Three things that I have learned….

1.    I learned that as a teacher, I have to ask myself four critical questions before I plan any activities that promote the learning of mathematics for the children. They are:

-       What is it I want the child to learn?

What is the objective behind the activity?

-       How do I know if the child has learned it? The teacher needs to use multiple assessment approaches to find out what each child understands, or may misunderstand. Child observation, docu­mentation of children’s talk, collec­tion of children’s work, and open-ended questions are positive approaches to assessing mathematical strengths and needs.

-        What do I do if the child is struggling with the activity? Is there something that the teacher has planned to ensure this child stays on the runway? Putting the child through repeated tasks will ensure that she does not veer off. That child will eventually start to take small steps and eventually take off when the time is right.

-       What about the activities for an advanced child?

A child may already know how to do the activity. The teacher must remember not to give her the same activity over and over again. Instead, she should plan for an activity that will promote higher order thinking.  

2.    The use of differentiated instructions from the concrete to pictorial to abstract (CPA) approach is important to cater to the different learning needs of children. Concrete materials allow beginning students to explore new concepts or extend their level of a concept that was learned earlier. Once a student is able to understand a mathematical concept, pictorial representation can take the place of concrete materials. The abstract method is only used after the child is familiar with the earlier two approaches.  

3.    The three big ideas in Mathematics: patterning, visualisation and number sense

Two questions that I have….

1.    How do I get parents involved in their child’s learning of mathematics?

2.    Is giving your child one-to-one tuition effective in promoting the learning of mathematics?

Use of technology

These days having a computer at home is a must. Children use it to retrieve information and play games. Used in school, it is also an effective tool to practise mathematical concepts and solve problems. The endless programmes and websites of games, virtual moving pictures and sounds enhance opportunities to learn important mathematics. Besides, it also makes learning mathematics more fun and effective compared to just solving these problems using a pencil and paper.  

Just like the computer, a calculator can “promote higher-order thinking and reasoning needed for problem solving in our information- and technology-based society” (p. 115). Calculator activities can help children become familiar with the various symbols on the keypad so that “more complex activities are possible.” Calculators can help to offload detailed computation hence allowing teachers to better focus on the development of flexible strategies and make more sense of other mathematic meaning and concepts. 

However, whilst the use of technology is an integral tool in expanding student’s ability to think about challenging mathematics, there must be some form of balance. In the end it still comes down to the mathematical skills that the child owns. It is up to the teacher to prepare the children to learn the process and take learning to a higher level. That is something technology cannot replace, well at least not completely anyway.

Lesson 4 - A Learning Point

I consider myself someone who knows my shapes pretty well. All my life I grew up with the simple notion that a square is just a shape with four equal sides and four right angles. A rectangle on the other hand, only has two equal sides and four right angles. After yesterday, I felt like a child with level zero visualisation. I only based my knowledge on the appearance of the shape – “it looks like a square” (p. 403) so it must be a square. However, Thursday’s lesson puzzled me yet again. Since when is a square a rectangle? I guess a square, like a rectangle is a quadrilateral with four right angles. Yet a rectangle can never be a square because it does not have four equal sides. Why didn’t my teacher tell me all this? Perhaps she may not have known it herself. Alright, so it is not too late to find out these things. After all, I am in this course to learn new things.

One thing I must take note of is to never put the two shapes together and ask children to circle or colour all the rectangles on paper. If a child picks out a square when I ask for rectangles, it is best to check with the child why he did that. Could it be that he picked up the square by accident because he was not sure of the shape or perhaps he really does have some prior knowledge about this shape from his parents or some other source.

As teachers, we must prepare children to learn skills such as patterning, visualisation and number sense. We must walk that journey with them to the end point. It is alright for children to get lost. It is alright for them to back track a little as long as we are with them providing that continuous support.

Lesson 2 and 3

Developing Early Number Concepts and Number Sense

Going back to the basics

Subitize? I heard this word for the first time on Tuesday. What is that? I found out that it is the ability to explore quantity before children can count. Once children can match counting words with objects, they will begin to understand that “the last count word indicates the amount of that set.” I never saw this as a problem that children might have and assumed that they always had it in them but little did I know. I mean how difficult can this be, right? It seems so simple and natural to look at the dots on a dice and just see the number but the book reveals otherwise. It points out that “Subitizing is a complex skill that needs to be developed and practiced through experiences with patterned sets” (p. 129). Why didn’t anybody tell me about this before? Well now I know better. I also found the idea of using dot plates interesting and easy to make. I intend to try this out with my class next week.

I just wanted to say that I am learning something new in class every day. Wednesday’s lesson was no different. I found out that equal parts of a fraction can be both the same shapes, or different ones and that it is wrong to say “three over four”, “three upon four” or “three out of four”. Instead, we should use three fourths or two fifths. All these little bits of information are a revelation to me! I am really glad this degree includes mathematics modules. Indeed, these past few days have been a learning journey for me.

Chapter 1

As mathematics education around the world undergo changes, it becomes inevitable that teachers have to be on top of things, mainly in the methods used to teach the subject as well as the content that is being taught. With the National Council of Teachers of Mathematics (NCTM) emphasising that “mathematics taught at each grade level needs to focus, go into more depth, and explicitly show connections” (p. 2), more changes can be expected over the next few years. Going “back to basics - reading, writing and arithmetic” (p. 1) and the work of theorist, Jean Piaget also helped to focus research on how students can best learn mathematics.

I have also observed significant adjustments in the way mathematics is taught in the schools in Singapore based on the curriculum of my two older children.  In fact, it has recently been reported that pupils entering primary school next year will be given more breathing room to grasp basic numeracy skills as the Ministry of Education (MOE) plans to drop part of the Primary 1 mathematics syllabus as part of its regular curriculum review.

It is important for schools to enforce and intertwine the six principles and standards for school mathematics in their programme. These principles which include

1.      Equity

      2.      Teaching

3.      Learning

4.      Curriculum

5.      Assessment

6.      Technology

will act as a guide and provide some form of direction for teachers of mathematics.  Of all the six principles, I feel that the teaching aspect is the most important. As more children of diverse learning abilities are integrated in schools, it is necessary for a teacher to firstly, find out what her student’s needs and understanding are before proceeding with ways to help them. I had a very poor understanding of mathematical concepts even in primary school. I do not ever recall my teacher having a keen interest and awareness in my individual development, and selecting “meaningful strategies” to support my learning. Math was taught one way - my teacher talked and I listened. There was not much hands-on or interaction amongst the students, let alone the teacher. She left me very much on my own to fend for myself. So yes, you can see why mathematics has never been my first love. However, I must admit the teachers of today are more proactive and responsive to the different challenges and uncertainties. Added emphasis is also now placed on teacher’s education, teaching approach, and curriculum content.  

A teacher of mathematics has to be flexible, and vary her knowledge of any mathematics content to fit the different learning styles of her children in the class. She must be equipped with the necessary strategies to counter any hindrance that might slow her down. She has to be diligent and persistent when faced with any challenging cases. Having a positive attitude and being ready for changes are also crucial skills for today’s teachers of mathematics.  More important, making time to be reflective and being self-conscious allows such teachers to relook at which areas need improvement or reflect on accomplishments and plan their own growth. I aspire to be that teacher and not let what happened to me in school years ago rub off on the children I am teaching now.

I must admit I was very sceptical when I saw teachings of fractions (decimal, percents, ratio), and algebraic thinking included in some of the sessions of the course outline. I asked myself why we needed to learn all this when I was only teaching in a pre-school. As I read on, I realised that these topics were parts of the five content standards. Each standard had a set of goals that was relevant to all grade bands but with different emphasis, and exclusive to only what students of that level needed to know. This provided me with a better picture about why my daughter could not initially do well in even very simple mathematical topics in primary school. Her lack of understanding of a concept, and the ability to connect new ideas to existing conceptual webs were the main reasons. These days she is much better, thanks to a teacher who helped her “think and reason mathematically” and “connect within and among mathematical ideas” (pp. 3-4).

 Chapter 2

Children’s interest in maths begins with me. I create a classroom environment that is conducive for children can take risks and share their mathematical ideas. I am aware that some children in my class take a longer time than others to grasp concepts. I use various tools and manipulatives as aides to represent the concept in the math corner for children to explore and enhance their learning.  I feel children pick up concepts faster when I use more concrete tools. Children are also encouraged to learn to evaluate their own ideas and those of others, scaffold each other’s learning, make decisions, test them, and develop reasoning and sense-making skills. I must admit I am still working on how to balance “productive struggle.” I often step in to show or explain to the children how to solve a problem too quickly.

               
The attached pictures show how using different tools and manipulatives can give children a clearer picture of various math concepts and help them understand patterning and matching numeracy to number words.

I seem to be able to relate to what Lesh and his colleagues said, “that children who have difficulty translating a concept from one representation to another also have difficulty solving problems and understanding computations” (p. 24). I had a very weak understanding of concepts e.g. I may be tested on the same concept but once the equation changed slightly, I could not solve it. My reaction was exactly what was stated on page 27 – “I can’t remember the way to do this type of problem.” Very seldom did I have a “can do” attitude (p. 28). I lack that perseverance and confidence. I did not understand the ideas and they did not make any sense to me. Hence, I feared the subject and tried to avoid doing my math homework or copied where possible.

I learn mathematics the instrumental manner – “endless list of isolated skills, concepts, rules, and symbols that must be refreshed regularly and often seem overwhelming to keep straight” (p. 29). As I look back now, I wonder if this inability to do mathematics came about because of my lack of interest or perhaps practice or could it have been a shortage of opportunities and adequate support from the teachers to learn mathematics.

The book talks about employing “invented strategies” (p. 27) and the use of flexibility to compute an answer which I totally agree. However, in the case of my daughter when she employed a different method to get her answer, her math teacher insisted that she use the way she taught her to get the answer. My daughter became confused and found it difficult to understand the method her teacher was using. She did not like the topic after that because of the “lack of retention and increased errors” (p. 27) when she used her teacher’s method. Overall, developing mathematical proficiency has more useful and worthwhile benefits for both the teacher and the student. It plays a significant role in art, science, language arts, and social studies. Indeed, knowing how to do mathematics indisputably connects one to the real world.

Lesson 1 Interesting and Inspirational

I have never enjoyed Mathematics as I did yesterday. Numbers always scare and confuse me. Yesterday’s lesson got me all excited! For once I saw numbers as something I could understand and relate to.

I was indeed intrigued by the first problem – which letter is the number 99? What was more interesting was the various methods that were used solve this problem. I saw patterns everywhere in this name game. Children are naturally drawn to patterns. I thought to myself, hey this is not so bad! I can do this!

I was inspired by the use of cards to teach children number words. Wait till I show this trick in class! I bet you they are going to be all in awe. They must be sure to think what an awesome teacher they have! Huh, little do they know that there is a trick to all this.

A few things struck me. One was to always use a noun behind every cardinal number. I never really gave it much thought until yesterday when I saw the reason behind it and why children so often misinterpret these concepts. Something else that struck me was the Concrete to Pictorial to Abstract (CPA) approach. I now have a clearer understanding of why the use of concrete materials is necessary when introducing various mathematics concepts to younger children. Yet another thing was the use of the word ‘mass’ more than ‘weight.’ It never donned on me that I had been using the wrong word all this while. I must remember to ask children questions such as ‘How much does this weight?’ or “How heavy is this bottle?’ when I teach the children the topic on measurement.

I must keep in mind to choose activities that promote the thinking and learning of mathematics in my classroom. I must develop an environment where children can ask questions, explore and investigate.