1 of x A brief history of problem solving in GCSE mathematics Andrew Taylor Copyright © AQA and its licensors. All rights reserved. Version 3.0

2 of x Before 2003 Copyright © AQA and its licensors. All rights reserved. Alternative to coursework 1.5 or 2 hour paper with two questions/investigations Marked on an adapted coursework grid Assessing Ma1: using and applying mathematics 20% weighting (same as coursework) Three strands to the objective –Make and monitor decisions to solve problems –Communicate mathematically –Develop skills of mathematical reasoning Version 3.0

3 of x Example: Higher tier 2000 Copyright © AQA and its licensors. All rights reserved. In a Fibonacci sequence, each term is calculated by adding together the two previous terms. For example, if the first two terms are 2 and 4 we obtain the following Fibonacci sequence: 2, 4, 6, 10, 16, 26, … (a)Explain how you would find the first two terms of the sequence above knowing that the fifth term is 16 and the sixth term is 26. (b)What are the first two terms of the Fibonacci sequence which has fifth term 11 and sixth term 18? (c)Starting with the fifth and sixth terms of any Fibonacci sequence show how you can find the first two terms. (d)It is possible to find the first two terms of a Fibonacci sequence if you know any two consecutive terms later in the sequence. Investigate. Version 3.0

4 of x Assessment Issues Copyright © AQA and its licensors. All rights reserved. Low grade boundaries Difficult to compare performance with coursework Taken by a minority (30% of NEAB entry) Lower overall outcomes for this cohort Danger of becoming formulaic (like coursework) Version 3.0

5 of x From 2003 to 2008 Compulsory coursework AO1(using and applying) also assessed in written papers (10% minimum) Three strands –Problem solving –Communicating –Reasoning Version 3.0 Copyright © AQA and its licensors. All rights reserved.

6 of x Example: GCSE 2003 Copyright © AQA and its licensors. All rights reserved. A circular photo frame is shown below. The diameter of the photo frame is 10 cm and the outer diameter of the frame is 18 cm. Calculate the area of the frame. Version cm 18 cm Not drawn accurately

7 of x Assessment Issues Few marks per strand Each strand atomised into many sub-statements –Eg for problem solving strand (about 4 marks per paper) Select own method Carry out multi-step task Break down into smaller, more manageable tasks Choose relevant information Identify information needed but not given Use a range of techniques Too little to affect teaching and learning Not a valid test of valuable skills Copyright © AQA and its licensors. All rights reserved. Version 3.0

8 of x From 2009 to 2011 No coursework AO1 marks increased to 20% Arguably more genuine problems appearing on some papers Not enough to drive curriculum change Copyright © AQA and its licensors. All rights reserved.

9 of x Curriculum pathways project (2008 to 2011) GCSE in Additional mathematics with a high proportion of reasoning and problem solving questions Assessment of functional elements within GCSE papers Development of new assessment objectives Approach to problem solving in timed, written examinations informed by close working with Leeds University Copyright © AQA and its licensors. All rights reserved.

10 of x Copyright © AQA and its licensors. All rights reserved. This diagram is made from 25 small squares and 16 large squares. What percentage of the diagram is shaded? Version 3.0

11 of x Assessment Issues Response to problems could be ‘all or nothing’ Mark schemes did not reward efficiency or elegance Additional GCSE without additional content not acceptable Copyright © AQA and its licensors. All rights reserved.

12 of x Current Assessment objectives developed from pathways project Now a significant and important assessment requirement Issues of consistency within and between awarding bodies Other, broadly equivalent, qualifications have different approaches Copyright © AQA and its licensors. All rights reserved.

13 of x Copyright © AQA and its licensors. All rights reserved. In a game, five darts are thrown at a target. To win, players must score 31. Show one possible way of scoring 31 with five darts. Version

14 of x Copyright © AQA and its licensors. All rights reserved. When a jug is full of water it weighs 250 grams. When the same jug is full of water it weighs 550 grams. How much does the jug weigh when it is empty? Version 3.0

15 of x A circle is drawn inside a square as shown. Show that the area of the circle is more than 75% of the area of the square. Copyright © AQA and its licensors. All rights reserved. Version 3.0

16 of x Copyright © AQA and its licensors. All rights reserved. The graph shows two straight lines. The equation of line A is y = 2  x. Work out the equation of line B. Version 3.0 line B x y line A O

17 of x Copyright © AQA and its licensors. All rights reserved. The diagram shows two circles C 1 and C 2. The center of C 1 is at the origin, O. The center of C 2 is at X (12, 9). The radius of C 2 is twice the radius of C 1. The circles touch at the point Y. The circle C 2 crosses the x -axis at A and B. Calculate the distance AB. Version 3.0 Y C2C2 C1C1 O AB X x y Not drawn accurately

18 of x From 2015 (2017 exams) Assessment objectives developed further and proportion of problem solving increased AO1: Use and apply standard techniques 40% H, 50% F AO2: Reason, interpret and communicate mathematically 30% H, 25% F AO3: Solve problems within mathematics and in other contexts 30% H, 25% F Copyright © AQA and its licensors. All rights reserved.

19 of x Assessment Issues Working with Ofqual and exam boards towards a common interpretation of what might be AO1,2 or 3 Each objective carries a number of sub-statements. It is not yet clear what all these mean and how they might be assessed No explicit mention of modelling but a number of statements that imply a modelling approach within AO3 Skills demand increase alongside significant content increase Copyright © AQA and its licensors. All rights reserved.

20 of x Thank you, Questions? Version 3.0 Copyright © AQA and its licensors. All rights reserved.