Mathematics Subject Knowledge Audit and Pre-course GuidancePrimary PGCE2016-2017

Welcome from the UWE primary mathematics team!

Welcome to your Primary PGCE. We appreciate that it may have been a while since you did much mathematics (and that it may not have been your first love in life even then). However, it would be a good idea to prepare yourself mathematically for your PGCE.

There are two 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/index.php/student-resources/past-papers/key-stage-2-ks2-sat-past-papers

We also provide you with a mathematics audit (at the end of this document) 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 (Teachers should be aware of the mathematical demands of the Key Stage above that in which they are working). Don’t be alarmed by it; have a go at the bits that you can and treat it as a formative exercise i.e. as a way of identifying areas to work on.

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 Khan Academy (https://www. khanacademy .org/ ) is also a good source of mathematics instruction. Another good source of information is the course text book (see below)

The single most useful thing that you can do to improve your mathematics is to make sure that you know your multiplication tables thoroughly.

Remember that 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 the vast majority of people who do a PGCE with us leave feeling more confident about mathematics and about teaching it.

Why mathematics subject knowledge is important

Teachers’ Standard Three (2014) states that all teachers, including trainee teachers, should demonstrate good subject and curriculum knowledge.

Teachers should have a secure knowledge of the relevant subject(s) and curriculum areas, foster and maintain pupils’ interest in the subject, and address misunderstandings.

If teaching early mathematics, (teachers should) demonstrate a clear understanding of appropriate teaching strategies.

(Teachers’ Standards, 2012).

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Mathematics subject knowledge (how to do the maths) is important because it informs pedagogic knowledge for mathematics (how to teach the maths). As a team, we identify and seek to combine these three aspects of knowledge for teaching primary mathematics:

general pedagogic knowledge (knowledge about how to teach)

mathematics subject knowledge (knowledge about how to do mathematics) pedagogic knowledge for mathematics (knowledge about how to teach mathematics)

This is very much in line with the thrust of the Williams Review of Primary Mathematics Teaching (2008):

“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. (2014) Mathematics Explained for Primary Teachers, (5th ed.) London: Sage.(The third, or fourth editions are almost 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 UWE library services.)

Other recommended readings.

To begin the process of thinking about mathematics and mathematics teaching, we would like to recommend a couple of short chapters and papers that you will hopefully find interesting.

The first is from Hughes, M., Desforges, C., Mitchell, C. and Carre, C. (2000) Numeracy and Beyond: Applying Mathematics in the Primary School. Buckingham: Open University Press.

The first chapter is available free as a pdf file from the following url and is well worth reading:

http://w.openup.co.uk/openup/chapters/0335201296.pdf

The second comes from Boaler, J. (2008) The Elephant in the Classroom: Helping Children Learn & Love Maths. London: Souvenir Press.

This is another superb book. Again, the first chapter is available free on-line and is well worth reading:

http://nrich.maths.org/content/id/7011/chapter1.pdf

Finally, we are recommending a research paper by Nunes and Bryant. The whole document is very long (so we don’t recommend you read it unless you are particularly interested), but the executive summary is only 5 pages and is essential reading. The whole document is available to download from:

http://dera.ioe.ac.uk/11154/1/DCSF-RR118.pdf

Sometimes the links to DCSF publications are moved. If you ‘Google’ DCSF RR118, you should find it.

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And finally …

Within all the busy-ness of our work, we try to find time to conduct small-scale research projects. This year, we have two projects running that you might like to be involved with.

Study 1: The first involves mathematics and specifically anxiety about mathematics. There are a number of research studies suggesting that large numbers of people have a degree of anxiety about mathematics. These people include teachers and trainee teachers. We are hoping that some of you might want to take part in a small-scale study looking at the possible benefits of writing about maths anxiety. A really interesting study has suggested that writing about maths anxiety helps to alleviate it.

We are running a small-scale study where we ask those of you who are a little anxious about mathematics to do a short piece of writing once each week for about 10 minutes.

Here are links to a couple of papers about this, which you might find interesting:

https://hpl.uchicago.edu/sites/hpl.uchicago.edu/files/uploads/TiCS%20Final_Maloney%26Beilock_2012.pdf

https://www.apa.org/pubs/journals/features/xap-0000013.pdf

Study 2: We strongly believe that great teachers come in many different ‘shapes and sizes’ i.e. that you don’t have to be a particular personality type to be a great teacher. While some of you will be natural extroverts, others of you may be more reflective and quiet. If you have a spare 20 minutes, you might be interested in watching this video:

We are interested in ‘quieter’ teachers, how their teaching practice develops, how they respond to the quieter children in their class. If you see yourself as a quieter, more reflective personality and would like to be involved in this study, please also get in touch.

If you would be interested in finding out about participating in either of our two small research studies (maths anxiety and quiet children), please get in touch using the e-mail below.

We hope that you enjoy getting prepared for your PGCE. If you have any questions about the mathematics part of your PGCE, please feel free to contact us (Marcus.Witt@uwe.ac.uk)

See you in September

Marcus Witt & Ben Wiggins(University of the West of England Primary Mathematics team)

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Primary PGCE Mathematics Audit.

The audit has been devised to help you to identify those areas of mathematics where you are confident and those areas where you need to develop your subject knowledge. Teachers should be confident in the mathematics in the Key Stage above that in which they will be teaching. As a result, much of the audit is at a higher level than you will routinely be teaching in Key Stage 2.

Don’t let this put you off. Have a go at the bits that you can and don’t let the other bits worry you. Do be honest, so that you can identify things that need to be revised. We can then help you when you arrive at UWE. You will need to give us the marks when you begin the course, so please keep the audit.

Section A: Number. CALCULATORS ARE NOT ALLOWED FOR SECTION A. You are allowed to carry out calculations manually (i.e. you don’t have to do it all mentally)

Question Your Answer Mark

1 487 + 848 =

2 813 – 564 =

3 3.6 x 3 =

4 36 x 43 =

5 £24 ÷ 5 =

6 9.2 – 4.85 =

Write down possible values for the question marks so that the statements are true

7 ? > 7

8 -5 < ?

9 ? < -3 < ?

10 -5 + 7 =

11 -5 – 7 =

12 -7 + 5 =

Calculate a value for each of the following

13 34

14 53

15 Find the Highest Common Factor of 48 and 72

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16 Give all the factors of 100

17 Write down 3 prime numbers between 10 and 20

18 28 = 22 x 7 as a product of prime factors

Express 63 as a product of prime factors

19 Find the Lowest Common Multiple of 24 and 18

20. You are given that 7 x 8 = 56Now write down the answers to the following calculations:

Answers Mark70 x 800

0.08 x 0.7

5.6 7

5600 0.8

21. The following table expresses the sale price as a proportion of the original price for three items. The proportion is expressed both as a simple fraction and as a percentage. Fill in the empty boxes.

MarkOriginal

PriceSale Price

Fraction in lowest

termsPercentag

e

£16 £4 1/4 25

£20 £16

£40 30

£24 75

Answers Mark

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22. a) Share 21 marbles in the ratio 4 to 3

b) 1 + 1 =3 2

c) 55% of 140

23.

Now add your marks for Section A. Insert the total here:

Section B: Algebra

24. Fill in the output numbers from this function machine: Answers Markn 3n2 - 540-3

25. Solve the following equations:

26. Create a general algebraic term for these function machines (i.e. so that you can work out the ‘OUT’ number for any given ‘IN’ number).

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Answers MarkA teacher earning £25000 a year is given a pay rise. The new salary is £26000. What is the percentage increase in salary?

Answers Mark2a – 7 = 19 a =

7y + 8 = 2y + 28 y =

The sum of two numbers is 35; the difference between the two numbers is 19. What are the two numbers?Starting from n = 1. Write down the first 5 numbers of a sequence whose general term is n2 + 1

IN OUT IN OUT1 3 2 8

2 7 4 14

3 11 7 23

4 15 8 26

5 19 10 32

n n

27. Simplify the following expressionsAnswers Mark

6a + 3b – 8a + 2b

3 (5e – 8)

m + 3n + 6m – 4n

28. 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.

Answer MarkWhat is the cost of hiring the video for 5 days?What is the daily additional cost for hiring the video?Write down a formula expressing C in terms of D

Now add all your marks for Section B. Insert the total here: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.

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Question Answer Mark

29If 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?

30Calculate the volume of the following solid figure: a cuboid with dimensions 7 cm, 10 cm, and 13 cm.

31Calculate the surface area of the following solid figure: a cuboid with dimensions 7 cm, 10 cm, and 13 cm.

32What is the average speed for a journey of 175 km undertaken in 2 hours and 30 minutes?

33.

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

Answer MarkA’B’ =B’C’ = A’C’ =

34. Calculate the area and perimeter of the following shape:

9

20 cm

12 cm9cm

30 cm

35. 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 Mark

ABCD

36. For each of the following statements, say whether it is true or false

Answer MarkAreaPerimeter

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True or False MarkEvery square is a parallelogramEvery rhombus is a squareEvery square is a regular quadrilateralEvery isosceles triangle has an obtuse angle

37. The shapes below are a regular hexagon and an isosceles trapezium. Calculate the size of the angles marked

Angle MarkAngle a = Angle b =Angle c =Angle d =

38. 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 Mark

A to B

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A to CA to D

Now add together your marks for Section C. Insert the mark here:

Section D – Data Handling

39. 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

Answer MarkMeanMedianModeRangeLower QuartileUpper QuartileInter-quartile range

40. When a ball is selected from a bag containing 3 black, 4 red and 5 blue balls, what is the probability that it is:

Answer MarkRedNot blueGreenRed or Blue

Now add together your marks for Section D. Insert the markhere:

Answers to audit questions.

Section A. All questions are worth 1 mark, unless otherwise stated.

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Question

Answer Question

Answer

1 1335 2 2493 10.8 3 15485 £4.80 6 4.357 Any number greater than 7 8 Any number greater than -59 First number less than -3,

second greater than -310 2

11 -12 12 -213 81 14 12515 24 16 1, 2, 4, 5, 10, 20, 25, 50,

100 17 Any three of 11, 13, 17, 19 18 7 x 3 x 3 or 32 x 7

19 72 20 56000, 0.056, 0.8, 7000 (4 marks)

21 4/5, 80%, £12, 3/10, £32, ¾. 6 marks)

22 12 and 9, 5/6, 77(3 marks)

23 4%

The Total for Section A is out of a possible 33.

Section B. Marks for the questions are stated.

Question

Answer Question

Answer

24 43, -5, 22 (3 marks) 25 a = 13, y = 48, 27 (4 marks)

26 4n – 1, 3n + 2 (2 marks) 27 5b – 2a, (or -2a + 5b) 15e – 24 (or -24 + 15e)7m – n (or –n + 7m)(3 marks)

28 £4, 20p, C = D/5 + 3(3 marks)

The Total for Section B is out of a possible 15.

Section C. Marks for the questions are stated.

Question

Answer Question

Answer

29 7.5 Km (1 mark) 30 910 cm3 (1 mark)31 582 cm2 (1 mark) 32 70 km/h (1 mark)33 A’B’ = 5 cm

B’C = 6 cmA’C’ = 10 cm (3 marks total)

34 Area = 330 cm2

Perimeter = 84 cm(2 marks total)

35 Shape A 1 (ref) and 1 (rot)Shape B 2 (ref) and 2 (rot)Shape C 0 (ref) and 2 (rot)Shape D 1 (ref) and 1 (rot)(8 marks in total)

36 TrueFalseTrueFalse (4 marks in total)

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37 Angle a = 120 Angle b = 60Angle c = 130 Angle d = 50(4 marks in total)

38 RotationReflectionTranslation(3 marks in total)

The Total for Section C is out of a possible 28.

Section D. Marks for the questions are stated.

Question

Answer Question

Answer

39 Mean = 8Median = 8Mode = 8Range = 15Upper Quartile = 5.5Lower Quartile = 10.5Inter Quartile Range = 5(7 marks in total)

40 1/3 7/12 0¾.(4 marks in total)

The Total for Section D is out of a possible 11.

The total audit score is out of a possible 87. The breakdown of the different sections is most important, as it will help you and us to identify where your mathematical strengths lie and where you need some additional support.

Whatever you score, please don’t worry, but DO keep the audit as we’d like to know your score when you arrive to begin the course. This will help us to support you in the best way possible. Thanks.

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