Welcome Numeracy and the Learners Welcome Back Session

Welcome Numeracy and the Learners

Welcome Back Session 4 ‘The Great Race’ (In threes…. )

üCollaboration üDialogue between students üWorking backwards üIdentifying redundant information

Swan’s Principle “Use co-operative group work”

Discussion • How many times does the average person’s heart beat in a lifetime? • How many people could comfortably stand in a room which measures 10 m x 10 m? • Outline a plan for a party for this group including all costs…

Key Questions Session 4 In order to move our students to their potential, what current numeracy teaching approaches (including communication approaches) and resources do we feel are effective? What might the impact of technology be on learner engagement, motivation and success in numeracy teaching and learning?

Realistic Maths Education

In RME, mathematics is developed by working with contexts which provide a source for generating mathematical models and hence a deep understanding of where mathematics comes from. https: //rme. org. uk/about-rme/what-is-rme/

Contexts can be taken from the real world, from fiction or from an area of mathematics that students are already familiar with – the important thing is that students are able to imagine and engage with them. https: //rme. org. uk/about-rme/what-is-rme/

The key features of an RME classroom are: *extended discussion of multiple contexts *development of students’ representations of contexts *focus on multiple strategies for solving problems *sharing, explaining and discussing strategies https: //rme. org. uk/about-rme/what-is-rme/

RME classrooms create a shift in “socio-mathematical norms” (Yackel & Cobb, 1996) which make them more inclusive – students are able to develop ownership of mathematics and engage confidently in discussion. They are encouraged to take their modelling as far as they can, while maintaining links to the context from which they are generated. https: //rme. org. uk/about-rme/what-is-rme/

How long has Ray got left to wait? How many megabytes (MB) are there left to load? How long has he been waiting so far?

• 10 minutes to install programme • A 40 MB download in total

Draw and shade 3 separate download bars to show progress after 5 min… 2 ½ min… 100% 40 MB 10 minutes

After approximately 3 minutes of now installing ‘Word’, this is what he notices… How long might this programme take to install? How much longer will it be before it is 100% installed?

Ray installs other programmes onto his computer. For each of the 8 programmes, work out the total installation time…

Ray’s method… Explain in detail Ray’s method for calculating the sale price… In what order did he fill in the numbers on the bar? Which retailer offers the best deal?

RRP £ 620 15% off sale Draw a bar and use Ray’s method to work out the sale price of the computer… How much more is the price at PC solutions compared to Computers R Us…

Ray notices that there is 9% off in the sale at Computers Unlimited. The RRP for Ray’s computer here was £ 580. Explain in details Ray’s method for calculating the ‘sale price’… What did he do first? i

Fish and Chips Material developed by Paul Dickinson, Steve Gough & Sue Hough at MMU

Thank you • Sue, Steve and Paul would like to thank all the teachers and students who have been involved in the trials of these materials • Some of the materials are closely linked to the ‘Making Sense of Maths’ series of books and are reproduced by the kind permission of Hodder Education

Note to Teacher This section uses the context of a chip shop to introduce simplifying, expanding, and factorising. This is a context that should be recognisable to students, and the intention is that they get involved in the context and make sense of the mathematics through this. Students may wish to drop the context as they come up with ‘quicker ways’ or ‘rules’, but the teacher’s role is to take them back to the context to check if their methods ‘work’. The basic idea is that an order can be asked for in different ways, and the cost can be worked out in different ways. So, for example, ‘fish and chips 3 times’ can be worked out by doing 3(f+c), or by 3 f+3 c (‘three fish and three lots of chips’).

Note to Teacher Students are asked to imagine themselves working at the shop, writing down phone orders, and then ‘bagging’ the orders. This leads to simplifying, expanding, and factorising expressions. The cost of different chip shop orders is used throughout as the mechanism for verifying the validity of individual solutions and more general methods. It is important that the teacher keeps reminding students that letters stand for the price of items and not the items themselves In trials of the materials, the biggest problems occurred if the context was dropped too early. The switch from ‘f’, ‘c’, etc to ‘x’ and ‘y’ is far from seamless, and teachers need to keep referring back to the context so that students can make sense of the more formal mathematics. The final part of this section looks at expanding and factorising, though the context is still there for checking answers.

At the Fish shop Fish Sausage Chips Peas Gravy £ 3. 20 £ 1. 30 £ 1. 40 £ 0. 80 £ 0. 50

At the Fish shop I do the price of 1 fish and 1 lot of chips, then times that by 3. I do the price of 3 fish, then the price of 3 lots of chips and add them up.

Lunchtime orders At lunchtime, people sometimes come in with big orders. Why do you think this is? One lunchtime, a full order is fish and chips 3 times, sausage and chips twice, fish and peas twice, and 2 extra portions of chips

Orders over the phone 1. 3(f+c) + 4(s+p) + 1 c + 2 p + 2(f+p). 2. 5 f + 4 s + 4 c + 8 p

At the Fish shop Fish Sausage Chips Peas Gravy £ 3. 20 £ 1. 30 £ 1. 40 £ 0. 80 £ 0. 50

Orders over the phone 3(f+c) + 4(s+p) + 1 c + 2 p + 2(f+p). Made simpler, 3(f+c) + 4(s+p) + 1 c + 2 p + 2(f+p) = 3 f + 3 c + 4 s + 4 p +c +2 p +2 f + 2 p =5 f + 4 s + 4 c + 8 p

Simplify 1. 2(s+c) + 3 (f+p) + 3(f+c) + 2 p + 3 c + 2 s 2. 2(f+c+g) + 2(f+c) + f + 3(s+c+p) + (s+c) + 2(c+g) + 3 c 3. 5(s+c+g) + (f+c) +2(f+c+p+g) +3(f+c+g) +2 s +2 c

Simplified orders Below are three ‘simplified’ orders. In each case, give two possible ways in which they may have been ordered over the phone 1. 9 s + 6 c 2. 12 f + 16 c 3. 6 f + 8 c + 4 s

Cancelling orders Sometimes, people phone through an order, then ring up a bit later and change it. On day, Jane looked at Azim’s notepad and saw 3(f+c) She came back to check it a few minutes later and saw that now on the notepad was 3(f+c) - (f+c) What do you think has happened here?

Cancelling orders The order was 3(f+c) - (f+c) What should Jane wrap up for the customer? On another occasion, Jane saw on Azim’s pad 3(f+c) – f + c Is this the same? What do you think 5(s+c+g) - 2(s+c) means? What is the simplified order here?

Wrapping it up People sometimes complain when they collect their orders if different bags have different things in them. One order was for 12 f + 16 c How could this be bagged so that each bag contains exactly the same?

Wrapping it up Try to do the same with these orders: 1. 3 s + 3 c 2. 4 f + 2 c + 2 p 3. 2 c + 4 s + 6 f 4. 3 s + 3 f + 6 p + 9 c 5. 12 s +9 c 6. 12 f +8 c +4 g

Doing the maths Remember that when we say ‘ 3 fish’ we are actually talking about the cost of 3 fish Expanded form How to say it in expanded form Factorised form How to say it in factorised form 3 f + 3 c 3 fish and 3 chips 3(f + c) 3 lots of fish and chips (or fish and chips 3 times) 2 f + 2 c + 2 p 6 f + 3 c + 3 p 2(f + c + p) 3(3 f + 2 c) 2 f + 4 c + 2 p

Using different letters Expanded form Factorised form 3 x + 3 y 3(x + y) 2(p + q + 3 r) 9 x + 6 c + 3 y 5(3 x + 2 y) 6 f + 4 x + 2 p

Summary • The context of ‘fish and chips’ can help you to simplify, expand, and factorise algebraic expressions, and to see the equivalence of such expressions. It can help to keep thinking about the context of fish and chips when you are solving algebraic problems

‘Making Sense of Maths’

Cognitive Load Theory Dylan Wiliam described Cognitive Load Theory as - “the single most important thing for teachers to know”, and I have to agree. (Craig Barton)

Classroom Culture “I never gave that much attention to the culture of my classroom. As long as my students were working hard, not misbehaving too much and ideally laughing at my jokes…” p 342>

Practice Testing and Distributed Practice “We rated two strategies – practice testing and distributed practice –as the most effective of those we reviewed because they can help students regardless of age, they can enhance learning and comprehension of a large range of materials, and, most important, they can boost student achievement. ” (Dunlosky, p 13>)

Teaching lower achieving students… “…surely it would be better to adopt a more informal approach? Let them play with numbers, let them experiment, let them discover. The problem was, they never really seemed to learn much this way. p 118>

When and why less guidance does not work… “Sure the teacher may give hints, but control and responsibility for what happens next and throughout the tasks rests firmly in the hands of the students. Is this necessarily a bad thing? Well, I fear it might just be. ” p 100>

Why struggle and failure aren’t always good… “For many years, I have been a firm believer in the key role of mistakes in the learning process…” p 76>

Mathematical Tech… What are the issues around using dynamic mathematical technology in mathematics learning?

Worked examples – the importance of choice… “I have now come to realise that the choice of examples are more important than anything else. ”

Dual Coding… Why every teacher should be using dual coding…

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Welcome Back Read: Roshenshine’s Principles of Instruction In terms of your own teaching, which principles do you find more challenging to implement? Share some of these potential challenges with a colleague…

Cognitive Load Theory Built upon two commonly accepted ideas: • There is a limit to how much new information the human brain can process at one time • There are no known limits to how much stored information can be processed at one time (Emerged from work of Sweller and colleagues, 80’s / 90’s)

Cognitive Load Theory The aim of cognitive load research is therefore: • To develop instructional techniques and recommendations that fit within the characteristics of working memory, in order to maximise learning

Working / Long term memory Working memory - where small amounts of information are stored for a very short duration, the limited mental state in which we think, average person can hold ‘four chunks’ of information in their working memory at one time Long term memory - is the memory system where large amounts of information are stored semi-permanently, ‘the big mental warehouse of things’

Cognitive Load Theory • Assumes that knowledge is stored in long term memory in the form of ‘schemas’ which are relevant to learning • They provide a system for organising and storing knowledge • Crucially for cognitive load theory, they reduce working memory load as a schema constitutes only a single element in working memory • The automation of ‘schemas’ reduces the burden on working memory because when information can be accessed automatically, the working memory is freed up to process new information