Conclusion (Part 2)

My Maths Metaphors

In Week 2, I mentioned about some metaphors that were given by the lecturer which I can relate to my maths learning experience. Having had to attend this unit for 12 weeks, I am now able to think of my own maths metaphors.

• Learning maths is like solving a puzzle. Be it a jigsaw puzzle or any other kinds of puzzles, in order to be able solve it, we need to gather the pieces of information and clues that we have.

• Learning maths is like participating in a treasure hunt where solving a problem depends on understanding the clues given.

• Learning maths is like going on a journey. Any journey has its starting point as well as its ending point. It goes the same in solving with any mathematical problems, if we try our very best to solve it, we will get the answer, it's only a matter of right or wrong.

My Personal Teaching Philosophy


Teaching Early Childhood Mathematics is not an easy task. One needs to have patience as well as passion in order to make the learning fun and meaningful. I realize that young learners should not be underestimated and lessons need to be interactive and expose them to language that is a step ahead of what they have already learnt.

After 12 weeks of workshops , I have begun to see that there are three main elements in teaching early childhood maths. The three main elements are:

• Literature That Teaches -This includes books or story books designed to teach maths to young learners

• Games That Teach -This includes hand-on activities as well as computer games

• Songs That Teach -This includes any songs and chants created to teach maths to the young learners

In order to cater for students' different learning preferences, I believe that these three elements should be included in teaching mathematics to early childhood students. Furthermore, with variety of teaching styles, learning can be maximized.

Week 12

Patterns & Algebra

Do early childhood learners learn about algebra? I would not have said yes, until I attended today’s workshop. Yes, they do! Taylor-Cox (2003) claims that it is never too early for young children to start thinking in terms of algebra. Instead, we do need to offer young children a solid foundation of algebraic thinking. Algebra has long served both as a gate and a barrier for students (Lott 2000). Therefore, we can prepare students to be successful in algebra if we begin teaching them to think algebraically in the early years.

The central ideas promoted in the national algebra standard for young children are (1) patterns, (2) mathematical situations and structures, (3) models of quantitative relationships, and (4) change (NCTM 2000, as cited in Taylor-Cox, 2003).

From the workshop activities that we have done, it could be seen that Algebra in the Early Childhood focuses more on patterns; recognizing, describing, extending, and translating patterns. Working with patterns invites young children to identify relationships and form generalizations (NCTM 2000, as cited in Taylor-Cox, 2003).

One of the activities that we did in the workshop was creating patterns. The purpose of this activity was to engage students in focusing on ‘What comes next after the pattern?’. Being able to describe these regular relationships leads to predicting how the pattern will continue. In my opinion, for young children, recognizing that patterns are predictable is an important Mathematical idea. In addition, having students to explain why that pattern comes after the next pattern is a good way to encourage students to interact as well as to think critically as it will enhance their prior Mathematical knowledge.

Creating patterns

Having to do this 'creating patterns' activity brought me back to the first few weeks of the semester where we learned about the beginning processes where students sort the objects according to their attributes and making pattern like 'repeating patterns' and 'growing patterns'.

Growing patterns

Reference:

Taylor-Cox, J. (2003). Algebra in the early years? Yes!. Young Children, 58(1), 14-21.

Week 11

Chance & Data

According to Geist (2001) every day children in the classroom actively work with raw data. They generate, organize and interpret data and they draw conclusions or make predictions on the basis of the data they have collected. Hence it is important to expose and teach the young learners about organizing and interpreting data.

In the workshop, we did sorting smarties according to their colour and comparing the amount of smarties between peers. We also get to predict the number of times the spinner (clips) land on the colour.

Making the smarties into a bar chart

Data collected when predicting the number of times the clips fall on a colour

The idea of using smarties to teach the young learners about data makes me realise that that I need to be more creative in my teaching and maximize the use of materials and resources in the classroom so that I will be able to get the students interested in what they are learning. This workshop has also proved to me that learning Mathematics can be really fun if we could incorporate good resources, materials and fun into the lesson.

Reference:

Geist, E. (2001). Children are born Mathematicians. Young Children, 56, 12-19.

Week 10

Measurement

As an adult, if we were asked a question on ‘how long is one minute?’ our answer would typically and certainly be ’60 seconds’. But what if we ask this question to the young learners in their early childhood, would they be able to give the same answer as us? Looking at a learning Mathematics in early childhood, what are other possible answers for this question? One of this week’s workshop activities is ‘Perceiving One minute’. We explored many possible ways for the young learners to perceive time.

In my opinion, it would be appropriate for young children to be introduced to non-standard units of time before they proceed to the standard units of time. It is because the idea of learning itself, where it should move from a lesser complexity of ideas to a more complex one. Hence in an early childhood classroom, perceiving one minute could be done with variety of simple variables like:

• No of beads can be threaded in one minute
• No of unifix cubes can be stacked in one minute
• No of pendulum swings in one minute
• No of marching steps in one minute

Another activity could be done with young learners to perceive time is looking at how a pendulum swings. Strings with different length are used in this activity. You can ask children to roughly create a data table as below:



When all the data is collected, let the students investigate the relationships between the length of the pendulum and the number of swings made. (e.g. the shorter the length of the pendulum, the greater the number of the swings made).

* In order to get a more accurate result of the number of swings, students could be asked to record them twice and then find the average.




One-Handed Demonstration Clock

Krech (2000) in his article, It’s time! talks about the use of one-handed demonstration clock to help students learn what ‘the space in between the numbers’ all about. This clock focuses merely on the hour hand in time-telling. The language used with this one-handed demonstration clock is an easy language such as:

• About nine o'clock
• A little past nine o'clock
• Halfway past nine o'clock

A one-handed demonstration clock

The minute hand can later be added to the one-handed demonstration clock, after the students have had plenty of practice with the hour hand. Krech (2000) believes that focusing on the hour hand on its own helps the students gain stronger overall understanding of how the minutes hand functions.





Literature That Teaches

This week’s workshop captured my attention as I got to know a few story books that are meant to teach Mathematics to young children. One of the books is:

The Bad Tempered Ladybird


The topics focused in this book are time and size. It uses non-standard units to measure the size (the size of the ladybird is used to compare the sizes of other animals. The terms used are like big and bigger.)

The fist time that we read the book, we asked the children to listen carefully and to focus on what the ladybug might have learned through its adventures. For the second reading, we asked them to listen and watch for patterns. On a chart paper, we recorded each pattern that the children identified (Kelly & Burke, 1998).

Back in Malaysia, during my time as an early childhood learner, I was not exposed to this kind of book. Even nowadays, the use of story books to teach Mathematics is still not very significant. As a future teacher, I'll take the challenge to promote this kind of literature to the Malaysian classroom.





References:

Kelly, M. G., & Burke, K.(1998). A matter of grouchy time. Teaching Children Mathematics, 4(7), 404-407.

Krech, B. (2000). It's time!. Instructor, 109(5), 16-17.

ExtraS

• Songs That Teach

This site contains a collection of songs and chants for teaching young learners Mathematics for various topics. Each song listed is included with its lyric and a preview of the song. To listen to the full version of the song listed, users need to buy the CD from the online store on the website. However, being able to listen to the song preview is already an advantage because the rest of the melody for the song can be created by the user itself (be creative!). Just a word of caution, do acknowledge the source if a lyric of song from this site is used in a lesson because it involves copyrights. (I myself would not like if my ideas get stolen!). Have fun!

*Other than the songs listed in the above website, I also found songs that teach from youtube. The first one is The Shape Song. It talks about the attributes of each basic shapes; a circle, square, rectangle and triangle. There are also visual representations for each shape. In my opinion, this song could be very useful in teaching young learners the properties of basic shapes because while recognizing shapes, children learn about the attributes of each shape.

*Another song Ifound on youtube is the Malaysian version song, entitled '10 Budak Hitam' (10 Black Kids). The original version of this song was not meant for educational purposes, but it has been modified to be one. This modified version of '10 budak Hitam' talks about 10 black kids who were playing together as big group at the beginning but later one by one get parted from the group. Basically, the song teaches students about subtraction. In between the subtraction processes, there are some moral values added to it.

• Games that teach

Below is one example of interactive flash games that could be found from the link above. This game help student further learning and understanding about angles in a circle. It may or may not be appropriate for the early childhood learners, it is all depend on the learners proficiency levels. The game is certainly appropriate for primary school children.

Week 9

Shape (Part 2)

I find that today’s session is the most interesting session so far. Why? Because today I have been introduced to a very useful Mathematics teaching instrument, which is a set of tangram. The tangram is an ancient Chinese geometric puzzle with seven pieces: 2 large mangles, 1 medium triangle, 2 small triangles, 1 square and 1 rhomboid (Bohning, 1997).

A set of tangram

I was amazed with the fact that this set of seven pieces of shape is a very powerful Mathematical instrument to teach geometry, especially to young children. When I was a kid, I remember to having had to do the puzzle using these seven pieces but I was not being explained about the relationships that these seven pieces of shape have. Back then, my focused was only to solve the puzzle. However, unlike a jigsaw puzzle where a piece must fit in only one way to complete a picture, the geometric tangram pieces can be arranged in many different ways to make figures of; a watchful cat, a soaring bird, a bubbling teapot and children who run and play (Bohning, 1997).

A figure of a watchful cat

Experiences with tangrams actively involve children as they develop the skills of a geometry vocabulary, shape identification, classification (Bohning, 1997). Clements and Sarama (2000) add that identifying shapes is important but the focus on the properties and the relationship should be strong. Children's reaction to tangramming is one of surprise and amazement, surprise that manipulating the geometric shapes can be so much fun and amazement that the seven pieces can be arranged to make so many different figures (Bohning, 1997).

After further investigation on the tangram set, we have also discovered that the seven pieces can be used to teach children about fraction (based on area of the shapes). For example,

• The area of 1 little triangle is equal = ½ the area of the square = ½ the area of the rhomboid.

Other than that, children could also be asked to create a rectangle using only 3, 4 or 5 pieces from a set of tangram.

A square with 3 pieces of a tangram set

A square with 4 pieces of a tangram set

A square with 5 pieces of a tangram set

As they are manipulating the shapes in the tangram set, children will develop a positive attitude toward geometry, furthering their spatial sense, and developing a basic understanding of geometric concepts and relationships (Bohning, 1997).

Reflection

It could be observed that young children learn Mathematics mostly through experiential learning. They observe, question and make mistakes. Hands-on activity like tangramming therefore gives children to solve puzzles and manipulate shapes without the restriction of one single right answer.

References:

Bohning, G., & Althouse, J. K. (1997). Using tangrams to teach geometry to young children. Early Childhood Education Journal, 24(4), 239-242.

Clements, D.H. & Sarama, J. (2000). Engaging young children in mathematics: Standards for Early Childhood Mathematics Education. London: Lawrence Erlbum Associates.

Week 8

Shape (Part 1)

This week’s lesson started off with classifying the 2D shapes (quadrilaterals). As we all have learnt, a trapezium, rhombus, rectangle and kite are all quadrilaterals. Some of them (e.g. rectangle and rhombus) can be classified in many categories under quadrilaterals. A rectangle for example is also known as a ‘square’ and a ‘paralellogram’ and can be classified as a trapezium because it has two pairs of parallel sides. Hence it is learned that any shapes should be classified according to its attributes/properties for example the number of sides it has. If shapes are classified by labeling, it will certainly confuse the students.

One of the activities that could be done to introduce young learners to 2D shapes is to get them each to cut out various shapes from a piece of paper and then sort the shapes according to its attributes/properties. Then have students to describe how they sort the shapes.

Have the students to sort the shapes and ask them how they would sort it.

Instead of cutting shapes out of papers, the teacher could also get the students to draw 2D shapes that they could find in their surroundings and then sort the shapes according to its attributes.

Further discussion about the shapes created by the students could be held in order to get students to recognize and notice the critical attributes of each shape. Question like ‘tell me more about the shape’ could be asked to stimulate students’ critical thinking.

Reflection

While discussing about the properties of quadrilaterals, we have come to realization that the US definition of a 'trapezium' is different from other countries. This might cause learners, especially the young ones confusion. The issue raised was 'could we have one standard definition of a trapezium?'.

*If we can have the standard measurement units for all over the world, why not?*

Week 7

Presentations

This week’s session was located for our individual presentations. The focus was on the first six weeks of tutorials and workshops. I was so impressed to see that each of us came up with different ways of presenting what we have learned for the past six weeks. Some did fancy online journals, some did Point Presentations and others had on-paper written journals. The styles of presentations were very different but everyone has shown that their philosophies of teaching Early Childhood Maths has started to developed week by week. As for me, I started to see that Early Childhood Mathematics would not just be introducing children to simple Mathematical concepts like shapes, space, calculation and time but also making it relevant so that children can relate them to the real world. Tucker, Singleton & Weaver (2002) describe an important learning principle in learning Mathematics:

“Students must learn mathematics with understanding, actively building new knowledge from experience and old knowledge.”

Reference

Tucker, B.F., Singleton, A.H., & Weaver, T.L. (2002). Teaching Mathematics to All Children: Designing and adapting instruction to meet the needs of diverse learners. New Jersey: Merrill Prentice Hall.

Conclusion (Part 1)

My Personal Teaching Philosophy


After six weeks being in EAB023 class, I believe that :

1. Mathematics in Early Childhood does not only involve numbers, it involves other mathematical concepts as well.

2. When it comes to their mathematics learning, all children should be treated as capable learners who know a great deal and who can learn a great deal more.

3. Teaching and learning of Mathematics in the early years should be as interesting as possible.

4. There should be a balance between accuracy and flexibility in developing computations of the early years learners. In solving a math problem, the process of solving the problem should be taken into account, rather than the answer of the problem.

5. There are a lot of mathematics games that teach and teachers should make use of them in order to assist learners in explorations of mathematical concepts.

Week 5

Mathematics Games

"Rather than teach Mathematics skills by drilling and rote memorization, teachers can plan rich environment and offer developmentally sequenced opportunities that allow children to explore mathematical concepts in the context of play." (Cutler, Gilkerson, Parrott & Bowne, 2003)



In relation the the above view of learning Mathematics, I myself find that learning Mathematics is so much fun with hands-on activities because we usually remember what we do rather than what we talk about. Furthermore, hands-on activities support young children's exploration of Math concepts (Cutler, Gilkerson, Parrott & Bowne, 2003).

Below are some links of online math games that I found on the internet:

1. http://www.priorywoods.middlesbrough.sch.uk/resources/programres.htm

(This site allows teachers/students to play Math games online or download the games on their computers and it is FREE)

2. http://www.apples4theteacher.com/math.html#numbersensegames

(Fun & interactive Math Games according to topics. Teachers might find some useful insights for their lessons from this site)

3. http://www.ababasoft.com/kids/index_math.html
(There are 12 free Maths games available on this site. Although it's written on the site 'Math games for school boys', but the games are suitable for both boys and girls and even the adults)



Reference:
Cutler K, Gilkerson D, Parrott S, Bowne M. (2003). Developing math games: Young Children, 58(1), 22-27.

Week 6

Calculators & Games

Calculators are very common tools used in Mathematics lessons but should them be used just for calculation? Never underestimate calculators! Besides using them for calculation purposes, they can also be used as tools for mathematics games. This was what we did in one of our lessons. Each and everyone of us were given a calculator and we were asked to display our favourite numbers on the calculators. We were then asked to group/arrange ourselves according to ascending and descending order of the numbers we displayed on the calculators.

There are other interesting games that a teacher can create in a Mathematics lesson and the games do not necessarily need to involve some fancy and expensive materials. A teacher just has to be creative and improvise any materials and resources which are available around them to make a mathematics lesson as interesting and meaningful as possible.


Below are some other Mathematics games that can be used in Mathematics lesson to help young children explore math concepts:

1. Subsitise Egg




2. Teddy Bear Race

3. Closer is Better

Week 3

Beginning Processes & Number

This week's lesson focused on developing computations in the early years. As the lesson went on, I managed to see that the processes involve in developing computations in the early years are in stages and the stages are:

1. Developing number concepts by matching one-to-one correspondence

2. Sorting by classifying quantity

3. Ordering by comparing more than 2 objects or more than 2 sets

4. Patterning by looking at visual representations

As we can see, these stages move from a simple process to a more complex process. This relates to a concept which Vygotsky classified as the Zone of Proximal Development where young learners are able to move to a more difficult level of tasks if they are scaffold by the adults (Morris, 2008).

In developing computations of the early years learners, I believe there should be a balance between accuracy and flexibility . Even for a topic as easy as counting, teachers should put emphasis on understanding of the one-to-one correspondence which is crucial in knowing how to count rationally, rather than teaching students memorization of number words in a given sequence. According to Unglaub (1997):

" Teaching a child to count isn't teaching her to understand math unless you teach her to touch another object for each additional number (2, 3, etc.) and other mathematical concepts involved in counting."

References:

Morris, C. J. F. (2008). Zone of proximal development. Retrieved August 30, 2008 from http://www.igs.net/~cmorris/zpd.html

Unglaub, K. (1997). What counts in learning to count?: Young Children, 52 (4), 48-50.

Week 4

Mathematics Syllabus Documents

This week's lesson started off with the comparison between Queensland Early Childhood Mathematics Syllabus Documents with some other countries' Early Childhood Mathematics Syllabus Documents (Malaysia, China and Norway). It has been so interesting for me to find out that in the Queensland Early Childhood Mathematics Syllabus Document, recognizing the images on the notes and coins is also included as a part of Mathematics lessons. It might sound like a very simple thing to be included in a lesson but it is actually a very important thing to learn. Learning to recognize images on the notes and coins involves a bit of history as well as general knowledge and this helps young learners to develop their thinking skills.

Mental Computation Strategies

Mental computation strategies are strategies used in dealing with math problems which involve bigger numbers. Some of the strategies are by using the 100 Charts and the 99 Charts and by using empty number lines. For each question given, there will absolutely be different kinds of suggested order of presentation of numbers. The teacher's role therefore is to encourage children to think of any possible solutions for the question given and then share and compare them with other students. This way, students will be able to see different ways that they can use to solve a particular question.

After attending this week's lecture and workshop, I have come to realize that in learning mathematics, the right answer is not the most important thing but the process towards getting the right answer is what matters the most.

Week 2

What is Early Childhood Mathematics?

When first time posed with this question, i associated Early Childhood Mathematics with introducing students to numbers ; counting 1,2,3 and so on and doing easy addition and subtraction processes. Some of my classmates shared the same view as mine. Here are some concept maps that they have created in response to the question of 'What is early Childhood Mathematics?'






However, as the discussions between my classmates and I progressed, I found that Early Childhood Mathematics does not only involve numbers, additions and subtractions. It also involves other mathematical concepts such as shapes, space, time, chances and data and so on. Some of my classmates shared this idea on their concept maps.





Towards the end of the class on that day, I have come to a consensus that Early Childhood Mathematics is not only about numbers, but it involves other Mathematical concepts as well.

As stated by Perry, Bob, Dockett and Sue (2002):
"It is important for us to not to consider young children as 'empty vessels' or even “leaky vessels” when it comes to their mathematics learning, instead treat all children as capable learners who know a great deal and who can learn a great deal more."



Math Metaphors

We were given some metaphors of what learning Mathematics is like. It took me some time to understand and relate some of the suggested metaphors to my mathematics learning experience. Among all of the listed metaphors given, I was able to relate some of them to my math learning experience; recipe , jungle, and jigsaw.

1. Learning mathematics is like learning a new cooking recipe where the teacher or books give step-by-step instructions and we just follow them.
- There are many formulas involved in learning mathematics. Just like step by step instructions in a recipe, the formulas direct us to find solutions for any math problems.

2. Mathematics is like a jungle because the ideas are all jumbled up.
- Different people have different ways and approaches to solve a particular math problem. We won't be able to see these differences unless we are given explanation about them.

3. Mathematics is like a jigsaw because the ideas fit neatly and beautifully together.
-When applying the right concepts and to the right problems, there are always definite answers in mathematics.

Reference:
Perry, Bob and Dockett, Sue. (2002). Ch 5 : Young Children's Access to Powerful Mathematical Ideas in English, Lyn D (ed), Handbook of international research in mathematics education, Mahwah, NJ: Lawrence Erlbaum Associates, pp.81-111.

Week 1

Unit Outline

Before I chose to enroll in this unit, I had a quick look at the unit outline which is available at QUT Virtual website.

'This unit aims to enhance your understandings, attitudes, values and skills in relation to early childhood mathematics. This unit will investigate teaching methods for developing concepts and skills for looking at mathematics as a unified body of knowledge' (QUT Virtual, 2008).

As mentioned above, by the end of this unit I expect myself to be exposed to various philosophies & beliefs of teaching mathematics in the early years, examined them, and finally developed my personal teaching philosophy.

Reference:
QUT Virtual. (2008). EAB023 unit outline. Retrieved August 27, 2008 from http://blackboard.qut.edu.au/webapps/portal/frameset.jsp?tab=courses&url=/bin/common/course.pl?course_id=_34648_1

Introduction

All the posts in this blog are based on the weekly lectures & workshops that I attended for EAB023 unit for this semester (week 1-week 6). My weekly reflections are based on my learning experience in the lectures and workshops and are also based on some additional readings related to the topics. Prior to that, I have created weekly concept maps as the outlines for my weekly reflections.