Building Effective Learning Strategies into a Mathematics Curriculum (Playing the Long Game) Jemma Sherwood Head of Mathematics and SLE, Haybridge High School & Sixth Form, Worcs. @jemmaths

Background Successful school. Watching and supporting teachers for many years. Progress of middle-achievers not as good as others.

Curriculum Sequencing Will Emeny, www.greatmathsteachingideas. com @maths_master

Curriculum – Y7

Curriculum – Y7

Curriculum – Y8

Curriculum – Y8

The Six Strategies for Effective Learning

Spaced (Distributed) Practice Dunlosky (2013) describes as “Implementing a schedule of practice that spreads out study activities over time”. Study in depth to increase quality of learning. Practise regularly to increase retention of learning. Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students’ learning with effective learning techniques: Promising directions from cognitive and educational psychology. Psychological Science in the Public Interest, 14(1), pp4-58.

Spaced Practice www.numeracyninjas.org

Spaced Practice www.mathsbox.org.uk

Spaced Practice – In the Scheme of Work   Unit 4 - The Cartesian Grid 4.1 plotting 2D coordinates in four quadrants 4.2 expressing number relationships algebraically 4.3 representing number relationships on a Cartesian grid 4.4 sketching graphs of straight lines and quadratics, with appropriate scaling, by finding a table of values 4.5 read/estimate values of variables from a graph (including quadratic, piecewise linear, exponential and reciprocal graphs), including solutions of simultaneous equations 4.6 Read gradient using 1 across, 'm' up/down and find y-intercept from a given graph, use these to give the equation in the form y = mx + c. Sketch a graph given the gradient and y-intercept (without plotting a table of values). Identify parallel lines from their equations, including rearranging. 4.7 Real-life graphs (of all sorts, including distance-time and velocity-time) 4.8 Finding speed from a distance-time graph, acceleration from a velocity-time graph and distance from a velocity-time graph   Unit 7 - Advanced Linear Graphs and Equations 7.1 Construct two linear simultaneous equations from a context and represent the solution graphically. 7.2 Find the solution to two simultaneous linear equations algebraically and check solutions 7.3 Calculate the gradient of a line using change in y/change in x 7.4 Identify equations of parallel and perpendicular lines. 7.5 Advanced straight line questions - finding equations given two points or a point and gradient. Solve problems related to this.

A Side Note on Lessons Thinking of “a lesson” can be detrimental. Sometimes things take part of a lesson. That’s ok. Sometimes they take longer than a lesson. That’s ok. It takes as long as it takes. E.g. TOPIC A consists of X, Y and Z. I am going to teach these over the course of 5 hours.

Interleaving Dunlosky (2013) describes as “Implementing a schedule of practice that mixes different types of problems, or a schedule of study that mixes different kinds of material, within a single study session.”

Creates a desirable difficulty Interleaving Necessitates spacing Creates a desirable difficulty Bjork, R.A. (1994). Memory and metamemory considerations in the training of human beings. In J. Metcalfe & A. Shimamura (Eds.), Metacognition: Knowing about knowing, pp. 185-205. Cambridge, MA: MIT Press.

Interleaving N2/N3 Four operations on positive integers and decimals Order of operations Bring in decimals N6 Negative numbers Bring in order of operations with negatives and decimals N7 Fractions Bring in order of operations with negatives, decimals and fractions

Interleaving From Median, by Don Steward

Retrieval Practice Trying to remember something you have started to forget. Often done through quizzing yourself – making yourself remember something without just reading it again. Quizzes must be low-stakes. It’s not a ‘test test’, it’s making yourself remember things.

Retrieval Practice Ten Quick Questions by CSF Software www.mathsbot.com Jonathan Hall @studymaths

Retrieval Practice Knowledge Organisers

Elaboration Explaining and describing things in detail. Dunlosky (2013) uses two techniques that come under this banner: “Elaborative interrogation: Generating an explanation for why an explicitly stated fact or concept is true.” “Self-explanation: Explaining how new information is related to known information, or explaining steps taken during problem solving” Sweat the small stuff.

Elaboration Make links wherever you can. Improve your own subject knowledge so you know when to elaborate and when not to. 𝑥 2 +8𝑥=10 compared to 𝑥 2 +8𝑥−10=0 4 2𝑥+3 =4(5𝑥+6) compared to 𝑥 2𝑥+3 =𝑥(5𝑥+6)

Concrete Examples The bedrock of maths instruction! How do you choose your examples? E.g. Solving equations: 𝑥+3=11 2𝑥−3=11 2𝑥−3=12 2.1𝑥−3.6=−12.4

Concrete Examples Rearranging a formula: Variation: 𝑥−𝑎=𝑏 𝑝𝑥−𝑎=𝑏 𝑝𝑥−𝑎 𝑐 =𝑏 𝑎 𝑥 =𝑏 𝑎 𝑥−1 =𝑏 𝑥 =𝑏 𝑎𝑥−𝑏=𝑑−𝑐𝑥 etc Variation: (𝑥+2)(𝑥+3) (𝑥+3)(𝑥+2) (𝑥−3)(𝑥+2) (𝑥+3)(𝑥−2) (𝑥+2)(𝑥−3) (𝑥−3)(𝑥−2) Taken from Craig Barton’s How I Wish I’d Taught Maths

Dual Coding Version 1: The question presented only in words.

Dual Coding Version 2: The question presented in words with an accompanying diagram (as it was in the examination).

Dual Coding Version 3: The question presented as an annotated diagram.

Dual Coding 5𝑥−3=19 +3 5𝑥=22 ÷5 𝑥= 22 5 5𝑥−3=19 +3 5𝑥=22 ÷5 𝑥= 22 5

Playing the Long Game A curriculum that exploits the natural hierarchy of maths +