Technology - 6th Principles in Mathematics

Who can deny the importance and the usefulness of technology in teaching mathematics in this 21st century?  I am one true dependent on technological tools to help me find my answers and using spreadsheet is my favourite way of computing datas and managing my accounts.  If without it, I am as good as brain dead. Check on the link  that will enthrall you with the many ways students can explore technology to support them in solving mathematical problems. www.nctm.org/uploadedFiles/About_NCTM/Position_Statements/Technology%20final.pdf

TPACK Framework

The TPACK framework (Walle, Karp & Williams, (2013), Figure 7.1 on page 113) shows the infusion of technology to the mix of technological, pedagogical and content knowledge. A framework to support teachers with the knowledge to provide an effective pedagogical practice using technology to enhanced their learning environment.  

No doubt, using technological tools in learning mathematics is useful,  but we have to bear in mind that "it cannot be a replacement for the full conceptual understanding of mathematics content.  Teachers strategic planning when and how the tools can and should be used is still essential for the support and to ensure that learning takes place.  

Though I am not favorable that parents nowadays indulge children with IPads and tablets without proper facilitation of learning, but I never once underestimate the power of internet resources.  I am certain that with proper guidance to access good websites, children will expand their knowledge and motivated to excel.  

Final Reflection

Learning Mathematics has always been my weaknesses and therefore teaching it has never been my forte.  Nevertheless, I have acquire plentiful of "Big Ideas" from this module and picking 3 is not difficult.  I could probably go on and on if not restricted to write only 3 learning points. 

So I would say below are some of the BIG IDEAS that I have learnt throughout the whole week:

1) The use of countable verses the uncountable nouns. Example, counting 1 chairs, 2 chairs are countable nouns but counting 1 chair, 2 tables are uncountable nouns.

From this lesson, I have learnt that the use of language interferes with the way Mathematics are taught to the children. In our context where our children are still in the emergent stage of learning mathematics, it would best avoid using interfering or distracting variables for counting that would create confusion.

2) Jerome Bruner's Concrete-Pictorial-Abstract (CPA) Approach

Learning the importance of this approach to sequence activities to help children learn mathematics is critical.  Using concrete to make relations to solving mathematical problems in real-life situations connecting to their prior knowledge is essential. Followed by using pictorial in diagrams, graph, drawings, as visual representation and finally abstract where workings with symbols to provide a shorter solution to problem solve the mathematical operations.

3) Differentiated Instructions

It had always been a fast forward experiences for our children when it comes to learning Mathematics.  We had never had enough time for differentiated learning as we have a syllabus to complete.  Teachers were not given time to reflect and exercise observation to notice the weaker students to help them to assimilate what was taught in the class, much less, give them the opportunity to explore the possibilities in getting answers. At least it happened in my school's setting.  But now, I would be able to help my teachers be aware of the CPA Approach to plan lessons that could accommodate learning for both struggling and achieving students with the intention of using mathematics to train their minds.

Most import message I gather from this course that my goals as an educator in mathematics are to be able to prepare children's mind to visualize, have good numbers sense, to see patterns and to develop their spirit to enable them to have the ability to see perspectives in search for possibilities.

My 2 questions for extended knowledge would be:

1) My school is beginning to see an influx of foreign students, especially from China who came with no  or little knowledge of English.  Most of them arrive to join us in the K2's level.  How is it possible for them to solve a Math story problems?

2)  Take the example of Patterns, is there an appropriate length of study for particular concept before we move on to the next? 

Learning to Accept that A Square is a Rectangle (19 July 2012)

The most impact full part of Thursday's lesson was being told that  "a square is a rectangle" with evidence on the websites.  I was dumb founded.  I just got to find out about this fact after living off half a century of my life.  All in the whole cohorts agreed that we weren't being told of the fact, so I am convinced that our teachers are not doing their job well in our times.  So for the past years we had not been fair in assessing children's ability to recognize square and rectangle.  We hadn't given them chance to explain their answers but quick to mark them down if their had recognize square as a rectangle or vice versa.  But nonetheless I was being pacified to know that it isn't necessary to let children about the definition of the shape and if I do, I would probably confused them.

Which brings me to understand the Van Hiele 5 Levels of geometric thought (Walle, Karp & Williams (2013) p. 403 - p. 406):

Level 0: Visualization - which we is the level our kindergarteners are where they can only classify shapes based on their appearance.

Level: 1: Analysis - the next level of learning which will occur when they are in P4 & P5 where children will learn that a collection of shapes goes together because of their properties.

Level 2: Informal Deduction - only taught in P6 onwards where children will observe shapes beyond their properties and and focus on logical arguments about it.

Level 3:  Deduction - students are taught in Sec 3 & 4 where they work with abstract statements about geometric properties and make conclusions from logic.

Level 4:  Rigor - The highest level of the van Hiele hierarchy, a mastery level where "there is a distinctions and relations between different axiomatic systems."  One that I, myself  could not explain deeper and I have not gone through this stage.

Nevertheless, through the class activities of the day, the main focus is to leave with the thought that I am here to learn NOT to teach only the content knowledge but preparing children to learn the content and moulding them to become creative thinkers to open their minds; to look at problems in different perspectives; and solving them in many possible ways. 

Day 2 & 3 (17 & 18 July)

Another two days of insightful learning in class, solving problems that I had never attempted before. While we attempted solving the problems, I noticed that we have yet to filter away from our didactic thinking which we were being moulded from young.  The activities were open-ended alright, but we were not trained to see it that way.  I hope by the end of this module, we will be able to do so.

The Math Trail is really interesting and I know children will definitely enjoyed it.  It had gotten all of us in a group to analyze, discuss and listening to each others opinions and perspectives.  Nobody knows the correct answers to all the problems but only working out on the possibilities.  A guess this activity will entice children to love mathematics.

Coming back to class, the fractions to me is a killer.  Though it did refreshed some memories but I find it tough.  I told myself to focus on the learning points when teaching the children is the main aim here. Examples like how to use appropriate mathematical terms when teaching, the precise ways of asking the questions and allowing children to look for probable ways to give their answers. 

What I find close related to what I am current practicing is from reading the text on chapter 8 since   its the emergent stage of learning numbers.  After we have adopted Growing with Math, teaching aides such as ten frames, missing part cards, number charts, etc... including the problem story approach in chapter 9 are no longer strange to me.  In fact, the learning of number concepts and number sense is what we have been practicing for the last two years. Children had been learning to count cardinal numbers, ordinal numbers, number bonding for addition, learn multiplication through number groupings and division through sharing.

However, the readings of these chapters had extended my knowledge with  how I could used the suggested activities, not only to bring children to next level of learning but how to use them for differentiated teaching as well.

Day 1 (16 July 2012) - Interest & Inspiration

The feeling was great walking into yesterday's class.  I have  heard plentiful about this module from my friends from the previous BSc classes.  I was expected to learn a lot in class and they were right, I did!

As I have mentioned before in my reflections that I was sceptical about my ability in understanding Mathematics, it's beginning to gradually diminished after last night's lesson.  My fear has began to fade a little and I am enjoying every minute of the lesson.  The math problems posted were intriguing and had all of us busy cracking our heads to find solution. 

The ultimate learning points from the lesson was geeting in-depth knowledge of how children learn Mathematics and learning the "BIG IDEAS" in Mathematics.   Of course having to know about the CPA helps me to remember how children learn depending on their development stages from concrete to pictorial to abstract. Not forgetting about variability to carefully provide appropriate materials for learning.

The Obedient Card activity has inspired me most to wanting to know more about how one mathematician could perform "Magic Trick".  I was impressed how the numbers could be arranged to created so much learning to take effect.  Being a non logical thinker, if not for my peers in class, burning our brain cells together, I would have given up the task trying to figure out how the cards should be arranged. I guess that was what it meant in the first two chapters of the textbook lesson which wrote about encouraging sharing of ideas to solve problems and accepting the different methods of getting an answer.   I felt less threaten working with my peers.  In fact I felt Math is fun, at this moment, and if that is how I felt, what more could children feel? 

What interesting is, I, a Math hopeless, was able to come up with a method to solve the first activity, which later some of my peers are using it and called it "May's Method". Awww... so sweet.  With this, I end my thoughts for the day and I will be looking forward to more excitement this evening and grasp as much learning as possible.

Chapter 2 - My Reflection

How so true that Maths and Science intertwined, one compliments each other.  Interesting though, but it always makes me go crazy about it.  For example, height, weight, length and sizes, which to teach first, so that I could make the connection?  It sure is mind boggling for me.

This chapter got me reflecting on how I, when I was a teacher, wasn't providing much opportunity for risk taking, reasoning and much less, sharing of ideas.  I lacked using mathematical terms to encourage active thinking and to boost engagement of activity.  I guess that was why many children had turned too dependent to be provided with solutions."I was too quick to provide answers." It had happened 10 years back then, imagine how much damages I have caused to the children under my care.

Now, history should not be repeated so I would be looking forward to learn the strategies to teach children to become problem solvers and critical thinkers with Mathematics.  At the same time, to impart my knowledge to my teachers, whom I believe, many are still practicing how I did it before.

In further reading of the text, when mentioned about the constructivist and sociocultural theories, I do agree that we need to have these two theories to guide us to see how children learn through constructing their own  knowledge with the facilitation of an adult.  These two must go hand-in-hand to make learning possible.

This reading also gave me additional insights of the importance of teachers and their understanding towards the implications of teaching mathematics in our K levels.  What we had done so far was building knowledge from prior knowledge, the rest of the recommendation such as:

• creating opportunities for students to interact with teachers and peers which allows them to be engaged in reflective thinking;
• encouraging multiple approaches for children to demonstrate their understanding through sharing  ways to solve a problem with different solutions;
• honoring diversity to recognise each learner is unique with different collection of prior knowledge and cultural experiences;

was hardly even practiced in our Math teaching environment, which we really had to work hard on it.

Further to help our preschoolers understand mathematics, of course, would be best with the use of manipulatives where they get to have hands-on experience, a visual and tangible way to construct  knowledge and understanding.  But sad to say that we, as teachers, have often done too much demo when conducting the lesson, mainly due to time constrain for analytical thinking to take place.

Lastly, I would confess that I do not have Logical-Mathematical Intelligence. So much so I wanted to achieve what I have said would be a major task for me?  Honestly I enjoyed watching and be impressed by people who could solve interesting mathematical solutions but I am the least bothered to find out how they have done it and gave up easily when asked to solve such problems.  I hope by the end of this studies, I can be enlightened and moved forward to provide better Math programme in the school to give children a smoother transition from Kindergarten to the Primary school.

Chapter 1 - Reflection

Teaching Mathematics to me is about developing thinkers and problem solvers, a task that will be ever daunting to me since I became a preschool teacher in this field.  I remembered myself as being so afraid and so poor in Math during my school days and now to be a responsible person to be teaching the young the subject.  “Hey May, are you in the right mind?” a question I often ask myself.  Nevertheless since I set my foot on it, I have to continue walking the path to learn how I can do it, how I should do it, and very importantly understand why I should do it.

While reading through the first chapter of the text, what I internalized most is the 6 principles, which gave me good insights into what I should be considering while planning a Math programme for children.  I felt well informed with the guide that looks into each aspect of support of learning and teaching simply by understanding the followings:

·      that all children should be given the opportunity to learn mathematics.

·      provisioning a coherent curriculum that builds around “big ideas” (pp. 2 & 7).

·  building teachers’ capacity to understand the mathematical content. before teaching the mathematical concepts to children; to ensure that teachers understand how children learn so as to be able to plan meaningful instructions for learning.

·     the importance of the learning process where children are able to assimilate instructions and learn with understanding to think and reason mathematically.

·      observing various techniques for assessing children’s learning.

·      finally, acknowledging the effect of using technology as a support for learning mathematics.

As being a curriculum planner, I literally would focus on the curriculum focal points.  The emphasis in this text is the importance of a coherent curriculum, which directs me to think of the current adoption of the math teaching approach from Growing with Mathematics (McGraw-Hill Educational Product).  

It had been two years since we have taught Math with its resources and followed as closely as possible to the approach but I do not feel there was much of a impact it had done for our children.  As I reflected now, it could be perhaps our teachers are not mathematicians and do not see themselves as one and therefore, our children does not turned out to be mathematically inclined.

Nevertheless, I would hope to be able to encourage our teachers after going through this module, to give them and myself the confidence to deliver meaningful math lesson just by being able to demonstrate persistence when conducting mathematical investigations; display positive attitude toward the subject; most importantly to be able to reflect on their own practices as suggested on page 10 from the text.