Final Blog - Final Thought after 8 cookies

It has been an enjoyable 2 weeks with Dr Yeah. Not because the workload is any easier but because of the real learning that have taken place and the confident in Mathematics that I am able to build up again after so many years...

In every lesson , I felt that I was taking on the identity of both a student and a teacher. As a student, I followed closely the instructions given  and allowed myself to explore and experience mathematics in a fun and brand new way. As a teacher, I am aware of the numerous modelling effort by Dr Yeah on how to conduct Math lessons to young children , the variety of the teaching materials available,  the proper math language to use, the do and don't , the scaffolding technique to get the children construct their own knowledge, the various friends that I get to know - Bruner, Diene, Van Hiele, Pick, etc.  Yes, I got it.

But one very important thing that I learned from this course is that

     the reason for teaching Math is not teaching Math itself.

And that Math is a tool for the
1. development of intellectual, logical reasoning, spatial visualization , analysis & abstract thought
2. development of numeracy, reasoning, thinking , and problem solving skills
3. opportunities of creative work and moment of enlightenment & joy.

This statement is profound! If I do  not get the goal correct, the effort that I am putting in is futile. If it means that I have to rethink my teaching philosophy and unlearn everything about teaching pre-school math, i will. In this hi-tech world whereby mathematical problems and computational skill can be easily solved and accomplished by pressing a few buttons on the keyboard. What is there left for us to do?

Our belief will determine our actions. So
                   let's crack some heads,
                         let's kill some brain cells,
                               let's persevere,
                                     let's out wit each other, and
                                             let's have some fun!
Cheers!
From Tr Anita

Blog # 7 Geometry

What is the (interior) angles of a pentagon?  Easy peasy, lemon squeezey. It is


                                                                    ( 5-2) 180 = 540  (degree)

But why is it so?  Oh, that was the formula given by my Math teacher when I was in Secondary 2.
But why is it so?  Oh, cause the number of non-overalp triangle you can draw from the same vertex of any polygon is always 2 less than the number of  its sides. and each triangle has a total angles of 180 degree. or you can put a dot in the centre of the pentagon and draw 5 lines towards the 5 angles and form 5 triangles,
180 X 5 - 360 = 540 (degree)

All these methods suddenly came to my mind.  I am trained to solve math questions  through formula, traditional  algoritthms. Not knowing that in order to deal with this problem, I have to reach level 2 - level 3 of  Geometric thinkinf according to the van Hiele's 5 levels ( level 0- level 4) of the hierarchy  model.  At level 2 (informal deduction),  the objects of thought are the properties of shapes and sometimes thinking about the relationship among these properties.  and moving towards the beginning of level 3 ( formal deduction) which starts to look at the relationships among properties of geometric objects.

Van Hiele's model  provides teachers with an insight of how children develop their geometric thinking and thus be able to use different strategies to help them move from one level to another.  Chapter 20 demonstrates how a teacher should nurture his/her children in geometric thinking by asking appropriate questions and using guided activities to help children construct and internalize their own learning. It is a systematic and  deliberate effort, it is the finest job of a Math teacher. To teach by not teaching .

Interesting website on geometry to share : http://coolmath.com/

Blog # 6 Whole Numbers

Chapter 10 to 12 are very useful for me as a Kindergarten teacher. We have practically two years to lay the foundation of mathematics for our young children and  we should do it right. As our friend Jerome Bruner suggested, teaching should start with concrete material ( enactive represetation ), then in pictorial forms  ( ikonic representation ) and finally in abstract terms ( symbolic representation ) . As I read through the chapter, I am mindful if I have been teaching numbers, tens and ones, addition and subtraction, etc., in a systematical way; and whether I have missed out any step that may be detrimental to children's understanding and learning process. I also ook out  for any class activities that I can use to reinforce the concepts.  It was a time for self reflection and do some serious stock taking.


Concepts  that are in place in preschool :

Ten and ones :
       Introducing the concept of 10s and 1s. First by counting in ones (unitary), then by grouping concrete materials into bundle of 10 and counting in 10s ( Base-ten).  Teacher will also match grouping by numerals, placed in labeled places and eventually written in standard form.

     BUT  we do not  expose children to equivalent or nonstandard base-ten approach. such as  that memtioned on p.189  ( Figure 11.1)   53 can also be represented by 3 tens and 23 ones.  In  fact, some teachers frown upon it when they see children represent tens and ones in this form, deeming it as unconventional , confusing, or not yet reach the undertanding of the tens and ones concept.  However, we should adopt Diene's idea of allowing variety in teaching mathematics, in the use of materials and in different ways of representing a math concept. THis kind of exposure provide opportunity for children to approach the concept in different angles and will eventually increase their uderstanding and facilitate the application   of the concept in the long run.

Introduction of traditional algorithm

     Teachers will definitely include  addition, subtration, multiplication and division  in the curriculum using the traditional algorithm. 

     But seldom will we encourage student-invented strategies in the math calculation. By observing children's explanantion and demonstration of their invented algorithm, teachers can have a glimpse of their processing skills  and understanding of concepts.  In fact, experts find that children make fewer error in self invented strategies, develop better number sense, forming basic of mental computation and estimation. Moreover, they are flexible and oftern faster methods of mathematic computation.  Thus, in Chapter 12 , the authors call for a delay in  the introduction of traditional agorithms but put in more time to understand how chldren derive they own solutions to the problems.

There are many more refreshing strategies fo teachng mathematic in the text. Happy exploring.

Tr Anita


  

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Blog #5 Technology

Technology is marching into our classroom. Like it or not, it is there. Our new generation is brought up with milk bottle in one hand and mouse in the other. Their mind is stimulated by colourful graphic and animated games even before they utter their first word.

I was amazed that there are so many softwares being developed and online site being created for developing and supporting various mathematical concepts and teaching. Tools for developing numeration? Check. Tools for developing geometry? Check. Tool for developing Probability? Check.Tool for developing algebraic thinking? Check.  Concept instruction? Check! Problem Solving? Check. Drill and Reinforcement? Check, check.

I suppose "resistance is futile". We as teacher should instead, embrace technology and use it as an effective tool to aid our teaching, making lessons more interesting and engaging. We need to be informed and be able to select suitable software that can enhance our lesson, critically review the effectiveness of each chosen program and make sure they meet the objectives we set to achieve.

                           Use technology to our advantages

The Teacher Resource listed at the end of Chapter 7 are treasure to every Mathematic teacher. I just spent more than 2 hours surfing and getting into the different websites. Warning! It is addictive!  I particularly like The Math Forum. (http://mathforum.org/ ).It provides a platform for interactions among teachers and it links to wonderful websites for different levels and different topics. As a K1 teacher, I am always looking for classroom activities to reinforce the math concept and this link that I found "Count Us In Game" has many fun games and  suggested activities for younger children. I will definitely bookmark it and try out in the coming term. Cheers

Blog #4 Which one come first?

In our centre, placement value or tens and ones is introduced to children in Kindergarten 1. Although children learn rote counting and recognition of numbers at an earlier age, most do not attach any value to the numbers, to attach a value to the number, teacher need to introduce the concept step by step.


Burner recommends to teach mathematics using the CPA approach. i.e. from concrete materials, to pictorial representation to the later stage of abstract representation using symbols. Diene also advocate variability in teaching mathematics i.e. teaching the same concept  from different angles and using different materials.
In this question, the teacher would show the children pictures of concrete object (sticks), the base ten blocks (  representation - proportionate ), and coins ( concrete representation - non-proportionate ) to help children visualize the value of 34 before moving into the more abstract notations.


In my opinion, the introduction of the 5 notations should be in the following order :


                                                              1) expanded notation

                                                              2) place value chart

                                                            3) tens and ones notation

                                                                4) number in numerals

                                                              5) number in words

It is arranged in this way because as we go down from each level,  the  notation becomes further away from the concrete and thus more and more abstract to the children.

1) expanded notation - children can  relate the counting of 30 and 4 in sticks and base ten blocks  

2) place value chart -  children will have to recognized that a set of ten cubes is represented by 1 ten instead

    of 10.

3) tens and one notation - children needs to convert the table format into a number words form

4) number in numerals - children need to internalized the concept to visualize 34 as a number with value, not

    just digits 3 and 4.

5) number in words - children will further convert the mathematics symbols into letter symbols. 

Some children can grab the concept ready while others may need more guidance and practices to fully understand. However, a systematic way to introduce mathematic concept will definitely help the children in their learning.

Blog # 3 Problem Solving and Environmental Learning

     "How fast can you count all the light balls on the decoration posts ouside the Cathay Building?"; "Can you map out the position of the llight balls?" ; "Is there any other way to count faster?" .

     These were the questions we posted to the children in our environmental learning assignment. A problem based approach to teach skip counting concept. It was quite an experience for me to bring Mathematics out of the classroom, or bringing the children out in that matter. Well, I propose fieldtrips for projects, I bring children out for science explorations and I do not hesitate to bring them to the playground  for outdoor play. But making a trip to learn Mathematics?  I need a paradigm shift, and  a shift for my principal  and co-workers too! 

     That Monday evening, we were there, literally counting the light balls ourselves, checking out the surrounding, trying to find other ways to reinforce the skip counting concept that we want to bring across to the children; imaging our children doing the exercise and taking on the challenge, scratching their heads to find a way to beat the stopwatch.  Then we found, railing in groups of 10, stairs, pebbles, fallen leaves, patterned floor slabs, feature wall... the list went on and on...  Yes, Mathematics is indeed everywhere in our environment. According to Jorome Bruner , teaching should start with concrete material, to pictorial to abstract symbols (CPA Approach) As such, environment is the best place to begin our Mathematic lessons.

    Problem based lesson needs even more planning. Teachers need to
1. know the children's prior knowlege,
2. establishe clear learning objective and decide on the environment feature that works best for the concept 
3. design problem task and questions and extended activities
4. assess  children's way to approach problem during the exploration
5. after the activities, discuss, justify and challenge various soluiton and summarize main idear and identify further problem.

One thing I like about problem based learning is its flexibility. There is no one fixed solution to the  problem and there is no restriction on how children should approach the problem. As such, children apply whatever resources and mathematics skills that they have internalized on the situation. It allows multiple entry points for the class regardless of their level of competency.Children who are more advanced in their mathematics ability will find it challenging as they can flex their mathematics muscles and dig deeper into the problem, trying out more advance concepts.  For the weaker children, they may not be able to work out the entire solution by themselves, but by observing how others approach the problem and derive the answer, they will be benefited from the peer scaffolding too.  Through their answers and explanations, I can better assess individual's understanding how the children approach the problem, and if they are apply the concept correctly. on the particular mathematic concept and make plan for the future activities. Besides, it was more fun and hands-on to the children, arousing their interest in mathematics and making it alive both within and without the classroom. 

Still not convinced, , anyone feel bored in Dr Yeap's class?

Rails that can be grouped in 5

pattern that can be grouped in 5 with chalk

small stones as counters for grouping in 5

    

Blog # 1 Reflection on Chapter 1 & 2

           Preschool teachers may feel inadequate in teaching phonics or shy away from music & movement,  but very rare you find teachers have problem in teaching mathematics. Afterall, how hard can it be to teach pre-school counting on and counting back, reading the clock, and forming ABAB pattern?  

          That was the thinking before I read the text for the Elementary Mathematics module. Chapter 1 and 2 set the standard and laid down the foundation of mathematic teaching for all professional educators. To teach mathematics, we need to know the mathematic content well, demonstrate persistence in the process of investigation, develop positive attitude towards the subject, ready to unlearn and relearn mathematics concepts and be reflective in one's teaching strategies and practices. It doesn't sound easy now.

         Just as educational  philosophy affects the deposition of a teacher, our believe in how mathematic is learned greatly affects the way we conduct our lessons. I personally think that both the constructive theory and the sociocultural theory explain well how children learn mathematics and the role of teacher in their learning. In the constructivist theroy, a teacher's responsibility is to provide a conducive learning environment which stimulates sensory experiences and increase exposure to physical surrounding. In the sociocultural theory, teachers need to know a child's current stage of cognitive development and scaffold him/her in the zone of proxmal develpment (ZPD). Mathematics should be introduced and taught in a systematic manner to help children build up their understanding of mathematic theories and allow self exploration of the ideas to achieve relational understanding  rather than emphasizing on the procedural proficiency that only resulted in mechanical steps of getthing the answers without understanding and connection with other mathematics concepts.

          As I reflected on my own teaching, I found myself
1. overly directive - children do not have much opportunity to exercise their thinking and problem solving   skills.
2. lack of a systematical and well thought through teaching strategies to guide children's conceptual  development
3. always hard pressed for time to scaffold individual especially those who are above class average.


          As preschool educators, we should not take teaching mathematics too lightly. Many of us are "victim" of mathematics drills in our early days of learning and thus missed out the opportunity to embrace mathematics and develop a love for the subject. We are here so we would not make the same mistake on our next generation.


  

Blog # 2 What did I learn about Mathematics?

     "The reasons of teaching Mathematics to young children is not the Mathematic itself ." and so Dr. Yeap told us during our first lecture on Elementary Mathematics today.

     Indeed we had a full four hours of lecture on Math but we did not write any formula, turn on our calculator, nor try to solve an algebra equation on paper. Instead we flexed our mind to break the code of the magic poker trick, play dice together with the primary one children and try to outwit our classmate to win the paperclip game. Our concepts on Mathematics  are recalled, we exercised hueristic skills, applied various thinking skills, activiated our metacognition mechanism and felt challenged and satisfied with our ability to find the solution to the problems. And all these are in fact the very essence of  Mathematics.

    The Ministry of Education of Singapore defines the Math Framework for Primary School Curricululm as following :

1. Mathematics is an excellent vehicle for the development and improvement of a person's intellectual, competence in logical reasoning, spatial visualization, analysis & abstract thought.

2. Students develop numercy, reasoning, thinking skill, and problem solving skills through the learning and application of Mathematics.

3. Math is also a subject of enjoyment & excitement which offers students opportunities for creative work and moment of enlightment & joy.

   Today, I am invited to look at Mathematics from a brand new angle. Mathematics is a vehicle, not the objectives of the teaching. Our goals are far beyond the conceptual and computational skills. Am I developing my children's intellectual ? Am I equipping them with problem-solve skills? Did I make them feel and look stupid by pushing Mathematics curriculum and methodology down their throats?

   With knowledge comes responsibility, my mathematics class will not be the same hereafter...


Cheers,
Anita
on Teachers' Day 2010