PGCE Primary and Early Years Mathematics Pre-Course Guidance and Subject Knowledge Audit 2017-2018

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Welcome from the UWE primary mathematics team! Welcome to your 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. Take the maths audit (see details below) to find out which areas of mathematics you particularly need to revise. You may then want to have a go at further Key Stage 2 SAT papers. 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 provide you with a mathematics audit (at the end of this document) that you should complete. It is based on Key Stage 2 SAT papers, and will be useful in showing you areas of mathematics that you need to develop. Please, do not 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. There are further instructions below.

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 (2012) 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). 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 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)

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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

Recommended (Primary 5-11):

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). Recommended (Early Years 3-7):

Haylock, D. and Cockburn, A. (2017) Understanding Mathematics for Young Children (5th ed.) London: Sage. Again, earlier editions are fine and cheaper. For people on the Early Years route (3-7), the Haylock book recommended above would also be useful. 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 link 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.

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:

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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 a project running that you might like to be involved with.

The study 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 and talking 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

If you would be interested in finding out about participating in our small research study), 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 , Ben and Gemma. (University of the West of England Primary Mathematics team)

Instructions for completing the audit (which can be found below). We are suggesting that you all have a go at this audit whether you are on the Primary (5-11), or Early Years (3-7) route. For people on the Early Years route, we appreciate that some of the mathematics in the audit is beyond what you will be teaching, but we feel it is important to understand the mathematics that the children you teach will go on to study. Having good, all-round maths subject knowledge will help you to make connections between different elements of maths, which will help you make concepts clearer for the children you teach. The audit is not a test, it is designed so that you (and we) have a sense of where your mathematical strengths are and which areas of maths you need to brush up, so that you can teach them effectively. Please do not be worried about it; try all the questions and use it to identify the areas you need to work on. If you want to print out the audit, so you can work on it, feel free to do so. However, there is no need to print it and it is quite a long document. You may want to look at the questions on the screen and just note down your answers on a piece of paper. Below the audit, there is a mark scheme, so you can mark your own work and derive a score. During our first maths seminar in September (which will happened during the week beginning the 11th of September – not during induction week), please bring your scores with you. You should have a score for each of the three sections of the audit. You just need to bring the raw score; no need to convert it to a percentage. We have gone through the audit carefully, and used it in previous years, but it is still possible that there are mistakes, or things that are not clear. If you have any questions at all, please feel free to e-mail Marcus (Marcus.Witt@uwe.ac.uk), or ask during induction week (week beginning 4th of September).

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PGCE Subject Knowledge Audit Primary 2017-18

Q1. Calculate the following: 52.85 + 143.6

1 mark

Q2. Calculate the following: 602 × 57

2 marks

Q3. In this tower, two numbers are multiplied to give the number above.

Write the missing numbers in the tower below to make it correct.

6

2 marks

Q4. Calculate the following: 816 ÷ 24

2 marks

Q5. A shop makes 100 sandwiches.

All the sandwiches are either cheese or tuna.

Some of the sandwiches also have salad with the cheese or tuna.

30 sandwiches have cheese with salad.

15 sandwiches have tuna without salad.

75 sandwiches have salad.

How many sandwiches have cheese without salad?

7

2 marks

Q6. Runa and Jon each start with the same number.

Runa rounds the number to the nearest hundred.

Jon rounds the number to the nearest ten.

Runa’s answer is double Jon’s answer.

Explain how this can be.

2 marks

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Q7.

There are 25 children in the lunch queue, including Nik.

Nik says,

‘There are twice as many children in front of me as there are behind me’.

How many children are in front of Nik?

2 marks

Q8. A dragon lived in a cave.

The dragon doubled in size every day.

After 20 days the dragon filled the cave.

After how many days did the dragon half-fill the cave?

After days

1 mark

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Q9. Look at this expression.

10y + 2

When y = 0.4, the value of 10y + 2 is an even number because 10 × 0.4 + 2 = 6

Write a value for y so that 10y + 2 is a prime number.

y =

1 mark

Now write a value for y so that 10y + 2 is a square number.

y =

1 mark

Q10.This photograph shows three Russian dolls.

The real-life height of the largest Russian doll is 13.5 cm.

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What is the real-life height of the smallest Russian doll?

2 marks

Q11 Look at this information.

Tom was born in 1988

Ben was born in 2000

Tom and Ben have the same birthday.

The ratio of Tom’s age to Ben’s age on their birthday in 2001 was 13 : 1.

What was the ratio of Tom’s age to Ben’s age on their birthday in 2003?

Write the ratio in its simplest form.

:

1 mark

In what year was the ratio of Tom’s age to Ben’s age 3 : 1?

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1 mark Q12. Arranging

Here are six number cards.

(a) Arrange these six cards to make the calculations below. The first one is done for you.

1 mark

(b) Now arrange the six cards to make a difference of 115

1 mark

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Q13. This design has one large square and two identical small squares.

The design measures 36 centimetres by 28 centimetres.

Calculate the length of a side of the large square.

1 mark

Q14. Lambs

On a farm 80 sheep gave birth.

30% of the sheep gave birth to two lambs. The rest of the sheep gave birth to just one lamb.

In total, how many lambs were born? Show your working.

........................ lambs

2 marks

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Q15. Shapes

The drawing shows how shapes A and B fit together to make a right-angled triangle.

Work out the size of each of the angles in shape B. Write them in the correct place in shape B below.

2 marks

Q16. Perimeters

Jenny and Alan each have a rectangle made out of paper.

One side is 10cm. The other side is n cm.

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(a) They write expressions for the perimeter of the rectangle.

Jenny writes 2n + 20

Alan writes 2(n + 10)

Tick ( ) the true statement below.

Jenny is correct and Alan is wrong.

Jenny is wrong and Alan is correct.

Both Jenny and Alan are correct.

Both Jenny and Alan are wrong.

1 mark

Q17 Square cut

The diagram shows a square.

Two straight lines cut the square into four rectangles.

The area of one of the rectangles is shown.

Not drawn accurately

Work out the area of the rectangle marked A.

......................... cm2 2 marks

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Q18. L-shape

What is the area of this L-shape?

Show your working.

........................ cm2

1 mark

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Q19. Cleo has 24 centimetre cubes.

She uses all 24 cubes to make a cuboid with dimensions 6 cm, 2 cm and 2 cm.

Write the dimensions of a different cuboid she can make using all 24 cubes.

_______________ cm, _______________ cm and _______________ cm

1 mark

Q20. This regular 12-sided shape has a number at each vertex.

Ben turns the pointer from zero, clockwise through 150°

Which number will the pointer now be at?

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Nisha turns the pointer clockwise from number 2 to number 11

Through how many degrees does the pointer turn?

°

2 marks

Q21.The following quadrilaterals all have a perimeter of 36cm.

Here is a table to show the length of each side.

Complete the table.

One quadrilateral is done for you.

Side lengths

square 9cm 9cm 9cm 9cm

rectangle 3cm

rhombus 9cm

kite 10cm

2 marks

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Q22. Views

Look at the two triangular prisms.

Isometric grid

They are joined to make the new shape below.

Isometric grid

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Complete the views of the new shape on the grid.

The first one is done for you.

2 marks

Q23. Seb has some cubes with a cross on each face and some cubes with a circle on each face.

He sticks five cubes together to make this shape.

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How many crosses and how many circles are there on the outside of the shape?

Number of crosses

Number of circles

1 mark

Q 24. Dice

The diagrams show nets for dice.Each dice has six faces, numbered 1 to 6

Write the missing numbers so that the numbers on opposite faces add to 7

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1 mark

Q25. Mice

Three types of mice might come into our homes.

Some mice are more likely to be found in homes far from woodland. Others are more likely to be found in homes close to woodland.

The bar charts show the percentages of mice that are of each type.

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Use the bar charts to answer these questions.

(a) About what percentage of mice in homes close to woodland are wood mice?

.......................... %

(b) About what percentage of mice in homes far from woodland are not wood mice?

.......................... %

2 marks

Q26. 500 children started a 20 kilometre sponsored cycle ride.

This graph shows how far they cycled.

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At what distance were exactly half of the children still cycling?

km

1 mark

Estimate how many children completed the 20 kilometre cycle ride.

1 mark

Q27. How fast you can type accurately is called your typing speed.

The regions of the graph show information about different typing speeds.

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Darren’s level of typing is elementary.

In 20 minutes he should be able to type between 500 and 700 words.

Jo’s level of typing is intermediate.

How many words should she be able to type in 20 minutes?

Between ...................... and ......................

1 mark

Kath’s typing speed is 30 words per minute.

What level is Kath’s typing?

Advanced Intermediate Elementary Beginner

Explain how you know.

1 mark Q28. Here are two spinners, A and B.

A B

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Hassan spins the pointer on each spinner. He adds his two scores together.

For each statement put a tick ( ) to show if it is certain, possible or impossible.

One has been done for you.

certain possible impossible

The total will be more than 15

The total will be an even number

The total will be less than 6

The score on A will be less than the score on B.

1 mark

Q29. Here is information about pupils in a class.

• The total number of pupils is 30

• 26 of the pupils do not wear glasses.

• A quarter of the pupils who do wear glasses are boys.

• There are 2 more boys than girls.

Use the information to fill in the missing numbers in the table below.

Number who do

wear glasses Number who do not

wear glasses Total

Number of boys

Number of girls

Total 30

2 marks

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Q30. Look at the information in these two pie charts.

Pupils in class 6K

Key:

Girls

Boys

Girls in class 6K

Key:

11 years old

Not 11 years old

Use the information in the two pie charts to complete the pie chart below.

Pupils in class 6K

Key:

11 year-old girls

All other pupils in the class

1 mark Well done. The answers and mark scheme are below. Please keep your audit score for each of the three sections and bring it with you to your first maths seminar in September (w/b 11th ). Thanks.

PGCE Primary Subject Knowledge Audit – Answers and Mark Scheme

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Area of maths Level Answer Points Score

Number and Calculation

1 5 196.45 2

2 5 34,314 2

3 5 Gives the three correct numbers in their correct positions, ie:

•

2

4 5 34 2

5 5 10 2

6 5 Gives a correct explanation with a number x such that 50 ≤ x < 55, or −5 < x < 5, as an example, eg:

• 53 to the nearest hundred is 100, and to the nearest ten is 50 and 2 × 50 = 100

• If it’s 50 or more but less than 55 it will round to 100 (nearest hundred) and 50 (nearest ten) and 100 is double 50

• 0 is 0 to the nearest 100 and 0 to the nearest 10 and twice 0 is 0

2

7 5 16 children in front of Nik 2

8 6 19 days 1

9 6 (a) Gives a value for y such that 10y + 2 is a prime number, e.g.

0.1, or ½ , or 1.7

(b) Gives a value for y such that 10y + 2 is a square number, e.g.:

2

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-0.1 or 0.2 or 0.7 or 1.4

10 6 9.6 2

11 6 5:1

2006

2

12 6 a) 543 + 621 or 641 + 523 and 514 + 236 or 216 + 534

b) 536 – 421 or 356 – 241

2

13 5 20 1

14 6 104 2

Total for Number and Calculation 26

Shape, Space and Measure

15 6 All four angles correct and correctly positioned, i.e.

2

16 6 Both Jenny and Alan are correct 1

17 6 42cm2 2

18 6 30 1

19 6 (a) Gives three integers other than 2, 2, 6 (in any order) whose product is 24, eg:

• 1, 1, 24

• 1, 24, 1

• 1, 2, 12

2

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• 1, 3, 8

• 1, 4, 6

• 2, 3, 4

20 5 5

270

2

21 5 Completes all three rows correctly, e.g.

• rectangle 3cm 3cm 15cm 15cm

rhombus 9cm 9cm 9cm 9cm

kite 10cm 10cm 8cm 8cm

2

22 6 Draws both views correctly using the grid, ie

2

23 5 8 crosses

14 circles

1 1

24 6 Gives all three numbers correctly for the first net, i.e.

Gives all three numbers correctly for the second net, i.e.

2

30

Both right for 1 mark

Total for Shape, Space and Measure 18

Data Handling

25 6 (a) 50 ± 2

(b) 55 ± 1

2

26 5 16

Answers in the range 180-190 inclusive

2

27 6 (a) Gives both correct values, ie

700 (or 701) and 1000 (or 999)

(in either order)

(b) Indicates Elementary and gives a correct explanation that places the speed clearly within the correct section on the graph, eg:

• 30 words in one minute is 300 words in ten minutes

• 30 wpm = 900 words in 30 minutes

• Darren is between 25 and 35 words per minute so she is the same as Darren

2

1

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Area of maths Score

Number and Calculation

Shape, Space and Measures

Data Handling

It is important that you use the information from this audit to inform your subject knowledge action plan, which will be explained in your first seminar. Please make use of your maths tutor as a source of support and suggested reading etc if you are unsure.

28 5 Award TWO marks for three rows ticked correctly as shown:

2

29 5 Completes all 8 entries of the table correctly, i.e.

... do wear glasses

... do not wear glasses

Total

... boys 1 15 16

... girls 3 11 14

Total 4 26 30

2

30 Divides the pie chart into two correct sectors and shades/labels correctly, eg

•

3 of the 8 sections shaded.

2

Total for Data Handling 13