Reflection - Class SIX

Arithmetic is where the answer is right and everything is nice and you can look out of the window and see the blue sky - or the answer is wrong and you have to start over and try again and see how it comes out this time. - Carl Sandburg

Mathematics was my worst subject back in school. I thought it had to do with my uncommon kind of common sense. But I realised as I went through the course that for the most part, my teachers were doing their job/their best, to deliver the lessons and results. And because mathematics spirals up from Primary school to Secondary school, students who couldn't cope with the learning of math actually didn't manage to make sense/connection out of what the teacher had taught. Then, it leads to losing self-esteem thus closing their doors on math eventually.

I finally understood why mathematics is a science of concepts and processes that have a pattern of regularity and logical order. In fact, it is actually just like mastering comprehension skills in attempting comprehension passages. All you have to know is how to infer, analyze and then problem-solve. I really appreciate that Dr. Yeap has enable me to see Mathematics in a new light - like How Big is a Foot actually puts the concept of measurement into simplest form for reading, conserving and understanding.

Are we, as educators, ready to change the world one child at a time, and help them learn mathematics the right way? Are we ready to help them think about problems and solutions rather than by helping them to do it the way we were taught? Are they merely empty cups waiting to be filled with concepts/knowledge without experiencing the whole fun of discovering learning?

My answer:
3 flights of 16 steps
1 flight of 14 steps
The height between each step is about 14.5cm

(3 x 16) + (1 x 14) x 14.5cm
= 48 + 14 x 14.5
= 62 x 14.5cm
= 899cm

Final answer: approximately 899cm.

Reflection - Class FIVE

Pure mathematics is, in its way, the poetry of logical ideas.
- Albert Einstein

I was sort of lost while going through the division of fractions, so I ended up working on an origami crane. But there was a sudden thought-provoking moment that got me thinking whether there was any mathematical sense in origami. So this was what I found:

Some classical construction problems of geometry — namely trisecting an arbitrary angle, or doubling the cube — are proven to be unsolvable using compass and straightedge, but can be solved using only a few paper folds. Paper fold strips can be constructed to solve equations up to degree 4. (The Huzita–Hatori axioms are one important contribution to this field of study.) Link: Origami & Geometric Construction

And then I further read on about Pick's Theorem that provides a method to calculate the area of simple polygons whose vertices lie on lattice points—points with integer coordinates in the x-y plane. The word “simple” in “simple polygon” only means that the polygon has no holes, and that its edges do not intersect. Refer to this article: [Pick's Theorem].

The 4 content goals for geometry, p.400:

1. Shapes and Properties
2. Transformation
3. Location
4. Visualization

Wow! So much for mere shapes of circle, triangle, rectangle and square. =\

Reflection - Class FOUR

So if a man's wit be wandering, let him study the mathematics; for in demonstrations, if his wit be called away never so little, he must begin again. - Francis Bacon, "Of Studies"

Bruner's CONCRETE-PICTORIAL-ABSTRACT approach is the basis for all Early Childhood educator. The notion of using concrete materials and giving concrete experiences is a long standing method of working with children.

To "teach" or present a concept to a child, we must first present a basis for the child.

Instead of merely presenting a rule, as many schoolbooks do, a better way is to teach children to visualize fractions, and perform some simple operations with these visual images or pictures, without knowingly applying any given 'rule'.

If a child is able to visualize fractions in his mind, they become more concrete - not just a number on top of other number without meaning. Then the child can estimate the answer before calculating, and evaluate the reasonableness of the final answer, and perform many of the simplest operations in his head.

Of course textbooks DO show fractions with pictures, and they DO show one or two examples of how a certain rule connects with a picture. But that is not enough! A better way is to make kids do lots of problems with fraction manipulatives - and DRAW fraction pictures for problems. That way they will form a mental visual model and can think through the pictures for simple problems.





As an educator, we would have to be aware of the end goal and work backwards to see the gradual steps that each child have to tale before attaining these goals.

Reflection - Class THREE

I know that two and two make four - & should be glad to prove it too if I could - though I must say if by any sort of process I could convert 2 & 2 into five it would give me much greater pleasure. - George Gordon, Lord Byron

Tonight we had a guest lecturer, Peggy Foo. In my opinion, the gist of the lesson today was Mathematical Investigation and Lesson Study.

I learnt that Mathematical Investigation has multidimensional content; it is open-ended, permitting several acceptable solutions; an exploration requiring a full period or several classes to complete; centered on a theme or event; and often embedded in a focus question. In addition, a mathematical investigation involves a number of processes, which include - researching outside sources to gather information; collecting data through such means as surveying, observing, or measuring; collaborating, with each team member taking on specific jobs; and using multiple strategies for reaching solutions and conclusions. The best part of it is that the teacher can carry this out at the beginning, middle or ending of the lesson.

Lesson Study is a big word, but is an IN-word for a teaching professional. I was googling about it online when I come across this page on [Lesson Study]. Lesson Study is a form of school-based professional development and teacher-initiated instructional improvement. From my best knowledge/understanding, it could be like the Learning Circle I am currently involved with back in school. We look at an aspect of our teaching as a group, then try to identify the cause of a problem (that may not be exactly a problem, but an area that we can improve on), then experiment with ways that we could improvise on the lesson or the way the lesson is carried out.

Reflection - Class TWO

The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions. - Ronald L. Graham

Over the many years of what our teachers have taught us, and the experiences that we have had to get us out of the rain, find where the berries are, and keep us from getting killed, we forgot or we never learnt that mathematics is fundamentally about problem solving and thinking. But the best of tonight's lesson was to rediscover the wonders of developing problem solving and thinking skills through

• Generalization (looking for patterns, relationships and connections)
• Visualization
• Communication (reasoning, justification, representation )
• Number sense
• Metacognition
  
  
  

More important than arriving at the right answer, a student should be taught the ways to engage in thought-processes to understand the problem and justify the rationale behind a solution they have found. Just like at a debate! I found the interconnection of the 5 elements even more clearly when I read through the [Primary Mathematics Syllabus]. There is so much more than just computation to Mathematics and getting the right answers the way your teacher have taught you. As an educator in Singapore, more than often, we forgot the essence of education is to educate children to educate themselves throughout their lives - because results are more important to some many parents.

Indeed, if our brains hadn't evolve, it would be better to look at things in a hundred thousand dimensions.

Reflection - Class ONE

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. - John Louis von Neumann

I always thought that mathematics is too complicated for a simple-minded me. Well, I think I am very wrong. It has never occur to me that mathematics "is more than completing sets of exercises or mimicking processes the teacher explains". I admit, when Dr. Yeap wanted us to figure out and problem-solve the Name Problem, I was rather reluctant to try. I was more of really waiting for the answers and the method to be given. But when we continued further to solve problems like Spelling Cards, it was like ahh-ha! There and then, I began to think that mathematics is like Kinder Surprise. Some people may be quick to figure the problem out, while others may need more time to see it through; essentially, it is how one chooses to perceive the "problem" and solve it to achieve the end goal. At the end of the process, you will realised that you have gained more than solving the problem and achieving the answer. Because if you can't fix the toy from the chocolate, you won't know what kind of surprise you're in for!

It's a Kinder Magic




I fully agree with Dr. Yeap on the point that teachers need to have a professional knowledge of what they are teaching. This would allow us to use the appropriate vocabulary to explain mathematical points/concepts when conversing with parents or other educators. (And to perhaps convince parents that we actually KNOW something!)

Chapter 2 - (The Language of Doing Mathematics)

explore. justify. construct. develop.
investigate. represent. verify.
describe. conjecture. formulate. explain.
use. solve. discover. predict.

(chapter 2. pg. 14)

Mentioned in the book, to create a setting where students are doing mathematics means a shift in the tasks given to the students and how classrooms are organized for mathematics lessons. It should involve creating a safe risk-taking environment where students should not be afraid to make mistakes, and ideally share their ideas and defend mathematical ideas. This thus, leads us to the role of a teacher in creating "the spirit of inquiry, trust and expectations" and helping children attain mathematical proficiency - (1) conceptual understanding, (2) procedural fluency, (3) strategic competence, (4) adaptive reasoning & (5) productive disposition.

I guess the most basic process for a teacher is none other than: Plan. Do. Reflect.

Chapter 2 - Reflection

Science is a process of figuring out and making sense. Mathematics is a science of concepts and processes that have a pattern of regularity and logical order.

These sentences on the first page of the chapter stopped me for some thoughts. It has never occur to me that mathematics could be that easy to understand. For one thing, I was never good with numbers. But I do realise, as I continued reading the chapter, that my perception of mathematics would affect the way that I would engage students in learning it, or worst, the environmental support that I may be able to offer them.

• A learning theory is not a teaching strategy, but the theory informs teaching.

I wondered why as a student, I have never thought or tried to see mathematics from a different perspective. Like the recent trend of applying differentiated instructions in classroom, I realised that mathematics could also be instructed this way. As a learner, I was afraid of maths because I could never understand why some topics have to be done in a certain way, and whether by solving the problem in some other ways would I be penalised or faulted. In a metaphorical term, learning should never be a cup-filling process. Students are not empty cups waiting to be filled. Rather, they should be provided with supportive environment to discover learning, and be scaffolded. Students should be encouraged to explore the different ways a question can be solved, and to realised that it is okay to make mistakes.

Chapter 1 - (Influences & Pressure on Mathematics Teaching)

State Standards and Curriculum determines what, when and how children will learn the mathematical concepts that is typically developmentally appropriate for them. It is a rather saddening issue because look at the number of tuition classes students in our local schools have to go through every week. They have remedial lessons, supplementary classes and tuition classes that almost generally drill them in mathematics. The only within the control of a teacher is the way in which he/she carries out the lesson and the methods of learning to make the lesson perhaps more interesting and engaging for the kids. That said, it is still not likely for them to spend more time in engaging students in fun activities during curriculum times because there is limited periods of lessons allocated for each academic subject...

I really wonder, do we want to just train kids in solving the problems correctly the way we want them to, or nurture them to be thinkers and engage them in thought-processes and discover learning for themselves?

Chapter 1 - Reflection

How many of us are born genius who can do mathematics without fear/doubts?

In order to become a mathematics teacher, one must have a passion for math and the perseverance to learn, do and practice math no matter how tough it might sometimes be. And like the words of Albert Einstein, "Pure mathematics is, in its way, the poetry of logical ideas". I am not an avid fan of mathematics. In fact, prior to learning how fun mathematics can be whilst doing my diploma, I would avoid mathematics. (I always thought it was a miracle that I managed to get B3 for E. Maths at the O's.) But I do like that in this chapter, mathematics is analysed right up to its simplest form - the most fundamental purpose and ways to instruct math. I particularly like the technology principle for school mathematics! Other than allowing calculators which makes numbers look friendlier, the use of computer applications or the latest iPhone app is a fun way to engage students (like me) in growing an interest and liking for mathematics.

For example - Math Bingo which is exactly what is sounds like. This fun app was developed by an elementary school teacher, and it shows.