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Mathematics Subject Knowledge Audit and Guidance PGCE 5-11; 7-11 School of Education Department of Primary Education A welcome from the Bath Spa University primary mathematics team We are looking forward to seeing you in September. Before we all meet then, there are some things that you can be getting on with, which will help to prepare you for the PGCE maths course. There are three important things to do: 1. Find out which areas of mathematics you particularly need to revise. A good place to start with this is by having a go at a Key Stage 2 SAT paper (or two). You can download past papers at: http://www.emaths.co.uk/KS2SAT.htm#Mathematics We also provide you with a maths audit that you should complete. This is more demanding than a Key Stage 2 SAT paper, but will be useful in showing you areas of mathematics that you need to develop. Don’t be alarmed by it; have a go at the bits that you can and treat it as a formative exercise (i.e. use it to identify what you need to revise). 2. Once you have identified areas that you need to work on, you can start the process of brushing up your subject knowledge. An excellent resource for this is the BBC Bitesize website. There is a Key Stage 2 and a Key Stage 3 site. Have a look and make a decision about where it would be best for you to begin. The Key Stage 2 material may superficially appear a little simple and the delivery is not aimed at adults, but the concepts are sound and the quizzes help you to know how much you have understood. The single most useful thing that you can do to improve your mathematics is to make sure that you know your multiplication tables thoroughly. There are hundreds of on-line games, which you could use to help learn your multiplication tables (and the associated division facts). Here are a couple of my personal favourites: http://www.arcademicskillbuilders.com/games/penguin-jump/penguin-jump.html http://www.arcademicskillbuilders.com/games/drag_race/drag_race.html The aim is for you to develop a deep understanding of simple mathematical ideas. This is what we will help you to do during the PGCE course. Looking at these materials and coming to the course in September having refreshed your mathematical knowledge will be very useful to you in this respect. Whatever your level of competence and confidence in mathematics, we would like to assure you that we have a proven track record for supporting people in developing their mathematics subject knowledge (SK) and confidence. “At the end of the mathematics SK sessions I feel secure and happy to teach mathematics to the children in my class. I feel that as a mathematician and a mathematics teacher, I have re- engaged with the subject and improved in both ability and confidence. My goal at the beginning of the year was to enjoy mathematics and inspire children to enjoy it. I believe that I have met that target and my development in mathematics has been my area of greatest progress.” Unsolicited comment in a recent assignment. 3. The third thing you could do is to find out more about the relationship between mathematical subject knowledge and pedagogical knowledge (these are not the same thing). This may seem an obvious link, but, (it may make you happy to learn), there is much more to being a good maths teacher than having really good subject knowledge. In fact, there is some evidence to suggest that people who struggle with the subject have some advantages when it comes to teaching primary maths, because they are better able to put themselves in the shoes of the children they are teaching. (People who find mathematics straight forward have other advantages when it comes to teaching though). Here is a link to a journal paper outlining some of the different kinds of knowledge that are involved in good primary mathematics teaching. Don’t be put off by the fact that it is from an academic journal; we will explain more about it once the course begins. 2 The article is called: The Knowledge Quartet: A tool for developing mathematics teaching. It is written by Tim Rowland. It is in the public domain and can be accessed here: http://www.maths-ed.org.uk/skima/Rowland%20Palermo%20paper.pdf Why mathematics subject knowledge is important The revised standards for teachers (DfE 2011) document states clearly that all teachers should ‘demonstrate good subject and curriculum knowledge’. This is elaborated further: ‘Teachers should have a secure knowledge of the relevant subject(s) and curriculum areas, foster and maintain pupils’ interest in the subject, and address misconceptions.’ Mathematics subject knowledge is important because it informs pedagogic knowledge for mathematics. As a team, we identify and seek to combine these three aspects of knowledge for teaching primary mathematics: mathematics subject knowledge (having a good understanding of mathematics) general pedagogic knowledge (knowledge about teaching generally) pedagogic knowledge for mathematics (specialised knowledge for teaching mathematics) This is very much in line with the thrust of the Williams Review of Primary Mathematics Teaching: “In-depth subject and pedagogical knowledge inspires confident teaching, which in turn extends children’s mathematical knowledge, skills and understanding.” (p9) Recommended books for mathematics subject knowledge The following books provide you with explanations and exercises which relate to primary mathematics and more advanced mathematics and will support you in developing your understanding and skills in mathematics. You should acquire one of the following. Previous experience has been that many people find our recommended book very helpful, but you may prefer the approach taken in another book. Recommended: Haylock, D. (2010) Mathematics Explained for Primary Teachers, (4th ed.) London: Sage. (The third edition is just as good and may be available second hand at a reduced price) You might also like: Cotton, T. (2010) Understanding and Teaching Primary Mathematics, Harlow: Pearson. (Note: This is available as an e-book through the BSU library services, which you will be able to access once you have registered in September.) In addition, there is a good book, which is aimed at the required subject knowledge for teaching in Key Stage 1. The book is: Haylock, D. and Cockburn, A. (2008) Understanding Mathematics for Young Children, London: Sage. You can preview both the Haylock books by simply typing the book titles into Google (and probably other search engines). You can then read selected pages to see if you like the style etc. 3 Optional pre-course mathematics subject knowledge support We are also pleased to be able to offer a short intensive course of mathematics subject knowledge support to complement your PGCE induction. The course aims to re-awaken understanding of primary level mathematics rather than to teach anything new. Attendance on this course is therefore optional, as many of you will already feel confident in your use of mathematics at this level. The course will run during and after your two-day induction as follows: Cohort 1 Monday, Sept 10th 4pm –6pm Wednesday, Sept 12th 9am – 12:30pm (with a mid-morning break for tea) Cohort 2 Wednesday, Sept 12th 4pm – 6pm Friday, Sept 14th 9 am – 12:30pm (with a mid-morning break for tea) The course will comprise three distinct sessions: The first (on the Monday/Wednesday at 4pm) will look at properties of number. We will cover fundamental ideas such as square numbers, prime numbers, factors and multiples. The second session (Wednesday or Friday at 9 o’clock) will look at shapes with a focus on symmetry and area. The final session (Wednesday or Friday at about 11 o’clock) will begin to prepare you for the TDA skills test in mathematics by looking at the relationships between fractions, decimals and percentages. The idea is that each session makes sense on its own. We are aware that, for some people, mathematics can be a source of concern and even anxiety. We are also confident that, during your PGCE course, you will become enthusiastic about mathematics (honestly!). This preparatory course will focus on boosting subject knowledge that you may have forgotten, or may not have fully understood when you learned it originally. It will involve some teaching input from us and the opportunity for you to do some mathematics. The course will be very non-threatening and will aim to give you as much targeted support as possible so that you can begin your PGCE year feeling more confident about your own mathematical abilities. Please remember, attendance at this short course is optional. It is additional to our full programme of support for teaching primary mathematics during your PGCE course. You should attend if you feel you need the extra mathematics subject knowledge support at this stage, perhaps because you have not used your maths recently or because you feel you only just secured the necessary GCSE pass. Otherwise we look forward to meeting you in due course. If you think that you would have benefited from pre-course mathematics support, but are unable to attend, we can assure you that there will be further subject knowledge support available once you begin your PGCE. If you are intending to attend these sessions, please let us know in advance by contacting m.witt@bathspa.ac.uk. Please say if you are on 3-7, 5-11 or 7-11 route if you know this. See you in September. Jill Mansergh, Anitra Vickery and Marcus Witt (Bath Spa University Primary Mathematics team) 4 AUDIT Section A: Number CALCULATORS ARE NOT ALLOWED FOR SECTION A (1) 487 + 848 = (2) 813 – 564 = Answer: Answer: (3) 3.6 x 3 = (4) 36 x 43 = Answer: Answer: (5) £24 ÷ 5 = (6) 9.2 – 4.85 = Answer: Answer: (7) Fill in the boxes with a number which (8) Solve. will make the statement true. 7 (a) (a) -5 + 7 = -5 (b) (b) -5 – 7 = (c) -3 (c) -7 + 5 = 5 (9) Calculate the value of each of the following: (10) Give all the factors of 100 a) 34 b) 53 Answer: (11) Find the Highest Common Factor of 48 (12) Write down 3 prime numbers between and 72 10 and 20 Answer: Answer: (13) 28 = 22 x 7 as a product of prime factors (14) Find the Lowest Common Multiple of 24 and 18 Express 63 as a product of prime factors Answer: Answer: 6 (15) You are given that 7 x 8 = 56 (16) The following table expresses the sale price as a Now write down the answers to proportion of the original price for three items. the following calculations: The proportion is expressed both as a simple fraction and as a percentage. Fill in the empty boxes. Answers 70 x 800 Fraction Original Sale in lowest Percentage Price Price terms 0.08 x 0.7 £16 £4 1/4 25 5.6 7 £20 £16 5600 0.8 £40 30 £24 75 Answers (17) a) Share 21 marbles in the ratio 4 to 3 b) 1 1 + = 3 2 c) 55% of 140 (18) A teacher earning £25000 a year is given a pay rise. The new salary is £26000. What is the percentage increase in salary? Answer: 7 Section B: Algebra (19) Fill in the output numbers from this function machine: n 3n2 - 5 4 0 -3 (20) Solve the following equations: a) 2a – 7 = 19 b) 7y + 8 = 2y + 28 Answer: Answer: (21) The sum of two numbers is 35; the difference between the two numbers is 19. What are the two numbers? Answer: 8 (22) Starting from n = 1. Write down the first 5 numbers of a sequence whose general term is: n2 + 1 Answer: (23) Create a general term for these function machines 1 3 2 8 2 7 4 14 3 11 7 23 4 15 8 26 5 19 10 32 n n (24) Simplify the following expressions: a) 6a + 3b – 8a + 2b = b) 3 ( 5e – 8) = 9 (25) Draw appropriate axes and label them x and y. Plot and label the following co-ordinates A = (4, 2 ) B = (-4, 2 ) C = (4, -2 ) D = (-4, -2 ) E = (0, 2 ) F = (4, 0 ) (26) In the grid below the vertical axis represents the cost C of hiring a video whilst the horizontal line represents the number of days D the video is hired for. The line graph represents the relationship between the two variables. (a) What is the cost of hiring the video for 5 days? (b) What is the daily additional cost for hiring the video? (c) Write down a formula expressing C in terms of D 10 Section C: Shape, Space and Measures In this section you must give the units for each answer. Give answers to 3 significant figures when appropriate. (27) If I walk a route which measures 7.5 cm on a map with a scale of 1:100000. How far will I actually walk in kilometres? Answer: (28) Calculate the lengths of A’B’, A’C’ and B’C’, if the linear scale factor between ABC and A’B’C’ is 0.4 Answers: A’B’ cm A’C’ cm B’C’ cm (29) Calculate the volume and surface area of the following solid figure: a cuboid with dimensions 7 cm, 10 cm, and 13 cm. Surface Area: Volume: 11 (30) Calculate the area of the following shape: 20 cm 12 cm 9cm 30 cm Answer: (31) What is the average speed for a journey of 175 km undertaken in 2 hours and 30 minutes? Answer: (32) How many planes of symmetry does a cuboid with dimensions 4cm x 4cm x 9cm have? Answer: 12 (33) For each of the shapes state in the table below: a) the number of lines of reflective symmetry b) the order of rotational symmetry Shape Lines of reflective symmetry Order of rotational symmetry A B C D (34) For each of the following statements say whether they are true or false a) Every square is a parallelogram a) b) Every rhombus is a square b) c) Every square is a regular quadrilateral c) d) Every isosceles triangle is obtuse angled d) 13 (35) The shapes below are a regular hexagon and an isosceles trapezium. Calculate the size of the angles marked a) b) c) d) (36) Use the terms Reflection, Rotation or Translation to describe how shape A in the figure below has been transformed onto i) Shape B ii) Shape C iii) Shape D Shapes involved Transformation A to B A to C A to D 14 Section D: Data handling (37) Give the mean, median, mode, range, lower quartile, upper quartile and inter- quartile range for the following set of test scores: 12 14 1 9 2 16 5 8 6 8 8 7 a Mean b Median c Mode d Range e Lower quartile f Upper quartile g Inter-quartile range (38) Which of the following variables are (39) When a ball is selected from a bag discrete and which are continuous? containing 3 black, 4 red and 5 blue what is the probability that it is: (a) A child’s height (a) Red The number of CDs a child (b) owns (b) Not blue The years in which (c) children were born (c) Green (d) Red or blue 15 (40) The probability of an event can be arrived at, or estimated, via the following approaches: a) A theoretical approach using equally likely outcomes b) Deriving a relative frequency from publicly available data c) Deriving a relative frequency by performing an experiment For each of the following events state whether a, b or c is the most appropriate approach to finding its probability. i) The sum of the numbers on two fair dice being 10 ii) A given drawing pin will land on its ‘back’ when dropped iii) It will snow in March. (41) Complete this list detailing all the outcomes when three coins are tossed together. (H stands for heads and T tails) HHH, THH, 16 Solutions for Initial Audit 1) 1335 2) 249 3) 10.8 4) 1548 5) £4.80 6) 4.35 7) Many solutions e.g. 9, 4, -7, 3 8) 2, -12, -2 9) a) 81 b) 10) 1, 2, 4, 5, 10, 20, 25, 50, 100 125 11) 24 12) 3 of 11, 13, 17, 19 13) 32 x 7 14) 72 15) 56000, 0.056, 0.8, 7000 16) 4/5, 80%, £12, 3/10, 17) 12 and 9, 5/6, 77 £32, 3/4 18) 4% 19) 43, -5, 22 20) a = 13, y = 4 21) 8, 27 22) 2, 5, 10, 17, 26 23) 4n – 1, 3n + 2 24) 5b – 2a, 15e - 24 25) 26) £4, 20p, C = D/5 + 3 27) 7.5 km 28) 5, 10, 6 29) 582 cm2, 910 cm3 30) 330 cm2 31) 70 km/h 32) 5 33) Reflective Rotational 34) a) T b) F c) T d) F A 1 1 B 2 2 C 0 2 35) a) 120 b) 60 c) 130 d) 50 D 1 1 37) a) 8 b) 8 c) 8 d) 15 e) 5.5 36) Rotation, Reflection, Translation f) 10.5 g) 5 38) C, D, D 39) 1/3, 7/12, 0, 3/4 40) a, c, b 41) HHH, THH, HTH, HHT, HTT, THT, TTH, TTT 17