6th Session – The Last Math Session & More Encounters with Math

At the start of the session, Dr Yeap stated the fact that we should not name a Math problem a problem sum (oxymoron). Instead Math problem should be called story problem or simply, word problem.

We revised doing a word problem with 4 ÷ ²/₃. An example of a word problem of this equation is:

If Jane shared 4 cakes with her friends and each of her friends received ²/₃ of the cakes, how many friends did Jane share the cakes with?

4 ÷ ²/_{3 =} 6

Jane shared the cakes with 6 friends.

Next, we went into assessments (which will form the essential part of our final assignment). Assessments are done to evaluate children’s understanding of Math concepts. Paper and pencil test is often the common method of assessment. Another method of assessment is oral test (interview); in which can be used as simply to assess if children understand the concept of time – instead of drawing the hands of the clock in worksheets. One should always remember that assessment should be valid or in line with the objective of the activity.

The story How Big is a Foot? By Rolf Myller is a simple story that can be introduced to children for the concept of measurement.

Tip of the day: If you want to lose weight, go to the moon.

With this, when teaching preschoolers on weight (focusing on non-standard measurement), teachers are encouraged to question children in this way:

How heavy is the teddy bear? (avoid any mention of weight – kg or mass)

We went to Bras Basah MRT to measure the height of from Ground floor to the basement, just by using a ruler.

** Solution to the MRT steps

16 steps X 4 sections X 13.5cm = 864cm (possibly incorrect!)

Dr Yeap emphasized on Bruner’s theory in teaching Math:

- concrete / enactive representative

- abstract

We did an activity to make a paper container to hold 15 beans to illustrate capacity. We thought our container could fit the 15 beans; however a group did the container so accurately that the container could exactly fit 15 beans. Hence, we should never underestimate the size of a container, especially when it contains water as there could be a large amount of water in one small container.

MATHEMATICS IS:

            Visualization               Number Sense             Looking for patterns

                         Communication                      Metacognition

To conclude this Math module, Dr Yeap read to us the story of a cocoon turning into a butterfly, as a representation on how we should allow children to fix their own Math problem and not spoon-feeding them as this will deter any future development in their growth.

Lastly, thank you Dr Yeap for making complicated MATH simpler!

5th Session…Bloom & Pick, Pick & Bloom

Fractions…again?!?! Now in problem sum…Didn’t we had enough yesterday???

Anyway, fractions continued and in conclusion I found out that fraction is:

- measurement number (quantity)

- represent proportion

We also learnt about Bloom’s Taxonomy.  Out of the six levels in Bloom’s Taxonomy, three levels were used in mathematics and they are knowledge, comprehension and application levels.

So enough on fractions and Bloom…we had better things to do visually.

Making shapes on dots! Really cracked the brain and came up with various shapes on the dots (including rhombus – which is NOT a triangle, heart & circle – which are INVALID as the shapes should all have straight lines). As we tried to figure out the area of the shapes on the dots, we come to understand Pick’s Theorem.

Based on Pick’s Theorem (area of figure drawn related to dots), the formula is as such:

Area (Area of figure) = i (dots inside) + p (dots in perimeter)

With that, the concept of tessellation was recalled. What is a tessellation? Tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. Another word for a tessellation is a tiling. Referring to the dictionary, it tells that "tessellate" means to form or arrange small squares in a checkered or mosaic pattern. Anyhow, there are five ideas of tessellation:

-         rotate

-         reflect

-         translate

-         stretch

-         shear

I noted the importance of NOT sending out artificial signals to children as these may lead to future confusion and insecurities. As children do things tight or wrong in life, always question them and get them to justify their answer. That is, I suppose, the best way for children not only to learn, but think out of the box too!

4th Session with Dienes (and his idea on Variations)

The first activity of the day was an interesting one where we had to think of 2 digits, put them together and subtract from the addition of the two digits. It is amazing how Dr Yeap was able to find out our answers just by knowing one number that we thought of. Well, at the end of it, I come to realize that there are indeed many connections between numbers and its pattern.

It is useful to note that things like money and kilograms are called continuous quantity and things like marbles and monsters are called discreet quantity as they are counted as a whole.

Zoltan Dienes stressed the importance of variations, where children learnt better with a variant of ideas on a topic. In this way, children get to explore the multiple ways of solving a problem; there will be no standardization (just like how we taught in school – don’t ask me why, it’s always like this!).

There are two types of division: sharing and grouping.

An example of division in sharing meaning is:

Three boys shared 12 cookies equally. How many cookies did each boy get?

An example of division in grouping meaning is:

12 cookies are put in groups of threes. How many groups of cookies are there?

From this session, there is one vivid idea that was stuck in this brain in relation to fraction: Being equal does not mean identical…(surely something that one should always ponder upon when encountering with shapes; or even other objects in this world, perhaps!)

3rd Session with Peggy…

Lesson Study (or kenkyu jugyo) is a teaching improvement process that has origins in Japanese elementary education, where it is a widespread professional development practice. Working in a small group, teachers collaborate with one another, meeting to discuss learning goals, to plan an actual classroom lesson (called a "research lesson"), to observe how it works in practice, and then to revise and report on the results so that other teachers can benefit from it (Wikipedia, 2011).

The above are the exact steps that Ms Peggy mentioned in the session pertaining to Lesson Study where it acts as a professional development tool or process that teachers engage in to systematically examine their own practices.

Differenciation is the focused word for this session. Differentiating instruction requires a degree of preparation. As a teacher, one should differentiate your instruction; in other words, teach using a variety of techniques and strategies that address the varying needs of all students. The teacher may have to scaffold the higher level learners yet accommodate to the struggling learners, all at the same time while focusing on the objective of the activity. Learning to differentiate instruction is a process. If the teacher truly wants to succeed, one should take it in small steps and constantly revise what does not work.

There were five learning points that we took away from the two case studies (videos) in relation to Lesson Study:

-         Design of task (Mathematical investigation)

-         Clear instructions and demonstrations

-         Effective questioning

-         Effective use of materials

-         Differentiation for different ability learners

The other focus for this session is the conservation of numbers through visualization. Using the unifix cubes and tangram are some activities that a teacher can implement in class to enhance children’s visualization skills. The unifix cubes lesson is to help children learn the idea of conservation of number. That is, no matter how the cubes are arranged, the number remains the same. Our group won with 25 structures using 5 unifix cubes each. YEAH!!!

With the tangram activity as homework, I could practice visualization with the rotations, reflections, or translations, and putting together or taking the tangram apart in different ways. This activity will definitely help children to develop their spatial memory and spatial visualization skills, to recognize and apply transformations.

Ms Peggy concluded the session requesting for us to write our feeling on the session. Mine was ‘Eye-Opener’.

2nd Session…The 1st Peg, Finally!

Lesson 7 – Making Largest Even Number was such a favorite. Definitely mine ‘cause that’s how I got myself a peg. This lesson itself is so similar to ‘tikam-tikam’ that I almost thought it’s all about luck. Oh well, luck is definitely part of the game (like how I got ‘6532’ in the correct squares) although probability or chance; giving numerical value, is the correct term of describing this game.

Thinking and problem solving are life skills in the 21^st century competencies and its 5 big ideas of what constitute mathematics are as follow:

1. Generalization
2. Visualization
3. Communication
4. Number Sense
5. Metacognition

Word for the evening…SUBITIZE…. that’s the ability of saying the number of items just by looking at it. Interesting, normal people will only be able to subitize small number of items.

We got to know two persons in relation to Math:

- Jerome Bruner (CPA approach; Concrete Pictorial Abstract Approach)

- Howard Gardner (multiple intelligence; and memory is NOT one of the intelligence. *So it’s ok to forget! Hee!*)

One important thing that I learnt in this session that all teachers should also note is that teaching Math is not about getting the answer, but going through the process of getting the answer, which may have varied ways to do so.

1st Session Surprises!

One thing that strikes me in the 1^st session is how Dr Yeap made us learn Math the way old-school teachers will never dare to teach the students. Full of surprises and so much brain juice squeezed!

A simple introduction of Dr Yeap’s name turns to be a Math problem where the objective is to find the 99^th letter in the name. Similarities in the position of the letters in the names were spotted with teachers with similar number of letters in their names e.g. 5-letters name like AISHA; the 99^th letter is S; which is the 3^rd letter in the name.

Familiar yet not-being-used-often terms like rote and rationale counting, cardinal, ordinal, nominal and measurement numbers are some uses of numbers that teachers will often use when teaching Math. In relation to ordinal numbers (which has two positioning; in space and in respect to time), it is most interesting to note the mistake that teachers and sometimes worksheets will asked students, “ Who is 3^rd in the race?” when the picture is showing runners running towards a finishing line. The correct way to enhance children’s understanding of ordinal numbers is to ask, “ Who is 3^rd from the finishing line?”

The pre-requisites to counting that all teachers should know pertaining to children’s early understanding of Math concepts are:

1. Children should be able to classify
2. Children should be able to rote count
3. Children should be able to do one-to-one correspondence
4. Children should be able to represent the number of item of the last number that they utter.

Besides the intriguing Spelling Card trick (and gave me the idea to ‘trick’ the K2 children earlier this morning; they were amazed! Even the class teacher was in awe!), it was such a boss moment last evening when we did the Arrange Five Numbers and at first try, I put ‘5’ in the middle and got all the numbers to equal to 10.

From this lesson and a refresher for me, the rule of odd and even numbers are as such:

Even + Even = Even

Odd + Even = Odd

Odd + Odd = Even

Frankly, the brain was overloaded with too much not-so-new yet very interesting information/ tricks/ ideas/ concepts (whatever you may call it) last evening. Yet, I can’t deny that I’m looking forward to more ‘Ah-Ah’ moment in the days to come!

Pre-Course Reading: Chapters 1 & 2

Math is NOT just counting 123...

As a preschool educator, one will surely delight in the young children’s zest for learning. From the outside, it might seem like our job is all about fun and games, in the hope that parents of young children will know (and appreciate) how we influence and model positive behaviors, shape instruction, cultivate optimism and positive attitudes about school and learning, boost self-esteem, and provide the foundation for their future in school and in the community.

On a personal note pertaining to Mathematics (and upon reading the two chapters), there is a need for me to confront some of my personal beliefs - about what it means to do mathematics, how one goes about learning mathematics, how to teach mathematics through problem solving, and what it means to assess mathematics integrated with instruction. This is a challenge as there is an initial dislike of the subject, resulting in a lack of confidence in my own math ability. However, as I read on, I began to understand that Math is not just about getting “one right answer” (as how I had learnt in school). Mathematics is about thinking and talking. This strategy will serve me well as an educator and the young children in becoming a society where all citizens are confident that they can do math.

Classroom discussion based on children’s own ideas and solutions to problems is absolutely “foundational to children’s learning” (Wood & Turner-Vorbek, 2001, p. 186). Because math is a multi-faceted subject, a child may be strong in some areas but have difficulty with others. In that respect, learning math is much like learning to read. Once we know where a child stands, play to the child's strengths while addressing the areas in which the child struggles.

In order to help children gain math literacy, we need to create opportunities for these preschoolers to learn by doing — engaging their minds, connecting with their senses, and tapping into their enthusiasm. Research reinforces the value of letting them learn about math through hands-on games and activities they enjoy. It is interesting to note that some children seem to be able to understand and engage in certain math activities without first having mastered other, simpler counting and math-related tasks.

With this knowledge, as my approach to math instruction evolves and expands, I will soon discover more opportunities to prepare the young preschoolers to succeed in math – and to learn to appreciate and enjoy it as well.

Reference

Van de Walle, J. (2006). Elementary and middle school mathematics: Teaching developmentally (7th Edition). New York: Longman.