CPA Approach On Tuesday, Dr Yeap introduced the Concrete- Pictorial- Abstract approach (CPA) that made a lot of sense in teaching young children Mathematics. He also explained how the lesson should developed over time when teaching number concepts to children. The materials used for the first lesson should be the same (e.g. 5 identical marbles). Then you can progressed to introducing 5 chalks of different colors. Since the focus is on numbers, Teachers should avoid providing materials of different sizes or length. For example, the chalks that come in different length will divert the children's attention. This is what we call distracting variables.
Subitize? Its meaning: toperceivethenumberof(agroupofitems)ata glanceandwithoutcounting
What the hell is that was my first reaction when Dr Yeap mentioned the word in class. As I read the definition of it in the textbook (Chapter 8, page 129) and a brief explanation from Dr Yeap, I understood what the word meant. I tried subtilizing with the dotted cards below. It was easy for me to subtilize the 2nd number card since the arrangement of dots are familiar (found on the die).
Development of Verbal Counting Skills As I was reading Chapter 8, suddenly I thought of what a mother to a child I used to teach tell me, "My child can count until 100". The scenario that I just mentioned is common among preschool teachers. Often parents failed to realize that their children are just rote counting but they have yet acquired the skill to connect this sequence in a one to one correspondence. Nevertheless, it's still important to expose children to the language of numbers from young. This can be done by exposing them to books and counting songs and rhymes.
Needless to say, I've learnt a lot from this module and it would be impossible for me to list all of them down.
Three things I've learnt
1. The Concrete-Pictorial-Abstract Approach (CPA)
Concrete components
include manipulative, measuring tools, or other objects the students can handle during the lesson.
Pictorial representations include drawings, diagrams, charts, or graphs that are drawn
by the students or are provided for the students to read and interpret.
Abstract refers to
symbolic representations such as numbers or letters that the student writes or interprets to demonstrate understanding of a task.
As a teacher to very young children, I am glad that I'm on the right track when comes to teaching Mathematics. When I planned for Math lessons, I made sure to include exploring with concrete materials all the time. For the topic on 'Exploring numbers', I introduced the counting rhyme 'One potato, two potato' and brought in real potatoes for them to explore. As an extension activity, they placed the potatoes one by one in the egg tray. Through the activity, I could see how children picked up the skill of one to one correspondence. After learning about this approach, it would provide as a guideline when I'm planning for different levels in future. Now I know what needs to be introduced first and last so children will be exposed to various ways in learning Mathematics.
2. Differentiated Instructions Throughout the pages in my notebook, the word 'differentiated' will surely appear everyday. From teaching geometry to whole numbers, Mr Yeap never failed to ask the students how we can differentiate the lesson for both advanced and struggling learners. That's where we teachers come in as we will surely have cases where children are either lagging behind or having the- child- who- knows- everything! It is sure helpful that Mr Yeap brought this up for every topic so that we'll be able to facilitate when the need arise. Most teachers I knew had this misconception that differentiated instructions will mean more work for them. I learnt that using the same material, you can simplify the lesson for the struggling learners and made the task doable for them. For advanced learners, you can get them to think further simply by restructuring your questions to make it a little complicated from the previous problem. All this is possible without doubling your workload!
3. Teaching children the right thing and using the right language to teach Mathematics As I've mentioned in my earlier blogs, I am looking forward to new discoveries in class every day. Because of these new discoveries too, I felt smarter and more knowledgable.
1. When teaching ordinal numbers, Teachers need to be mindful on how she used it to indicate position and space. For example, "Ali is 1st from the finishing line".
2. Instead of using the word 'weight', Teachers should use 'mass' instead. This is because the unit for weight is Newtons (N) and not in kilograms. Dr Yeap suggested putting the question or sentence differently like "How heavy are you?" or "How much do you weigh?"
3. The textbook suggest not to use the language 'take away' as children might use the word wrongly. This is significant for mastering subtraction facts. Instead, use the word 'subtract' or 'minus'.
4. When you say 'divide', you also mean 'equal'.
5. It's not 3 out of 4 but 3 fourths
5. The use of 'less' and 'lesser' less- refers to quantities (countable) lesser- refers to qualitative (uncountable) The right way: 2 is less than 3
6. When teaching measurement, teach children the language to connect quantity with numbers. When using non- standard unit, use 'about'. For example, "The length of the table is about 6 ice- cream sticks.
7. Pi is not equal to 22/7 or 3.14. Instead, put it as "Take pi to be...."
Two questions that I have
1. I'm just wondering if a child displays good Mathematical skills at an early age, will that mean he will excel in Mathematics as he grows up? Will a child who's poor at Mathematics during his preschool years will eventually catch up when he goes to primary school?
2. Are there any recommended resource book in teaching Mathematics for early childhood educators in the market? For example, examples of how you can integrate technology when teaching Mathematics or ideas for related Math games.
I searched for 'use of technology in teaching math to preschool' in Google and was drawn to the result titled, Early Childhood Mathematics: Promoting Good Beginnings. In the write-up, technology is one of NTCM (National Council of Teachers of Mathematics) principles for school mathematics. It cited that technology is essential to teach- ing and learning Mathematics. It influences the mathematics that is taught and enhances student's learning. Click on the link if you are interested to read further http://www.naeyc.org/files/naeyc/file/positions/psmath.pdf
There are so many free applications that you can download for smartphones and tablets. Often, I watched how children were engaged in playing with these applications that they were oblivious to their surroundings in public places. I am guilty of it myself but I made sure to download educational applications for my child to play with. For example, from this application called 'kids Shapes', my child is able to connect shapes with his environment like how he pointed out that the clock is circle and the towel is a rectangle. He is able to identify the number 7 first as that's the number he always pressed on the remote control to watch his favorite channel, OKTO! So as you can see, technology is not that harmful if used selectively and incidentally. Likewise for teachers, computer technology is a good example where Teachers can use to enhance learning when teaching Mathematics. Teachers can select reliable softwares and determined how
best to incorporate computer use in the day-to-
day curriculum for children’s learning
experiences to be rich and productive. However, this action requires thoughtful and informed decision-making on Teacher's part.
As I've been working mainly with younger children (18months- 4 years old), there are limitations on how I can integrate technology in the curriculum. However, I used the computer to show them counting songs and rhymes on youtube. For example, after introducing the song 'Five little speckled frogs', children were able to dramatized the song themselves without much Teacher's intervention. They listened to the music and pretended to 'jump' into the pool. On another lesson, i use OHP to teach them shapes. They flashed blocks of the four basic shapes and see the enlarged image on the wall. I can still remember how children enjoyed the activity and they kept visiting the corner again even after the lesson ended.
The child pointing out to the circle on the aw
Integrating technology in my Math lesson
Let me end my blog with these inspirational quotes:
"There can be infinite uses of the computer and of new age technology, but if the teachers themselves are not able to bring it into the classroom and make it work, then it fails."
- Nancy
"It is important to remember that educational software, like textbooks, is only one tool in the learning process. Neither can be a substitute for well trained teachers, leadership and parental involvement." - Keith Kruger, CEO of the Consortium for School Networking (CoSN)
After a long time since polytechnic days, I am actually in a class learning Mathematics again! On my way for class in the taxi, I was hoping that the day will pass really fast so that I will be able to go back home to attend to my newborn. I was praying that the class better do justice to my sacrifices since it's only been two weeks since I gave birth.
Verdict after first lesson: I would not trade anything else for today. It was a day of making discoveries and solving number problems.
Dr Yeap cited a quote from Albert Einstein, “Many of the things you can count, don't count. Many of the things you can't count really count.” I was thinking hard on the meaning behind it and finally I came to an understanding on what it's all about. Like love, it can't be measured and it's supply is infinite. There is no way we can measure how much we love something or someone. I believe a measurement of love is measured by one's personal consciousness.
Out of the activities that we did today, I particularly enjoyed the card game. He introduced the game like how we would do it with our preschoolers. I could imagine children getting intrigued by the 'magic show' and wanting to know the secret of being able to do the same. Needless to say, Dr Yeap made us think of how it is possible. Together with my group mates, we brainstormed and came up with different solutions. It was all about trial and error and finding the pattern. It was getting interesting when we are so close to finding the answer. Both Sara and I watched in awe as Atiqah spelled out the name of the numbers. We shrieked with delight as Atiqah flipped the right card. It sure felt good when you are able to 'crack' the problem. I can't wait to share this 'Magic card game' with my colleagues and children in school!
How should Mathematics be experienced by the learner?
I agree with the author when he said that we tend to feel proud of our accomplishments when solving tough problems. Therefore, persistence, effort and concentration are important in learning Mathematics. It sure felt good that you were able to find the answer after a long struggle trying to understand the question. The learner should generate strategies for solving problems, applying those approaches, seeing if they lead to solutions and checking to see whether your answers make sense. It's always inspiring to see children find solutions to the problem themselves and often I used this observation as part of my assessment on that child. Learners should be given opportunities to interact with each other and with the teacher. Classrooms should be conducive for learners to provide structures and support to make sense of Mathematics in addition to what they already know. Learners should be given opportunities to build connections between what they know and what they are learning. More time should be given for learners to explore Mathematics. Learners should not be judged when they make mistakes instead, treat it as opportunities for learning. Flexibility and assistance should be given to learners who are new to the concepts. Learner's ideas should be valued and included in classroom discussion of the Mathematics. Growing up, I hardly felt that I experienced Mathematics the way I should have. It's either a right or wrong answers although there are times where I was lucky to earn 1 or 2 marks in problem sums that weigh 6 marks.
Coincidentally, I chanced upon this blog about a Teacher who posted his experience of learning and teaching Mathematics.
Using bear counters for the lesson on sizes (small, medium and big)
Using straws to thread for lesson on length (short and long)
Using toy cars to learn counting
Students have different understandings of concepts. When children are given more ways to think about and test an emerging idea , they have a better chance forming and integrating it into a rich web of concepts and develop relational understanding. Figure 2.11 on page 24 showed five different representations of mathematical ideas. There are pictures, written symbols, oral language, real- world situations and manipulative models. For the age group that I'm working with, oral language plays an important part in the Math curriculum. When teaching numbers, they are exposed to different counting songs and rhymes where they picked up language of numbers. I could also relate to understanding Mathematics through real life situations. For example, as an extension activity to learning shapes in the classroom, children went outdoors to look for circular, squarish, triangular and rectangular objects. Manipulatives are evident in any classroom in my school. For example, there are bear counters for children to use when learning colors, sizes and numbers. Different age groups use it differently. There are also other learning materials for children to use placed at the Math corner. For example, ice- cream sticks or toothpicks where children explored with for counting and making shapes.
Before I start reflecting on the chapters that I have read, allow me to share this quote above. As an early childhood educator that worked closely with young children, I am proud that I'm teaching and exposing them to Mathematics at such an early stage of their life. The fact that I am their first Math teacher made me even proud of the profession that I'm in for the last 8 years.
Numerous standards movement have taken place throughout the years.
1989- Curriculum and Evaluation Standards for School Mathematics
1991- Professional Standards for Teaching Mathematics
1995- Assessment Standard for School Mathematics
2000- Principles and Standards for School Mathematics
2006- Curriculum Focal Points
and finally 2010- Common Core State Standards (presented by Council of Chief State School Officers, CCSSO)
My first impression when I read about the different standards movement that were published was how Mathematics education has evolved throughout the years. This reminds me of an experience that I had 6 years ago when I taught tuition to a Primary 6 child who is sitting for her PSLE. Often, I had to do the sums myself first and checked the answer sheet before teaching the student. I found myself flipping through her Maths textbook looking for ways to teach her problem sums as I didn't want her to be confused with the different teaching styles. I don't remember much emphasis was given to diagrams during my time but times have changed. It took me a while to get used to the new Math standard and sad but true, I was learning alongside with her too. But one thing for sure, all this movements have brought positive transformation not only to the U.S. but the world too.
Principles and Standards for School Mathematics (2006)
The Six Principles:
- Equity
All students deserve an opportunity to learn Mathematics
- Curriculum
Children should see it as an integrated whole and not a collection of bits and pieces
- Teaching
Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well
- Learning
Based on two fundamental ideas ( Learning Mathematics with understanding and Students learn Mathematics with understanding)
- Assessment
Teachers can better make the daily decisions that support student learning by gathering data of their understanding of concepts and growth in reasoning
- Technology
It influences the Mathematics that is taught and enhance students' learning
I stumbled upon this video of Conrad Wolfram presenting his idea of teaching kids math through computer programming. It will definitely sparked more interest to learn Maths for the children but on a long run, I'm worried of the side effects it may bring.
Becoming a Teacher of Mathematics
The statement above clearly showed how important and vital the role of a Math teacher. You will shape Mathematics for the students you teach. Not only curriculum standard have gone through various transformation but a lot has been said about what a Math teacher should possess.
- Knowledge of Mathematics
- Persistence
- Positive Attitude
- Readiness for change
- Reflective Disposition
Being a teacher to two year olds this year, I believed some of the characteristics above are either already in me or something that I picked up along the way. We are required to write reflections every week where I will be reflecting on lessons where objectives were met or that required a lot of careful and individualized planning. I felt that the reflection process has made me grow and develop into an effective teacher. When I was told by the Centre Director that I will be taking Toddlers this year, I was skeptical about it. The first thing that came to mind, "What am I going to teach them?" Therefore, having the positive attitude that you can do it and open to changes made me accept the challenges that comes with it. Looking back, I am one proud teacher when my 2 year olds sang counting songs and rhymes to their parents at home.
