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Blog Entry 8 - Reflections about the course

I agree with the following statement jointly issued by NAEYC and NTCM in 2002: "All children need an early start in Mathematics." Just as important that all children got exposed to language learning early in their life, a strong mathematical foundation should start from young too. Mathematics should not be taught as a level of knowledge, but rather, it should be taught for understanding so that children who learn it are developing in themselves a range of cognitive competency, that allow them to deal with the many demands of the outside world.

I remembered as mentioned in Lesson 4, Dr. Yeap introduced to us the 3 meanings of Addition ad Subtraction: Part-whole, Change Meaning, and Comparison. It suddenly dawned to me that I had not quite paid attention to this, or rather, I had not even knew about it. While studying the 3 meanings in detail, I noticed that each meaning actually takes the children progressively in developing understanding about addition and subtraction from simple to complex: Part-whole-->Change meaning --> Comparison.

Honestly speaking, I do feel "silly" at some points of time during class, for the fact that I took a longer-than-others time to understand and grasp the logic behind solving the various mathematical tasks we were assigned. However, when I finally understood them, I reflected and realized that I am actually taking myself through a journey of critical thinking moments - exactly what teachers should bring children through! Many mistakes that teachers commit when teaching mathematics: Teach for the sake of getting a solution, and not teach for understanding.


Most impactful lesson

Of all the classes, the lesson on geometry at the last lesson had the greatest impact on me - I was very amazed at how geometry learning can be so complex but at the same time, it is also easy if one had grasped the understanding of it well. Geometry is not just about seeing shapes as what it is, but rather understanding what it is, so as to be able to apply its understanding in a variety of contexts.

Using tangrams to create shapes

Using the geo-dots to create shapes with differentiating sides and areas

I had never tried the Sudoku game before: The sight of it "numbs" my thinking. The challenging part was not doing the cubes right, but strategizing the numbers in a way that they do not repeat. Although I had found it challenging, it was fun!

I feel that the neglience of creative Mathematics education has resulted in many children not knowing how to approach the subject positively - I was one of them. I remembered my Secondary school maths teacher coming to class only with a marker in his pocket and started to teach by writing on the board - it did not helped me much given the fact my forte is not in Maths!

The whole class on Elementary Mathematics taught me one thing that I found very valuable: Mathematics is not about formulas, it is about the development of understanding and conceptualizing mathematical content. As simple as it may be to some people, however, I feel Mathematics is a very abstract area - one needs to "see beyond" the given sums or texts to be able to apply the relevant logical thinking and approach the solution with understanding.

"In mathematics I can report no deficience, except it be that men do not sufficiently understand the excellent use of the Pure Mathematics." ~Roger Bacon

Blog Entry 7 - Geometry

The understanding of Geometry takes upon a combination of understanding about shapes and having good spatial sense. "Spatial sense can be defined as an intuition about shapes and the relationship among shapes......They are able to use geometric ideas to describe and analyze their world." (Van De Walle, Karp & Bay-Williams, 2009).

The study of Geometry involves skills that allows individuals to create good connections with the world around them. Everything in the environment is geometric - The Esplanade is oval, the long stretch of road on the highway is rectangular, and the lift buttons are circular. The development ot geometric thinking goes through developmental stages, as described by Van Hiele: He has decribed the levels of geometric thinking according to "what we think and wat types of geometric ideas we think about, rather than how much knowledge we have" (Van De Walle, Karp & Bay-Williams, 2009). Van Hiele's theory of geometric thinking is listed as:

Level 0: Visualisation (classes of shapes)
Level 1: Analysis (Propertities of shapes)
Level 2: Informal Deduction (Relationships among popertities)
Level 3: Deduction (Deductive systems of propertities)
Level 4: Rigor (Analysis of deductive systems).

The child's ability to sort out shapes according to its physical property puts him at Level 1 of Van Hiele's theory: Properties of shapes.

Finding the interior angles in a pentagon:

Finding the interior angles of a pentagon requires that the individual has developed at Level 4 of Van Hiele's theory: Rigor.

A Therefore, the formula of working out an interior angles of a pentagon is:

1. A pentagon is made up of 3 triangles:


2. The interior angles of each triangle in a pentagon adds up to to 180°.

3. Therefore, the interior angles of a pentagon is worked out as:
3 x 180° = 540°

Teaching children geometry requires that they be involved in a variety of critical thinking and making connections with the environment. The reading of the chapter has allowed me to develop a new set of understanding about Geometry, and even better than I did before. I am creating in myself a new set of understanding of the topic, and how I can create more constructive learning opportunities about Geometry. Click here for more information on finding the interior angles of the various geometric shapes.

Blog Entry 6 - Teaching of Number Sense

What came to my mind at the mention of Number Sense? Counting. Recognition of numbers. Knowing its quantity.

Number sense simply means making sense of numbers; and it does not end by just knowing how to count - it is the ability to make sense of numbers by creating meaningful connections with the environment. Howden (1989) defined number sense as: "good intuition about numbers and their relationships." It is a process of making sense of numbers by being able to visualize its use in a variety of contexts.

Teaching number sense requires that the teacher has a great deal of understanding on how the topics pertaining to number sense are taught. It is not something that can be taught by reading and planning: The teacher may have good theoretical knowledge about a topic but is unable to execute the delivery of the topic effectively. With this, I would like to emphasize on the fact that teachers should pre-teach themselves the topic that they want to teach children, and put themselves into the shoes of the children to anticipate how they may approach the activity. The following are some listing of number sense activities that were taught, and not taught in my experience as a preschool teacher; and why:

Activities in practice:
1. Count by Ones
A commonly facilitated practice that I have been doing, it teaches children to concentrate on the quantity and at the same time, develops their one-to-one correspondence counting. Children are encouraged to point to each manipulative/object/item that they are counting, so as to focus their attention on the task. One of my favourite practice to enhance the children's ability to count by ones is the use of songs and rhymes.

2. Two-digit Number Names
Children are taught the number names for single-digit numerals first, before they move on to learning two-digit number names. However, a challenge when I teach children two-digit number names: Single-digit number names are straight forward; 1 is one, and 2 is two. However, certain two-digit number names can be quite confusing for the children to acquire - 5 is five, but when it is in the tens value, it becomes a two-digit number name fifty. As the children generally note a pattern in the spelling of the number names from single-digit to two-digit, having them conceptualize a new knowledge not applicable within their "general knowledge" may create some disturbances as the children try to accommodate the new knowledge to be learned.

Activities not in practice:
1. Partitioning strategies in teaching Multiplication
I have not introduced this strategy before, as I find it quite confusing for the children to master. The children can easily get confused with the requirements of the task if understanding of Multiplication is not strong enough.

2. Doubles and Near-doubles
Unless the child is absolutely good with his addition, otherwise, it would be too difficult for the children to conceptualize the skills to work on multiplication tasks such as Doubles and Near-doubles, as it requires the accomodation of multiple addition tasks.

Blog Entry 5 - Technology

Should technology be inculcated into mathematics education and replace the use of activity sheets that children does?

Will the use of technology make children more reliant on external tools and forsake the use of their cognitive ability to solve problems?

According to the NCTM, technology is an essential tool for learning and teaching mathematics. Technology should not be seen as a substitute for the figure of a teacher, nor should it be seen as a substitute to replace any learning experiences. It should instead be utilized as an alternative approach to teaching mathematics, to enhance the quality of children's participation during the process of learning. As mentioned by Van De Walle, Karp & Bay-William (2009), technology enlarges the scope of the content students can learn and it can widen the range of problems that children are able to tackle.

Technology has become so readily available that I must honestly admit even as an adult, I portray an over-reliant attitude on technology to "perform" tasks for me. Instead of mentally calculating the costs of things I have bought (which is possible in some situations), I would rather use the calculator. Instead of manually recording my monthly expenses on pen and paper, I do mine using excel - I've succumb to the convenience of technology in today's world!

As I was reading Chapter 7 on technology, one thing that strike me the most is the use of computers in today's world, as with more and more children are gaining easy acess to the use of computers to access the virtual world. However, it is also the adults responsibility that when introducing the use of technology to children, they should be supervised so that they do not "abuse" the use of it. I particularly like the patterning activity as posted on the NLVM's website. The activity is very interactive and the way it lines up the pattern sequence in a curved line instead of straight suggests and show children that there are many ways of presenting knowledge. As Van De Walle, Karp & Bay-Williams (2009) suggests, "the user of a well-designed tool has an electronic 'thinker toy' with which to exlore mathematical ideas."

Website with interesting Maths activities:
http://www.coolmath4kids.com/

Blog Entry 4 - Place Value

"Place-value understanding requires an integration of new and difficult-to-construct concepts of grouping by tens with procedural knowledge of how groups are recorded in our place-value scheme, how numbers are written, and how they are spoken" (Van De Walle, Karp & Bay-Williams, 2009, p. 188).

The main difficulty that children face when learning place-value concepts is the ability to conceptualize the idea of how numbers are formed. Teaching of place-value concept should move from simple to complex, which allows for children to assimilate their prior knowledge in attempts to accomodate with the new knowledge taught. The class was asked to brainstorm on what is our sequence of approach towards teaching place-value, and the following is my stance:

1. Number in Numerals

Children should be re-introduced to the numerals as an individual whole, not part-whole (e.g. 3+1 = 4), including the number "0". In reinforcing this concept, I would get the children to count the quantity of the abovementioned numerals with manipulatives, as those numerals denotes the "ones" value they would encounter when learning about place-value.

2. Place Value Chart

Introducing the place-value chart allows the children to assimilate their prior knowledge of numerals as a whole, numerals in words, number concepts; and put together to accomodate a new concept of how numbers are "broken up".

3. Tens and Ones

Using unifix cubes, I would introduce the numbers by "breaking" it up and get the children to count the "tens" using a colour, and the "ones" in another colour. This is to enable the children to conceptualize visually that for example, 38 is made up of 3 tens and 8 ones.

4. Number in Words

5. Expanded Notion

I perceive the Expanded Notion as the most difficult step to master as it involves assimilating, accomodating and then assimilate again. It requires that the child reognizes the numeral as a whole, "cut" it up mentally, and then conceptualize the numerals as a representation of a whole number again. For example the numeral 38, I would write the number 30 on a card, seperated into two different columns, and then on another card, write the number 8 and get the child to cover up the 0, which makes 38.

Blog Entry 3 - Problem Solving (Environmental Task)

Problem solving: What does it mean to problem solve? Is it finding the solution to a problem, or, acquiring understanding of how and why the problem is solved the way it is?

I personally agree with the latter. One needs to capture an understanding of how the problem is solved, rather than solve for the sake of meeting the end-product requirement (finding a solution/solutions). If a child is taught to approach solving with the perspective of solving the problem without understanding why, then the process spent on solving problem is not meaningful and constructive, without a purpose for learning.

As Van De Walle, Karp, and Bay-Williams (2009, p. 33) suggest, "Teaching through problem solving requires a paradigm shift... she is changing her philosophy of how she thinks children learn and how she can best help them learn." The presentation of preparing content for children to experience problem solving experience/experiences should be planned purposefully so that children are engaged meaningfully in the process. With this, the teacher plays a great role in facilitation the process, by providing opportunities for the children to develop a gradual continuum of assimilating and accommodating their prior and present knowledge.


Environmental Task
The class was tasked with the responsibility of creating an "environmental task" where we were asked to create the teaching of a mathematical concept through the environment. My group had decided to focus on the concept Units of Measurement, by expanding the children's experience on the topic through the use of non-standard units of measurement to measure the length/circumference of objects/structures in the environment.

The children are involved in problem solving opportunities with the teacher's facilitation by:

1. Asking the children Open-ended questions
- "What do you think can be used to measure this object?"

2. Involve children in brainstorming ways of measuring the objects/structure
- "Asking children to brainstorm on the different ways of using their bodies to measure the circumference/length of the objects/structures.





The following are some of the places we have visited, and the intended teaching content pertaining to Units of Measurement:

1. Getting the children to count how many arm lengths are required to measure the length of the structure.

2. Counting how many foot-long is a stretch of step on the stairs.

3. Counting how many children are needed to go round the different structures.

"No problem can stand the assault of problem solving." ~Voltaire

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Dear Dr. Yeap,

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Thank you!

Blog Entry 2 - My reflections in the first lesson

I was blown off my feet. Totally. For a long, long, time, I actually began to develop a love for Maths! Somehow, I had expected the lesson to be just as typical as other modules I had taken - but, NO! It was different, very different.

The first lesson was an eye-opener for me to be acquainted with the Mathematics world. My theoretical perspective of Mathematics prior to the class: It is just a subject that involves numbers, maths concepts; and Maths is just what it is. Teachers teach Maths to children so that they are equipped with the knowledge of the formulas and methodologies required to be able to do Mathematics well.

As mentioned by The Singapore Primary Curriculum Mathematics Curriculum Syllabus (2006), "Mathematics is an excellent vehicle for the development and improvement of a person's intellectual competence in logical reasoning, spatial visualisation, analysis and abstract thought." I had not given much thought to this until I reflected on the tasks that we worked on during class. For example, the Dice Trick: As a child (even ourselves as we worked on the task and attempt to identify the sequence/trends in the number patterns) works on such an activity to pick out the trends in the numbers, he or she does not quickly identify the solution, but instead go through a process of piecing together all possibilities in order to identity the best possible way and explanation for the activity.

Dr. Yeap has mentioned in class that children nowadays are less inquisitive, and I think that is exactly why teachers should facilitate their thinking process so that they develop an inquisitive nature! Children need to be facilitate so that they develop a set of critical thoughts behind the logic of how mathematics formulas work, and why. Mathematics formulas is not stationary: It can be twisted, changed, and understood with a set of well-facilitated experience in place for the children.

The first lesson has changed my perspective on the purpose of interacting with Mathematics - It is not to solve problems by strictly applying formulas; but to develop and enhance a child's cognitive competence so that he or she is equipped with awaken intellectual abilities to deal with the demands of the society.

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." ~John Louis von Neumann

Blog Entry 1: Reading Reflections of Chapter 1 & 2

I remember this incident back in Secondary 4 when I got my Maths preliminary results and my Maths teacher told me: "Xiang Ling, at least you failed decently this time round!" I honestly admit I was never good in Maths, and I can never remember myself getting excited about doing Maths - the last time I ever sat down and did Maths (other than teaching preschoolers preschool-Maths) was 10 years ago!

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.

Building on their existing knowledge of AB pattern by getting the children to form AABB patterns with the same group of materials

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

References:
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