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Chapter
2: Exploring What It Means to Know and Do Mathematics

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__Reading Reflections__

It was interesting to read that "mathematics is the science of concepts and processes that have a pattern of regularity and logical order." I had never thought of mathematics in scientific terms before this. The use of language when doing mathematics, creates an awareness of higher level thinking in children as they use conceptual verbs to problem solve and derive strategies and answers. Through the use of explicit language, students are able to make connections and see patterns in a problem as they engage in productive struggles - "students must have the tools and prior knowledge to solve a problem, and not be given a problem that is out of reach, or they will struggle without being productive; yet students should not be given tasks that are straightforward and easy, or they will not be struggling with mathematical ideas" (Walle, Karp, Bay-Williams, 2013, p.15)

Doing mathematics involves the science of pattern and order to understand and apply concepts. Once students are able to see these patterns, they are able to make connections and engage in their productive struggle. Besides this, using technology, such as a calculator makes counting accessible for students who can't count yet. Making predictions, drawing pictures or even using objects to solve problems are some basic ways to do mathematics.

Mathematics theories are rooted in the constructivism and sociocultural theories of Piaget and Vygotsky. As theorized by Piaget, "Integrated

*networks,*or

*cognitive schemas*are both the product of constructing knowledge and the tools with which additional new knowledge can be constructed."

(Walle, Karp, Bay-Williams, 2013, p.19)

According to Vygotsky's sociocultural theory, ,

*semiotic mediation*is used in classroom interactions through the use of diagrams, symbols, representations to make meaning which falls within the ZPD (Zone of Proximal Development) of the learner.

Therefore, the mathematics teacher needs to:

- build new knowledge from prior knowledge
- provide opportunities to talk about mathematics
- build in opportunities for reflective thought
- encourage multiple approaches
- engage students in productive struggles
- treat errors as opportunities for learning
- scaffold new content
- honour diversity

Students should also be encouraged to think aloud hypothetically when they have solved a problem. Getting them to recall and explain how the problem was solved would present them with opportunities to recall and build upon their existing knowledge. When the students are able to understand and make sense, they also feel confident in their ability to understand and do mathematics which will increase their performance level.

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