REFLECTION - FINAL THOUGHTS

Before attending this module, I find that Mathematics a subject that is rigid and normally, I will follow the lesson plans to deliver the lessons accordingly. I did not have any idea on the Primary School Mathematics Curriculum framework as well as its rational. I have no idea that Mathematics can be taught in a ‘fun’ manner as I learnt through ‘drilling’ in my Primary years and maybe that is the reason why I do not like Mathematics since young.

After attending this module, I have gain insights on ways that we could make Mathematics interesting for children and Mathematics isn’t what I see as ‘boring’ and ‘rigid’. Mathematics is indeed an excellent vehicle for the development and improvement of a person’s intellectual competence in logical reasoning, spatial visualization, analysis and abstract thought as children will develop numeracy, reasoning, thinking skills, and problem solving skills through the learning process. Mathematics is also a subject of enjoyment and fun. I feel that we, educators make a difference in the lives of the children and we also play a significant role whether the child likes or dislikes the subject. And hence, it is about time that we reflect on the ways we teach mathematics and how to let children have ‘fun’ and at the same time learn the concepts that we have plan instead of making them dislike ‘Mathematics’ the way I did.

I agree with Dr Yeap that we must constantly remind ourselves not to ‘spoon-feed’ children with answers but scaffold them by asking leading questions in order for them to look for ‘answers’ themselves and constantly reflect on our practices. Teachers should enhance children’s mathematics learning when they ask questions that provoke clarifications, extensions, and development of new understandings.

Although my knowledge is still far from complete (due to the limited time that we have on this module), I now have a fuller picture on how to plan for developmentally appropriate activities for preschoolers to acquire and the practices to promote their understanding and most importantly, in a ‘fun’ manner so that children can learn the concepts and at the same time enjoy the process and eventually like this subject. I feel that this knowledge, however, is not yet in the hands of all preschool teachers in a form to effectively guide their teaching. It is not surprising then that many teachers who learn Mathematics through ‘drilling’ have the same initial mindset as mine and mathematics makes only fleeting, random appearances in their classrooms. Some may give mathematics adequate time in the curriculum but attempt to cover so many mathematical topics as they have to follow the curriculum.

I have learned that in planning for new investigations and activities, teachers should think of ways to engage children in revisiting concepts they have previously explored. Such experiences enable children to forge links between previously encountered mathematical ideas and new applications. (Spiral Curriculum-building on knowledge)

Apart from that, extended investigations will offer children excellent opportunities to apply mathematics as well as to develop independence, persistence, and flexibility in making sense of real-life problems. During the activities, children encounter many mathematical problems and questions and with teacher guidance and facilitation, they think about how to gather and record information and develop representations to help them in understanding and using the information and communicating theirs to others. Thus, educators need to plan for children’s in-depth involvement with mathematical ideas. Teachers should ensure that the mathematics experiences woven throughout the curriculum follow logical sequences, allow depth and focus, and help children move forward in knowledge and skills.

I have applied some hands-on and challenging activities which I have learnt in this module in my class and I am happy to see that the children are working together to problem solve and I could see their sense of satisfaction when they finally solved the problem. (This is how we as learners feel when we are trying to solve the problems in class.) Hence, I believe that it is up to the teachers to make “Mathematics come alive” to captivate children’s interest while teaching them concepts.

Children exploring with different ways to form "square" with tangrams

Throughout the early years, children notice and explore mathematical dimensions of their world. They compare quantities, find patterns, navigate in space, and grapple with real problems such as balancing a tall block building or sharing a bowl of coco crunch fairly with a friend and Mathematics does help them make sense of their world outside of school and helps them construct a solid foundation for success in school later on.

I think preschool educators should relook in their curriculum so that they are inline with those in the Primary School Mathematics Framework. I believe most preschool educators are not familiar with the Primary Mathematics Curriculum framework as well as the rationale. It is time for us to reflect to see if we are teaching children sufficiently (building a strong foundation for The development of mathematical problem solving ability) or are we stuffing them with too much ‘surface’ information (learning a bit of here and there).

I feel that Mathematics is too important to be left to chance, and yet it must also be connected to children’s lives. In making all of these choices, effective teachers should build on children’s informal mathematical knowledge and experiences, always taking children’s cultural background and language into consideration and support children’s learning by thoughtfully and continually assessing all children’s mathematical knowledge, skills, and strategies. Using those information and insights gathered from assessment, teachers should plan and adapt teaching to help individual children to grasp the concepts.

Our role as a teacher is to facilitate children’s learning by letting them learn Mathematics by doing Mathematics. (Van De Walle et al, 2009, p.33). For children who cannot communicate ideas, our role as a teacher is to scaffold by modeling and using alternate medium to aid them.  They should recognize that even young children invent their own mathematical ideas and strategies and that their ideas can be quite different from adults and thus it is important for us to constantly reflect on our teaching practices so that we could strengthen children’s problem-solving and reasoning processes. Opportunities should be given to children to allow self-exploration...

And lastly,  I have a total of 5 cookies....

Reflection on Chapter 20 - Geometry

In chapter 20, the content goals for geometry include shapes and properties, transformation, location as well as visualization. I am not good in the topic on Geometry since young. During the activity on finding the interior angles, it took me a while to figure out the interior angles in a pentagon as I had to recall what I had learn two decades ago about isosceles, equilateral triangles, the properties of the shapes like pentagon, rhombus, parallelogram etc.

I have learned the five hierarchies of ways of understanding spatial ideas by the Van Hiele theory of geometric thought both in class as well in Chapter 20. Another thing that struck me in class is that “Diamond is not a shape”. What have I been teaching children all these while?

I began to reflect my teaching practice on “the 2Ds and 3Ds shape” after the lesson. Whether am I guilty of confusing children when I am showing them concrete items? Did I use the ‘right’ material?

I began to realize that sometimes, it is not easy for a child to visualize a certain concept if he or she has not reach a certain level of Van Hiele’s Theory and teacher must facilitate their learning by providing the ‘right’ material and giving ‘specific’ instructions. In Chapter 20, it is stated that the levels of Van Hiele level is sequential and to arrive at one level, children must move through prior levels. Children need to be given opportunities to explore and interact with content and these experiences the teachers provide are the single most important factor in moving children up in this development ladder.




An unforgettable activity

 


Children using tangrams to form a square during free play.



Reflection on practice-Number Sense

I strongly agree with the NCTM Standards that “As students work with numbers, they gradually develop flexibility in thinking about numbers, which is a hallmark of number sense…. Number sense develops as students understand the size of numbers, develop multiple ways of thinking about and representing numbers, use numbers as referents, and develop accurate perceptions about the effects of operations on numbers” (p.80).

In my centre, our Mathematics curriculum touches on the following:

1) Number sense – counting, recongintion of numbers, more and less,
adding and subtracting, ordinal numbers, number patterns, grouping,
sharing
2) part-part-whole
3) More than/Less than
4) patterning set recognition

 

Depending on the children’s level of understanding, the class teacher will reinforce the topic until most children have grasp the concept as after every topic, the teacher has to assess the children. Teachers are always reminded to provide more hands-on activities and practices related when they teach a particular concept. It relates to what we have learned in class about Jerome Bruner’s “CPA Approach” as the Nursery children are learning Mathematics concept through songs, rhymes, books as well as concrete materials. They learn by manipulating with concrete materials and do not have worksheets to do whereas for the  Kindergarten children, they move from concrete materials, pictorial to abstract.

I think that the concept of anchoring numbers to 5 and 10 is not a common topic found in some Preschool Mathematics Curriculum and apart from that, not many preschool centres encourage the use of calculators in teaching numeracy and many still stick to the traditional method by teaching children to add by doing ‘finger counting’.

In my centre, the K2 children are taught on the use of Doubles (count in twos, fives, tens) and immediately after children have learned numbers 1 to 20, a lot of emphasis is place on adding and subtracting numbers within 20. However, the Mathematics Curriculum did not focus much on place value as well as estimation. In Chapter 8 of the textbook, there are many examples of hands-on activities for me to refer to when I need to teach numbers. However, there is a need for me to slowly modify or invent my own activities for the children under my care so that they could enjoy the process when learning about numbers.

It is said that we should not introducing symbols (“+”, “-“ etc) to preschoolers, however, I think that most centre are doing so. So, when is the best time to introduce the symbol to them?

The activities that are not common practice in pre-school:

1) Doubles and near-doubles

2) Anchoring numbers to 5 and 10

3) Usage of calculator

Learning the concept of place value using base ten materials

Children using cubes to do addition

Reflection on Chapter7: Using Technology to Teach Mathematics

After reading this chapter, I believe that technology has become an essential tool for doing mathematics in today’s world as it can be used in a variety of ways to improve and enhance the learning of mathematics. As NCTM Position Statement (2008) highlights in its standards, technology can facilitate mathematical problem solving, communication, reasoning and proof; moreover technology can provide students with opportunities to explore different representations of mathematical ideas and support them in making connections both within and outside of mathematics.

Today, the emphasis on teaching content is indeed greater than in the recent past. These changes have set a new challenge for the preschool teacher; knowing what to teach and how to present it and use multiple opportunities during the day to help child build competencies in math in a fun manner. The uses of technologies such as computers have much more to offer than drill and practice. I feel that the basics of using computers and the uses of technology are to communicate, to learn new information, to solve problems, and to create. Apart from that, it also stresses social skills, and can be used in conjunction with all parts of the constructive learning process.

The use of calculators mentioned in this chapter strike me most. Some people may argue that even though in the BC days (before calculators) walking scorers didn’t use them, and is there a need for technology? Calculators appear to be tools for adults to use as we wish but not for children to use in learning mathematics for most parents. I remembered that I was only allowed to use a calculator in Secondary School. However, it is not the case now as a Primary 5 child is able to use a calculator to aid them in Section B and C of a Mathematics Test.

From the link, it is stated that at grade 2, guided work with calculators can enable students to explore number and patterns, focus on problem-solving processes, and investigate realistic applications. After reading this link, I believe that calculator can be used as a tool to teach the concept of place value and counting on. However, it is essential for students to study mathematics for an hour a day under the guidance of teachers who enjoy mathematics and are prepared to teach it well. Do all our teachers have this “belief”? Frankly, before reading this chapter and reading the information on the links, I am quite against the idea of using calculator. After reading, I feel that the challenge for us is to investigate how calculators may be used as tools to think with rather than as tools to replace thinking. Several questions came to my mind as I was reading this section:

• What will the use of calculators mean for teaching mathematics in preschool context? (Fun way of learning numerals for preschoolers)
• How do calculators and similar technologies influence children’s developing knowledge of mathematical processes?
• Will children be mindlessly using these technologies? Or are they thinking about mathematics differently?
• What is the minimum mathematical knowledge needed before a child can use calculators to meaningfully explore mathematical understandings of specific concepts?
• Can technology catalyze a re-conceptualization of the content and methods of teaching mathematics?

Technology such as The World Wide Web, CD-ROMs, and other media disseminate vast quantities of quantitative information and it has indeed unlocked new ways to allow children a glimpse into an analytical world and to acquire the tools to explore it. The use of technology in the classroom, whether it is by computer, overhead projector, interactive whiteboard or any other electronic device that allows children to interact with the mathematical problems at hand. Teaching mathematics in a technology classroom requires more than simply using mathematics with technology. It also requires teacher to design the lesson to focus, motivate, and highlight the mathematics in a meaningful way.
I do agree that software might provide tools that enhance children’s imagination and interest and it may serve as a reinforcement of grasping a certain concepts. However, I feel that it’s so important to focus on the intended mathematical objective (e.g. searching for a pattern in a sequence of numbers) and then allow children to use a tool (i.e. calculators) on the mathematical work, such as computation.

In the news, more Primary Schools in Singapore will receive funding from Government to renovate their classroom and there will be learning corners as well as computers and interactive whiteboard in each classroom. (News on 5 dated 11th September 2010) There has been a rapid increase in the usage of computers in schools now and I believe that technology should not be overlooked in the curriculum. Using technology can also help to prepare the children for the high tech lifestyle of the real world. Maybe, if we view technology from a broad perspective, we will be able to see how it can be integrated into all aspects of the preschool classroom. Technological advances have changed and I believe that it is still changing many aspects of society and education in Singapore.

These are the links that I particularly like:
Video:
1) http://www.learner.org/vod/vod_window.html?pid=873
The activities shown in the video is engaging and I have gain new insight on ways to introduce and reinforce the math concept on “Place Value” in a more interesting manner.

2) Math Buddies (http://www.learner.org/vod/vod_window.html?pid=872)
The activities involve hands-on activities and the usage of common materials for Kindergarteners to experience the numbers 1 through 50. Activities are useful for me to reinforce number sense and numeration, communication, connections.

3) Problem Solving (http://www.learner.org/vod/vod_window.html?pid=992)
This video illustrates students investigating and learning mathematics through problem solving and the teachers share their approaches and observations.

Another activity that I like was "Developing Geometry Understandings and Spatial Skills. This is a fun and interesting activity and I can’t wait to conduct this activity with the children.

SEQUENCING LEARNING TASK FOR PLACE VALUE (8th September 2010)

In my opinion, after the pictorial representation using popsicles and based ten materials, I would the sequence the five tasks as follows:
• Task 1: Place Value Chart
• Task 2: Number in tens and ones
• Task 3: Expanded notation
• Task 4: Numeral
• Task 5: Number in words

In the earlier lesson, children are shown concrete materials that showed 3 groups of ten and 4 ones and hence, I feel that it will be easier for children to learn place value chart as it represent what they have seen and learnt earlier using concrete materials and hence will be more meaningful for them . After that, I will introduce children to numbers in tens and ones as it links to what they have seen on the place value chart. Upon understanding the place value using pictorial and concrete materials, I will take them further by exploring 3 tens into numerals (in abstract form). The concept of ‘tens’ and ‘ones’ will be introduced to them in numerals. By now, children will be able to understand that 3 tens is equivalent to 30 and they should be able to visualize that 30 and 4 is the same as 3 tens and 4 ones. I feel that groupings by 10 should be matched with place value and eventually written in standard form and hence, the next task is to show children "number in numerals". Let children know that 34 is equivalent to 30 and 4. By now, children should be able to visualize "34" as 3 gropus of tens and 4 ones. The last task will then be showing them "numbers in words".

Reflection on Environmental Task 6th September 2010

I strongly agree that it is important for teachers to understand that mathematics is to be taught through problem solving rather than memorizing formulas and methods and problem-based activities should be the vehicle which the desired curriculum is developed and the learning is the outcome of the problem solving method. After reading the chapters on problem solving; I realized that teachers teaching mathematics needs to have a paradigm shift and need to change their philosophy of how they think children learn. It is so different as compared to the way I learnt Mathematics during my time. Teachers must plan and develop interesting problem-based activities that could engage children in a fun and interesting manner and allow them to verify and relate their strategies as this process will allow them to grasp and understand mathematics on a deeper level also known as meta-cognition.

Many questions raced through my mind the moment we received the environmental task. How can we create meaningful and engaging context in a ‘real’ environment? What to select an appropriate problem-based task in a natural environment and at the same time, teaching children how to problem solve? Which age group are we targeting? Initially, we were merely brainstorming on the content area and where to conduct the activities that is suitable for preschoolers.

The content area that we brainstorm includes non-standard measurement, shapes, colours and numbers. Next, deciding where to have the activities is a big ‘headache’ to us as we have to think of safety reasons & boundaries. We need to clear and focus on our task as there are many things that environment can teach and the type of instruction to be given.

Eventually, after much consideration, we have decided to use the supermarket as it relates to their daily life. The task involved the children using different combinations of products, found in the supermarkets, to come to a total value of 5. The problem in the task for the children would be trying to come out with as many combinations of 5 as they can with their partner. Using the value card provided, the children need to work with their partner and justify with their partners how they can derive to the answer by matching the product to the number value. The group with the most combination wins the game. After deciding on the rules of game, our group went into the supermarket and started playing the game. We were giggling and playing happily in the supermarket.

After this activity, I learned that learning Mathematical concepts need not be directive as children are actively constructing their knowledge when they are involved in the learning process. The challenge to me now is “How to think of more interesting ways to teach Mathematical concepts in a fun manner so that children can enjoy the process of learning.”

Reflection of Chapters 1 and 2

Mathematics was never my forte since young as my teachers of Primary and Secondary School made me do tons and tons of practice to master a concept. I believe many people of my era would reckon that Maths is boring as they suffer the same fate as me. However, my liking for Maths changed when I met my A. Maths Teacher, Mr. Lim. He was very patience with me and took great pains to explain to me the concepts. Hence, I believe that it is the "TEACHER" who makes a difference in the lives of the children and they are the ones who will make the children 'like' the subject that they taught.


After reading Chapters 1 and 2, I began to realize that there were actually theories, standards and principles involved when teaching Mathematics. Although the NCTM standards and principles mentioned is not the Singapore Maths Curriculum, however, I feel that Mathematics Curriculum does evolved around those principles, standards and theories. I strongly believed that teacher is the one who is able to create an interest in children for learning mathematics and the different approach you adopt will affect the way children learn. It is thus important for teachers to ‘like’ the subject before they teach as they will be able to think of creative and interesting ways to teach mathematical concept and engage children in critical thinking and problem solving and at the same time, making mathematics “fun’ to learn.

Hence, teachers should constantly reflect on what and how they teach based on their observation and assessment of children and at the same time have a network of teachers who share ideas and different methodology of teaching concepts.