Technology in teaching mathematics

I never thought of technology as an important part of a mathematics curriculum. Not until I completed this 24 hour course on Elementary Mathematics and read through Chapter 7 - Using Technological Tools to Teach Mathematics. The authors of the text suggests that teachers should "consider technology as a conscious component of each lesson and a regular strategy for enhancing student development" (P. 113). I decided to to try out the applet that was provided in the link to Pick's Theorem. I started making the different polygons that I drew on the Geo board provided in Ban Har's class. It became more interesting as I challenged myself to make variations of polygons of the same area. (This was so much quicker than drawing it on paper and having to calculate the area! =p)

Here are the 6 variations I came up with:

I also found an interesting video, on the role of technology in mathematics education by Jeremy Roschell, director of the  Center for Technology in Learning at SRI International and Ken Koedinger, professor of human-computer interaction and psychology at Carnegie Mellon University.  

What I have learnt in 24 hours.

In a span of 24 hours, I gained a great deal of knowledge on how mathematics played a significant role in supporting children's development holistically. In preschools, I feel that we tend to place alot of in the aspect of language and literacy development, so much so that mathematics is often somewhat 'forgotten'. It may not be the case for others but that is the case for me. Hence, I am glad that I went through this course as it gave me a different perspective of mathematics education.

Here are the three big ideas that I learnt after the precious 24 hours:

Big Idea 1: The 4 main uses of numbers:

- Ordinal numbers: 1st, 2nd, 3rd... ...
- Cardinal numbers: Using numbers for counting purposes Eg: 1 dog, 2 dogs, 3 dogs...
- Norminal numbers: Using numbers as a name Eg: Bus No. 14
- Measurement numbers:  Using numbers to measure Eg: 4 grams, 5 kilograms...


Big Idea 2: What can be counted and what cannot be counted.

Saying the noun after each number as you count. "1 apple, 2 apples, 3 apples, 4 apples..."
This helps children when they eventually learn addition and subtraction. 2 apples and 5 oranges will never become 7 apple oranges just like how 2x + 2y will never become 4 xy.


Big Idea 3: The CPA sequence (Bruner, J.)

According to Jerome Bruner's reasoning theory, the CPA (Concrete, Pictorial, Abstract) teaching sequence reflects how children learn mathematical concepts. When teaching mathematics, always begin with concrete representations before moving into pictorial and the abstract.
I guess if I was taught fractions through the CPA teaching sequence, I would have understood it much better! After having the concrete experience in using the patterned blocks  provided for Problem 18, I could finally figure out how one fourth of two thirds is also one sixth of a whole! The whole concrete experience made a great difference in learning!

The ONE thing that is New to me

So a square is a rectangle and a rectangle is NOT a square. Now THAT was something new to me. I am sure one of my mathematics teacher told me that before but somehow I just didn't get it. Anyway, I spent my entire journey home looking back at the four sessions that we have been through, trying to figure out the ONE thing that was new to me, that struck me during these four sessions. I thought really hard and realized that I brought home something new about mathematics after every session. Every single problem that was thrown to us, taught me something I didn't know about a Mathematical concept. I am starting to wonder how I managed to score B3s for both my E and A-maths for my O-levels. Did I really master the facts of mathematics???

With that being said, I went through all my notes and decided to go with the ONE activity that left the deepest impression. I would say, the card trick! What a fantastic way to teach children about numbers! What I really like about this particular activity was how the teacher could modify it to teach different mathematical concepts. From spelling numbers to the language of ordinal numbers, the concept of left to right, there is just so much children can learn from this activity.

Developmental Nature of Basic Facts Mastery

I was particularly drawn to Chapter 10 of the text where the author discussed about the developmental process on how students gain mastery of the basic facts in Mathematics. The three phrases in the process of learning facts are as follows (p. 172):

Phase 1 - Counting strategies: Using object or verbal counting to arrive at an answer. (That was exactly what I did when we were asked to figure out the 99th letter in Ban Har's name!)

Phase 2 - Reasoning strategies: Using known information to logically determine an unknown combination. (I know that 4 x 12 = 48, so 4 x 13 is 4 more, which is 52. ) 

Phase 3 - Mastery: producing answers efficiently (fast and accurately) (Being able to provide the answer almost immediately. What is 9 x 9? It's 81!)

I am now wondering if the reason I did well in my Secondary school mathematics was most likely because I was pretty good with memorizing and figuring out which formula to use to solve each particular problem. I don't think I really understood the problem itself. One great example would be when Ban Har posted us with the problem How many halves are there in 3/4? I stared at the problem for the longest time and thought it was 6. It turned out that I was giving an answer to another question - How many eighths are there in 3/4?. It was only after explaination that I realized that the initial problem was simply 3/4 divided by 1/2. If the problem was posted to us as  ¾ ÷ ½, I knew what to do right away - ¾ x 2 = 1½ . But is that really the Mastery of facts? I was able to solve it not becuase I understood the mathematical content. I was able to solve it because I was 'trained' to do so by my mathematics teacher - Don't ask why. And that was probably why I never did well in fractions. I skipped the entire reasoning part of the mathematical content.

Mathematics is not only about giving the right answers. It is also about the process one has to go through when solving the problem. There is no ONE correct way to solve the problem. In all, for children to be able to gain mastery of the basic facts in mathematics, they need to be given countless of opportunities in using effective reasoning strategies in the classroom.

Take for example 51 - 7. Using the standard subtraction algorithm doesn't require much thinking as long as you know they way to do it. But think about the many reasoning strategies that one can use to solve the problem without the use of the subtraction algorithm! I can count down to 50 and take 7 way (43) and add 1 more to 44. Another child may decide to use the Take from the 10 strategy, 50 is 10 + 41. 10 - 7 is 3. Adding 3 back to 41 would also make 44. I am glad to know that they have decided to move away from teaching children to use the standard algorithm to solve problems. Because like what Ban Har says, that's what something brainless like a calculator can do. Not that algorithm should not be taught at all but it should come in only after children master the basic facts in mathematics. We as teachers, need to recognize the importance of promoting the use of effective reasoning strategies to help children gain mastery in mathematics.

An interesting and inspirational session

I have to say that it was a really interesting 4 hour session on Mathematics last night. The one big thing that really struck me was how a good mathematics teacher should welcome the many possible methods to solve ONE mathematics problem. We all know that at the back of our mind, somehow or rather, we grew up in a system where students should ONLY use the method that was taught by our teacher to solve the problems. If we ever tried a different method and arrive at the correct answer, sorry, there goes your marks. 

The one thing that inspired me was the various Mathematical concepts that could be also taught through the three different lessons we went through. What's more is that these activities could be used with young children and even adults like us! It gave me a bigger picture on how a teacher could differentiate instructions in a classroom. 

My experience as a learner
Reading through chapters 1 & 2 brought back memories on how my hatred for Mathematics turned into love during my secondary school days. I have always dreaded solving Mathematics problems but my parents are really good at it. My dad coached me personally in Elementary Mathematics (E-Maths) when I was first introduced to it in school. He showed me how solving Mathematical problems could be a joy. When I got the hang of it, I started to fall in love with it! But things took a turn when I took up Additional Mathematics (A-Maths) during my third year in school. Having two Mathematics subjects at the same time, I started to dread the classes, doing tests and examinations. What made things worse was that my close friends were also weak in this subject and we were always behind the others in class. We were at the brink of giving up when this particular Mathematics teacher took over during our fourth year in school. His lessons were very interesting and engaging. He would also use every other weekend to coach some of us who were weaker in the subject. We saw his love in teaching Mathematics and we started to develop our passion for it too. We eventually saw gradual improvements in our grades and I graduated from school with Bs for both subjects!

I cannot help but agree with Walle, Karp and Williams that “families’ and teachers’ attitudes toward mathematics may enhance or detract from student’s ability to do math” (2010, p.9). For me, it was my dad and my teachers’ attitude towards Mathematics that eventually triggered my interest in it. That also explains why I decided to take up a Diploma in Banking and Financial Services upon graduation. It took 3 years for my love for numbers to turn into hate and eventually decided that I preferred the simple ABCs and 1+1.

And THAT was my love-hate relationship with mathematics.

My experience as a teacher

I have taught Mathematics to the 5 and 6 year olds, through the use of Growing with Mathematics programme in the previous childcare centre I worked with. I would simply follow through the lessons that were written, using the resources provided by the programme. I thought to myself “Boy, teaching Math was easy!” I have to confess that I have never once heard of the Principles and Standards for School Mathematics (NCTM) or the Common Core State Standards that were mentioned in the Chapter 1 (P. 2-7). It was only after reading through the chapters that I realized the Growing with Mathematics programme addresses the NCTM Standards stated under the Principles and Standards for School Mathematics. I have been teaching Mathematics to the children with a programme that addresses the NCTM Standards without knowing what the standards were up till today! Looking through the five content standards stated gave me a clearer picture of how children should learn and use mathematical knowledge.

Another thing that really struck me after reading through these chapters was how a teacher can make use of the classroom environment for learning and teaching Mathematics to the children. The following photos show a few of the Mathematics activities that I did with the children in my classroom:

The children learning the concept of sequencing through dramatizing the growth of a plant.

The children learning about numbers through reciting number rhymes.

I started reflecting on my classroom environment and how I could have created even more opportunities for children to construct their own learning in mathematics.Mathematics should not be taught only during the allocated timeslots with the use of a Math programme alone, it can be done anytime throughout the day in the classroom.