EDU 330 Elementary Mathematics - Pre-course reading, Chapters 1 & 2

Reading Reflection – Chapter One

Teaching Mathematics in the Era of the NCTM Standards

v  Important tools for teacher in order to teach Math efficiently - knowledge of mathematics and how student learn mathematics.

v  Teaching mathematics should not only focus on doing exercises and testing, rather it should be more holistic - focussing on mathematical thinking, reasoning and problem solving skill.

v  A each level, children's learning should be on certain focus, go in more depth and there should be connections.

My Reflection - from my understanding, I feel that children's learning of mathematics should be in the form of spiral curriculum. That is, at each grade level, teacher should build upon children's knowledge on mathematical concepts from their previous experience. Hence, it will help to form concrete understanding on the various concepts taught.

Principles and Standards for School Mathematics

The Six Principles

1. The Equity Principle

Regardless of personality traits, background or physical challenges, all children should be given the equal opportunity and guidance to learn mathematics. This support or guidance should be equipped with high expectation for all the children.

My Reflection - From my understanding, to have high expectation for all the children and to support, sound similar to Vygotsky's theory of Zone of Proximal Development. Meaning to say, teachers should set high expectation for the children to acquire as much knowledge as possible through scaffolding.

2.      The Curriculum Principle

Teachers should enforce the understanding that mathematics learning is as a whole. In other word - through the various mathematical skills, it can be integrated into other forms of learning from other disciplines.

3.      The Teaching Principle

For effective teaching, teachers must focus on the “KWL” – what children know, what children need to know and finally how they have learned it.

4.      The Learning Principle

Learning with understanding. Building upon new knowledge based on previous knowledge gained – spiral curriculum.

5.      The Assessment Principle

Using assessment as the mean to gauge children’s level of understanding. It also serves as a mean for teachers to plan future lessons and ways to further enhance children’s learning.

6.      The Technology Principle

In this IT savvy Era, children could be exposed to mathematics via the use of technology. For instance, by using computers, children could learn the different mathematical skills.

The Five Process Standards

The Five Standards are inter-related and should be viewed as a whole. Problem solving is essential in mathematics and children must be able to provide logical reasoning for their answers. They should also be able to communicate and explain their ideas and make connection to past experiences, real situation and linking to other areas. Finally, children should be able to use symbols, chart, graph and such to express their ideas.
 

Six Major Shift in the Classroom Environment

These shifts are necessary in order to allow children to develop mathematical understanding.

The Teaching Standards

Of the seven teaching standards, I would focus more on three of the standards, namely,

v  Standard 4 – Learning Environment

To me, learning environment plays a vital role in ensuring that children are able to maximize their learning potential based on that particular subject. With the appropriate materials, space, time and such, children will be able explore the concepts that they are learning in depth.

v  Standard 6 – Reflection on Student Learning

Reflection and follow-up with plans of actions are something that teachers should do without. Hence, I make it a point for all my teachers to do daily reflections on children learning. From these reflections, teachers will have better information pertaining to every child’s learning performance before deciding on the next plan of action.

v  Standard 7 – Reflection on Teaching Practice

Besides reflecting on children learning, teachers also must reflect on their teaching style and the content covered. It is essential to do so for the purpose of improving their teaching method and also to gain more insights about their teaching.

My Reflection

In a nutshell, I have gained better understanding in teaching mathematics. I totally agreed that the teaching process should not only focus on practices and assessment, rather, it should be more of in depth learning of the various concepts – dealing with concrete materials and real-life experiences before going to abstract, be able to do reasoning, make meaning and not forgetting to be able to connect the learning to other disciplines – to name a few.

 

Figure 1: Real – life experience of dealing with money (concrete learning).

  

Figure 2: Moving on to symbols.

Reading Reflection – Chapter Two

Exploring What It Means to Know and Do Mathematics

v  The “how”, “why” and “what” of teaching mathematics.

v  It is a science of concepts and processes that have a pattern of regularity and logical order.

My Reflection – During my school days, doing mathematic was in a very regimental form. We were asked to do the sums written on the board and “no questions asked”. At the end of the class session, we submitted our work and teacher would just mark it right or wrong. No proper explanation was given should the answers were wronged and honestly even if I got a few of the sums right – I did not actually know the logic behind it. Having gone through this “meaningless” mathematic experiences, I am determined that the children in my centre should not go through the same process.

What Does It Mean to Do Mathematics?

Mathematics should be for the purpose of their lives skills such as ability to handle money, problem solve, time management, planning and achieving their goals in lives and for effective work performance – just to name a few.  In order to create an environment where children are encouraged to do share and defend mathematical ideas, teachers have to ensure that the classroom is organized for mathematical lessons.

The Language of Doing Mathematics

In the traditional way, mathematic jargons revolved around the words “plussing” and “doing times”. Under the Principles and Standards by National Council of Teachers of Mathematics (2000), the collective verbs describe more of the authentic work of doing mathematics.

These verbs require higher level thinking and involved meaning making and figuring out. It is essential that every idea introduced during mathematics teaching should be understood by the children. By using these verbs, children learning will be more meaningful as they are able to make sense of the concepts taught.

My Reflection – I tried asking my teachers to use some of the verbs (investigate, discover, explore and predict) in replacement to the traditional words in mathematics and observed children’s reaction. It was interesting as initially children were taken aback and took quite awhile to grasp what was actually going on. After awhile, we realized that the children were more relaxed and enjoyed the session more as they felt at ease and were less rigid in their knowledge-acquiring process.

Picture 3: Children trying to contruct word from the puzzles based on the picture given - thus meeting the objective that mathematics concepts should be applicable to ther disiplines.

 

Figure 4: From this Percussion activity, children were able to apply some of the mathematical languages such as describe and represent. They were able to describe the different things used as percussions, the colors, shapes and sound it made. And they also learned that simple household things can serve as representation of musical instruments.

Productive Classroom Culture

As stated by O’Conner and Anderson (2003), the teachers would have set a in place a powerful context for student learning should the teachers are able to produce productive classroom culture.

What Does It Mean to Learn Mathematics?

With all honesty, I found it very challenging in my attempt to try out the sums. Exactly as stated in the text, I also questioned myself as to why children should be doing these types of sums – how does these sums help them in their pursuit of knowledge?

Constructivist Theory

The theory revolves around the fact that children are not blank slate, rather, creators of their own learning.  In the assimilation process, new concepts that are introduced will fit in together with the prior knowledge of the child. With this new information, it expands the existing network. As for accommodation, when new concept could not fit in with the existing network, thus the brain has to revamp the existing schemes to incorporate new ideas.

My Reflection – After reading this theory and analyzing figure 2.8, I have a better understanding of how the Constructivist Theory works. Upon reflection, I realized that indeed it is true that we always rely on our past experiences to make connection when we encounter new ideas. And it is also very true that we try to accommodate new ideas and move on when such ideas we have never come across before and we could not connect it to our past experiences.

Sociocultural Theory

A strong believer of Vygotsky, I am especially interested in his Zone of Proximal Development Theory.  I believed through scaffolding, children will be able to maximize their learning potential. Apart from mathematics and other academic components, my teachers usually scaffold children’s learning in other areas as well such as dressing up and packing their own bag independently.

For instance, in the beginning teacher would assist the child in dressing up and packing of her bag after showering. After a while, when the child becomes competent, teacher will slowly move away and child will do it independently.

 Figure 5: Once child is competent and able to perform task independently, teacher will cease the scaffolding.

Implications for Teaching Mathematics

Regardless of theories, be it social constructivist or cognitive constructivism, both should be interwoven as active classroom discussion based on children’s own perspectives and problem solving solutions is definitely the foundation to children’s learning. (Wood & Turner-Vorbeck, 2001, p.186, cited in Van de Walle, 2006).

What Does It Mean to Understand Mathematics?

As stated in both sociocultural and constructivist theories, learners make connections of their new ideas to the existing ones. Due to each individual learning ability, learner making connection of the ideas will vary depending on their individual understanding.

My Reflection – In my own word and based on my understanding of the paragraph, the gist of the content is that the way each learner makes connection of ideas will depend on the depth of their understanding. Hence, even if the same concept or idea is posed to a group of children, each of their understanding will be in the form of a spectrum depending on their how extensive or narrow their prior experiences are.

Mathematics Proficiency

v  Conceptual understanding – knowledge about the relationships of a topic or the basic ideas

v  Procedural understanding – knowledge of the rules and the procedures used to carry out mathematical processes. Also, the symbolic representation of mathematics.

Five Strands of Mathematics Proficiency

Benefits of a Relational Understanding

In order to be able to teach effectively, teachers have to put in lots of effort and equipped with the knowledge to educate the children so that they are able to make connections of their learning.

v  Effective learning of new concepts and procedures – the more robust children’s understanding of the concepts, the more connections they are able to built – connecting it to the existing conceptual ideas they have.

v  Less to remember – Knowledge should be in the form of “Big Idea” or as whole rather than isolated concepts.

v  Increased retention and recall – retrieving information is possible when learner has the concept connected to an entire web of ideas.

v  Enhanced problem-solving abilities – in a rich network where concepts are embedded, transferability is greatly enhanced so is problem-solving.

v  Improved attitudes and beliefs – when learner is able to understand the concepts taught and they make sense to him/her, the feeling of “I can do it!” surfaces thus enhancing the learner’s positive mindset towards mathematics.

Conclusion

This reading reflection has taken me to a fairly in depth journey in understanding mathematics. When I was a student, mathematics was one subject that I always had feared on. Even now, though I have somewhat gained better insights to mathematics, I am still apprehensive as to whether I am able to grasp the numerous concepts behind it. With all honesty, I find the textbook to be rather complex and challenging to understand and internalized. Nevertheless, I am full of vigor to learn as much as I can from the upcoming classed so that I will be able to share my knowledge with the teachers in my centre – definitely for the benefit of the children!