Why I like Mathematics and why Students should Discover and Create their Own

I gave a presentation yesterday to some Year 11/Grade 10 students about why I like maths. I tried to steer clear of the applications of mathematics in technology and science and wanted to give a presentation that made sense from their reference frame – hence if I was giving a presentation to younger students, older students or parents I’d probably choose different things to talk about.

Whilst I think that technical skill and learning content is important in everything we do and aids discovery and creativity, I do think that we don’t do enough in school mathematics to expose students to discovering and creating their own stuff. That’s the fundamental aspect of the talk.

At the start of the presentation, I showed two clips (Clip 1 – if you’re stretched for time watch from 1:50 onwards and Clip 2) about the new Karate Kid film and asked them for their opinions on why he became so frustrated and why he didn’t enjoy learning Kung Fu. The answers are obvious:

1) He didn’t have a clue why he was putting a coat up on a peg and couldn’t see how this would help in his life.

2) Continuous drills of doing the same thing over and over again is not Kung Fu – it’s practice!

In the second clip, even though he knew he’d continue to get bullied, he decided that it wasn’t worth the effort and nearly quit. It was only when he was able to apply skills to a scenario and “Do Kung Fu” that he began to understand and was genuinely shocked at what he’d been able to achieve – he obviously had no idea that he was able to do that and just needed Mr.Hang to provide the right opportunity for him to be creative.

After that, I asked them to read the second page from A Mathematician’s Lament by Paul Lockhart. It’s the part where an Art teacher has a nightmare about observing a class that is solely based on rote learning, drills and no creativity. I said that I felt that my own maths Education was quite similar to that and a student said that this is a common perception about how maths is taught in schools.

After that I started talking about a few examples which I felt would illustrate the playful and creative side of mathematics to a class of 15-16 year old students. Unfortunately we only had 45 minutes so we couldn’t explore things together, it was more about my explorations. It’s highly edited due to the 500MB capacity of vimeo. [By the way, I don't have a clue where I plucked out the 90% statistic towards the end - extra planning was needed here I think!]

Creating and Exploring Mathematics: How many different directions can we take a topic in 15 minutes?

After reading Bryan Meyer’s thesis (which I highly recommend you take the time to look at), I decided to do something I’m ashamed to admit I rarely do – actually that might be a lie – I don’t think I’ve ever done what I just did. Before I get into it, here’s a quick bit of context…

Bryan Meyer mentions an interesting lesson/sequence of lessons in which he asks the students to explore consecutive numbers in anyway they feel like exploring them. As you can probably imagine, some of the students were completely stumped – they had no idea what he was asking them to do. The students were simply too accustomed to being told exactly what to do and when to do it and had no sense of how to create ”mathematical artwork” and play around with mathematics. Essentially, they didn’t know how to create their own mathematics – only how to learn some of the vast array of content which other people have created and possibly apply some of that knowledge to problems given to them by the teacher.

In a Mathemtician’s Lament, Paul Lockhart thinks this is a travesty. He asks the important question, “Why don’t we want our students to learn to do mathematics?”

I could get into a discussion here about external assessment issues, pressure on teachers or teacher perceptions of learning mathematics. Instead, I decided to ask myself the following question:

When have I ever sat down and explored an area of mathematics simply for fun?

I tell my students that mathematics is fun, creative and playful but then I rarely practice what I preach. Instead, I spend time learning about mathematics which other people have created and occasionally add to that by filling in some of the gaps myself - I rarely just mess around with a random topic for fun. Please don’t take this post as confirmation that I’m all about students solely creating their own mathematics and never learning anything that other people have created. I believe in a very balanced approach to a curriculum in which a particular teaching and learning approach during one activity isn’t the best way to do it for another activity. Direct instruction may work for one thing but then exploring a concept using an applet may be better for another. It just depends on the topic and the students you’re working with.

Anyhow, I felt is was highly hypocritical of me to preach the creativity of mathematics and rarely explore things for myself. Hence, I sat down at the table this morning  and purposefully explored consecutive numbers for no other reason than to see what I could come up with and what direction it took me. I didn’t stop when I’d come up with something, I just asked another question which took me in a different direction. After 15 minutes, I was quite amazed at where I’d got to and what I was thinking about.

I stopped at the 15 minute mark and immediately wondered where other people would be at this point – what wonderful mathematics would come out of a discussion between a room of teachers who sat by themselves and explored consecutive numbers for 15 minutes? What would we learn from each other?

It’d be bloody brilliant if there are any teachers out there prepared to spend 15 minutes thinking about consecutive numbers to see what they’ve come up with and where they’ve got to. If anyone does this, I’d be very interested to discuss it. I’m giving a talk next week to our Excellence Challenge students titled,  ”Mathematics: Why some people fall in love with it.” It’d be great to discuss the playful nature of mathematics and use this activity as a springboard to expose the students to the true nature of mathematics.

Update: Here’s my 15 minutes of play.  It’s a bit tidier than my original sheets but had to re-write with bigger handwriting so that the photos came out okay. I was pleased to re-discover the integral test for divergence of the harmonic series. I remember learning this about 10 years ago and not having the foggiest where it came from – great to actually discover it for yourself. Infact, when I checked whether my reasoning was correct – I found the exact same reasoning on wikipedia! Outcome: Must do 15 minutes of mathematical play each week.

Introduction to Fractal Geometry 2: Julia Sets and the Mandelbrot Set

The Mandelbrot Set – created with Geogebra

It’s possible, or more like probable, that the most mysterious shape ever to arise from the study of Mathematics is that of the Mandelbrot Set. It’s amazing how such an intriguing shape could arise from purely mathematical pursuits. The shape of the Mandelbrot set (often likened to a Beetle), is highly complex yet it is derived from such a simple equation.

Before I get into the post, it’s important to give huge credit to Professor Lee Stemkoski who works at Adelphi University. He figured out a simple and effective way to paint the Mandelbrot set in Geogebra using a lovely little trick – have ten points trace the fractal instead of one! The pictures in this post are a result of some small tweaks to his method.

Any discussion of the Mandelbrot set should start with Julia Sets. It was Gaston Julia (1893-1978) and Pierre Fatou (1978-1929) who began the study of Julia Sets during the First World War. They were essentially interested in whether, when you continually apply a rule to a number, it gets closer to a particular finite value (it converges) or whether it goes off to infinity (it diverges).

As an example, take the function f(x) = x². If I put x=2 into this function then I get 4 out.

f(2)=2²=4

and if I put what came out, back into the function, then in this case I’ll get f(4)=4²=16, then f(16)=16²=256, and so on. As you can see, putting 2 into this function to start with shows that it diverges off to infinity.

However, if I put x=0.5 into the function then it will converge and get closer and closer to zero. (f(0.5)=0.5²=0.25, f(0.25)=0.25²=0.0625, …)

The thing that scares students about Julia Sets is that Julia and Fatou were looking at whether complex numbers converge or diverge. This isn’t a problem because complex numbers really aren’t that complex at all. They arose out of a simple choice to see what happens if we allow square roots of negative numbers to have solutions (weird to think of at first but then doing that turns out to make lots of calculations much simpler). See the Khan Academy section on Complex Numbers if you’d like to learn more about them before reading on.

The function Julia and Fatou were inputting complex numbers into is written below:

f(z)=z²+c

where z and c are complex numbers, c being a constant.

So, like the example above, you take a complex number, say for example, z=1+i, choose a constant, say c=0.3 and put these numbers into the function f(z)=z²+c.

f(1+i)=(1+i)²+0.3=(1+i)(1+i)+0.3=1+i+i-1+0.3=2i+0.3

Then of course put 2i+0.3 into the function again, and keep going until you see whether it diverges or converges. This is what happens if you continue the process (the continual process of inputting values into the function is called iteration) :

1+i → 0.3 + 2i  → -3.6 + 1.2i  → 11.9 – 8.7i  → -66.6 – 206i  → -38019 + 27472i  →

6908220936 + 2088930939i  → etc.

and by the way, after six more iterations, the number is so big that geogebra cannot compute it. To give you some idea of how big these numbers get, after 4 more iterations, the number is so big that it can’t even fit across the laptop screen in font 12. So it’s pretty obvious that this number diverges off to infinity.

In fact, for most of the input numbers (points on the complex plane), the value diverges off to infinity. However, it’s interesting to look at how quickly they diverge. On a very simple level, you could colour a point black if geogebra can compute it after 25 iterations (i.e. it’s not too big to fit on the screen) and red if geogebra can’t compute it.

Hence,  the point z=1+i we inputted above would be coloured black when c=0.3 but the point z=0.2-0.2i for example would be colored red. Indeed, if we did this for every point in the complex plane, we get this picture in geogebra:

Note again that all the points which are black here may diverge, they just don’t do it as quickly as the red points. The above Julia set is called disconnected – which is self explantory really because the black parts aren’t all connected into one object.

Here’s what happens if you do it for different values of c (the constant).

For c = -1

For c = -0.8 +0.2i

The two examples above are connected Julia sets.

Isn’t it amazing how a simple equation, changed ever so slightly, can result in such amazingly interesting and complex shapes! I also love the fact that using the applet, you’re essentially unveiling the pictures bit by bit which adds suspense to the mysterious beauty behind these objects. It’s a real shame that Julia and Fatou never got to see these objects in all of their beauty due to lack of computing power.

So you might be wondering how the Mandelbrot set is related to the Julia sets?

The Mandelbrot set is essentially a map of the Julia sets. If the Julia set for a particular value of c is disconnected, then we’ll paint the point c black, if it’s connected we’ll paint the point c red (see top of post). If we do this for all values of c then we get the Mandelbrot set (Wolfram Alpha shows this quite nicely – if you type in “Julia set c = 0.3″, then you’ll see the Julia set and this point on the Mandelbrot Set). I tweaked Professor Stemkoski’s geogebra file just to show that we can make it more interesting with Geogebra. Basically, I painted the points that converge (or diverge slowly) pink, points that diverge (but do it ‘quite’ slowly) blue and points that diverge quickly black. Here’s what you get.

Some people ask what the point of the Mandelbrot set is? Does it have any real applications? From what I’m aware of, it currently doesn’t. It does show however that shapes exist in the mathematical world which don’t exist in nature – how interesting!

Here’s the link to Professor Stemkoski’s youtube video explaining how he put the initial file together.

Here’s the links to the geogebra files (Julia set painter and Mandelbrot set painter) - you HAVE to play with these – it gives you a very, very, very small sense of how Benoit Mandelbrot must have felt when he saw the first printed image of the Mandelbrot set in 1980.

New Sunflower Applet: Fibonacci and the Golden Angle

Here’s a new applet I created (simply because I haven’t made one in a while and thought it’d be fun). Look at how different angles result in different sunflower configurations by clicking the picture below.

Why is the golden angle 137.5 degrees? If you’re struggling to figure it out based on the standard derivation of the golden ratio then there’s a few videos on youtube that’ll help.

Anymore ideas for teaching averages?

I was looking at some TES resources the other day trying to find new ideas for teaching mean from a frequency table. I found a few good ones so this post is just a collection of ideas for teaching averages, both from the TES (slightly extended or modified) and things I’ve picked up over the years. This was originally just supposed to be private blog post but I decided to make it public just to see if anyone else has more ideas to bring the topic to life or make it relevant.

1. Fun: I read a post recently (but have no idea where?) with an idea based on the hilariously funny “Is it Alan or Steve” youtube video. Show the video to students and ask them to record how often the words Alan and Steve are said. Ask the class for their answers and then discuss how we could obtain a more reliable result with the data we have. Lead them into calculating the mean, median and mode of the class results and discuss which is the most appropriate. (I’ll get rid of this explanation and post the link to the blog post as soon as someone can tell me where it is)

2. Fun: Averages for Ducks – this was my first post over 15 months ago (Funny to look back at it now). It’s a similar idea to the one above, tell the students that you were followed to school by a flock of birds and they keep flying in and out of the room. Duck under the desk and tell the class to get down because you can hear them coming. Flash a sheet with 15-25 ducks on it and ask for class estimates for how many ducks there were on the sheet. Lead again into discussion on mean, median and mode (here’s a quick sheet I made for it).

3. How do you get students to understand that the mean isn’t always the best average to use? You could:

• Look at the average wage of employees at a small business with one manager and 7 employees. Artificially make the manager’s wage high so that it skews the mean to make it non-representative for the sample. Alternatively, discuss ignoring outliers.
• Similar scenario. Look at the average house prices in a small village which has say 10 small houses and a mansion. Again, median may be a better choice here over a skewed mean.
• A silly one. An alien comes down to planet Earth and bombards you with questions about the human species. One of them is: ”What’s the average number of arms on a human being?” Giving the mean in this case will result in you giving a number like 1.99999999999 instead of 2. The mode would be a better average to use.

4. Modified from a great TES worksheet: Imagine you work for a big mobile phone company. How could you determine how satisfied your customers are with a new phone? Talk through possibilities with the class before introducing them to the data. Show them a phone on www.amazon.com with viewers ratings – discuss reliability of the data. Then find mean, median and mode from frequency table to get representative figures. Do this for a number of different phone brands to compare. You can see the numbers for the Iphone below.

6. Extended idea from a great TES worksheet: Which Drving Instructor to use? Show students a PowerPoint slide with 3 driving instructors and the prices they charge for a one hour driving lesson. Ask them which instructor they’d pick and why. Lead them into discussing other things they might want to know about driving instructor quality. Show the students information on how many lessons people needed to pass their driving test with different instructors (in the form of a frequency table). Ask the students to find the mean, median and mode number of driving lessons people needed to pass with each instructor and hence estimate how much they would have to pay in total for lessons. Go onto asking which would be the best average for the driving instructors to use to advertise themselves in a newspaper/website.

7. Generating their own data – whilst I did see this on a blog recently, it is also something I did 2 years ago with a class. We set up the question, “Who has quicker reactions, girls or boys?” There’s a great, but very addictive, applet for this on the nrich site. First make sure that the whole group are using exactly the same controls to obtain reliable results. Get each student to do 30 trials, record their results and then find the mean of their individual results. Stick the mean scores into a grouped frequency table and find girls and boys averages from there.

As a side note, we used the same data later on in the year to look at whether increased trials affected reaction time, analysing correlation and lines of best fit (regression).

8. Discuss class size averages in different Countries – how might this affect learning? Or alternatively, average life expectancy on www.gapminder.org.

9. Even though it’s barely worth mentioning because it’s so obvious – sport averages are always good to go with. Get students to find averages based on their favorite sports (or books, or films, etc.) for homework. Alternatively, get them to find out some crazy average facts such as the facts from this website – Example - ”The average person eats almost 1500 pounds of food a year.” - I’ve found this to be a great way to jaz up units of measurement.

It’d be great to hear more ideas if anyone has any.

Should students award themselves stickers once they’ve understood a topic?

I’ve read some interesting posts recently regarding reward stickers when students achieve mastery of a topic (See posts written by James Gurung and Bruno Reddy). I would’ve loved this as a kid. In fact, I loved getting badges in Scouts for exactly the same reason (I remember getting the first badge for my attempt at making a bird box – I think the Scout Leader gave me the badge out of sheer pity after the third attempt).

I’ve been umming and ahhing over this reward based system over the past week though. Here’s why…

Just because I would’ve loved it, that doesn’t mean it’s the best way to keep all of my students motivated. Dweck has, of course, carried out a large amount of research on the effects of extrinsic rewards on fixed mindset orientations and intrinsic motivation. This research shows that rewards such as house points, merits, gold stars or stickers can lead to pupils choosing to avoid challenge, create excuses for failure, see ability as fixed and not incremental, give up in the face of difficulty and become upset when faced with failure. Furthermore, students are more likely to be competitive and develop a mindset in which they are less likely to try when there is no tangible extrinsic reward.

Dweck recommends that we try to promote an intrinsic motivation to learn by setting interesting activities which increase a student’s likelihood of engaging in the activity because they want to find an answer and want to improve for learning’s sake (not because they want recognition), praising effort and persistence rather than ‘ability’ or speed of calculation, giving feedback which moves the learning forward.

Whilst I fundamentally agree with Dweck, I also struggle with the implication that  extrinsic motivation and intrinsic motivation are mutually exclusive ideas.

When in fact, I think we’d all agree that they are intrinsically linked and depend on a numbers of factors such as what the activity is, whether free choice is involved, who is presenting the activity (it may be an extrinsic motivation to please a particular teacher), how the student has been brought up (Parent: “You must do well in school to get a good job” – extrinsic reward) and societal/cultural views on education. Hence the diagram for a particular pupil may, at one point in time look like this.

Don’t get me wrong, I’m not saying that a person’s motivation can’t be changed over time and I’m certainly not saying that there is a ‘correct formula’ that we can apply to every student – I just think there are a huge number of reasons why a person is motivated they way they are. The problem is that if one day we suddenly decide that we want to promote an intrinsic motivation in all of our students then we’re surely going to run into some issues. As mentioned above, we’re competing against many factors such as how they’ve been brought up to view education or whether they’ve been given extrinsic rewards in the past by primary teachers and expect this to continue, or indeed whether other teachers in the school give lots of extrinsic rewards and the students expect this to happen in your lessons as well. Hence, simply putting an immediate stop to extrinsic rewards in a classroom may not produce the results we’re hoping for.

So how does all of this relate to mastery orientated stickers when students have understood a topic?

One thing I haven’t yet mentioned is that these particular stickers, whilst acting as an extrinsic reward, could help to promote independent and self-regulatory skills in the student (as James Gurung mentions). The student is constantly reflecting and regulating how well they are doing on each topic and have a more concrete sense of which topics they still need to improve upon. This to me does not sound like a bad thing. They aid other important skills we want our students to develop.

What’s my conclusion:

I think that that promoting intrinsic motivation and growth mindsets is extremely important in education. However, I don’t think this is something we can all immediately switch to and obtain fantastic overnight results. There has to be a slow, school-wide, on-going process to aid students to become independent learners with a growth mindset. My feeling is that these stickers would be brilliant for the lower years of secondary school (UK: Year 7 and 8/US: Grade 6 and 7). I think I’d use them as “basic skills” stickers so that students know that they’re going into an an exam with the basic content skills needed to solve problems (in the same way that Bruno Reddy recommends). I believe they would help students practice the skills of self-regulation at a young age so that when it comes to taking external exams, they already have the skills to revise effectively. I would also say that regular conversations with the group (and parents) regarding why they are getting these rewards and how it will help them now and in the future are appropriate and necessary.

What are other people’s thoughts on this? Agree or disagree? I’d really like some second opinions before I consider making and buying them.

The Mathematics of Rainbows

I’ve always been pretty fascinated by rainbows – anyone surely has to admit that it’s truely amazing how nature can construct such a beautiful image. After many years of wondering, I finally decided to spend the evening learning about them. The problem is that when I started to watch a few videos and read a few websites, they gave information without full explanations (is there anything more infuriating?). Don’t get me wrong, the videos are great for about 90% of the information but I felt like a bit more digging was needed to fully grasp the mathematics behind them. Thus this post is for all of those people that want to learn about the formation of a rainbow but feel like explanations are missing out some of the vital steps in the process.

I would recommend that before you read this post, you watch the video that I started with. It’s made by the Open University for one of their Undergrad Maths Courses so I suspect that it may deliberately have left stuff out so that the students could investigate (if this is the case, they certainly hooked me in!).

Then pop over to this website I read after watching the video. It’s fine for a quick overview but my first question after reading it: Where did they get all of those angles from?

Question 1: When do rainbows occur?

Easy one: When we have cloud, rain or spraying water and UV light.

Question 2 (considerably longer answer): How do rainbows occur?

Rainbows occur due to a combination of refraction and reflection. Let’s start by looking at what happens when light hits a water droplet.

The light hits the water droplet, at angle of incidence (i), and is refracted towards the normal line at angle of refraction (r). Just to note, since the light is now travelling within the water droplet, which is a denser medium, it slows down. When it hits the back of the droplet, it is reflected (in fact, some of the light is reflected and some is refracted at a boundary, but I am only concerned with this particular path of the light for now). It is then refracted again upon leaving the water droplet.

It may be of some use to start by analysing the angle at which the initial light beam from the Sun is turned through when it comes back out of the water droplet.

Notice that the first step was to draw in two isosceles triangles. I could then figure out how far around I needed to take the initial beam to get out of the droplet. Using vertically opposite angles, we get the first deviation of i-r, then continuing round we end up with a fully rotated angle of:

i – r + 2π – 2r + i – r = 2π + 2i – 4r

However, this isn’t the actual rotated angle – i.e. the angle the ray has turned through. Below is a picture of how far the ray has turned through.

The deviated angle that I found was infact the one below.

Hence all I need to do is take off π radians from 2π + 2i – 4r to give R(i,r) = π + 2i – 4r

We now have a function for how far the light beam has been rotated round from when it enters the droplet to when it leaves. The problem is that the function has two variables in it - I’d much prefer to work with a one-variable function. Luckily, Snell’s law of refraction (a physical application of the sine rule with speeds as opposed to distances) will help us with this.

Snell’s Law:      sin (i) = n sin (r)

where n is the refractive index of the medium, given by:

n = speed of light in first medium/speed of light in second medium.

The speed of light in air is approximately equal to the speed of light in a vacuum. The speed of light in water is about 75% of that in a vacuum. Hence, in this instance,

n ≈ 1/(3/4) = 4/3

So rearranging snell’s law with n = 4/3 gives:   r = arcsin(0.75sin(i))

Our rotation function now only involves one variable:

R(i) = π + 2i – 4arcsin(0.75sin(i))

Looking at the graph of the function above, we have a stationary point at (1.04 radians, 2.41 radians) which means that when the angle of incidence (i) is 59.6 degrees, the angle of rotation (R) of the sunlight is 138.1 degrees. I wonder if this point has any significance? Indeed something interesting is happening at this point. I happen to have calculated the supplementary angle to the “critical angle” all the other websites mention but don’t show how you get. This critical angle is often stated at around 42 degrees (180 degrees – 138.1 degrees).

Why is this angle so “critical”? It seems that when light hits the water droplet creating this angle, we get a higher intensity of light than at any other angle. This is because the gradient of the curve near the stationary point is small (smaller than other points on the curve), so we have a range of values for the angle of incidence around the stationary point which give a similar rotation angle of the beam (It took me some time to realise this but I think this idea is better displayed in diagramatic form at the following video between 5 minutess 20 seconds and 6 minutes 30 seconds ).

I could go further and talk about why the colours always follow a particular pattern or how double-bows appear, or in fact why it is lighter underneath the rainbow than above it, but these are things you can find out from other sources.