MATHEMATICS PORTFOLIO
26803351031240Mathematics Subject Knowledge Audit and Pre-course GuidancePGCE Early Years2016-201700Mathematics Subject Knowledge Audit and Pre-course GuidancePGCE Early Years2016-2017Welcome from the UWE primary and early years 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: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: also provide you with a mathematics audit (at the end of this document) that you should complete. It 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. 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 () 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 importantTeachers’ 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).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 knowledgeThe 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. and Cockburn, A. (2008) Understanding Mathematics for Young Children. London: Sage and/or Haylock, D. (2010) Mathematics Explained for Primary Teachers. 4th ed. Sage. The Haylock and Cockburn book is more directed towards the early years, but the Haylock book is an excellent source of subject knowledgeYou 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: 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:, 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: the links to DCSF publications are moved. If you ‘Google’ DCSF RR118, you should find it. 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: 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 SeptemberMarcus Witt & Ben Wiggins(University of the West of England Primary Mathematics team)Early Years 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. No calculators for this section please. You are allowed to carry out calculations manually (i.e. you don’t have to do it all mentally)QuestionYour AnswerMark1487 + 848 =2813 – 564 =33.6 x 3 = 436 x 43 =5?24 ÷ 5 =69.2 – 4.85 =Questions 7, 8 and 9: Put the following in order from lowest to highest72.07; 2.7; 2.68; 0.72; 2.06898?; ?; ?; ?; ?94; -2; -10; 17; -510Round 76 to the nearest 1011Round 326 to the nearest 10012Round 5.7 to the nearest 113Round 5.38 to the nearest 0.114Give all the factors of 10015Write down 3 prime numbers between 10 and 2016What is the lowest common multiple of 6 and 9?Now write down your total score for Section A:Section B: Problem solving, reasoning and mathematical thinking. Write down the numbers represented by the symbol to make the calculations correct. 17( + 1?) x 2 = 1018(0.3 + ) = 119(2 + ) x 9 = 6320(3 + 4) x = 5621( x 3) x 6= 5422(2? + 3?) x 5 = In the following questions, write down the missing numbers in the sequences235, 9, , 17, 21, , 29 … = = 241, 3, 6, 10, , 21, , 36, … = = 25-5, , -1, , 3, 5, … = =260.28, , , 0.31, 0.32 = =27Sarah had a bag of cherries. She ate 5 cherries, then gave half of what she had left to Liam.Liam ate 5 of his cherries, then gave half of what he had left to Amy.Amy got 2 cherries.How many did Sarah have to begin with?28Tom thinks of a number. He works out that 40% of his number is 160. What was his original number?29An iced cake costs 10p more than a plain cake. Jane buys 4 of each and spends ?2 in total. How much does an iced cake cost?30George thinks of a number. He divides it by 4 and then adds 25 to the result. He ends up with 36. What was his number?31.You are given that 7 x 8 = 56Now write down the answers to the following calculations:AnswersMark70 x 8000.08 x 0.75.6 7Now add up your score for Section B Section C – Shape, Space, Measures.32. Which of these shapes are rhombuses?2621915126365F00F1715135114300D00D96266068580B00B9588569215A00A367031139701C00C161036035560E00EAnswerMark33. For each of the patterns (A-D) below, state the number of lines of symmetry and the order of rotational symmetry.27743151295400085026572390002983865196215B00B12503151104900086042511049000316865300990A00A27743156286500850265438150014884407239000612140196215C00C95504019621500372872072390003004820120015D00D9029701200150038290505715004238625571500PatternLines of symmetryOrder of rotational symmetryMarkABCD34. Fill in the grid below concerning the properties of the shapes given. Put a tick or a cross in each column for each shape.5322570736600063627073660004179570121285D00D3027045121285C00C181737026035B00BAEShapeIs regularHas at least one right angleIs a quadrilateralMarkABCDE35. Using the information given below, calculate the lengths of A and B. 883920831850039795458318500 45 cm 20 cm3179445111760A00A3722370140335B00B2007870450850088392045085003131820260350088392092710003493770641350031318201403350020078701403350088392014033500LengthMarkAB36. Given that the perimeter of the rectangle below is 50 cm, work out its area. 883920349250032575510287010 cm0010 cmAreaMarkNow add up your score for Section C Section D – Data Handling and Probability37. For some reason, the children recorded the number of cars of different colours, going past the school gate. Their findings are shown below. 626745139700002171701250952500025016078206350000645795635000021717012509520000200321754511557000396049517716500807720177165006553206286500217170124460150001506457956223000217170124460100001002417445622300064579562230002171701143005000506362706159500398145850900000387477066675Other00Other233172066675Yellow00Yellow313182057150Green00Green149352057150Red00Red70294557150Blue00Blue62674575565002093595113665Car Colour00Car ColourQuestionAnswerMarkHow many more red cars than yellow cars passed the school?Approximately how many green cars passed the school?Which colours had more than 200 cars?Approximately how many cars passed all together?38. Look at the train timetable below and then answer the questions. Bus A Bus B Bus PPerth06:3507:20Then at the same time each hour until20:04Freemantle06:5507:3520:19Port ArthurArr 07:12Dep 07:20Arr 07:50Dep 08:05Arr 20:35Dep 20:41Morecombe07:4608:2521:00Freetown07:5308:3221:06Canary Island08:1908:5521:25Ormskirk09:0209:3222:00QuestionAnswerMarkWhat time does Bus A arrive in Canary Island?How long does it take bus B to go from Freemantle to Ormskirk?Jim wants to go from Morecombe to Freetown. He arrives at the bus stop at 11:07. How long must he wait for the next bus?What time will he arrive in Freetown?Which bus waits for the shortest time in Port Arthur?39. When a ball is selected from a bag containing 3 black, 4 red and 5 blue balls, what is the probability that it is:AnswerMarkRedNot blueGreenRed or BlueNow add up your score for Section D Answers to audit questions. Section A. All questions are worth 1 mark, unless otherwise stated. QuestionAnswerQuestionAnswer113352249310.8315485?4.8064.3570.72; 2.0689; 2.07; 2.68; 2.78?; ?; ?; ?; ?9-10; -5; -2; 4; 17 108011300126135.4141, 2, 4, 5, 10, 20, 25, 50, 100 (2 marks for all; 1 mark for any 6 or more correct)15Any three of 11, 13, 17, 191618Section A is out of a possible 17 marks. QuestionAnswerQuestionAnswer17 = 3 ? 18 = 0.719 = 520 = 821 = 322 = 302313, 25 (2 marks)2415, 28 (2 marks)25-3, 1 (2 marks)260.29, 0.3 (2 marks)2723284002930p30443156000; 0.056; 0.8 (3 marks)Section B is out of a possible 21 marks. 32A, B, C, D 2 marks for all 4, 1 mark for 3. 33.PatternLines of symmetryOrder of rotational symmetryMarkA11(2 marks maximum)B11(2 marks maximum)C02(2 marks maximum)D02(2 marks maximum)34.ShapeIs regularHas at least one right angleIs a quadrilateralMarkAXX(3 marks maximum)BXX(3 marks maximum)CXX(3 marks maximum)D(3 marks maximum)EXX(3 marks maximum)35.LengthMarkA5cm(1 mark)B15cm(1 mark)36. Area = 150 cm2 (1 Mark)Section C is out of a possible 28 marks.37.QuestionAnswerMarkHow many more red cars than yellow cars passed the school?150(1 mark)Approximately how many green cars passed the school?220 (+/- 5)(1 mark)Which colours had more than 200 cars?Red & Green(1 mark)Approximately how many cars passed all together?930 (+/- 20)(1 mark)38.QuestionAnswerMarkWhat time does Bus A arrive in Canary Island?08:19(1 mark)How long does it take bus B to go from Freemantle to Ormskirk?1 hr 57 mins(1 mark)Jim wants to go from Morecombe to Freetown. He arrives at the bus stop at 11:07. How long must he wait for the next bus?18 mins(1 mark)What time will he arrive in Freetown?11:32(1 mark)Which bus waits for the shortest time in Port Arthur?Bus P(1 mark)39.AnswerMarkRed1/3 or 1 in 3 (1 mark)Not blue7/12 or 7 in 12(1 mark)Green0(1 mark)Red or Blue9/12 or ? or 75% or 0.75(1 mark)Section D is out of a possible 13 marks.The total score is out of 81. Whatever you score, please don’t worry, but DO keep the audit as we’d like to know your score for each of the different sections when you arrive to begin the course. This will help us to support you in the best way possible. Thanks. ................
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