Observing maths You are required to observe and write about, from the following aspects of mathematics covered in Block 2, four activities involving children working mathematically: 1 counting and understanding number 2 place value 3 using number words and/or symbols 4 fractions, decimals, percentages, ratio or proportion 5 properties of number operations 6 mental calculation strategies 7 written calculation methods 8 measurement 9 properties of shapes or transformation and symmetry 10 handling statistical data. The mathematical aspects set out above are arranged into four sections, A, B, C and D, as shown in the table below. Specific details of what you are required to write for your four chosen activities are set out below. You must choose one aspect to observe and write about from each section. For example, you may choose to write about aspects 2, 4, 7 and 10 from sections A, B, C and D respectively. Aspects A 1, 2, 3 B 4, 8 C 5, 6, 7 D 9, 10 In planning to complete the ‘Observing maths’ part of your Maths Workbook, we strongly encourage you to try to carry out as many of your four observations as possible in your school setting. This will enable you to draw on your knowledge of the context, in particular the children’s prior learning, and also to discuss the activity with the children and practitioners in school. This may provide you with a greater depth of insight than it is possible to obtain from watching a video. We encourage you to try to observe activities across a range of age groups. This is not a requirement but will help you to develop your appreciation and understanding of children’s mathematical learning across the primary years as a whole. You can make use of one or more of the practice-setting videos for your observations of any of the four sections if you are unable to arrange to observe a ‘live’ activity in school. Your selection of activities will therefore comprise one of the following three possibilities: four ‘live’ observations in school a combination of observations of ‘live’ and ‘video’ activities four observations from the practice-setting videos. For in-school activities, you may find it necessary to complete your observation in school before you carry out the relevant reading online and in the Maths Reader. This will be fine. You will be able to write down details of the school-based activity in this document, and then complete your analysis once you have done the necessary reading and online activities. There are five sections to complete for each of your chosen aspects. Please read the guidance below, in ‘Structure’, for what to include in each section. We have also provided a completed example, which you should refer to as a guide to the level of detail required. We do not specify a minimum or maximum word length for writing up each observation because the ‘Observing maths’ part of the Maths Workbook is in the form of a portfolio rather than an academic essay. We suggest that around 600 words per activity will enable you to fully meet the requirements for TMA 02. Please remember that your completed ‘Observing maths’ section will form a part of your submission for TMA 02, for which you will be required to show the following: how your own mathematical subject knowledge has developed how this will or could impact on your work supporting children in school. You need to include sufficient detail to demonstrate both of these. This means that when assessing what you have written about your observations, your tutors will be paying particular attention to the following sections: key mathematical ideas and vocabulary role of resources/mental imagery reflections and next steps for children’s learning references: you should include in-text references as you would for an essay to make links to relevant reading, e.g. Haylock and Manning (2019); or NRICH (2016). This is important to show how your subject knowledge is developing. At the end of the assignment, include a reference list that contains the source details for everything you have referred to in both Part 1 Observing maths and Part 2 Reflecting on your learning in Block 2.] [Structure ‘Live’ activities in school may involve a group of children or an individual child. For ease of expression, guidance here refers generally to ‘children’. Each of the five practice-setting videos shows an activity with a group of children. There are five sections to complete for each aspect; these are set out below. Activity State what the intended learning outcomes were for the activity. Individual schools may have their own term for learning outcomes for example, ‘learning intention’, ‘we are learning to’ or ‘learning objective’). For ‘live’ activities you will need to liaise with the class teacher to ascertain the learning outcomes for the activity, or produce appropriate learning outcomes if you plan the activities yourself. Describe what the children were doing. You should include the age or year group of the children and describe the role of the teacher or other adult in the activity. Children’s learning Describe what you believe the children learned from the activity, or what they found difficult to grasp, if appropriate. Indicate the evidence that led you to form this conclusion. This might include what the children did, wrote or said. Key mathematical ideas and vocabulary Outline the key mathematical subject knowledge and vocabulary involved in the activity, with reference to any relevant reading you have carried out. This may be your online reading for Block 2, Haylock and Manning (2019) (the Maths Reader) or include other online sources, books, resources in school, national curriculum or guidance documents. You may need to refer to chapters from the Reader beyond those you have been directed to read in the module activities. You must acknowledge the sources of your reading. • Where appropriate, identify any links with other areas of mathematics. Acknowledge the source for any definitions you include. Role of resources/mental imagery Briefly discuss the role of resources and/or mental imagery in supporting children’s understanding. If there was no evidence of this, you should state that this was so and suggest how resources or imagery might have helped, if appropriate. Again, you should refer to your relevant reading. Reflections and next steps for children's learning • What went particularly well in the activity, and why do you think this was the case? • What (if anything) would you suggest might have been done differently, and why? [Example of a completed section 7. Properties of number operations Activity Maria (teaching assistant) worked with three eight-year-olds to represent multiplications (e.g. 5 x 3) as arrays. Learning objectives: - to understand that multiplication can be carried out in any order - to improve rapid recall of multiplication facts. Maria’s role: - to encourage the children to use a range of relevant vocabulary - to assess their understanding that multiplication can be carried out in any order. The class teacher indicated this practice is key in improving the children’s fluency in multiplication as they move towards key stage 2 (DfE, 2013, p. 25). No additional resources provided. The children drew the arrays in their maths books. Children’s learning All children produced an appropriate diagram of an array from a multiplication presented in written form, e.g. for 5 x 3: XXXXX XXXXX XXXXX They explained this as ‘five lots/sets/lines’ of three, although Jo used the term ‘five rows of three’ due to her confusion over the terms ‘rows’ and ‘column’ rather than misunderstanding of what ‘5 x 3’ meant. All children knew that the order of the numbers could be changed to give the same total, and could draw an appropriate array: XXX XXX XXX XXX XXX Nayeem said, ‘All times tables can be done the same both ways. The answers are the same.’ The children agreed that the two arrays each had the same number of crosses: ‘It’s just like the crosses have been turned round’ (Jo). Key mathematical ideas and vocabulary 1 Multiplication as repeated addition (W8 S2) – this relates to the idea of ‘lots of’ or ‘sets of’ (Haylock and Manning, 2019, p. 142). 2 Image of multiplication as a rectangular array (Haylock and Manning, 2019, p. 144). The NRICH (2016) website explains some other useful contexts for arrays (e.g. learning multiplication facts, exploring factors/prime numbers). 3 Commutative law of multiplication – i.e. that multiplication can be carried out in any order (Haylock and Manning, 2019, p. 155). 4 Key vocabulary: ‘lots of’, ‘groups of’, ‘sets of’, ‘times’, ‘multiplied by’. 5 Jo’s comment about the number of crosses staying the same demonstrates a grasp of conservation of number (Haylock and Manning, 2019, p. 362) Role of resources/mental imagery The image of the array itself is powerful: it shows at a glance that, 3 x 5 is the same as 5 x 3. A peg board would be a good ‘hands-on’ resource to allow children to explore factors – e.g. ‘How many ways can you find to make 24?’ (Haylock and Manning 2019, p.164) Vocabulary cards could have been useful to ensure the children were familiar with the range of language that can be used to describe arrays e.g. row, column, table. Reflections and next steps for children’s learning From the children’s responses the use of arrays seems an effective way of demonstrating the commutative law. The children liked producing a picture for their answer, and this helped them to achieve the first objective. I was interested in Haylock and Manning’s (2019, p. 143) learning and teaching point that although 3 x 5 means ‘5 lots of 3’, there is no need to make a fuss over how children say or represent it, as it is much more important that they understand that multiplication can be carried out in any order. However, Jo’s misunderstanding suggests that the range of vocabulary used has the potential to confuse. Children need to develop a mathematical vocabulary to be able to talk multiplication and thus develop their knowledge and understanding (DfE, 2000, p. 1) Look for opportunities to develop the children’s understanding of ‘rows’ and ‘columns’ possibly in data handling or through other subjects (e.g. science, geography). Use the image of a multiplication square to encourage the children that, because of commutativity, they already know most of the ‘difficult’ times-table facts (e.g. 7 and 8).] 
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