I would like to reflect upon the systematic approach to get children understand the concept of counting to 10 using the storybook.

Get children to:

1) predict the story “Two ways to count to 10” based on the given words.

2) write the predicted story in 5-10 lines given the helping words below:

eg. Leopard King, contest, feast, threw the spear, looking for a prince, cleverest animal

3) If children are unable to write the story, get them to say out verbally using the given words.

4) Teacher reads the story to the children.

5) Retell the story in round robin through pictures or text.

We were then asked by M/s Peggy Foo to brainstorm ways in which this task can be differentiated for different learners (e.g. those children having difficulties versus those children having higher ability).

Possible answers:

– to get the average students to find different ways to make 10

– the weaker students could be given ten frame and counters to help them visualise

– the advanced learners could be asked to find all the different ways to make 10

Carol Ann Tomlinson, advocate for differentiated instruction theory, believed that the process of “ensuring that what a student learns, how he/she learns it, and how the student demonstrates what he/she has learned is a match for that student’s readiness level, interests, and preferred mode of learning” (Tomlinson, 2001)

Adapting Tomlinson’s guidelines for ideas using Differentiated Instruction for which, technology can be used to:

◾Clarify key concepts and generalizations.

◾Emphasize critical and creative thinking.

◾Engaging all learners is essential.

◾Provide a balance between teacher-assigned and student-selected tasks

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Despite the various types of triangle; scalene, isosceles, equilateral, it is 180 degrees.

-Use of protractor

Cut the three edges of a triangle and put it on a straight line. if it is on a straight line, it is 180 degrees.

Thus,visualisation and meta cognition skills enables easier problem skills.

Piaget stated that, “Children construct their own knowledge in

response to their experiences.”

Thus when children are given concrete materials, they are able to visualise their experience in getting to know why a triangle is 180 degrees than to learn how to get the answer.

Dr. Yeap mentioned that “Mathematics is an excellent vehicle for the development and improvement of a person’s intellectual competence..”

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1) Instrumental understanding: rote or ‘learn the rule/method/algorithm’ kind of learning (which gives quicker results for the teacher in the short term), e.g. writing 10 would be understood as “This is how we write 10” in instrumental terms.

2) Relational understanding: a more meaningful learning in which the child is able to understand the links and relationships which give mathematics its structure (which is more beneficial in the long term and aids motivation), e.g. writing 10 would be understood as “This is why we write 10 like this (in terms of place value)” in relational terms.

Nowadays children are exposed to relational understandings to get children to be aware of the semantics that answer may be the same but the meaning is different. For example: 3×2=6 and 2×3=6. But 3×2 means 3 baskets of apples and 2×3 means 2 baskets of apples. On the other hand, an example of Multiplication comparison in a problem sum would be : I have 3 cookies. My friend has twice as much. How many cookies does my friend have?

I agree to Dr. Yeap’s saying that it is important for teachers to go through with children the journey to get to the destination and not merely teaching them through rote or rule learning.

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I would like to reflect upon the three main ways that teachers could adapt when teaching this subject.

Firstly, teachers should allow children to explore the materials. They could scaffold and model their teaching. But it is important to note that explaining to children is not a good strategy to help children to learn.

Secondly,Teachers could use Bruner’s CPA approach which states that it is best for children to learn with concrete objects first, followed by pictorial then abstract.

Jerome Bruner’s theory of “CPA approach benefits all students but has been shown to be particularly effective with students who have mathematics difficulties, mainly because it moves gradually from actual objects through pictures and then to symbol”

(Jordan, Miller, & Mercer, 1998).

Thirdly, teachers should provide variation of the same task. Dienes’ theory relates specifically to teaching and learning of mathematics rather than teaching and learning in general. It consists of four principles:

1. Dynamic principle: Preliminary, structured activities using concrete materials should be provided to give necessary experiences from which mathematical concepts can be built eventually. Later on, mental activities can be used in the same way.

2. Constructivity Principle: In structuring activities, construction of concepts should always proceed analysis.

3. Mathematical Variability Principle: Concepts involving variables should be learnt by experiences involving the largest possible number of variables.

4. Perceptual Variability or Multiple Embodiment Principle: In order to allow as much scope as possible for individual variations in concept-formation and to induce children to gather the mathematical essence of an abstraction, the same conceptual structure should be presented in the form of as many perceptual equivalents as possible.

(Adapted from Dienes, Z.P, 1971 Building up Mathematics,4th ed.)

With the above methods, children will learn to subitize when given the opportunities whether its intentionally or unintentionally.

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I enjoyed using 10 frames with the beads in my learning of whole numbers and the best part is that it is easy to make. Otherwise an egg-carton could also be used. I feel that 10 frame is a fun way for children to learn it as it gives them the opportunity to learn by manipulation and concrete materials. It allows them to see things visually, to look for patterns, and to provide an awareness of number sense.

I was surprised to know that a simple 10 frame and beads could provide children with so much of learning in a fun way. Children are able to learn number bonds, space values, one-to-one correspondence, conservation, addition and subtraction, etc.

Children learn because they construct knowledge themselves. Thus it is important for teachers to build the platform for them.

As Dr. Yeap mentioned, “You cannot imagine well if you have never experienced. Imagination is visualization which is connected to construction.”

Below is a useful clip showing how to get started using ten frame.

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The tangram provided me with a great insight of teaching children to use the pieces to form various orientation of shapes. For example: to use 2 small triangle and 1 square piece to create a rectangle and to use 4 triangles to make a rectangle. The seven shapes in the tangram (five triangles, one square and one parallelogram) provides concrete learning experience to children.

It is stated that “When children are actively involved, they are manipulating with real objects, exploring, reflecting, interacting, making decisions and communicating with other children and adults to construct knowledge and ideas from their experiences.” (MOE, 2007).

I would like to reflect on Dr. Yeap’s saying that, ” If input is knowledge, we don’t tend to forget but if input is information, we may forget if we do not revisit it for sometime.”

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The text provided me with good insight on ways to build my instructional skills that are significant to effective teaching. I learnt that the most important tools needed to be an effective Math teacher are : the teacher’s knowledge on Mathematics and the ways students learn the subject. I agree to the fact that what it means to know and do mathematics and about how students make sense of mathematics will affect the teacher’s instructional approach. Thus, teachers play an important role in shaping mathematic for the students.

According to NCTM, “*Learning mathematics is maximized when teachers focus on mathematical thinking and reasoning*”

(NCTM, 2009, n.d.)

Curriculum documents (standards) have a significant influence on what is taught, and even how it is taught. I too believe that curriculum should not only seek to complete many topics but also to sustain students to learn the math concepts in depth and with understanding.

The classroom environment in which children are doing Mathematics should be one in which they are allowed to be risk takers, sharing and defending mathematics ideas. The students are seen to be active learners in solving problems and the teacher facilitates by asking questions to ensure that students are making the connections and understand the task that they are exploring.

Moreover the verbs as quoted in the text book such as compare, explain, construct, explore, describe, solve, are said to provide opportunities for higher level thinking and encompass “making sense” and “figuring out.” It engages students in doing Mathematics.

I agree that engaging students in “positive struggle” is what help students learn Mathematics. The focus is on students applying their prior knowledge, testing ideas, making ideas, making connections and comparisons, and conjectures. Thus, students must have the tools and prior knowledge to solve a problem that is slightly beyond the capabilities of the students, It makes learning goals feel attainable and efforts seem worthwhile.

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