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The most significant impact this module has on me

Wow! I am thrown off my chair after the first lesson. Today my mindset toward teaching and learning Mathematics has changed drastically. Firstly, understanding the outcomes of doing mathematics, secondly the approach to teaching Mathematics and lastly Mathematics can be fun.

Outcome of teaching Mathematics to the children as stated by MOE; is the vehicle for the development and improvement of a person’s intellectual competence and it is applicable in everyday living and in the workplace. Mathematics is a subject of enjoyment and excitement. This will be my goals for teaching Mathematics.

Many theories were covered in this module and the one strike me most is Jerome Bruner’s CPA theory. This is especially applicable for preschoolers as they need the concrete experiences and this ties to the three big ideas in teaching Mathematics; that is visualization, patterning and number sense.

I was never interested in Mathematics least being challenged to attempt to take on Mathematics as being fun. But this module helps me to visualize patterns in Mathematics and it was challenging seeing it. Once found, a sense of achievement is felt. Once the feeling of accomplishment is felt, one will build confidence in future's attempts and that will lead to perseverance of wanting to continue to attempt and the resilience to want to try although failing. This module has opened me to the use of technology and I am now addicted to playing Mathematics games. See how I use ‘playing’ and not doing Mathematics because it’s fun.

Geometric thinking

 It was through this exercise that I visualised that a pentagon was made up of 3 triangles and the properties of each triangle is 180° therefore making a pentagon 540°. Then I tried dividing the pentagon into a triangle and a trapezium and added up their interior angles (180° + 360° = 540°), and I have 540° too.

I was taught in school to remember the individual polygons angles; I was not brought through on how we can visualise and understand the parts making up of these polygons.

I went on trying out with the rest of the polygons and realised a pattern in the calculation of the interior angles of any polygons, see pattern below:

shapes	sides	Sum of interior angles
triangle	3	180°
quadrilateral	4	2 x 180° = 360
pentagon	5	3 x 180° = 540°
hexagon	6	4 x 180° = 720°
heptagon	7	5 x 180° = 900°
octagon	8	(8 - 2) x 180° = 1080°
nonagon	9	(9 – 2) x 180° = 1260°
decagon	10	(10 – 2)  x 180° = 1440°
Any polygon	n	(n-2) x 180°
50 sided polygon	50	(50 -2) x 180° = 8640°

Common practice in pre-school teaching on whole numbers

The common Mathematics practices in preschool emphasised on rote counting to start off and moving to addition, subtraction, sharing and grouping which come under computation. There is also the use of language such as ‘more than’, ‘less than’, ‘one more’ and ‘one less’ to be consider in problem-solving.

Direct modelling using manipulative or drawing along with counting to represent the meaning of story problem is expected of the preschoolers. Most of the time, the teachers will demonstrate the solution and the children merely copy the steps usually without internalising the understanding and methods to the solution of the problems.

As cited in Van De Walle (2008), Research indicates that students using methods they understand make many fewer errors then when strategies are learned without understanding (Gravemeijer & van Galen, 2003; Kamii & Dominick, 1997).

Well, with this in mind, I may want to empower the children to invent their own strategies to addition, subtraction, problem solving and etc thereby providing them with opportunity to internalizing their understanding of number sense.

I would want to introduce using the ten-frame cards to help children extend the make 10 idea to larger numbers, as seen in p220. This can be done first with single digit numbers than moving on to double digit number.

The other would be the patterns on the hundreds chart (p200). This provides children with opportunities to see patterns in the chart. Taking into consideration that patterning is one of the big ideas in Mathematics teaching and learning.

Using Technology to teach Mathematics

The one thing that strikes me in chapter 7 is that calculators can improve attitudes and motivation.  According to the authors, research results reveal that students who frequently use calculators have better attitudes towards the subject of Mathematics (Ellington, 2003).

Recalling the experience I had doing Mathematics in school, I realised that we never had that opportunity  to use calculators and much time was spent  on getting the sum correct, so much so that we  are left with very little time to solve the problems.

Coming back again to kindergarteners, I believe they need to grasp the foundation of concepts first before allowing them to use the calculators for checking and exploration. This will empower them and encourage them to want to learn Mathematics.

Technology has made practice problem-solving so interesting as they have made it fun and most of the time one is placed in the position of having an opponent thus posing one with a challenge to win. In that process, one is learning new strategies and thinking of ways to outwit its opponent.  It is non-threatening and fun, which is what children love.  

I enjoyed playing the ‘Slam Ball’ in the Illuminations and plan to outwit my opponent for ‘Dig it’ in my next game @ Illuminations - Calculation Nation http://calculationnation.nctm.org/Games/Default.aspx

Lesson 4 – Sequencing the transition from concrete to abstract

According to Bruner’s CPA approach, children will need the concrete experiences where they are given the sticks to bundle into tens and ones. By doing so, they are able to say they have 3 bundles of tens and 4 ones.


The next step is to allow the children to transfer their knowledge of having 10 sticks in a bundle to the base ten block, doing so they have acquired a new idea and they will be able to do away with the real concrete and rely on the base ten blocks in the future.

The next step is to transfer the value of the base ten blocks to non-proportional materials like the coins. Each ten cent coin represents each base ten block. This is a higher level of understanding for the children to acquire.

Children who have the knowledge of using non-proportional materials are ready to move on to using abstract that is representing with number on the place value chart.

Knowing where to place the values on the place value chart, the children will be able to read and write them in tens and ones notations.

Next they will need to expand on the notation, that is to be able to tell that 3 tens is 30 and 4 ones is 4, thus able to move to numerals (34) which will represent 3 tens and 4 ones.

The next step, they can be introduced to number in words which spelled out the numeral; thirty-four.

The five tasks are in the following sequence:

1. Place value chart

2. Tens & ones notations

3. Expanded notations

4. Numerals

5. Numbers in words

Lesson 3 – Problem Solving & Environment-based tasks

According to Van de Walle, J. (2009), it’s important to understand that Mathematics is to be taught through problem-solving. The learning outcome is the problem-solving process. Chapter 3 & 4 introduces that we should take into consideration children’s prior knowledge and build on it through engaging them in tasks that are problem-based.
As we plan the pictograph lesson using the environment, we keep Bruner’s CPA approach in mind. The following are some of our reflections:

• Concrete experience
The sculpture provides the concrete experiences where the children have the prior knowledge of shapes and count to identify and count the number of shapes. As we counted the spears we were lost as to where we started counting from. This will pose a challenge for the children; they will need to come up with a strategy as to how they would not count the spears twice, which is to know where their starting point is. Another challenge was to count or not to count the inner spears.

• Pictorial representation
They are to transfer their concrete experience into pictorial representation as they use the circle to represent the spears on the pictograph and etc.

• Abstract
They will need to put numbers to the pictorial representation that is to record in numbers and do a comparison.

Expending from the above activity, I was able to connect with Bruner’s principles on:

• Readiness
Instruction must be related to the experiences and contexts that make the students willing and able to learn.

• Spiral organization
Instruction must be structured so that it can be easily grasped by the student.

• Going beyond the information given
Instruction should be designed to facilitate extrapolation and or fill in the gaps.

I found the follow link a reinforcement to our lesson.

http://www.youtube.com/watch?v=uCS7GO0fkc4