Planning, preparing and thinking about level 2

WMA 21 November 2011

Introductions

Anne Lawrence Centre for Educational DevelopmentMassey University Palmerston Northa.lawrence@massey.ac.nz

http://ced-mxteachers-news-site.wikispaces.com

Looking back and looking forward

• Level one: What have we learned? • Level two: Many possibilities for courses• Implications for teaching mathematics

Getting startedLooking back: what have we learned?

Level oneAssessing

Teaching Externals

InternalsLearning

Numeracy

Unit standardsAchievement stds

Getting started• The graphs & patterns had two questions on discontinuous step

functions . There was nothing like this in the one exemplar provided and our students had no idea how to do it. Furthermore, as these were the first two questions a lot of our students gave up and did not attempt the bits they could do.

• The Trig & Geometry standard had a 3D question when the standard specifically states " use trigonometric ratios and Pythagoras' theorem in two dimensions" and out of all the angle properties, there were 5 questions about cyclic quadrilaterals!

• Why write an exam that is substantially different from the exemplar?• Why have the different material as the first two questions?• Why put the familiar material at the end of the question? • Why have so many questions on cyclic quads?

Level one exams: teacher feedback

Getting startedLooking back at our courses

Level one

What needs changing?

What did we do well?

What were the successes for students?

What didn’t work well for students?

Getting startedLevel two coursesMany possibilities Discuss, explore, explain….

Getting startedLooking forward

Level two

What else could we consider?

What do we want to keep?

What changes do we need to make?

What will we do to support our students?

Getting startedTop course AO ASTerm one:Investigating relationships

M 7-1 co-ordinate geometryM 7-2 graphs M 7-3 sequences and series M 7-6 algebraM 7-7 equations

2.12.22.3

Term two: Variation

S 7-1 Statistical investigations S 7-2 Make inferences S 7-3 Evaluate statistically based reports S 7-4 chance

2.92.12

Term three: Investigating change

M 7-2 graphsM 7-9 gradient functions M 7-10 differentiation & anti-diffn..

2.7

Level two courses… one possibility

Getting startedLevel two courses… another optionMiddle course AO ASTerm One – Applying Mathematics

M7-2 graphsM7-4 trigonometric relationshipsM7-5 networks M7-8 simultaneous equations

2.22.52.14

Term Two – Statistics

S7-1 statistical investigations S7-2 Make inferences S7-3 Evaluate statistically based reports:S7-4 chance

2.112.122.14

Term Three – Investigating relationships

M7-1 co-ordinate geometry M7-2 graphs (Exclude log and exponential functions)M7-3 sequences and seriesM7-6 algebraic expressionsM7-7 trigonometric equations (Exclude trig functions)

2.12.6

Getting startedLevel three courses … one possibilityTwo maths courses with common first semester

Content

semester 1 – Core maths Graphs and functionsAlgebra

semester 2A – Modern Mathematics

Critical pathSystems of equations Linear programming

Semester 2B – Calculus DifferentiationAnti-differentiation

Getting startedWhy learn mathematics?

Getting startedWhy learn mathematics?• Learning to think mathematically• Logical thinking• Problem solving• Understanding concepts• Developing mathematical skills• • •

Getting startedThinking and Understanding…

• Conceptual understanding demands knowing what, knowing how, knowing why, and knowing when (to apply).

• In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations.

Getting startedA typical maths classroomSketch a graph(s) showing time spent on practice exercises, and time spent on problem solving during a typical unit.

Getting startedA typical maths classroomMost of the time students are doing practice exercises. Some time is devoted to problem solving activities but they are usually at the end of the lesson or topic.

Getting startedJust a bunch of rules to remember• Is it surprising that students think of mathematics as

a bunch of rules and procedures?

• What should the graph look like if we want to change the way students think about mathematics?

Getting startedThinking comes from problem solving

We need a good source of rich tasks that we can use for teaching (yes, for assessing too).

Two Classrooms•Tasks that provide students with the opportunity to develop and show their reasoning and mathematical thinking. •Tasks which allow for student choice about the mathematics and the problem-solving strategies they use to model and investigate situations. •Tasks that require students to call on their mathematical knowledge and search for a pathway to the answer.

Worthwhile tasks

Sources of worthwhile tasksNCTM http://www.nctm.org

• Horseshoes in flight

Balanced assessment http://balancedassessment.concord.org • The catenary problem

Mathematics assessment project http://map.mathshell.org• Lines and rectangles

NRICH http://nrich.maths.org • Tangled trig graphs

Getting startedExploring tasks• Jigsaw activity:

• Go to expert groups to explore a task• Your task is to report back to your ‘home’ group about

the task you explored in your expert group.

• Key questions:• Where would this task fit?• How can I extend the task?• What prompts, probes, scaffolding would be useful for

students who are stuck?

Getting startednrich: eg Tangled graphs

Here is a pattern I made with some graphs of trigonometric function.

http://nrich.maths.org

Getting startedTangled trig graphs (cont) 1. The purple line is the graph y=sinx. Can you identify the

coordinates of the points where it crosses the axes and where it reaches its maximum and its minimum values?

2. How could I make the red graph from the purple graph? Can you work out the equation of the red graph?

Getting startedTangled trig graphs (cont) 3. The green graph has equation

y=sin2x. Can you describe how to make the green graph from the purple graph? How does the transformation of the graph relate to the way the equation has changed?

4. Using these ideas, can you work out the equations of the other graphs I have drawn?

5. Imagine you had a graphical calculator but the sine button is broken. Can you draw the same patterns using the cosine function instead? Explain how you can transform a cosine graph into a sine graph.

Getting startedMathematics Assessment Project (MAP)

http://map.mathshell.org

Getting startedBalanced assessmenthttp://balancedassessment.concord.org

Are you a Maths Problem Solver?PROBLEM?In your own words what is being asked?• Read and reread. • Sketch initial diagram.• Explain the problem in your own words.• What sort of answer am I looking for?

EXPLOREMake sense of the problem and connect it to what I have learned.• Investigate/Play with what is given in the problem.• What methods from my toolbox can I use?• What else do I need to figure out?

SOLVEChange the words into a maths problem and solve it.• Organize and sort my exploration.• Construct useful diagrams or models.• Turn words into labels.• Apply methods from my toolbox.• Record my steps.

EXTENDHow do I show relational and extended abstract thinking?• Explore the problem further.• Identify patterns, relationships and exceptions.• Abstracting • Reflect on my solution.

CONCLUDEFinishing it up.• What is my final answer?• Is my solution communicated clearly? • What are the success criteria?

Solving problems? Use the PESEC cycle!

QuickTime™ and a decompressor

are needed to see this picture.

Putting your own spin on this

Think about any standard topic• Recast the content as questions that students

can explore• Resist the temptation to tell students the

content . Believe that students can investigate and derive relationships and mathematical concepts.

Consider finding x and y intercepts

• How do you currently teach this?

Consider finding x and y intercepts

• Ask students to look at a graph and identify the intercepts• Ask students why these points may be of interest to

someone trying to interpret the graph• Create a context for a graph that requires interpretation of

one of the intercepts• Ask students to create their own rules and to compare these

to the textbook rules. Why do the rules work?• Ask students to graph a “messy” equation that’s difficult to

graph: something like: 33x + 14y = 231

Looking back and looking forward• Reflection on level one• Thinking about level two courses• Why learn mathematics?• Teaching through problem solving:

• horseshoes in flight - http://www.nctm.org • the catenary problem - http://

balancedassessment.concord.org • tangled trig graphs - http://nrich.maths.org • lines and rectangles - http://map.mathshell.org

Thank you

http://ced-mxteachers-news-site.wikispaces.com

Two Classrooms• use contexts familiar to the students; what is problematic is

the mathematics rather than the situation • encourage students to use intuitive solutions as well as

knowledge and skills they already possess • can have several solutions • challenge students to use the strategy that would highlight

the depth of their understanding of the concept involved • allow students to show the connections they have made

between the concepts they have learned

Good problem solving tasks…

Two Classrooms

•How can we ensure all students can access these tasks?•How can we provide support and still maintain levels of cognitive demand?

Accessing tasks

• What support do students need to tackle demanding tasks?

• What scaffolding is appropriate?• What prompts would teachers want to give?• How can we provide support without

reducing the cognitive demand?

Supporting, Scaffolding, Prompting…

Getting startedMAP eg Lines and rectangles

Lines and Rectangles

37

Line segment SP has equation y = 2x + 3.

P

S

R

QFind the equationsof the line segmentsforming the otherthree sides of the rectangle.

Lines and rectangles (cont)

Getting startedFind the curve assumed by a loose chain hung freely

from two fixed points.1. Look at a chain of that kind or imagine its shape. Trace its shape on a piece of paper. Measure it and analyze its values at different points. Write an expression which represents this function, and explain why you chose this expression.2. For a long time this curve was thought of as a parabola. Then it turned out that the precise description of the hanging chain can be given by the combination of two exponential functions: .

The values of this function are very hard to compute without calculators. It is therefore useful to find out how good an approximation to the catenary shape other functions provide.

Balanced assessment eg The Catenary problem

Getting startedGraph both functions: the function you have picked as your representative function and the given function f(x). Sketch the two functions on the same coordinate system. Describe the differences between them in the region of -5 < x < 5:a. For how many different values of x do they intersect? b. Consider the functional representation of the slope of each function; compare the two. (How many times do the slope functions intersect?)c. Suggest another method to compare the two functions.3. Devise a way to present how close the two graphs are for any value of x (any width of the hanging chain). Explain your decisions.4. Extension:Devise a way to estimate the length of the catenary without using any measurement tool.

The Catenary problem (cont)

nctm eg Horseshoes in flight http://www.nctm.org

1. Which expression is the most useful for finding the maximum height of the horseshoe, and why is it the most useful expression?

2. What information can you conclude about the horseshoe’s flight from other equivalent expressions? Explain your answers.