Facilitating young children s understanding of the equal sign

Transcription

1 Facilitating young children s understanding of the equal sign Dr Thérèse Dooley, St Patrick s College, Drumcondra and Aisling Kirwan, Holy Family National School, Rathcoole, Co. Dublin

2 Talk Tools Task Teaching mathematics well requires attention to three aspects: Tasks, Tools and Talk (Askew, 2012)

3 Any teaching, including the particular case of mathematics, actually teaches far more than the content: children are learning much more than just mathematics in mathematics lessons. They are learning a lot about themselves, about their peers and their relationships. (Askew, 2012, p.xvii)

4 Talk Children talking about their mathematical thinking is identified as an important way for them to make their thinking visible (Fuson et al., 2005) It involves encouraging and supporting children s communication, and their initial efforts to engage in reasoning and argumentation The teacher has a key role to play in providing a model of the language that is appropriate in a particular mathematical context Language as a tool for developing children s understanding of concepts, strategies and mathematical representations Challenge of eliciting talk about mathematics with young children not to be underestimated (Report no. 18, Chapter 2)

5 Tasks Open-ended tasks support student thinking and exploration. Differentiation can be facilitated by providing the same basic task to all students and taking individual needs into account (e.g., extra supports, extension activities etc.). Productive task engagement requires that tasks are closely linked to a student s current level of knowledge and understanding but are just beyond his or her cognitive reach. Tasks can remain cognitively challenging throughout a lesson if emphasis is placed on ways of thinking rather than on correct procedures, if sufficient time is allocated to completion of the task and if there is a continued emphasis by the teacher on justification and explanation. (Anthony and Walshaw, 2007 (selected) in Report no. 18, Chapter 2)

6 Tools Learning environments that are rich in the use of a wide range of tools (including digital tools) support all children s mathematical learning. Tools including both physical artefacts and symbolic resources are an integral aspect of human cognition and activity. Among the forms of representation that children use to organise and convey their thinking are concrete manipulatives, mental models, symbolic notation, tables, graphs, number lines, stories, and drawings (Langrall et al., 2008). (Report no. 18 Chapters 2/3)

7 Cobb (2007) sees teaching as a coherent system rather than a set of discrete, interchangeable strategies. It encompasses four elements: a non-threatening classroom atmosphere instructional tasks tools and representations classroom discourse (Report no. 18, Chapter 1)

8 Background and Context The rigid view that first class children had of the equal sign was evident in their attempts to solve missing addend equations. E.g: 4 + = 10 Participation in lesson study with my M.Ed group led to the planning and teaching of a lesson where Cuisenaire Rods were used to aid children to better understand the equal sign.

9 Format of the research Research took the form of a Teaching Experiment whereby the dual role of teacher-researcher was undertaken. Study took place in a mixed first class of 28 pupils. Fourteen lessons were carried out over a four week period.

10 Issues research attempted to address Children s misinterpretation of the equal sign as a do something symbol (Frieman & Lee, 2004). Inability to develop a relational understanding of the equal sign, that is, seeing the connections between both sides of the equation as a result of this misinterpretation (Warren, 2006). Misunderstanding of unconventional equations due to such a limited view of the equal sign. E.g: 6 = 3 + 3, 10 = 10, 7 1 = 5 + 1, 3 =

11 Data Collection Audio-visual recordings Children s reflective journals Researcher s reflective journal Work samples Field notes Pre, post and delayed post tests

12 Why Cuisenaire Rods? Simple, uncomplicated pieces of apparatus with a close relationship to number (Trivett, 1959). No research to suggest that Cuisenaire Rods had been previously used to explore the equal sign. Weight balances had been used to some success, but problems arose in making the link with number ( Warren, 2007).

13 Cuisenaire Rod Tasks Task 1: Find a combination of rods that are the same length as an orange rod. Task 2: Make a train equivalent in length to a train with the following combination of rods; light green, black, pink and dark green. Task 3: Make a train equivalent in length to a train with the following combination of rods; orange yellow and pink. Use at least four different rods. Now try it using as few rods as possible. Task 4: Investigate whether or not an orange rod (10) is equal to the following combination; dark green (6), pink (2) and pink (2). Task 5: Is 7 = 7? Investigate using the rods. Task 6: Is 6 = ? Investigate using the rods. Task 7: Can you use the rods to find the missing addend for the following equation; = 4 +

14 Use of Pre, Post and Delayed Post Tests Tests based on one carried out by Baroody and Ginsberg (1983) whereby children were asked to correct conventional and unconventional equations. Pre and post tests made up of the same format: the children were asked to decide if a number of equations were true or false and to give a rationale for their answer. Delayed post test administered a month after the teaching experiment (TE) and consisted of some true/false questions and missing addend equations for the children to solve.

15 Pre-Test Findings 12 items on the test, one conventional equation in the form a + b = c, the other 11 were unconventional equations in the form a = b + c, a = b, a + b = c + d. 24 of the 26 children marked the conventional equation as being correct. The unconventional items were marked as incorrect by the majority of the class. False items such as = were marked as incorrect by the majority of children, but this was as a result of their structure being weird or strange rather than through understanding the relationship the equal sign establishes between both sides of the equation.

16 3 = It doesn t make sense It is false because the sum is sepose to be = 3 or = 3 It is the rong way around You mixed it up its supposed to be = 3 It is false because it is odd and weird The equal sign can not go in the middle

17 Focus Group Six children, two higher achieving pupils, two middle achieving pupils and two lower achieving pupils. Following the pre-test, it was established that all the children in the focus group had a do something view of the equal sign. The children had difficulty interpreting the unconventional equations as a result of this misunderstanding.

18 Lessons 1-4 Focused on equivalence and allowing children time to explore the properties of the rods colour, length etc. Tasks given whereby children had to find equivalent rods to match a variety of rod combinations. Tasks became progressively more difficult and children had to build trains of rods of specific lengths.

19 Lessons 5-8 Focused on recording the rod combinations, firstly on blank paper to allow the children to concentrate solely on recording the rods without having the additional focus of recording numbers. After initial recordings, rod combinations were recorded on squared paper, with the association of the rods with number being the focus. Children began to record sentences to describe the rod combinations initially, with the colour names of the rods eventually being replaced by their respective numbers. E.g: white 1, red 2 etc.

20 Initial rod recordings on blank paper 1

21 Initial rod recordings on blank paper 2

22 Initial recordings on squared paper

23 Recording rod combinations with words, numbers and the equal sign

24 Blue (9) and pink (2) equals orange (10) and white (1)

25 Yellow is equal to a white and a white and a white and a white and a white. 5 =

26 Lessons 9-11 Focused on examining a variety of unconventional equations using the rods, as well as missing addends. Aimed at allowing the children to make a deeper connection between the rods and number. The equal sign indicated the relationship between equivalent combinations of rods.

27 Missing addend: = 4 +

28 Lessons Focus on moving the children away from the rods and put the emphasis on number. The children worked through correcting various unconventional equations without the rods. Continued to solve for missing addend by focusing on the relationship the equal sign made between both sides of the equation. Children invented balanced equations of their own.

29 Children s invented equations: Making the move from concrete to symbol

30 Results of Post-Test Significant improvement overall in results from pre to post-test. Some children started to show that they had made the transition from concrete to symbol, and had begun to develop a relational understanding of the equal sign. Important to note that not all children made such a transition, and still had confusion around the equal sign.

31 Balancing both sides of the equation

32 Showing an understanding of the relationship the equal sign makes between both sides of the equation.

33 Transition from concrete to symbol not yet made

34 Post-Test: 3 = It is true because it is just backwards There is 3 = and that is 3 = 3 3 is equeal to because three is equeal to 3 It is true it is just backwards It is like = 3 Equal because a white and a pink is equal to a light green

35 Delayed Post-Test Took place a month after the initial posttest. Very little change from the results of the post-test. Most children seemed to show that transfer of learning had taken place. The majority of children successfully solved the missing addend section which was not included in the post-test.

36 Concluding Findings A lot of children developed a more relational view of the equal sign after the teaching experiment. Children became more accepting of unconventional equations, and began to invent their own ones. Most children could successfully balance equations after the teaching experiment. The rods were used as referents by the children, especially when solving missing addend problems. Some children made the move from concrete to symbol, and were better able to read and interpret equations as a result. Not all children made this transition, and some were not as yet clear in their understanding of the equal sign.

37 Limitations of the Study Small scale teaching experiment Study conducted over a short period of time, therefore there is scope for further development and consolidation of learning if this was to be conducted over a longer timeframe. There is further scope to extend children s understanding of the equal sign to include as a substitute for as well as understanding the need to balance both sides of the equation.

38 Implications This teaching experiment could be implemented at a more basic level for the junior classes before the formal introduction to symbol. A focus on understanding was paramount to this TE, and moving the focus away from the textbook was highlighted as a result. The use of appropriate manipulatives in the classroom is recommended to aid children to move from concrete to symbol. Such developments do not occur in a linear fashion, nor do all children make the move.