From Graphs to Maps?

This is the final post about the Maths behind the Concrete Project. In a previous post I described how we used graphs and words to describe relationship between variables (such as distance to factory) and how dirty the air was.

The next step was to move from a graph (which gives information as you move along a 1-dimensional line) to a map (which gives you information as you move about a 2-dimensional plane). We tried to make something like this.

I, in my role as mathematician, created a whole load of data for the students to plot on a graph. A year ago I would have thought that this is cheating, but now I focus on the purpose of the project. The aim was not for the students to use Excel to efficiently calculate, but to understand how to plot information and draw conclusions from graphs. Leave the boring stuff to the teacher.

We spent a two double lessons on this. The first lesson was a complete success because:

1. The numbers involved were easy to understand (we started with lots of simple examples, with made-up numbers)
2. Colouring is fun! The exercise was novel 3. I poured a lot of energy into the atmosphere of the class
4. There were four adults in the room
5. I modeled the process very clearly on the board  Most focussed lesson of the year for this student. He loved the clearly explained process.

The second lesson was a complete failure because:

1. The numbers were now very small. (I toyed with multiplying all of them by a factor of 10 to make more manageable, but wanted the students to get a sense of just how small the concentrations in the air were. Maybe this was a mistake, understanding 2.3  x 10^-19 might be too challenging) 2. The students were gradually losing motivation in the project. The concrete factories were no longer going ahead – so why should we continue to work on this project? Fascinating issue, which clearly highlights a potential pitfall of PBL – it could encourage students to only learn when they are solving a genuine problem.
3. There were a number of students who were very vocal that “this wasn’t really maths”. Because the maps are unfamiliar and involve colouring, there surely musn’t be much deep learning going on. Students often surprise me with their very traditional opinions on what learning looks like.

In the end, teachers followed students. There was no genuine need for the maps to be completed, so we didn’t push for them. Instead, we learnt about sequences and finding the nth term – back to the worksheets that the students recognise as “proper maths”.

Pictures WITH words Sum of odd numbers is a square number

One extreme is the fabulous book Proof Without Words (Geogebra version here), that prides itself on beautiful but mysterious pictures that reveal mathematical structures without any words. In the Concrete Project however, we have been focussing on being able to articulate your understanding through words.

Stage 1: sketch a predicting graph and talk through as a group why you have chosen that particular shape  A pollution scientist throws herself into sketching graphs with the students

Stage 2: Formalise the sketch using data Stage 3: Attempt (and struggle) to describe and explain the shape of the graph in your English lesson. Stage 4: Sketch other graphs with similar shapes, to show non-verbal understanding of a graph with a maximum Which one is the odd one out? Excellent work on the scale of the graph. Talking points about unusual shapes

Stage 5: With scaffolding, try again to describe and explain the graph Mountain graphs Step-by-step descriptions, no real explanation. I have spoken to this student and (I am convinced) he completely understands the concept. However, when he writes it down the understanding is invisible.

Stage 6: After repetition, repetition, repetition, students begin to be able to describe and explain the graphs. Warning: have the students now just memorised the correct sentences?

Stage 7: Tell 1 million people about your understanding on BBC London.

One key difference between the Maths teacher and the English teacher:

• The Maths teacher is satisfied a pupil understands the concept if they are able to wave their hands around vaguely at the right time and draw graphs to show how variables are related. Implicit understanding hinted at.
• The English teacher is satisfied when they are able to write or say full sentences that detail how the variables are related. Explicit understanding stated.

Is this a difference between individual teachers’ preferences, or something deeper about types of understanding in different subjects?

Why is describing a graph so difficult?

A graph is a visual way of linking two variables together – how the movement of one affects the movement of another. The combining of two separate concepts (distance and air pollution for example) requires sufficient working memory and multi-step thinking. A single point on a graph represents two points on separate scales. Students who would be able to deal with one scale might struggle to make the leap to two scales. Students who understand intuitively how changing one variable affects another may therefore struggle to sketch the correct graph.

Graphs have been used to represent data in a way that we would recognise only since the early 19th century. The idea of using one point on a graph to represent multiple numbers only started with Descartes in the 17th century.  The late blooming of graphs as a way of thinking in the history of maths is a sensible pointer to the fact that conceptually they are a tricky beast.

Why are chimneys like party poppers?

We are nearing the final stages of the English and Maths project, where a bunch of Yr9 students are using their understanding of how pollution spreads to convince the local residents and the planning officials that building 4 factories in the heart of the Olympic Park is a bad idea. Jess wrote about the project planning here, and the press attention here.

The mathematical foundation for the argument against the factories has constantly shifted over the past term. Both frustrating and rewarding. Here is the journey, starting in October.

1. Initially I assumed that any theoretical understanding of pollution would be difficult for a nurture group, probably delving into multi-variable differential equations.
2. Therefore, we at first planned to record levels of pollution in the local area, and use this data to predict what the pollution might be like after the construction of concrete factories. We were very kindly given a class set of tags that paired with phones to give data. Unfortunately the data was not detailed enough, logistically it was difficult to go out and record high quality data, and not every student had a phone. The dream of students independently going out and using their phone to change the world was somewhat dashed. That said, a few students do continue to use their tags and record the pollution in their daily lives – small win.
3.  If measuring air pollution was not going to work, would we have to go back and understand the theoretical models? A few weeks of emailing around pollution scientists in London and treading water in lessons ensued. Elsa, from Southbank University, came and spoke to the class, and taught me about the Gaussian Plume Model. We had found an equation that could map out spread of pollution. 4. First attempt: students to understand the equation and substitute in the relevant variables (for example, distance away from concrete factory and speed of wind). I made a beautiful but hopelessly complicated flowchart to aid this process. 5. The equation was too unwieldy. Next step: I plug the numbers into the equation (or sometimes students use a ready-built calculator to do it themselves, and the students plot the graphs and analyse the results. Plotting graphs with very small and very big numbers involved revealed great teaching points about rounding and scale.  We discovered that the students were having difficulty understanding the variables affecting how dirty the air that you breath in is. For example, what does “distance from chimney to my location”? Building on an idea from Jess I made a double-lesson analogy of Chimney as party popper, with the exploding paper representing the pollution.  Distance to factory, distance to side of factory, and height of chimney, become real. Conclusions

• Through this tortuous journey of dead-ends and frustrated ideas we (accidentally?) exposed the students to a variety of mathematical ideas. Had we gone straight to Stage Five: plotting graphs and analysing them, then there would have been no data collection, no understanding of equations, no algebraic substitutions.
• The fact that we teachers genuinely had no idea of how to solve the problem gave us intellectual excitement throughout the project, useful in pumping up the students and our lessons.