Learning mathematics involves learning a way of thinking. It involves learning important, powerful mathematical ideas rather than a collection of disconnected procedures for carrying out calculations. But also it entails learning how to generate those ideas and how to justify to oneself and to others that those ideas are true.

Given our original contention that children’s implicit knowledge about arithmetic can provide a foundation for learning algebra, the findings from this study emphasise that not all children are currently able to draw upon learning experiences that support the bridge between arithmetic and algebra. Descriptions from this study indicate that, despite the intent of curriculum developers and teachers, many students have learned arithmetic in a way that is not conducive to the enrichment of structural understanding. We maintain that instruction that focuses on procedural learning will have limited flow-on effect for the development of number sense and algebraic reasoning.

Number sense is a way of thinking rather than a body of knowledge and skills. To foster the development of mathematical thinking, and algebraic thinking in particular, we suggest that students need more productive modes of instruction than are currently in operation. Until students genuinely understand arithmetic at a level at which they can explain and justify the properties they are using as they carry out calculations, they will lack a secure foundation for much of algebra. Teachers can foster early understandings of the commutativity principle by asking students to solve word problems involving, for example, four and three more, and three and four more and asking them to compare and discuss results. Futhermore, a variety of games and activities can be introduced to highlight the commutative principle and the same-sum concept. Baroody and Standifer (1993) suggest the “Game of Ten”. They describe the game in this way:

The “Game of Ten” can be used to practice the various combinations summing to 10. The game is played with a regular deck of cards without face cards. The cards are placed face down in a pile, and the first card is turned over by the dealer. The person to the left of the dealer turns over the next card and places it next to the first uncovered card. Play continues in this manner until a player uncovers a card equal to ten or a card that if combined with any previously uncovered card (or cards) equals ten. The player then announces his of her find and collects the card or cards. For example, if a player uncovers a three, and a seven was uncovered earlier, the child would announce; “Seven plus three equals ten”. The game ends when the deck of cards is exhausted. The winner is the person with the largest number of cards. (p. 90)

Teaching is a very situated activity, and teaching which takes as its goal the development of students’ mathematical understanding proceeds from an acknowledgment of what students already understand. This requires that teachers assign more importance to engaging in continuous learning about the development of mathematical understanding of the students in their own classes than to their preordained instructional programme. As many educators have noted, in order to engage in instruction which supports mathematical sense-making, teachers need to focus their attention away from their own pedagogical effectiveness, and attend instead to the interactive processes of meaning construction, by creating intellectual communities with the students in their classrooms (see Wood, Scott-Nelson, & Warfield, 2001).

When the goal is not the communication of teacher knowledge but the devolution of a useful problem, management of learning, sensitivity to students, and mathematical challenge become the characteristics of teaching (Jaworski, 1994). Within the inquiry classroom, learning is recognised as both an individual and a social activity. The very social nature of children’s learning provides multiple opportunities for rich interactions with others that substantially contribute to learning, to the extent that conceptual understanding benefits considerably from dialogue and collaboration with others. Using their own statements, as well as those of their peers and teacher, as thinking devices, enables students to acquire a deeper understanding of mathematics (Knuth & Peressini, 2001). Instances of disagreement arise from the diverse ideas generated by discussions that emphasise students’ thinking and reasoning.

Students need to explore and develop with others the language used to describe and express mathematical situations. Wood (1999) notes that it is the explicit and sensitive manner in which the teacher initiates and establishes the expectations for learning through participation which enhances learning with understanding. In creating the conditions for personal meanings to become available for classroom discussion, norms for explaining and for listening are established which in turn provide a context for the development of mathematical thinking. The social structures which the teacher creates in the classroom, the working definitions of what counts as knowledge, and the processes whereby knowledge is assumed to be acquired all influence the ‘mathematics’ that a child learns. The everyday patterns of interaction and the norms that are constituted contribute to children’s beliefs about the nature of mathematical knowledge and the ways in which one learn and uses mathematics in everyday life (Boaler, 2000; Yackel & Cobb, 1996; Yackel, 2001).

Findings from research indicate that creating a context for argument in classrooms requires that teachers establish an expectation for students to articulate their thinking. However, for articulation to be meaningful to all the students in a class, there must be a common basis for communication. Teachers need to:

• Establish a network of mutual expectations for student participation in the mathematics classroom – students should be expected to participate in the examination, critique, and validation of their mathematical knowledge through reasoned discourse (Wood, 1999).
• Establish specific expectations for the children’s behaviours that enable engagement in disagreements with one another. Here a central issue is the distinction between criticism that is personal and criticism that is about mathematical ideas (Lampert, & Blunk, 1998).
• Define the participating role for the listener. As listeners, participants need to do more than pay attention and listen politely; they are expected to take an active role and take responsibility for assisting others in making sense of mathematics. Listeners are expected to follow the thinking and reasoning of others to determine whether what is presented is logical and makes sense, and where necessary, voice disagreement and provide reasons for disagreeing. Additionally, listeners are expected to indicate when they did not understand an explanation or contribution and ask clarifying questions (Gravemeijer et al. 2000).
• Create time and space in discussions so the children can experience participation.
• Use appropriate tools (with young children these will usually be manipulative materials) in order to provide common referents for discussion. Later, notations can provide a common basis for discussion, helping students to clarify their thinking.

The teacher plays a crucial role in establishing the sociomathematical norms of the classroom and hence in establishing the mathematical quality of a student’s argumentation. Since students do not always know what constitutes a quality response (Bicknell, 1998), it is the teacher who must facilitate the negotiation of an appropriate response. Teachers can help students become aware of the mathematical properties that they use by providing students opportunities to make explicit their understanding of why number properties such as commutativity and identity ‘hold good’ and supporting participation in forms of conjectorising and generalising about the properties of numbers. Here the primary strategy for negotiating with the class as to what would count as a justification is to focus the discussion on the relationships between numbers under the operations and allow the students to create compelling arguments about whether or not the pattern always holds. Lampert (1990) notes:

The teacher has more power over how acts and utterances get interpreted, being in a position of social and intellectual authority, but these interpretations are finally the result of negotiation with students about how the activity is to be regarded (pp. 34-35).

Students will not use forms of generalisation until they are aware of the status of that knowledge and the value of community verification. The most useful motivator towards the achievement of this is an investigation of a situation that would lead to different conjectures by different students, and the resolution of conflicts which motivate generalities. Teachers can help students focus on the numbers and operation, and the relationship among them, as mathematical topics that are interesting in themselves, not just as a means to calculate an answer. For example, as early as Year 1, children have implicit knowledge of the commutative property of addition (evidenced by counting on). What needs to happen more frequently in classrooms is that these, and other basic properties of number, are made explicit. Carpenter et al. (in press) suggest that we need to make generalisations the explicit focus of attention in order that:

• They are available to all students.
• Students understand why the procedures they use work the way they do.
• Students can apply them flexibly in a variety of contexts.
• Students recognise the connections between arithmetic and algebra and can use their understanding of arithmetic as a foundation for learning algebra with understanding.

In order to make generalisations explicit, children need to have a language to talk about them – they need to develop skills in describing mathematical ideas clearly and precisely. Initially, as children start making conjectures about mathematical ideas that they think are true for all numbers, they frequently are not precise in their expression of ideas. However, knowing the technical vocabulary, such as commutative property or distributive property¸ is not as important as realising how such a property is used. When students are ready, the teacher should introduce the mathematical terms to help students’ make more precise explanations.

Of note in this current study was the dearth of quality responses employing reference to generalisations of number properties (see section 3.x). An example provided by Carpenter and colleagues (in press) illustrates the richness of explanation and justification possible from a young child:

If we have 3+5, it would be like this [makes a collection of 3 counters and to the right of it a collection of 5 counters]. If it were 5+3, we can just move these over here [moves the collection of 5 to the left of the collection of 3]. See that’s 5+3, but it’s still the same things. We haven’t changed the number. It’s like that with any two numbers. You can move them around like that, but you still have the same number of chips.

What is significant about this example is that the student at first supports the conjecture with reference to ‘the sameness of 3+5 and 5+3’, but then offers a generalisation of the commutative property for addition of all numbers and provides a further warrant related to the moving around [rearranging] maintaining the same number of counters. She demonstrates a process in which she represents the first sum and transforms that representation to represent the second sum. Thus, although the argument as presented involved specific numbers, the form of the argument was general and indicated how the same general process could be applied to any two whole numbers.

In another example from a study by Bastable and Schifter (cited in Kaput, 1999) students clearly illustrate the progression of moving from a demonstration of the commutative law of multiplication involving a specific example to a generalisation for all numbers. After working with array representations in class one student decided she could prove it – holding up three sticks of 7 Unifix cubes, she said:

See, in this array I have three 7s. Now watch, I take this array [picking up the three 7-sticks] and put it on top of this array [turns them 90 degrees and places them on the seven 3-sticks she has previously arranged]. And look – they fit exactly. So 3 x 7 equals 7 x 3, and there’s 21 in both. No matter which equation you do it for, it will always fit exactly.

At the end of this explanation, both students eagerly explained another way to prove it:

I’ll use the same equation as Lauren, but I’ll only need one of the sets of sticks. I’ll use this one [picks up the three 7-sticks]. When you look at it this way [holding the sticks up vertically], you have three 7s. But this way [turning the sticks sideways], you have seven 3s. See?…So this one array shows both 7 x 3 or 3 x 7.

Although both students had used a 3 x 7 array to explain their points, the final simpler representations convinced other students in the class of the general claim that ‘it would work for all numbers’. In these examples we can clearly see that even though young children do not have the formal language of mathematics to articulate their generalisation they are able to convince themselves and others of the conjectures through using a very concrete solution, in combination with natural, informal language. In these cases the students used cubes and sticks to generate their ideas, to show one another their thinking, and to justify claims that were clearly theirs, not their teacher’s. In these examples the concrete solution provides a basis for generalisation: This challenges the commonly held assumption that abstract solutions are more sophisticated than solutions involving physical objects.

Using this case study, Kaput (1999) argues that this activity involved more than the children’s development of the concept of commutativity of multiplication: The students were actually constructing both the very idea of multiplication (although only two aspects: repeated addition and array models) while beginning to develop the notion of mathematical justification and proof (p. 138). Thus, by understanding how children know that a mathematical answer is true, teachers can help students develop natural ways to verify and convince themselves and others, using the kind of reasoning that will eventually lead to mathematical proof.

Mathematical understanding takes a long time to develop (Anthony & Knight, 1999; Pirie & Kieren, 1992), and the kind of mathematical thinking that can provide a foundation for learning algebra must be developed over an extended period of time starting early in a child’s schooling. The focus of any numeracy programme for young children should not be to teach algebraic procedures; but it should clearly develop ways of thinking about arithmetic that are more consistent with the ways that students have to think to learn algebra successfully. To continue with an artificial separation of arithmetic and algebra would be to deprive young students of powerful ways of thinking about mathematics (Carpenter, Franke, & Levi, in press; MacGregor & Stacey, 1999).

Specific areas of focus within the introduction of arithmetic suggested by Carpenter et al. (in press) include:

• The meaning of the equal sign (Falkner, Levi, & Carpenter, 1999).
• Development of relational thinking (using true-false and
  open number sentences).
• Conjectures about number operations such as commutativity,
  distributive laws, identity elements, etc.
• Number sentence problems with several variables,
  e.g., x + y = 7, or s + s = 8 (including a focus on justifying that all solutions have been found and expressing conjectures using algebra notation).
• Representation of conjectures using symbols
  (e.g., a + b = b + a; a x 1 = 1).

Within all of these contexts the goal is not merely to teach students appropriate conceptions of the use of the equal sign or the distributive property, for example – it is equally important to engage them in thinking flexibly about number operations and relations and in productive mathematical argument. Cognisant of Kaput’s (1999) warning that “failures to teach for understanding are most often the result of breaking the link with meaningful experience” (p. 137) we reiterate that the goal of these activities with young children is not to teach algebraic procedures per se, but rather to provide the opportunity for children to relate these algebraic equations to the arithmetic they understand, and in the process, further their understanding of the arithmetic. As such, the development of these aspects of students’ mathematical thinking should not be perceived as one more topic area to teach – ideally – mathematical thinking should be an integral part of teaching arithmetic.

Students in the study did not employ array models for justifying the commutativity of multiplication. This raises serious questions about the use of multiple representations, including the array model, for the teaching of multiplication. Given appropriate problems there is ample research to suggest that young children are able to solve multiplication problems informally, by representing them in a number of different forms (Mulligan & Mitchelmore, 1996). Initial strategies include direct modelling with counting and grouping skills, and use of strategies based on addition and subtraction. However, since the rectangular array reflects the formal underlying structure of multiplication, is should be employed as a model for multiplication problems.

Given the difficulty many children had in providing a representation of multiplication per se, the difficulty some children had in distinguishing a model for 2 x 5 from 5 x 2, and the total absence of array models presented by students in this study, we are led to question whether students have been provided with sufficient contextually based problems involving multiplication. A clear implication is that children need to be provided appropriate experiences to encourage a range of mathematical representations of multiplication. Context problems which evoke the various representations, including rectangular configurations, need to be given a prominent place in classrooms in order that advanced multiplicative thinking might be developed. Students need to experience working through problems modelled by arrays (e.g., There are 4 lines of children with 3 children in each line. How many children are there altogether?), together with problems involving cartesian products (e.g., There are 3 sizes of ice cream cones and 4 flavours. How many different choices of ice-cream cones can you make?) Approaches advocated in the current Numeracy Development Project (Ministry of Education, 2002b) support the development of array models and acknowledge the importance of commutativity property.

While in the initial stages it is important to situate any representations within the child’s informal practices, ultimately students must move beyond physical models, partly because such models do not easily represent all multiplicative situations and partly because these models become inappropriate with rational numbers. With sufficient experience the teacher should then pose problems that gradually, but explicitly remove physical prompts or supports, and encourage students to form mental images (Sullivan, Clarke, Cheeseman & Mulligan, 2000).

While in the past there has been an assumption that the mathematics taught and learned in primary schools is relatively straightforward, it is now increasingly recognised that this is not the case (Ma, 1999). The mathematics involved is, in fact, conceptually complex, and we are aware from work in recent numeracy projects (Higgins, 2001; Thomas & Ward, 2001) that many teachers have, what may at best be described as, a tentative knowledge of learning progressions associated with early number. Joanne Mulligan’s extensive research assessing multiplication and division strategies notes that multiplicative concepts are often not well understood or well taught by teachers at primary and secondary level (Mulligan, 1998). Sullivan et al. (2001) also contend that teachers have over-relied on physical representation of multiplication involving repeated addition at the expense of array type representations and use of appropriate mental images.

It has also been noted that many New Zealand teachers need guidance in order to develop the skills necessary to assess the validity of a mathematical argument or method of solution (Bicknell, 1998). What is needed, and is indeed a feature of current professional development programmes in numeracy, is increased opportunities in both pre-and in-service programmes to address such topics as the structure of the base-ten numbers system, the meaning of the basic operations, the logic of rational numbers, and the links to algebraic thinking. In exploring these topics, teachers must be given experiences that support the development of richly connected mathematical concepts. It is precisely such connections among concepts and their contexts, and the ability to use them flexibly and creatively which are called upon in the classroom (Askew et al., 1997; Ma, 1999).

As well as a sound content knowledge base, Schifter (2001) argues that teachers need additional mathematics skills in order to be able to:

• attend to the mathematics in what one’s students are saying and doing;
• assess the mathematical validity of students’ ideas;
• listen for the sense in students’ mathematical thinking, even when something is amiss;
• identify the conceptual issues the students are working on.

Teachers with such knowledge are not only able to ask children to explain and compare how they solve problems, and understand the strategies the children describe, they are able to pose questions that require them to think more deeply about the mathematics underlying both the strategies they were using and the problems they were solving.

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