What I have Learnt So Far.....

Our incredible math lecturer asked us to reflect on our six days of learning. Let me begin by saying that this module, till now, is the most mentally exhausting but extremely enlightening one for me and of course I have learnt so many big ideas such as looking at patterns and grouping etc. Other than this, I have also learnt about how I can help children in learning mathematics so that they can understand and enjoy this subject.

1.    Mathematical ideas that I was supported in discovering on my own:

The pattern of numbers in my name in finding the 99^th letter, learning about fraction by manipulating the pattern block and so much more. I’ve learnt and remembered a lot during the past six days mainly because I have to evaluate my own ideas about mathematics. We were encouraged to search for different ways of thinking and problem solving, moreover, we were given opportunities to talk about our thinking. We were like ‘children’ in the classroom, learning about how children learn mathematics. And I now know that part of our job as a teacher is to help children make sense of what they are doing in mathematics.

2.    The Importance of Rich and Rewarding Mathematic Environment

Every lesson in this module began with a problem that was not so tedious (considering that we are now pursuing our degrees) but got us thinking hard. I like to work near people but be able to move at my own pace. When I feel satisfied in my own mathematical exploration, I don't care whether I am ahead of or behind those around me. I don't want to be told how to do it, I want to find it on my own. This is something I have to teach my children as well. There is the problem on fraction which I only can manage to understand when I got home. I find that some adults and children are hard on themselves when they make mistakes or they can’t get the answers. I think it is vital that as teacher, I have to provide a supportive math environment so that children would have no time berating themselves but to put their attention on enjoying learning math. It is really hard to continue doing math when one is feeling stupid and left behind.

Question

What do you think of using interactive whiteboards for teaching math? I have the concern that having these whiteboards makes the classroom more teachers centred and less children involvement. Such as the video you have shown us during the first few lessons. When the children are using manipulative and even pencil and paper tasks, they appear to have talk and discussion among themselves. But they are all focused and quiet when the teacher is teaching through the interactive whiteboard.

The Problem That I have Problem With

Not everyone likes fractions. In my younger days, I remember hearing my classmates’ complaints. I also remember teachers that didn’t explain them well. I also know of many adults who don’t understand why they have to take one eighths multiply by one quarters which they say that the answer would be so insignificant anyway so why bothers. As for me, I did not bother to think about all these, I just knew that I had to score well for my subjects to go into a good institution and so fraction was a no problem topic to me. If the paper requires me to find how many one thirds are there in one half, I know I just need to take one divide by one thirds. If I have to compare fractions, I know I have to make the denominators the same. I can recognize what are proper, improper and mixed fractions. But can I visualize how many one thirds are there in a half, I can’t. Do I know how the word ‘denominator’ came about? I don’t know. Basically, I am doing fractions without understanding it.

The fraction problem posted by my lecturer on lesson 3 was how many one halves are there in three quarters. Within a minute, I could get the answer by dividing. But it took me many hours to understand it from the models. I couldn’t even get it in class and I had to spend at least one hour after I reached home to figure it out. And seeing the model really make sense of all the fractions stuff that I have learnt throughout my schooling days. Thus fraction needs to be taught basing it on visual models so that children will get a firm grasp of the concepts, before simply memorizing the various steps. This is one good example to show that the purpose of learning mathematics is to understand the world. Such as in fractions, it is supposedly to teach me to be able to visualize, to have a mental picture and to be able to apply it in life. The other day, my brother just told me that if I want to take good photographs, I have to stick to the techniques called ‘The rules of third’. Even in photography, there are fractions. 

So really, learning is meant for understanding, not for the sole purpose of a challenge since I have heard many others claim they like mathematics for the challenge. We educate everyone because knowledge is good, we don't do it to make our lives harder.

Using Technologies in Teaching Math

I have a love and hate feelings towards technologies. I find them scary; something without a brain and yet so much cleverer. On the other hand, I love them because they are undeniably helpful and make life so much easier. (Remember those days without handphones!)  I don’t want to avoid technologies because it is now a big part of our life. NCTM states that “Technology is an essential tool for learning mathematics in the 21st century, and all schools must ensure that all their students have access to technology.” For this section, I am going to share and reflect upon the ICT Master Plan introduced and developed by MOE.

The ICT (Information and communication Technology) Master Plan’s vision is to enrich and transform the learning environments of out student and equip them with the critical competencies and disposition in order to succeed in a knowledge economy. A few of the schools are already using these programs to enhance students’ learning. The number of schools which includes primary schools will be increased by 2015.

When I first knew that there would be such program in our primary schools, I felt uncomfortable. I was afraid that teachers would be replaced and children would be emotionless for they are taught by brainless technologies. However, after reading more about it, I was surprised to know that the program focuses on the importance of the empowerment of schools and teachers. Principals, teachers, the industry experts and the parents are all involved.  The teachers are the heart of the innovations; the use of technology cannot replace conceptual understanding, or problem-solving skills. In this kind of innovative classroom, teacher has an added responsibility to make knowledgeable decision so that students can use technologies in a meaningful and an effective way. Hence a greater emphasis is in placed to ensure that teachers get professional training in the relevant area.  

This makes me wonder how all this would affect or influence the Early Childhood Education since we are preparing our pre-schoolers for formal education. I believe that as a teacher, I must remain open to learning new technologies. Hopefully, before any of these implementations in the Early Childhood classrooms, there would be sufficient training for the teachers so we will be able to implement them effectively in a balanced instructional program.

Reflection On Chapters 12 & 13

I still remembered my astonishment when my eight year old daughter showed me how she did her addition computation. I couldn’t understand a thing and so did her. And because I couldn’t understand, I didn’t know how to teach her and in the end, I resorted to the tradition algorithms method. After reading chapter 12 and 13 of the book ‘Elementary and middle school mathematics’, I was surprised to learn that the teaching of mathematics have moved towards teaching children to compute mentally. Children are encouraged to use ‘invented’ strategies to do addition, subtraction and even multiplication and division. It is beneficial because the person using his or her own ‘invented’ strategies can then understand how the answer is derived rather than merely following the steps in standard algorithms. Upon reflecting, although I was taught to use algorithms to do math computation in schools, I believe that I did not quite develop mathematical thinking or understanding. Because despite hours of instruction and practice, I often fail to apply them correctly in problem solving situations. And hence, I grew up thinking that that mathematics is a collection of mysterious formulas and procedures that have to be memorized and practiced.  

I believe that in supporting children to develop their own strategies, children can be more aware of the place values and have less conceptual errors. Moreover, there is a real understanding through creating their own strategies, hence they are more willing to solve unfamiliar problems in the future.

The process of developing invented strategies needs support from the adults. Both teachers and parents then play an important role. Teachers have to allow ample time to guide children, facilitate the sharing of various solutions, listen carefully to their ideas, and make explanations about the math ideas presented in the invented strategies.

Reflection (Lesson One)

"The real voyage of discovery consists not in seeking new lands, but seeing with new eyes"

Marcel Proust

While my husband was driving home yesterday night after my class, I told him I was extremely tired.I have been using my brain and thinking very hard for the almost four hours during my first lesson on this module.  Although I was mentally exhausted, the class turned out to be lots of fun. For the first time, I saw creativity in Mathematics. I know it sounds rather odd but that's what I feel. The class started with a simple problem and everyone of us tried to use different methods to solve that problem. No right or wrong methods, is a free country afterall! (That's what Dr. Yeap like to say). The reasoning and communication that the class provided were creative. We discovered the various solutions to the problems. It was very insightful to me to see that how we can focus on the communicative aspect of Mathematics. I was totally absorbed in trying to solve those problems while at the same time I felt the excitment in the discovery alone as well as exploring together with the class. 

Reflection Two

   "Mathematics is more than just the right or wrong answer, it is about solutions."

The Process is as Important as the Product

One of the reasons for people to dislike mathematics is that it is too rigid for it usually has only one correct answer to a math problem. However they seem to forget the process of thinking and satisfaction one can get after being able to solve a mathematic problem. Van De Walle, Karp & Bay-Williams (2013) stated that “[d]oing mathematics includes using justification as a mean of determining whether an answer is correct.” The reasoning and justification is as important as the answer in learning mathematics. Being able to reason is essential to understanding mathematics. It allows learners to be able to understand how mathematics makes sense. This is such important information for teachers since I have observed that many of our children are memorizing when they could be understanding. It is interesting to read how as a teacher, I can provide rich environments for learning mathematical reasoning by encouraging children to present their thinking to the class. For the pre-schoolers and kindergarteners, I would expect their explanations to be in their own language and often will be represented verbally or with objects.

Using Manipulative to Learn Mathematics

I agree that Mathematics is an abstract subject for young children. It is mostly numerals and does not seem to represent any meaning if physical materials are not used to support the development of mathematic concept or procedures. Using manipulative to teach Maths is common in kindergartens. Young children learn by hands on activities. You cannot just open up to a page in any book and expect them to know something without some kind of introduction first. That really goes for any level that you teach, but especially with early childhood. Children need to see it in front of them before they can see it on paper. Also, hands on activities make the lesson much more interesting However I am glad to read from the book that we should allow the children to explore the tools with appropriate guidance. Upon reflecting, very often I am over assisting by requesting them to follow what I am doing. Allowing children to learn mathematics concepts with tools seem like very easy, however I have seen children who initially find it difficult to coordinate their counting with the movement of their hands. This is an important foundation stage. In order for children to perform higher order calculation such as addition and multiplications, children must first be able to recognize and create examples to represent numbers.

Reference

Van de Walle, J.A., Karp, K.S. & Bay-Williams, J.M. (2013). Elementary and middle school mathematics. Teaching developmentally. (8^th ed.). NY: Longman.

Reflection One

What comes to mind when you think about math? For me it evokes memories of frustration and failure. Nevertheless, I am not letting my attitude towards Math detract my children from learning this subject. On the other hand, I am determined to apply interesting approaches to teach this subject so children could really enjoy it.

Singapore Curriculum Framework for Mathematics

The framework set by MOE for Primary School Mathematics states: “The development of mathematical problem solving ability is dependent on five inter-related components, namely, Concepts, Skills, Processes, Attitudes and Metacognition.” These different components should fit together to help children gain skills in using mathematics to solve both abstract and real world problems. Some aspects of our framework are similar to National Council of Teachers of Mathematics (NCTM) but there are also differences. I personally find this framework complex but well thought out because it does not only emphasize on the content but also the processes and the outcomes of learning Mathematics. It is glad to know that our framework takes into consideration of the attitude of the learners. However children and even young adults often complain that mathematics is hard. I also understand that attitudes are shaped by what children’s experiences with learning math are like. Hence it is vital that mathematics is learnt in fun and meaningful ways. Therefore care and attention has to be given when designing lessons for children in order to build an appreciation for this subject.

The Importance of Mathematics in Kindergartens

While we get a lot of attention for our Singapore Math from primary grades onwards, it is also important to look into teaching early math concepts to our kindergarteners and preschoolers. The framework for Singapore Kindergarten emphasizes on nurturing each child holistically and the emphasis related to Mathematics is on Numeracy. Quoted from the book by Van de Walle, Karp & Bay-Williams (2013), “The process standards should not be regarded as separate content in the math curriculum.” This simply means that children should see that Mathematics plays a part in other disciplines such as arts and languages etc. Nonetheless I feel that I grew up learning Mathematics as a separate subject. This would be a strong reminder for me as a teacher to integrate Mathematics into other areas of learning. In this one week course, I hope to learn ways to introduce basic mathematical concepts to our youngest group of children.

References

Ministry Of Education. (2006). Mathematics primary syllabus. Singapore: MOE

Van de Walle, J.A., Karp, K.S. & Bay-Williams, J.M. (2013). Elementary and middle school mathematics. Teaching developmentally. (8^th ed.). NY: Longman.