Transcript of "Think Board Interview, Recommendations and Reflection"
SEMESTER 1 2011 EDUC8502 TEACHING MATHEMATICS IN EARLY YEARS Assignment 2 What do children know about numbers? Due: Friday May 13th 2011 Sharon McCleary 19113469 Unit Co-ordinator: Associate Professor Christine Howitt Tutor: Ms. Clair Kipling
What do children know about numbers? Background The purpose of this report is to present findings and recommendations arising from an interview with Gabi, a Year 2 female student, conducted on 1st April 2011 at Hollywood Primary School in Perth, Western Australia. The duration of the interview was approximately 40 minutes. It focussed on determining what the child knew about numbers, using Think Boards as a strategy to promote communication about her number knowledge and connections across various modes of representation. The Think Board is a recording format that allows the student to express their understanding of a concept in various ways (i.e. using stories, symbols, pictures and real-‐life representations). It gives valuable insight into the connections the student has formed between enactive (concrete objects), iconic (pictures, diagrams) and symbolic (words, symbols) representations of mathematical concepts (Frid, 2004), shedding light on the individual student’s process of mathematical meaning-‐making and providing a useful method of identifying future learning areas within the child’s zone of proximal development (Krause, 2010). Introduction The interview commenced with a relaxed discussion about numbers, aimed at making the student feel comfortable, developing rapport, and determining her general disposition towards numbers. The book “10” by Vladimir Radunsky was read, and the child was introduced to the pre-‐made Think Boards. She was asked to choose 3 numbers within set ranges to represent in different ways on the Think Boards, and guided through each of the sections. She displayed genuine excitement about numbers and was eager to participate, initially asking if she could use larger numbers outside the given ranges.
Teaching Mathematics in Early Years EDUC8502 Gabi selected ‘15’ (recommended range of 11-‐19) as her first number, and 80 as her second (recommended range 50-‐100). After completing the second Think Board she realised she would have difficulty representing larger numbers and chose ‘2’ as her third number. Attempts to persuade her to choose a higher number resulted in her choice of ‘20’ as her third number. She was clearly outside her comfort zone when larger numbers were suggested and it was not appropriate to challenge her further that particular day. The resources provided for the child to use included various sets of counters, environmentally available natural materials such as leaves and stones, coloured pencils, stickers, stamps, and lead pencils. The three Think Boards are included for reference in Appendix A, B and C respectively, along with photographs of the ‘real’ items used on each Think Board. Student Profile: Analysis Think Board One: ‘15’ Gabi was able to represent the number ‘15’ correctly in symbolic form, as can be seen in the “symbol” section of Think Board One in Appendix A, where she wrote ‘15’ using the correct pencil grip and number-‐writing formation. When asked to represent ‘15’ using real objects, she hesitated, asking “15 of anything?” This demonstrated an understanding of ‘number’ as an idea that describes things in a group, independent of what is being counted or labelled (Demant, 2008). She proceeded to collect and count 15 leaves from the surrounding gardens, initially counting in 1’s, then collecting groups of two and skip counting (“7,9,11”), before collecting a group of four and counting-‐on to arrive at 15. She then verified there were 15 leaves using rational counting (Cathcart, 2011): making a one-‐to-‐one correspondence between each leaf and the sequential number name as she placed it on the Think Board. This revealed a solid understanding of the principles of counting identified by Gelman and Gallistel (1978), namely the Stable Order Principle, One-‐to-‐one correspondence, Cardinal principal, abstraction principle and the order-‐irrelevance principle (Compton, 2007). Sharon McCleary 3
Teaching Mathematics in Early Years EDUC8502 Gabi initially had difficulty representing ’15’ pictorially, and was unable to respond to prompts requesting her to think about instances of this number in everyday life. However, after further explanation (i.e. ‘3’ could be represented by three little pigs or a triangle), she produced an example relating to the ‘real’ section on her Think Board, drawing three flowers with five petals each (See Picture Section of Think Board One). She counted each petal individually, then stated “5+5+5 equals 15”. This shows she successfully decomposes and recomposes numbers, and has an internal concept of multiplication as repeated addition of equivalent groups, consistent with the second level of conceptual development for multiplication representations given by Thomas (Thomas, 1997). The Story section of Think Board One indicates she has sound knowledge of the standard classroom number practise of creating and representing word problems using conventional symbols (i.e. 14+1=15). It also shows she was building meaningful connections for the context of this particular Think Board as she engaged with the activity, as her symbolic (story) and enactive (real) representations related to the same theme (i.e. garden). During this part of the interview, Gabi demonstrated good early number sense, a solid understanding of counting and the beginnings of calculation. Think Board Two ‘80’: Think Board Two (Appendix B) shows ‘80’ represented in a non-‐standard form in the Picture section: seven longs, nine units and one separated unit; the place value chart was suggested. Gabi was unable to create an equivalent representation of ‘80’ when requested. She did not recognise ‘9+1’ could be traded for a ‘10’ and represented by an additional long, displaying confusion even when this was explicitly demonstrated and stated. This indicates she has not fully abstracted the concept of a unit of ten (Gray, 1999); she is operating within the extended stage of structural development described by Thomas (2002), using the sub-‐system of units to form her understanding of the base-‐10 system. This developing understanding of the base-‐10 system is also apparent in the Real section of Think Board Two where she has used seven bananas to represent seven tens and ten random fruit counters for the remaining ten. Sharon McCleary 4
Teaching Mathematics in Early Years EDUC8502 Both sections indicate she can partition the decade and represent the number accurately, but reveal a limited understanding of grouping and place value concepts. They also indicate strain on her working memory since she finds it difficult to consider the discrete parts and the whole number simultaneously in part-‐part-‐whole relationships (Gray, 2000). This may result from repeated classroom experiences of partitioning ten, and shows she has not conceptualised groups of ten as a unit, or visualised the pattern of tens making up 100. Gabi did not provide authentic real-‐world connections in the Story section of Think Board Two, indicating her limited awareness of real-‐world contexts for this number. Think Board Three ‘20’: Gabi initially chose ‘2’ for this Think Board, stating “I’ll pick an easier number, ‘2’. It’s my Birthday!”. This indicates her awareness that the previous representations had been difficult, and shows she is capable of building authentic real-‐world connections for numbers with which she is familiar and comfortable. She proceeded to use the birthday connection with the number ‘20’. Examination of the Picture section of Think Board Three shows that she drew twenty cupcakes to represent the number, linking this drawing to her Story section by showing the 17 cupcakes separated from the “3 new cupcakes”. The Real section of this Think Board reinforces this link by representing the ‘17’ using bananas and differentiating the ‘3’ using bunches of grapes. This shows her understanding of part-‐part-‐whole relationships, however, as can be seen in the Story section of the Think Board, she represents her number sentence incorrectly as “19+1=20”, again indicating some confusion with part-‐part-‐whole relationships. When asked to write the number sentence corresponding to her story, she produced the “17+3=20”, as shown on Think Board Three. Again, she did not use a place value chart. This demonstrates it is not a natural part of her expressive repertoire; she thinks of multidigit numbers in terms of units and is operating within the first layer of the number system (Geist, 2009). (842 words) Sharon McCleary 5
Teaching Mathematics in Early Years EDUC8502 Recommendations: The main areas Gabi requires support in are: • Developing the underlying conceptualisations involved in grouping in tens and place value operations. • Consolidating her number sense for multidigit numbers, initially up to 100. These areas have been identified using observations from the interview and prioritised using the WA Curriculum Framework (WA Curriculum Council, 2005), First Steps Documents (Willis, 2004) and The Australian Curriculum, Mathematics (Australian Curriculum, Assessment and Reporting Authority [ACARA], 2010). They represent the foundation for developing understanding of our numeration system and higher-‐level concepts of number, including estimation and computation (Cathcart, 2011). WA Curriculum Framework: (Curriculum Council, 2005) Gabi has predominantly achieved Level 2 of the WA Mathematics Curriculum Framework (Curriculum Council, 2005): “Understand Numbers (N6.a.2): Reads, writes, says and counts with whole numbers beyond 100, using them to compare collection sizes and describe order.” Understand Operations (N7.2): Understands the meaning and connections between counting, number partitions, addition and subtraction; uses this understanding to represent situations involving all four basic operations. Calculate (N8.2): Counts, partitions and regroups in order to add and subtract one-‐and two-‐digit numbers, drawing mostly on mental strategies for one-‐digit numbers and a calculator if numbers are beyond the student’s present scope.” (WA Curriculum Framework Progress Maps Mathematics Outcomes Overview: Number, 2009) In her Think Board representations (See Think Boards Two and Three), Gabi partitioned the last decade, demonstrating she thinks of numbers as part-‐part-‐whole relations. There was little evidence of her understanding the regrouping: she seemed to create the seven tens from procedural knowledge as she was unable to explain the Sharon McCleary 6
Teaching Mathematics in Early Years EDUC8502 base-‐10 grouping concepts behind the procedure, demonstrating a lack of relational understanding (Cathcart, 2011). She also revealed limited number sense for larger numbers, ‘20’ and ‘80’, relying on counting in units (rather than grouping tens) to represent these numbers, indicating she has not fully internalised the concept of grouping in tens to facilitate more efficient counting. Therefore activities emphasising counting, grouping, place value and number patterns up to 100 should be introduced. The Australian Curriculum: (ACARA, 2010) The Australian Curriculum Year Two elaboration requires students to “Recognise, model, represent and order numbers to at least 1000”, and “Group, partition and rearrange collections up to 1000 in hundreds, tens and ones to facilitate more efficient counting.” (Australian Curriculum, Assessment and Reporting Authority [ACARA], The Australian Curriculum, Mathematics, 2010). It would be difficult for Gabi to build number sense for numbers up to 1000 as required by The Australian Curriculum since she has not yet consolidated grouping, place value and number patterns for numbers under 100. Grouping by tens is fundamental to the place value system, and a thorough understanding of place value is necessary for the development of higher-‐order number sense and operations (Reys, 1989). Therefore, Gabi would benefit from consolidation of the Year 1 outcome “Count collections to 100 by partitioning numbers using place value.” (ACARA, The Australian Curriculum Mathematics, 2010). This is the earliest curriculum outcome which she is not confidently able to demonstrate, and it has therefore been prioritised in order to minimise misconceptions and build a solid foundation for future work. First Steps in Mathematics Documents: (Willis, 2004) During the interview, Gabi displayed several characteristics typical of the First Steps Quantifying Phase (Willis, 2004), automatically selecting counting as a strategy, skip counting leaves when constructing Think Board One and realising it would give the same result as counting by ones. She was able to write number sentences matching Sharon McCleary 7
Teaching Mathematics in Early Years EDUC8502 the semantic structure for each of the Think Boards, producing small number addition problems. Each Think Board demonstrated her tendency to think about number in terms of part-‐part-‐whole relations (e.g. 19+1=20 Think Board Three), typical of a child in the Quantifying Phase (Willis, 2004). A key element of this phase is conservation of number, which Gabi demonstrated when re-‐arranging counters without having to re-‐count them. This indicates she is developmentally able to deal with abstract symbolic activities and can mentally manipulate numbers represented by symbols with a real understanding of what she is doing (Charlesworth, 2007). She demonstrated this confidently for smaller numbers (i.e. ‘15’ Think Board One), but did not display an understanding of the place value symbols used to represent larger numbers and would benefit from more concrete experiences constructing systems of 10’s. This would consolidate her understanding of the Base-‐10 patterns and place value representations up to 100, and eventually translate to larger numbers. Recommended Activities: The following two activities have been designed to give exposure to these outcomes: Activity 1: Build a 100’s Chart using Tens-Frames. First Steps in Mathematics - Number: (Willis, 2010): Understand Whole and Decimal Numbers Key Understandings (Willis, 2010, pg 52): “KU5 There are patterns in the way we write whole numbers that help us remember their order.” Reason About Number Patterns (Willis, 2010, pg 242) “KU 5 Our numeration system has a lot of specially built-‐in patterns that make working with numbers easier.” Materials: Lead Pencil, coloured pencils, paper, die, two different coloured counters, 10 ‘tens frames’, a 100’s chart cut into strips of 10 (i.e. 1-‐20, 11-‐20, 21-‐30, etc). Sharon McCleary 8
Teaching Mathematics in Early Years EDUC8502 1. Roll the die and use coloured counters to fill in the tens frame. Alternate the colour of counters used for each roll of the die. 2. When the first tens frame is completely full, write the corresponding number sentence using the coloured counters to assist. 4. Trade the completed tens frame for the first row of the 100’s chart, and colour the numbers corresponding to the coloured counters. 5. Continue this process until all 10 tens frames have been completely filled, and the entire 100’s chart has been generated. This activity capitalises on Gabi’s ability to partition ten (shown on Think Boards Two and Three), integrating visualisation to assist recognition of the part-‐part-‐whole relationships within the tens frame (McIntosh, 1997), but extending her thinking to the next level of counting, where ten units are grouped together and ‘ten’ becomes the iterable unit (Jones, 1994). Studies show that imagery is used extensively in the construction of mathematical meaning, with Presmeg (1986) identifying five main types of visual imagery: concrete, pattern (relationships), memory, kinaesthetic (involving muscular activity) and dynamic (Thomas, 2002). This activity utilises concrete, pattern, kinaesthetic, and memory imagery to reinforce connections between verbal, imagistic and formal notation systems of representation (Goldin, 1987). It strengthens the connections between the concrete counters and the conventional symbolic representation by writing the corresponding number sentence, providing explicit links which encourage mathematical learning. It also introduces the idea of trading ten units for a single entity of ten, allowing the student to use their previous constructs of the system of 1’s to develop an understanding of the Base-‐10 system and the patterns within the 100’s chart. This concept of grouping is a crucial part of the numeration system and understanding base-‐10 and place value. Sharon McCleary 9
Teaching Mathematics in Early Years EDUC8502 The activity gradually builds multi-‐digit number sense by reinforcing the structure and order of the 100’s chart. This is an essential pre-‐requisite for understanding larger numbers up to 1000. Activity Two: Jelly-Bean Party Bag Game (Estimation, Counting and Grouping using Place Value Mats) First Steps in Mathematics – Number: (Willis, 2010): Understand Whole and Decimal Numbers Key Understandings (Willis, 2010, pg12&60): “KU1 We can count a collection to find out how many are in it. KU6 Place value helps us to think of the same whole number in different ways and this can be useful.” Materials: Large bag of painted beans (e.g.97), party bags, place value charts, die. 1. Ask student to estimate how many jelly beans are in the large bag. 2. Roll the die, explaining the place value chart by representing single-‐digit numbers as individual beans in the 1’s column. 2. Explain once there are 10 jelly-‐beans they can be put into a party bag in the 10’s column: this will help us count faster. 3. When the bag is empty, ask student to count using the party bags, and write the number in the place value chart, comparing it with their estimate. 4. Ask how many jelly-‐beans in each column to encourage partitioning of this number and demonstrate the difference between face value and complete value. (i.e.90+7=97). This activity balances challenge and success, providing a meaningful, real-‐life context for counting. Sharon McCleary 10
Teaching Mathematics in Early Years EDUC8502 It encourages estimation, which is an effective way of developing number sense (Reys, 1989), and allows direct links from the enactive processes of counting and grouping to the written/symbolic place value representations of number. Failure to understand place value systems often stems from an inability to differentiate between face value and complete value, since the same number can represent several values (Varelas, 1997). Focusing on the semiotics aspects of the written place value system during the activity helps students differentiate between face value and complete value, and assists conceptual understanding of place value, which is crucial for developing higher-‐level number concepts. Rubin and Russell (1992) state counting, grouping, estimating and notating are essential in developing representations of the number system (Thomas, 1994). The activity utilises a game format for problem solving to promote automaticity and consolidate key concepts by encourage justification of mathematical ideas in a social context, thereby improving mathematical fluency (Geist, 2009). Fluency, Reasoning and Problem Solving are Proficiency Strands within the Content Structure of The Australian Curriculum (ACARA, 2010). (1378 Words) Critical Reflection: The interview process made me realise children’s mathematical thinking is highly personal and very different to adult thinking (McIntosh, 1997; Sfard, 2005); their number concepts are limited by their developmental stage and real-‐world experiences, their own ability to make mathematical sense out of these experiences, and to communicate these understandings effectively. Gabi’s interview responses indicated her conscious exposure to numbers was mostly confined to the classroom, since she had difficulty providing authentic examples for the Think Board representations. I was surprised how difficult it was for Gabi to make real-‐world connections and realised the links between classroom mathematics and the real-‐world need to be regularly and explicitly made to become meaningful for children. The Think Boards challenged her to consider number concepts using different modes of representation, and enabled her to explore connections between these representations. I observed Gabi actively trying to make meaning in true Sharon McCleary 11
Teaching Mathematics in Early Years EDUC8502 constructivist fashion as she engaged in the activities -‐ similar to Wilkerson-‐Jerde & Wilensky’s (2011) description of mathematical learning as the process of building a network of mathematical resources by establishing relationships between different components and properties of mathematical ideas. I realised children are only able to reveal their knowledge if they are given the opportunity to do so, and classroom activities need to be open-‐ended to allow them to demonstrate and explore their own mathematical thinking without placing limitations on it. The teacher’s role is to provide opportunities for deep understanding of concepts and make clear links between the concept and the conventional mathematical symbols, allowing semiotic meaning making without stifling their inherent mathematical thought processes. In order to achieve a deep understanding of mathematical concepts and achieve autonomous learning, children must be allowed to reinvent mathematical concepts in their own minds (Kamii, 1984). During the interview, communication was pivotal in encouraging further learning. Talking about the Real and Story sections of the Think Boards made Gabi’s thinking visible (Whitin, 2000) and generated further learning opportunities; it motivated learning in the child, and teaching in the adult (Sfard, 2005). This was a clear demonstration of Gabi constructing knowledge in a social context (Vygotsky, 1978), where communication clarified and consolidated her thinking. It also demonstrated how meaning arises from the tension (Radford, 2011) between the child’s inner understanding of mathematical ideas and their functioning in a shared sociocultural world of semiotic systems (Fried, 2011). Given that very young children are still developing knowledge of mathematical language and conventions, and the limitations of interpreting their external representations, it is important to observe them on multiple occasions, through various representational modes, using active listening, and interpreting gesture, pictures and symbols to determine their mathematical understandings and assist them towards achieving autonomous learning. (446 words) Sharon McCleary 12
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Teaching Mathematics in Early Years EDUC8502 Thomas, N., Mulligan, J. & Goldin, G., (2002), Children’s representation and structural development of the counting sequence 1-‐100, Journal of Mathematical Behaviour, 21, 117-‐133. Varelas, M. & Becker, J., (1997), Children’s Developing Understanding of Place Value: Semiotic Aspects, Cognition and Instruction, 15(2), 265-‐ 286. Vygotsky, L., (1978), Mind in Society, Harvard University Press, Cambridge, MA. Whitin, P. & Whitin, D., (2000), Math is Language Too: Talking and Writing in the Mathematics Classroom, National Council of Teachers of English, USA. Wilkerson-‐Jerde, M. & Wilensky, U., (2011), How do mathematicians learn math?: resources and acts for constructing and understanding mathematics, doi: 10.1007/s10649-‐011-‐9306-‐5. Willis, S., Devlin, W., Jacob, L., Powell, B., Tomazos, D. & Treacy, K., (2004), First Steps in Mathematics: Number (Book 1 & 2), Rigby, Australia. Sharon McCleary 15