Mathematics is all about making sense of events of patterns and order. It is a series of activity of generating strategies, seeking solutions and evaluating whether it make sense. In Chapter 2, I learnt 3 big ideas:  

• the meaning behind doing Mathematics
• importance of connecting existing ideas to new skills (blue dots and red dots)
• learning theories of Piaget on constructivist theory and Vygotsky's sociocultural theory.

What really got me is the importance of the teacher's role and how opportunities are essential for allowing the children to assess their prior knowledge to build on new knowledge- assimilation and accommodation.  I have a Math corner set up in my classroom. The children explored various materials provided there and have the opportunity to discuss and share ideas with one another. I think this is a great way for the children to take on new learning on top of the existing ones that they already acquired. This is what the blue dots and red dots are about. 
It is also interesting to learn how the two theories (constructivist and sociocultural) work together to enhance the children's learning process. I feel that the children should adopt active learning and the teacher must be present to facilitate and give the support needed throughout the process. 

Tools and materials should be used when an emerging idea is taught as children these days are visual learners. Figure 2.11 on page 24 showed a representation on how each tool aids the development of new concepts. "The more ways children are given to think about and test an emerging idea, they better chance they will correctly form and integrate it into a rich web of concepts and therefore develop a relational understanding." (p. 24)
Lastly, the use of technology-based tools are mentioned. It is important to keep our children aware of the technologies that are created to make learning more effective. However, is using the calculator advisable for pre schoolers and primary school kids? The text states that "it is important to  include calculators as a tool." (p.26)   I thought calculators are use mainly in secondary schools in Singapore. Will our young children use calculators to calculate once they discover the convenience and choose the easy way instead of doing mental calculations?