As what Van De Walle, Karp, and Bay-Williams (2009) say, a teacher's beliefs about what it means to know and do mathematics and about how children come to make sense of mathematics will affect how the teacher approach instruction. The teacher is a figure in the classroom who is responsible for imparting knowledge through meaningful ways that will create a lasting impression in the minds of the children he or she teaches. If a teacher believes that the number 4 can only be obtained by adding 2 + 2 and children are "brushed off" when they suggest other possible ways of obtaining the number 4 (e.g. 3 + 1), then the teacher would have "killed" the children's interest towards learning Mathematics. The reason being simple: The teacher believes him or herself is the authoritative figure who teaches, and children who are learning must listen and take in whatever is being taught.
Although Maths was not my best subject at school, I had loved teaching my group of children the subject because this is one area of learning where I could guide and direct the children towards self-explored learning; where they are scaffolded along a process-driven acquisition of mathematical knowledge. "Learning mathematics is maximized when teachers focus on mathematical thinking and reasoning (Van De Walle, Karp, & Bay-Williams, 2009)." Instead of telling children 2 + 2 =4, it would be more effective if children are guided through a process of logical thinking by getting them to work with hands-on manipulatives and materials of the same set in the following way: Count 2 sets of 2 buttons per set and place them in 2 green baskets, followed by pouring the buttons into another larger green basket, and have them count the total buttons. This would help child visualize the actualization process of 2 + 2 =4, instead of being fed with the answer.
Chapter 1 also details the following Principals and Standards for School Mathematics which I find very directive and purposeful when guiding teachers in planning for mathematics teaching: The Equity Principle, The Curriculum Principle, The Teaching Principle, The Learning Principle, The Assessment Principle, and The Technology Principle.
Chapter 2 has highlighted many theories that explains the acquisition of mathematical knowledge and understanding. As what the Constructivist Theory suggests, "Assimilation occurs when a new concept 'fits' with prior knowledge and the new information expands an existing network (Van De Walle, Karp & Bay-Williams, 2009)." Take for example a child's ability to perform a series of AABB pattern would very much be built on his or her prior experience with performing an AB pattern. With the prior knowledge and experience of being able to perform an AB pattern (prior knowledge), the child is able to assimilate this existing network and fit it with the new concept - AABB pattern, and perform a series of AABB pattern independently by applying the understanding and logical thinking he or she has acquired in the process of forming an AB pattern.
Getting child to work with hand-on materials such as stationary to form AB pattern sequence.
Providing children with different coloured ice-cream sticks to form their AB pattern sequence
The era of mathematics education has evolved with the continuous revamp of the local curriculum. The approach of "Teach Less, Learn More" approach (Ministry of Education, 2008) suggests a differentiating approach in today's curriculum teaching as opposed to the traditional ways many years back. It also the duty of the teacher to provide a successful environment for learning where children feel empowered to make their own decisions and voice out their ideas and offering explanations (Van De Walle, Karp & Bay-Williams, 2009).
"The essence of mathematics is not to make simple things complicated, but to make complicated things simple ." ~S. Gudder
Ministry Of Education, Singapore (8 January, 2008). More support for schools' "Teach less, learn more" initiatives. Retrieved 10 September, 2010 from http://www3.moe.edu.sg/press/2008/pr20080108.htm
Van De Walle, J., Karp, K. S. & Bay-Williams, J. M. (2009). Elementary and middle school mathematics: Teaching developmentally (7th ed.). New York: Longman