Reflection on Session 6

Assessment Tasks

Tonight, the class learned about assessment.  By assessing children, we can find out whether they have procedural (computation), conceptual (meaning), or conventional (which needs explicit teaching) understanding.  It is important that children know the real life meaning.  We ask ourselves: Do they have conceptual meaning?  Are they using procedural methods to do equations?  Or are they able to use strategies to overcome a weakness?

In an education system where competition is present, people will need formal assessment to differentiate what they are measuring.  Assessment that we use have to be :

- valid (measure what they are supposed to measure)

- reliable (the same performance always attract the same score)

The 3 types of assessment tasks are:

* paper and pencil test

* oral test

* group or individual test

Paper and pencil task cannot measure everything; it may not show whether a child is able to use visual method.  To find out more, teachers can also use oral test via simple talking to find out more and understand the chilld better.

Story Time

It was so refreshing to hear a story during a math lesson :)

Dr Yeap read us a story 'How Big Is A Foot' by Rolf Myller to introduce the concept of standard units for measurement.  We explored units of measurement for time, weight, length, distance, volume and capacity.  The class went to the nearest MRT station with our rulers and we were to find out the height from the ground to the top of the finished floor.  We stretched our muscles by climbing up the stairs, counted the steps, and measured the height of each step. 

Doing  a math task with the real stairs was really more meaningful than working out a word problem on paper.  I visualized the stairs as a right-angled triangle, discussed with my group, and we came out with the following calculations:

Flights of stairs in total = 4

Number of steps in the first flight = 14

Number of steps in each of the next 3 flights = 16

Total number of steps = 14+48 = 62

Height of each step = 14.5 cm

Height from the ground to the top of the finished floor

= 62x14.5 cm

= 899 cm

I would like to share a video about a teacher letting her class experience how big a T-Rex was by using their bodies as non-standard measurement.  The children had fun as they got involved in their physical experience.

(Source: "Learning Math Early - It's Big!" by EriksonNews from Youtube)

It is important that children know how to use representation.  As Jerome Bruner put, it is 'enactive representation'.  It is something that you can act on to embody an idea.  "Bruner states that what determines the level of intellectual development is the extent to which the child has been given appropriate instruction together with practice or experience.  So- the right way of presentation and the right explanation will enable a child to grasp a concept usually only understood by an adult" (Mcleod, 2008).

Another acivity we did tonight was creating a container that could hold 15 beans.  There was much thinking and problem solving to make one that could fit exactly.  An activity like this can be used as a tool to assess students as a group.

There are 3 levels of learning ability:

- totally cannot

- partial

- complete

After the assessment, teachers can come up with some remedial actions to help children to at least have partially developed idea.

Throughout the whole math module, I have learned that math is about thinking and reasoning.  It  is not merely calculations, computations, but rather, intellectual competence.  As teachers, we need to give children opportunities to visualize, communicate, see patterns and relationships, and make connections to generalize.  Children should be encouraged to talk their thoughts, expressing orally or in written form, be it diagrams, words, symbols, tables, or graphs.

At the end of this module, I cannot say that I like math, but the interest is budding.  It is so different now as compared to the beginnig of the module; I disliked math because of my unpleasant past experiences with my school days teachers.  I am still not good at math, I'm sure Dr Yeap can tell by the quizzes I have done.  However, my journey with math does not end with the end of the module.  Knowing that math is not a destination but a vehicle for me to develop my intellectual competence, I will continue to explore and learn.

Dr Yeap ended this session with a very good caterpillar story "How To Make Sure That A Butterfly Cannot Fly".  It was powerful and made us ponder about the kind of education system we want our children to have. Do we want to teach them the easiest way out by showing them the short cut, or teach them life skills?  Would we rather provide them a fish, or teach them to be a real fishermen? There is really no short cuts in learning; the decision is up to us.

Thank you so much Dr Yeap, for all the wonderful math sessions! 

Reference: Mcleod, S.(2008). Simply psychology.  Bruner.  Retrieved 1 Sept, 2011 from  http://www.simplypsychology.org/bruner.html



Reflection on Session 5

Division of Fractions

Tonight, the class looked into division of fractions in a visual way, and we also learned that fractions are used as:
- measurement numbers (in quantity and must have a unit, like cm, or $)
- proportion numbers (represent proportion and no unit is required)

Some of the tasks we did in class were:
Example 1, Out of 3/4 (3 fourths), how many 1/2's are there?

Method 1: by calculation
                 3/4 divided by 2/4
                 =(3divided by 2) divided by (4divided by4)
                 = 1 1/2 divided by 1
                 = 1 1/2

Method 2: by visual
                
3/4 divided by 1/2 = 3/4 divided by 2/4.
                
That is to say, how many 2/4 can you see inside 3/4?
                
The answer is 1 1/2




Example 2, In 2 fifths, how many tenths are there?

Method 1: by calculation
                
2 fifths divided by 1 tenth
= 4 tenths divided by 1 tenth
= 4 divided by 1
= 4

Method 2: by visual
2/5 is the same as 4/10 so we drew 10 units and shaded 4 of them.
From the picture, it is very clear that the answer is 4.

 I think the usage of pictures and models are excellent ways of helping children to understand the concept of division in fractions.  Problems that seem to be complicated can become easier to comprehend.

In Bloom's Taxonomy, there are different cognitive levels (cognitive porcesses):

Level 1 - Knowledge Level (eg. know what is the value of 5/7 -1/2)
Level 2 - Comprehension Level (able to use situation after you comprehend it)
Level 3 - Application Level

"The major idea of the taxonomy is that what educators want students to know (encompassed in statments of educational objectives) can be arranged in a hierachy from less to more complex.  The levels are understood to be successive, so that one level must be mastered before the next level can be reached" (Huitt, 2011).

Knowing this framework will enable teachers to assess students' cognitive level.  We can also design higher order level questions to facilitate and challenge our students for more extensive and elaborate answers.  Below are some questions that can be asked under the categories defined by Bloom (Source: Types of questions based on Bloom's Taxonomy.  Retrieved 30 August, 2011 from http://www2.honolulu.edu/facdev/guidebk/teachtip/questype.htm) :

Knowledge level:
-remembering
-recognizing
-recalling identification
-recalling information
 * who, what, when, where, how...?
 * describe

Comprehension level:
-interpreting
-translating from 1 medium to another
-describing in one's own word
-organisation and selection of facts and ideas
  * retell ...

Application level:
-subdividing something to show how it is put together
-finding the underlying structure of a communication identifying motives
-seperation of a whole into component parts
  * what are the parts or features of...?
  * classify ... according to ...
  * outline/diagram ...

  * how does ... compare / contrast with ...?

  * what evidence can you list for ...?

Area and Measurement

We were asked to draw as many squares as we could on a piece of dotted paper. We then made comparisons and found out how many times the smallest square was smaller than other squares.  We also drew different shapes by joining 4 dots.  The class squeezed out their brain juice to think of different solutions to find out the area of different shapes.  Some of the methods we used were:

1) cut and paste to compare

2) using halves (eg. dividing the square into halves)

3) using subtraction way to minus away unwanted areas

4) counting dots inside the shapes

For peculiar shapes, I find it easiest to use the subtraction method to take away the unwanted parts.

It was fun to see how many ways we could come out with, and Dr Yeap gave us time to let us think and discuss.  In this lesson, I learned not to be eager to show students what the answers should be.  Adults tend to give out exernal signals (eg. frown, smile, thumbs-up, etc.) for right or wrong answers.  Children who are dependent on these artificial signs will have difficulty in independent thinking and decision making.  We need to let them have the internal ability.  We should ask them questions even if they have the correct answers, and let them explain and justify themselves.

Thank you Dr Yeap, for being so patient with us.  You have set a good example of how a math teacher should be.

Citation: Huitt, W. (2011). Bloom et al.'s taxonomy of the cognitive domain.  Educational Psychology Interactive.  Valdosta, GA : Valdosta State University.  Retrieved 30 August, 2011 from http://www.edpsycinteractive.org/topics/cognition/bloom.html[pdf]

Reflection on Session 4

Tonight's lesson started with a Mind Reading Game with digits 1 to 9.  When we told Dr Yeap the first number (out of 2 numbers) we had in mind, he was able to tell us the final difference!  How freaky it is that somebody can read your mind!  Everything came to light when we compared and made connections among the number patterns.  Dr Yeap's further explanation with the base ten materials made it even clearer for us to see how it worked and why it did not matter what the second number was.  Indeed, every idea has a pictorial and concrete version, and it dawned on me that the CPA Approach by Jerome Bruner is applicable to adults as well.

We do not play with a fraction of a marble

Next, we went on with addition and subtraction.  We learned that quantity comes in 2 forms: continuous and discreet. 

Money and kilogrammes are of continuous quantity and they can be expressed in decimal and fractions.  Items like marbles will be of discreet quantity.  For example, we do not use 1/4 on a marble as we see it as a whole; 1/4 marble is considered a broken marble.

Bascially, there are 3 situations for addition and subtraction:



Change situation

In change situation, there is an initial quantity, a change in quantity, and a final quantity.

Example, Tommy has 37 marbles. He gave Peter 19 marbles.  How many marbles has Tommy left?

Here, we can see a lapse of time, and the unknown is the final quantity.  Take note that the unknown can also be the initial, or the changed quantity, depending on the question.

Part-whole situation

In part-whole situatiion, there are at least 2 parts and the whole.



Example, There are 37 students in a class.  19 students are present.  How many students are absent?



When using manipulatives to teach part-whole situation like the one above, care has to be taken that we do not physically take the manipulatives (which represent part) away from the whole.



Compare situation

In compare situation, there are 2 quantities that are being compared. 

Example, I have $37.  I have $19 more than you.  How much do you have?

Often than not, we teach children that 'more' or 'together' means 'add'.  How wrong! We cannot apply this in the problem above.  This is a bad 'keyword strategy' that can cause misconception!  Math is not about repetition; it is therefore essential that we let children select the operation.  Teachers must give variation after teaching addition and subtraction.  This is important because we want to consciously expose children to variation to the same idea. 

Children learn well when you give them variant idea.

- Zolton Dienes

Following addition and subtraction was time for fraction and some paper folding acitivities.  Hmm... nothing to do with origami, though.
Do you know that words like 'fourth' and 'fifth' are nouns?  When it comes to fraction, we should say, for example, 'two fifth', and not '2 upon 5', or '2 out of 5'.  I'm guilty of saying that to my students.  Well, many of us do that but that is not the correct way.  Children cannot relate when they are asked to add '2 upon 5' to '3 upon 5'.  If we are to prepare children well, we have to consider the language that we use. 

In fraction, being equal does not mean it must be identical.

Consider these:

Do they look identical?

They may look different but yet represent the same amount. 

Now let us look at fractions at 3 levels:

Concrete Level

A portion can be cut out for comparison.

Pictorial Level

These are equivalent fractions. 1/8 and 1/8= 2/8 = 1/4

Abstract Level

We can work out the formula and prove that area A is equal to area B.

And now... division time...

We have learned that there are 3 meaning in addition.  For division, there are 2, namely sharing and grouping meaning.

Grouping - Example, in 12, there are 3 groups of 4 i.e. 12 divided by 3 is 4.

Sharing - Example, when 12 cookies are shared among 4 children, each child will get 3.

                               12 divided by 4 is 3.

During class, we learned to solve this problem:

In 3 fourths, how many fourths are there?

Who would have thought that 3/4 divided by 1/4 can be so easily solved by simply looking at a picuture?

Yes, and the answer is 3!

There was a lot to learn in tonight's lesson but it was so enlightening!  Thank you Dr Yeap, for teaching us simple methods to understand.  They are so much more meaningful than formulas that I can never understand.  :)

 

Reflection on Session 3

In today's session, we had Ms Peggy as our guest lecturer.  The class learned about Lesson Study as a professional development tool, and we examined 2 case studies of centres using Lesson Study. After watching the videos, we came up with 9 areas that can help to make up good teaching:

- sitting position

- level of engagement/involvement 

- use of materials/manipulatives

- flow/sequence of lesson

- classroom management

- communication (teacher-pupil, pupil-pupil)

- questioning techniques (number of questions, types of questions)

- attitudes/disposition of teacher

- differentiation

From the 2 case studies, I see that mathematical investigation is beneficial.  With planned activities that are divergent in nature, students have opportunities to use efficient materials or manipulatives to explore and experiment mathematical ideas and situations in many ways.  While clear instruction, demonstrations, and effective questioning are important, teachers should also observe and offer differentiated tasks to challenge higher ability students, and scaffold to help the weaker ones. Some examples of differentiated tasks are, to let students describe their strategy and do justification, and even let them document the steps they have taken to arrive at their solutions.

  

Tonight, I also learned a fun and engaging way of teaching children number conservation: constructing different structures with 5 unifix cubes each.   When children take part in this acivity, they will be able to see that even though the structures could be different, the number of cubes remained unchanged.  As I involved in the constructions, I thought of and tried out as many designs as I could, making sure that the designs would not repeat themselves.  What I did was visualization - thinking about the shape of the structure mentally and represent it with the materials on hand.  As Van De Walle, Karp & Willams(2010) describe, visualization "involves being able to create mental images of shapes and then turn them around mentally, thinking about how they look from different perspectives - predicting the results of various transformation" (p.429). For a child of a higher ability, a challenging task will be to draw perspective view of the block structures.

  

7-piece square

Another good activity for visualization is forming shapes with tangram (set of puzzle shapes in 7).  It helps us to develop our ability to think and reason in geometric contexts.  As the number of required pieces of tangram increased, the formation of square and triangle became more difficult.  It was challenging, but fun.  I am sure children will also enjoy this activity as they learn.

7-piece triangle

Reference: Van De Walle, J., Karp, K. & Bay-Williams, J. (2010).  Elementary & middle school mathematics.  Teaching developmentally (7th ed.).  Boston, MA: Allyn and Bacon

Reflection on Session 2

In tonight's lesson, I learned that when planning a math lesson, there are 3 questions that I can ask:

- what is it that I want to the students to learn?

- what if they cannot?

- what if they already know it?

When students are not able to do a task, we have to think about intervention.  However, if they are of a higher ability, we can take the chance to challenge them with differentiated instructions.  For example, in the 'Take 1 or 2' game (drawing sticks) whereby the person with the last 1 or 2 straws on hand is the winner, we can ask the higher ability child to think:  Can   he see any pattern in the observation? What strategy will he use to win?  Would he like to have 15 sticks when it is his turn to pick? 





Is 15 a good number for you?
The winner can have all the bread sticks

When I watched the Holiday Game video, I could see how Dr Yeap facilitated the students in their analysis.  Once the children saw the pattern of the dice game, they could unlock the mystery.  Though it took them quite a while to do it, the supportive teacher and meaningful activity were able to hold the children's interest and helped them persevere.  The interesting task kept me thinking as well.  Never have I known that a dice, having 6 faces that bear dots from 1 to 6, always give a constant sum of 7 when we add the dots of the opposite sides together.  As such, for 2 dice that  are placed together, when the dots on the opposite sides are added to the dots on the interfaces, it will come out with a constant sum of 14.  For example, if the opposite sides of the 2 dice are 1 and 6, the sum of dots on the interfaces can be calculated as follows:  1+?+6=14

Dots on opposite sides	Total sum of dots in the middle
1 and 6	7
3 and 5	6
2 and 5	7
4 and 2	Try workinjg this out
5 and 4	Try working this out
6 and 3	Try working this out

Through this activity, I can help students to look for patterns and  to develop number sense. What an interesting way to teach three 1-digit number addition!

When children learn math, they learn through:

- procedure

- conceptual understanding

- conventional understanding

It is sad that I learned math without conceptual understandinjg with I was young.  The memorizing  of times table had really caused me much frustration and agony.  Teachers are not to teach children machine skills; what is needed is decision making and number sense. Instead of correcting children all the time, we should  let the children learn how to self-correct.  Lessons learned from first hands experience are lessons ingrained for life.  It is important that children possess human competency to survive in this world.  The CPA Approach by Jerome Bruner can be adopted:

Concrete, Pictorial, Abstract

First, children will construct through concrete objects.  Next, they  proceed to the usage of pictorial to represent concrete, and lastly the usage of abstract ideas to make connections and to figure out. 

To sum up, the 5 key concepts that we teach in math program are:

- Generalization (patterns, relationships, connections)

- Visualization

- Communication (language, representation, reasoning, justification)

- Number Sense

- Metacognition (dealing with informatin, knowing weakness and overcome it)

What matters is not what we teach, but how we teach them.  It is essential that teachers provide the opportunities for quality instructions.  We have to remember that math is a vehicle for the development of thinking,  NOT a destination.

Reflection on Session 1

Before class today, I saw Dr Yeap going through some papers and working on them at Kopitiam near SEED Institute.  Seemingly, he was working on some problem sums which he had prepared for class use.  That frightened me quite a bit as I could anticipate the amount of problems we had to solve in class.

However, I had a pleasant surprise that the first math lesson was actually enjoyable.  The lesson started with Dr Yeap inviting us to use his name to count to and fro, and guess the 99th letter.  I am not strategetically inclined so I simply counted on.  Before long, I realized that I could not go on like this; there must be a simpler way.  After Dr Yeap's prompting, the class shared their ideas, and I started to see patterns and relationships among the numbers. As I applied the strategies to find the letter on the 99th position of my name, it became easier.  I could see that the numbers in the first row are different by 12, and the digits in the 'ones' place are different by 2.  In no time, I was able to tell that the answer was 'u'.  My classmates' sharing also helped me to see that the alternate numbers in second row are multiples of 12.  Hey, math has suddenly become more interesting! :)

The task on the cookies owned by 3 kids explained which level a child is at while he or she is doing counting and addition.  The levels of counting are: counting all, counting on, and counting on the application of commutative property of addition (5+7=7+5). Knowing all these levels enable teachers to know whether children have known, for example, how to use the '10' strategy, conservation of numbers, or whether they have already attained the level of communtative property of addition.  Also, as teachers assess children's counting, we need to know that there are 4 pre-requisites:

- ability to classify

-ability to do rote counting

-ability to do one-to-one correspondence

- ability to appreciate that the last number uttered represents the number of things in the group.

 In the Spelling Card Trick (poker cards), Dr Yeap showed us an interesting way to get into number spelling, and the using of  logical thinking to arrange the cards in the correct order so that we will be able to call them out in sequence.  It was fun, and I see that as we tried to problem solve, we were actually actively constructing our knowledge at the same time.  I dropped down my ideas on a piece of paper and experimented with the cards.  I find that visual aids and manipulatives are excellent tools to help me explore, experiment, and think better. 

Today, I realise that math is not solely about numbers, formulas, and procedures; we need thinking skills,  discussion and communuication.  With these, we share ideas and we can understand better. How I wish my previous math teachers were like Dr Yeap, to teach in such a way that it inspires and probes me to think further.  Often, teachers simply focus on students' inablility to solve a math task and forget to reflect  the way we present the tasks to the students.  For instance, in the event of patterning, teachers have to bear in mind the elements of patterns and remember to state the terms clearly to the students before they get started.

After today's lesson, I begin to see that math can be fun and interesting, and that math concepts can be taught in a way that is easily understood.  Teachers have to remember  to provide well-facilitated expereiences and classroom environment that  encourage students' engagement in higher level thinking.  Of course, we do not want to forget to be open and be appreciative of different possibilities to approach a problem.

My New Journey with Math

Reflection on Chapters 1 and 2
Math is like a mysterious lady, so deep and sophisticated that I cannot fathom.  We are merely acquaintances - math has never been my forte.  However, after reading chapters 1 and 2, I realize that as a teacher, I have to first love maths so that my interest can rub off on my students.

When I read the principles and standards released by NCTM, I find it good to have such guidance and direction for math teachers as we plan our lessons.  Under The Teaching Principle, it states that "teachers' actions are what encourage students to think, question, solve problems, and discuss their ideas, strategies, and solutions" (NCTM, 2000, p.18).  This is so true.  As I recall my primary school days, my math teacher scolded me "stupid" when I could not solve the math problem on the chalkboard, and she laughed with the class as I went back to my seat in tears.  Worse, my secondary school math teacher labeled the whole class as "numbskulls" and "nincompoops".  Name callings definitely did not spur me to think or solve problems as I had no conceptual understanding in the first place.  Extreme awful feeling of defeat and humiliation had caused me to lose my self-respect as a math learner, and my budding interest for math was completely nipped off.     

 I agree that "effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well" (NCTM, 2000, p.16).  As a teacher, I ensure that I do not follow the footsteps of my math teachers; now I see that I also need to have the heart to understand my students, and be enthuiastic in facilitating them to develop a positive self-concept about their ability to learn and understand mathematics.

 It worries me a bit, though, to know that high-quality education needs teachers to (1) understand deeply the mathematics they are teaching; (2) understand how children learn mathematics; and (3) select instrucional tasks and strategies that will enhance learning.  With my bad experience with math and my limited interest in it, will I do a good job as a math teacher?

I have to face the fact that we have already entered into a new mathematics era; gone were the days of meaningless drillings and usage of formulas without understanding.  According to the 5 Process Standards, children have to learn how to problem solve, reason and prove, communicate, make connections within and among mathematical ideas and to the world, and also use representation to express mahematical ideas and relationships.  Well, it is time that I refresh and start on a new journey with math. I hope to kindle an interest for math, and be able to apply inquiry-oriented approaches to help my students focus on mathematical thinking and reasoning.  Besides this, I also have to inculcate technology into my teaching, as according to The Technology Principle, technology influences the mathematics that is taught and enhances students' learning (NCTM, 2000, p.24).  It seems that I have a lot to learn, but I am willing to try, bearing in mind that as a teacher of mathematics, not only must I have a depth of mathematics knowledge, I also have to be a good model for persistence, to display positive attitude, to be ready for changes, and to have a reflective disposition.

 So how do I get started?  I have to know what is doing mathematics.  The authors explain that "it means generating strategies for solving problems,applying approaches, seeing if they lead to solutions, and checking to see if your answers make sense" (Van De Walle, Karp, & Bay-Williams, 2010).  To help students share and defend mathematical ideas, the classroom environment has to promote risk-taking, and mathematical tasks are to be worthwhile.  I must admit that I am lacking in creating this kind of environment as I am a traditional teacher who explains too much.  Chapter 2 has introduced four features of productive classroom culture which I can adopt.  As mentioned in the chapter, construtcivist and sociocultural theories emphasize learners to build connection among existing and new ideas.  To help learners to become mathematically proficient, they should be allowed to reflect, use prior knowledge, have social interactions, and experiment in problem solving.  Classrooms that embrace culture for learning can empower the students to develop a sense of autonomy to play an active role in learning, and in testing out ideas and strategies.

Teachers play a part in creating a supportive environment and in making math lessons fun and exciting.  It is important that we are passionate to provide experiences that provoke students' thinking, reasoning, and reflection.  I am looking forward to learn from the coming math module to better myself both as a math learner and as a teacher.

 "Believe in kids!"

- John A.Van de Walle