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

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

Different types of tools and manipulative can be used to represent concepts and test knowledge and understanding. These include websites, geo boards, base-ten blocks, spinners, number lines, unifix cubes, counters)

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.

Chapter 1: Teaching Mathematics in the 21st Century

Reading Reflections

"In this changing world, those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their future." NCTM (2000, p. 50)

It is important for teachers to be able to focus on mathematical thinking and reasoning as they prepare to help students learn mathematics. When teachers understand and apply the Principles and Standards for School Mathematics in the school curriculum by integrating it with other subjects, they provide a rich and meaningful mathematics education for the children.

Mathematics is not just about knowing and calculating numbers, but  learning it "with understanding, actively building new knowledge from experience and prior knowledge." NCTM (2000, p. 20)
It requires children to base their learning on the Five Process Standards which include:

• Problem solving
• Reasoning and proof
• Communication
• Connections
• Representation

As the teacher uses these processes in her mathematics classroom, there is coherence and comprehension taking place. Children learn to draw knowledge from prior experiences and make connections to new ideas through reasoning and communications as they problem solve.

Teachers who understand the pedagogy behind teaching mathematics to young children, strive to create an environment that will offer and encourage all children with equal opportunities to learn. this can be achieved through the use of different media, such as technology, reasoning, communications, and reflections. As children interact with the different media available, they become exposed to using strategies to derive answers rather than focus on only "one right answer".