Self-Differentiated Graphing

 

Continuing my thoughts on using technology with graphing (see previous blog post). Students choose which family of graphs to draw. Drawing multiple lines on one coordinate grid encourages thinking about similarities and differences.

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Scaffolding

One student draws quadratic curves (excellent BIDMAS  and substitution practice) while her neighbour draws straight lines through the origin (excellent for understanding what a graph is and how to use coordinates accurately). Both feel that they are doing the same task, bought into it through choice of difficulty, and able to help each other.

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I like this type of mixed ability 

 

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Floor Functions

Every Sunday afternoon I spend a happy hour in a cafe with Desmond, a student preparing to study Maths at university. We struggle in the darkness, wading through difficult problems. It is beneficial for both of us. This week, a problem from UKMT (3.5 hours, 4 questions to think about).

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Progression of the problem:

  1. Guess it has something to do with odds and evens, and try that.
  2. Give up
  3. Calculate the first few values by hand
  4. Create a graph of the function on DesmosUntitled picture.png
  5. Realise that the number of factors of n is somehow important
  6. Claim that  Twin Prime Conjecture and Mersenne Prime Conjecture are both true
  7. Get sad when our phone tells us neither has been proved yet
  8. Talk lots about numbers being dragged up or down

 

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Incomprehensible

I really enjoyed applying the skills I had been trying to teach at school – chunking the problem, drawing a picture, making links. We started to understand the problem a bit, but definitely were not near to a solution. See strategies at end of this post.

 

Final stage: ask girlfriend, who happens to be doing a PHD in Maths, to solve it for you. In her words:

“I thought I would want to use numbers that I could understand the factors easily. I realized that if I understand the factors of n, then I don’t know anything about the factors of n+1. So I could do it in a straightforward way.

And your graph showed that it increased on a large scale, but not on a small scale. So I guessed, and knew I would want to approximate above and below”

Knowing to play around with powers of 2 shows great intuition, learnt from many years of practise.

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Problem solving at School 21:

 

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The School 21 Problem-Solving Toolkit
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My attempt at linearising the problem-solving process

Structured Exploration, Proactive Self-checking

When I began to use Desmos and Geogebra in maths lessons, the tasks were lacking in structure. “Click on this link and have a play” I would say optimistically, hoping the students would be self-motivated enough to wonder and ask questions themselves. Varied success. Here are two first attempts at providing specific purposes for technology in the Maths classroom.

Structured Exploration

Simpy put: “Click on this link, have a play, and then do this subsequent exercise”. Allows teacher assessment, students given some (but not total) direction. Examples:

Proactive Self-checking

Here are three levels of marking:

  1. Teacher holds the answers, and marks herself. Teacher has good knowledge of student’s understanding, but the delay between question and feedback can be long, and this is time-consuming for teacher.
  2. Student is given the answers somehow (back of textbook, on a separate slide…) and passively checks their work against it. Immediate feedback, but often lack of cognitive depth in feedback
  3. Student works out the answer in a different way. This level encourages building of links between separate areas of maths, and allows the student to continually assess their progress. Let’s call it Proactive Self-Checking.

Here are some examples of proactive self-checking (do you think it is important enough to need capitalisation?):

When practising plotting points from an equation, students can compare their hand-drawn graph to a desmos graph.

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Desmos doesn’t have that weird kink in the line – what have I done wrong?
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The line on Desmos is going up but mine is horizontal – something is wrong!

This is a bit obvious, no real links made. How about encouraging the use of area models to check expanding brackets?

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I might use a more efficient method to calculate, but by using this method to check I am constantly reminded of the link

How about starting to understand simultaneous equations,to check that I solved any equation? If I am using bar modelling or the balance method to solve my equation, then being able to link the answer to a picture is deep stuff!

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2x + 5 = 3 has a solution at x = -1

How about using pictures to make sure you can simplify expressions?

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I don’t know what I have done wrong, but at least now I know I need to ask a peer for help!

Your next steps:

Think about providing structured exploration for angle rules?

Design proactive self-checking for your next lesson?

Use this collection of exercises as an intro to Geogebra with your class?

Search the excellent Geogebra database or Teacher Desmos Site for ideas?

Try this challenge?

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Playing with STEP

Thinking about STEP – excellent transition from School to University Mathematics.

Spent an hour in a cafe working with Desmond, a student who is doing STEP this year and is about to go to Oxford to fill his brain with prime numbers.

Here are some links:

We struggled with this question, mostly conceptually, for a pleasant head-scratching hour.

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When R is (2,1), the situation is shown below.

  • The blue line is the scenario that gives the smallest sum of distances OP and OQ
  • The red line is the scenario that gives the smallest distance PQ

We both strongly wanted the blue and red line to be the same line! It turns out they are not… Strange.

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Pictures WITH words

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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

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A pollution scientist throws herself into sketching graphs with the students

Stage 2: Formalise the sketch using data

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Stage 3: Attempt (and struggle) to describe and explain the shape of the graph in your English lesson.

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Stage 4: Sketch other graphs with similar shapes, to show non-verbal understanding of a graph with a maximum

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Which one is the odd one out?
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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

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Mountain graphs
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Step-by-step descriptions, no real explanation.
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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.

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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.untitled-picture
  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.untitled-picture
  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.untitled-picture

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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. untitled-picture

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Distance to factory, distance to side of factory, and height of chimney, become real.

 

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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.