Mathematics Developmental Continuum P-10

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The Development of Deductive Reasoning

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Evident in a mathematical argument

From a very young age, children learn to justify their arguments in a social setting, for example, “I want to do that because [older brother or sister] is doing it”. In a student’s social environment, justification has the purpose of convincing someone, whether or not the reasoning is valid. In other words, the authority for the justification lies with the student and is therefore subject to challenge by others who also regard themselves as having authority.

As students move towards deductive reasoning in the context of mathematics, the language of reasoning may be the same (for example, words such as ‘so’ and ‘because’), but it is the nature of the evidence that is different.

Evidence that is commonly offered in support of mathematical reasoning is:

  • authority of a person (we need to move students away from this)
  • offering examples (inductive reasoning)
  • deductive reasoning from known and agreed facts.

Offering examples is one of the most common ways in which students will present arguments. For example, many students will explain why you annex a zero to multiply by ten by giving examples. They might note that 2 × 10 = 20 and 3 × 10 = 30 and 12 × 10 = 120. These students should be encouraged to look for reasons why ‘annexing a zero’ will always work (for whole numbers). The examples do show that it sometimes works, but a logical argument that it always works for whole numbers can be found by analysing the place value of the digits in 3 and in 30, etc. If the examples are analysed carefully, the essence of a logically correct mathematical argument can be constructed, even by children working towards Level 3 or 4. They can see how 34 (3 tens and 4 ones) becomes 3 hundreds and 4 tens and how annexing a zero shifts each number into the higher place value column. See Number Slides.

In fact, even very young children can reason deductively. For example, a child can work out that 3 + 5 is equal to 8 because 4 + 4 = 8 (and 3 is one less, 5 is one more, so it balances).

The nature of acceptable justification obviously changes with the students’ conceptual understanding. For example, in Years 5 to 7, it is acceptable to say that the sum of the angle measures of a triangle is 180 degrees because “we tore off the corners of a triangle and together they made a straight line and we know that a straight line is 180 degrees ”. At this level, they do not have the geometric knowledge to say more than the property appeared to hold for all the triangles that the class happened to test. As students’ geometric knowledge increases, they can move beyond the corner tearing to an argument based on an understanding of the relationship between the angles associated with parallel lines and a transversal. This will be a deductive argument based on agreed principles.

In summary, students at every level can reason deductively about some things. They will need to reason from examples or from demonstrations with models about some things where they do not yet have adequate conceptual understanding. However, when the concepts are in place, deductive reasoning is the goal.

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Mathematical language and explanations

The language of mathematics is often critical in students’ ability to explain and justify their reasoning. Students who do not have the necessary mathematical vocabulary are limited in verbalising their reasoning “I know it’s true but I can’t explain it”, “I don’t know the right words”. Being able to use correct mathematical language is a prerequisite to students being able to justify their reasoning.

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Making a classroom a mathematical community

The purpose of a mathematical argument is also to convince, but the argument must be based on evidence accepted by the mathematical community. In the early years the mathematical community is the class of students and their teacher. If a student claims, for example, that “7 is bigger than 9”, the mathematical community of the classroom would be able to argue that this was not true, perhaps by counting out loud and noticing that 7 comes before 9. As students become older, the mathematical community broadens to include wider influences (such as textbooks and the  VCE examiners), but the general principle that the purpose of mathematical argument is to convince the mathematical community is a helpful guiding principle for teaching.

Some research studies suggest that students develop a greater understanding of developing proofs based on deductive reasoning if they are given the opportunity to engage in argumentation and conjecturing as part of the proving process. Mariotti, Bartolini Bussi, Boero, Ferri, and Garuti (1997) assert that successful proof construction is dependent on continuity of reasoning, or ‘cognitive unity’, between producing a conjecture and constructing a proof of the conjecture, with the process of argumentation creating a bridge between statements made during conjecturing and statements used in the proof construction.

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Geometry software and proof

Interactive geometry software such as The Geometer’s Sketchpad or Cabri Geometry is particularly appropriate for conjecturing activities in geometry. The dynamic imagery associated with these software environments enables students to recognise key features that remain invariant in a geometric figure, to develop conjectures based on these observations, and to test conjectures quickly and accurately. It can help them to follow a chain of deductive reasoning to prove the conjecture.

Laborde (1998a) asserts that dynamic drawings offer stronger visual evidence than a single static drawing: “A spatial property may emerge as an invariant in the movement whereas this might not be noticeable in one static drawing” (p. 117). She notes that when students are engaged in problem-solving tasks in dynamic geometry computer environments, “a critical point of the solving process is the visual recognition of a geometrical invariant by the students, which allows them to move to geometry” (p. 120).

Laborde (1998b) reports that observations of students working on a geometry problem in pencil-and-paper and dynamic geometry environments showed that the problem made sense for the students only after they were able to visually manipulate their screen construction. Laborde argues that in a pencil-and-paper environment, students’ movement between the spatial-graphical and theoretical domains is restricted, whereas the software environment promotes links between the two domains.

Scher (1999, p. 24) asserts that dynamic geometry software can influence the style of experimentation and reasoning so that “the boundary between deductive reasoning and dynamic geometry becomes blurred: the software finds its way into the proof process”.

References:

Laborde, C. (1998a). Visual phenomena in the teaching/learning of geometry in a computer-based environment. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21 st century (pp. 113 - 121). Dordrecht , The Netherlands : Kluwer.

Laborde, C. (1998b). Relationships between the spatial and theoretical in geometry: The role of computer dynamic representations in problem solving. In J. Tinsley & D. Johnson (Eds.), Information and communication technologies in school mathematics (pp. 183 - 194). London : Chapman Hall.

Mariotti, M. A., Bartolini Bussi, M. G., Boero, P., Ferri, F., & Garuti, R. (1997). Approaching geometry theorems in contexts: From history and epistemology to cognition. In E. Pehkonen (Ed.), Proceedings of the 21 st Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 180 - 195). Lahti , Finland : PME.

Scher, D. (1999). Problem solving and proof in the age of dynamic geometry. Micromath, 15 (1), 24 - 30.