Developing reasoning in algebra in years 7-11 Jennie Golding President-elect of The Mathematical Association MA Conference Oxford 7-9 April 2016

Algebra treasure hunt What algebraic and other skills and understandings are needed? (issues of closure, communication, informal as well as formal reasoning, variable used) How can you adapt to other algebraic learning needs? This session focuses on provision for relatively ‘slow graspers’

Problem solving Fluency Reasoning Communication

Aims for algebraic reasoning Build on a secure algebraic understanding of number to Understand and engage with algebra as a natural mathematical language in which they are fluent Appreciate that algebra can be used to express relationships between quantities (so can become less hard work than number!) and so to predict and explain mathematical and other situations Understand that letters might represent an unknown, a variable, a label, a parameter, a constant…and that ‘=‘ also has a variety of meanings Set up and work with equations and expressions as ‘clues’ to numbers or descriptions of situations, and formulae, functions and graphs as representing relationships between two or more variables, including through technology Nunes et al 2009; Watson, Jones and Pratt 2013

What can teachers do? Build up algebraic understanding of number eg from 100-squares, number lines, calendars, and patterns and relationships Build confidence with a variety of different sorts of number Build fluency, reasoning, communication and mathematical habits and processes on a day to day basis - within and beyond maths Use algebra and algebraic thinking across the curriculum Work towards genuine PS in classrooms on a regular and frequent basis: there’s a wealth of evidence that will also enhance deep conceptual understanding, flexibility, reasoning and affect. (Lester, 2002) Teach domain-specific metacognition: teacher modelling is powerful (Mevarech and Kramarski, 2014) Cooperative learning methods can enhance reasoning, fluency, communication and problem solving since they require students to articulate thinking, use mathematical language, work within ZPD, provide elaborated explanations, and be involved in conflict resolutions

Sources of problems/building up reasoning

The power of algebra: Is it true? If not, why not? If true, when? or always? Can you convince me? The sum of two odd numbers can sometimes be odd. The product of three whole numbers is never the same as their sum. If the sum of the digits of a number is divisible by 3 then the number is divisible by 3. The product of two odd numbers is odd Algebra is often an alternative to lots of hard work: ‘mathematicians are lazy’

Encourage informal solutions.. Maths teachers and spiders: 10 heads, 56 legs. Insects and spiders: How many of each? 11 heads, 80 legs. How many of each?

Introduce algebra across the curriculum Make-24, eg 1,2,3,5 2,4,4,8 3,3,8,8 1,7,8, x 2 (for what value of x?) 1,2,6, x 2 -3x+5 (for what value of x?)

Confidence with notation: always, sometimes or never true? (Improving learning in mathematics, Standards Unit ‘box’ ‘Mostly algebra’ or n+5 = 11q+2 = q+162n+3 = 3+2n 2t-3 = 3-2t3+2y = 5yp+12 = s+12 4p > 9+pn+5 < 20 2( x +3) = 2 x +3 2(3+s) = 6+2s X 2 > 4 X 2 = 5 X X 2 > X 9 X 2 = ( 3 X ) 2 c+3 = d+3

Link expressions, words, tables, shapes

Shift the focus – use rich question stems -and ask students to write own questions

Magic squares

What do you see? What do you know? Make links. When is algebra appropriate or useful? (

Similarly, magic squares…

Premier league algebra (2015 table) What do the different headings mean? Can you re-create the table in Excel using formulae for ‘Pts’ and ‘G.D.’? Can you find a scoring system that would knock Man City off the top, or save Wolves from relegation? How might this scoring system affect play?

Reasoning with expressions

And so to word problems susceptible to algebra… Mark set out for town with a £20 note and £1.50 in change. His bus fare in each direction is 50p. He wants to buy a drink that costs 98p, and then to buy 3 CDs. What’s the most he can pay for each CD, on average?

for-gold-problem-solving