stu Hasic's Quiz Boxes.

Download Stu's Quiz Boxes at http://quizboxes.com/

Quiz Boxes offers a Jeopardy! style game with questions of increasing complexity organised into categories, with a high stakes question at the end. Students love this game - and with careful planning and implementation (you will need to design the questions) it makes for a terrific fun period with high levels of engagement and gets students doing a lot more maths revision than they might have otherwise intended :-). There are many ways you can use Quiz Boxes so I would like to share an approach I have found that works well for classes of all levels of maths achievement. 

Quiz Design

• Choose categories that students are interested in.  Current hot topics are "One Direction", "Justin Bieber", "Beyonce", "You Tube Hits" and "In the Movies".  Find whatever your class is interested in. Once they play the game, they will suggest topics to you. Since I don't know that much about One Direction, I go to Wikipedia and collect the factual information I need. Find some obscure information for the harder questions. Your students will be amazed you know something so detailed about One Direction - and infuriated most of them don't know it.  I like to use student interests for half the categories, and use more explicit math topic categories for the rest.
• Work maths into the "non-maths" categories. For example, my third question on One Direction was "What percentage of One Direction are boys?". OK - it is a simple question - but it reinforces the idea that 100% means "all". One question I found generated interesting responses was "How many records has Beyonce sold?" - which gave a good opportunity to explore estimation. Another One Direction question: What is the name of the band member who is last in alphabetic order?"  Again - it's easy, but it gets some mathematical thinking happening.
• Make the maths category questions easy at the start You want students to engage with the maths categories. I always start with easy questions - if you make them too hard, students will turn off - it's not a game any more. I save the harder questions for the 800 and 1000 point questions. I make the end-game question a more challenging - but doable - math question on the current topic.

The Quiz editor in Stu's Quiz Boxes.
 I find I can reuse the quizzes across many grade levels,
 making this an efficient use of lesson preparation time.

Playing the game

This game is so much fun, and the students get so excited, it's essential to have a management strategy.

• Every group gets a chance to answer the question. This is perhaps the biggest change I make to playing the game: I don't have a "first-answer-wins" approach. In a classroom of 30 students, it's impossible to work out who gave the first answer and the noise levels are impossible if you go this way. Instead every group has a mini whiteboard to write their answer (you could use just a sheet of paper).  Once I see a group has a quality answer (doesn't have to be correct - just interesting), I yell "2 minutes" and give all the other groups time to complete. When I call "time up", we look at all the answers and every group that has a correct answer gets the points.
• Encourage group checking of answers. As the questions get harder and are worth more points, I ask each group to ensure everyone agrees on the answer before presenting it. This gives the group a chance to teach the content to each other. It's wonderful to see students try to convince each other their answer to a maths question is correct.
• Noise level management. This is hard because it's so exciting. Never have you seen a class so interested in knowing what 8% of $200 is! As the noise level rises you'll have to calm the class down.
• Prizes. I confess to motivating with a very small chocolate prize. I give one to every student at the end and don't buy into "but we won...." discussions - as far as I'm concerned everyone is a winner if they participated :-)  Waving the packet at the start of the game gets their attention - but it's amazing how quickly the students forget about the chocolate and become obsessed with winning game points.

Special thanks to Stu Hasic who so kindly donated Quiz Boxes to the education community.  I highly recommend you try Quiz Boxes with your classes. And over time you will develop a bank of quizzes which you can share with other teachers in your faculty - or maybe even at Stu's website.

Practicalities

Here's what you need:

• A data projector (or an Interactive White Board)
• A copy of Quiz Boxes - free download from Stu's web site
• A pre-prepared quiz. It can take a good hour to design a quiz, but you will find you can reuse quizzes across many year levels and they stay current for several years.  You might like to challenge your class to design quiz questions for a category - although this will take some time and planning.
• Students arranged in groups - maximum six groups for Quiz Boxes.
• Mini-whiteboards OR a pad of paper per group.
• Solid walls between you and the classroom next door. Close your windows :-)

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.

Images of Integration

Slice and dice: that's how I think about the calculus topic of Integration - take something complicated, slice into increasingly fine slices, then put it all back together. In my quest to encourage my students to see this theme in the wonderful world around them, here is a selection of images I used this term to help show the idea, generously made available by people around the world through a Creative Commons License on Flickr. If you're taking great photographs - think about sharing them under Creative Commons - a wonderful resource for teachers to help inspire students.

Bay St, St Louis Bridge by Alaskan Dude on Flickr

La Agora, be el.manu on Flickr

L'Hemisferic by el.manu on Flickr

Tower by timtom, on Flickr

Untitled, by SymoO, on Flickr

The idea of looking for visual representations inspired one of my students to take a photo of the magnificent Neuroscience Research Australia under construction across the road from our school - which just screams at me "Area under the curve!" every time I walk past it.

Neuroscience Research Australia building 2012 - under construction. Photo by J Yu - used with permission.

This is part 3 of a series of posts on teaching Integration.
Part 2:  Exploring Inequality - an entry point to calculus

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

Slicing and Dicing

To my way of thinking, the topic of Integration is all about 'slicing and dicing' - thinking about what happens when you take an object and chop it into increasingly thinner slices, then put those slices all back together again. Here's a fascinating and gruesome hook I used in my senior mathematics class this year to consolidate* the theme of "slicing and dicing": What would happen if you sliced up a human being?

Warning: This content is only suitable for a senior class, and you should warn students there are medical images coming up. There won't be any blood, but it might affect sensitive students and the dissection of human bodies may not be culturally appropriate in your classroom.

First we start in reverse, using a scene from one of my favourite science fiction films "The Fifth Element"

Then let your students know the images of the human body used aren't computer generated, but actually come from The Visible Human Project. Cue in this video clip:

My students were grossed out and fascinated - and then asked to see it several more times! It took them a while to come to terms with the fact the images weren't generated using a medical scanning device, but by actually slicing up a body. Lots of questions followed!

Depending on time and if you think this is a good idea or not, there are some websites where students can use an online Java application to dynamically explore the data by selecting their own slices in any orientation and see the resulting image created by reassembling the original slices to your specification. 

Here are two websites I found worth exploring:

http://visiblehuman.epfl.ch/index.php

http://visiblehuman.epfl.ch/index.php

Where to next? Many options for discussion about: 

• the mathematics and computation required to reassemble the data so that different views can be constructed.
• the ethics of using bodies from condemned prisoners for science.
• the value of the data from The Visible Human Project - there were scientific as well as ethical criticisms of the project.
• Recent advances in 3D printing technology to "print" biological components using layers of living cells. A long term goal is to print transplant organs using cells from the donor. A quality video from ABC Catalyst program at http://www.abc.net.au/catalyst/stories/3618385.htm (starting at 00:03:00).

One of my students later told me the data from The Visible Human Project is also used in a (rather violent) Japanese manga film Gantz.

(*) I used this lesson idea in the middle of the topic sequence. For my first Integration lesson, I went down a different path - but that's for the next post!  Part 2: Exploring Inequality

Still going ...

Photo: "Yellow" by darkmatter CC-BY-NC-ND
http://www.flickr.com/photos/cdm/84202849/

It's been a very long and tough final school term. I'm still running the "marathon" - albeit limping on some days. Ran headlong into some very steep hills (teaching Mathematics Extension 2 for the first time, in addition to teaching Mathematics Extension 1 for the first time... madness!). Combine this with the normal teaching load, writing over a hundred school reports and accumulated sleep deprivation - not good. Running too fast, too hard - feels like I've done a year's work in a term.  In recovery mode now - still hundreds of end-of-year papers to mark but only a few weeks to go!

Like all marathons though, the experience is amazing - the views incredible. Lots of teaching ideas share in this blog once my energy levels are restored.

The blue shark of full mastery

"Sir, does mastery count for more than the test mark?", asked one of my Year 7 students this week. I beamed back - "YES!"  Slowly but surely, I'm weaning this class to look beyond their scores ("You got 95%! I got 98%!" ... yes - it's a high achieving class :-) ) and focusing on mastery.  Recently I have been making little mini-report cards which I staple onto the end-of-topic test paper:

My classes now have a symbolic language for achievement levels : the red dolphin stamp is 'Not Demonstrated' and 'Starting Out', the orange seahorse is for 'Progressing', and the orange killer whale is for "Mastery". If you get Mastery for all the standards, you also get the blue shark.  I find the visual imagery helps focus on achievement of the standards.  And it doesn't just work for Year 7 - even my Year 12 students like the blue shark.

My goal with these mini report cards is to make the standards and the student's achievement of those standards prominent - the topic test score is there, but it doesn't dominate the feedback.  Why? Because even in this high achieving class, a score of 90% means there is something students can improve on - and I want to focus on that specific item. I try to write a helpful comment, focusing on the standards that need work and some ideas how the student can advance that standard. While the students are looking at their test, I walk around the class and try to chat to every student about their achievement in terms of the standards and what we can do to raise them (that can be hard with 28 students in 30 minutes!).

These little report cards though reveal a deeper change in my approach to Standards Based Grading....

SBG: Where I'm at now

Time pressures have taken their toll on my loftier goals of high precision SBG implementation - and I have found I'm migrating closer to what Frank Noschese calls "Keep It Simple Standards Based Grading.  Now that I have simplified the system, I find it also makes it clearer and more approachable to students.

Less standards per topic - especially for junior classes. My lists are still too long for senior classes - mainly because I am trying to cover all the syllabus points (there are a lot!). 

Achievement levels: I'm happy with the language of my achievement levels "Not Demonstrated/Starting Out/Progressing/Mastery" - I believe they give clear and honest feedback without being discouraging - they don't say 'you failed' - they say 'you're not there yet'. I'm not comfortable with a simple Yes/No binary decision because I want the levels to support my goals for student motivation and engagement - to reinforce they are on a learning path - I want to recognise their 'progress so far'. A sheet full of 'No' results isn't going to encourage lower achieving students.

The role of quizzes: I have effectively stopped using quizzes for grades.  Woah - that's a big departure from the SBG ethos! Why? Because I believe that meeting standards once in a quiz isn't enough : the student has to retain the standard. So for me, the end of topic test does matter. If a student could demonstrate the standard in quizzes during the topic, but can't demonstrate them at the end of the topic, I think there is a problem.  But I haven't abandoned quizzes - on the contrary, they are a key part of my formative assessment strategy. I still give regular quizzes and use the dolphins, seahorses, killer-whales and blue sharks to give feedback during the topic. I do record the quiz results to help direct my teaching of the topic. But the difference is quizzes taken during the teaching of a topic don't count toward grades. I save that for the end of the topic. If a during-topic quiz shows me a few students need help on a specific standard, I give them specific support. If I see many students need help on a specific standard, then I alter the teaching the next day and put this standard in the next quiz for the whole class. So I don't do repeat attempts on quizzes, and I don't try to juggle grades based on quizzes and quiz retries.

The role of topic tests: I use the topic test to decide the level of achievement for each standard and report this to students with their topic test mark. I do this by grouping test questions against standards - either explicitly in the test design, or working backwards from a preexisting test. This does mean marking takes longer, but it gives much more useful feedback than a single test score. The results should not be a big surprise because the quizzes have been giving the student feedback along the way.  Retry attempts happen after the topic test, I give students the chance to improve their topic grade by taking quizzes or alternate tests for specific standards. That's how they can change their topic grade. In my grade book I have the topic test result (which stays constant), and an array of standards achievements which can be updated by retries.

Topic test result is recorded, along with initial end-of-topic achievement of standards.
I use red-orange-green traffic light indicators to quickly spot areas of concern.
Students can improve their standards results after the topic test by taking quizzes.

The final grade: I blend the topic test (snapshot in time result), with the standards achievement levels (which students can change through post-test quizzes) - giving more weighting to the standards indicators than to the topic test results. Why? Because I want students to have the opportunity to raise their grade through further effort. This reduces test anxiety and redirects the learning focus to mastery. 

And back to the blue shark .... stamps are fun - kids (and teachers!) love them. And when it comes to assessment, having a discussion about whether you got a seahorse, a killer whale or a blue shark - well it just takes some of the sting out of assessment and helps everyone realise the symbol or the grade isn't what's important : it's working towards mastery that counts.

Woot! A blue shark!
I get my animal stamps from www.allyoucanstamp.com 

A note on my constraints: I am the only teacher in my faculty using SBG - so I have to maintain the topic test results to allow for comparison across classes. With middle level classes, my grading system has to be consistent with other teachers (since we rank across the cohort) so my grades have to come exclusively from the tests. Perhaps one day I might be able to convince my colleagues to allow retries for the grading in these classes. For the senior years there are statutory regulations on assessment policy which are sadly high-stakes, single-attempt only assessments. So for the higher level classes, I can only use SBG to guide my formative assessment.  My hope is that this translates into the summative assessment results.

Your thoughts? Have I oversimplified SBG? How could I improve this approach?