Powered By Blogger

Wednesday, April 20, 2011

‘Mathematical Philosophy’ - My beliefs on how to teach mathematics effectively

Mathematics is “not my cup of tea.” It was irritating to know that I would have to sit through a series of lectures that would improve my skills in teaching mathematics. My first thought was that I wouldn’t learn much from the lectures. How wonderfully wrong I was!

Mathematics is defined as ‘the study of the measurement, properties and relationships of quantities and sets, using numbers and symbols. (n.d. 2004) This definition is a classic example of people’s notion about mathematics – boring, scarey and only for ‘super-brains’.

For countless generations, “mathematics teaching and learning practices…centre on memorisation of facts, and practice of pre-set meaningless procedures, which promote a view of mathematics as lacking creativity, imagination, or critical thought.” Sandra, F., and Len, Sparrow,. (2007).

It was expected of students to not only learn by rote the times tables and equations but also to work on algorithms without really understanding the logic or reason behind what they were doing. Students were not allowed to ask questions as to why they learnt what they learnt and why there was no other way to do what they did. Any questions were either met with stoic silence or a curt ‘it’s always done this way’ remark. This was how I learnt mathematics during my primary as well as secondary education. No wonder, by the time I reached Higher Secondary, I didn’t understand anything about what I was doing in class.

I believe I would have continued hating mathematics if it weren’t for my tenth grade teacher who inspite of limited resources and no access to the latest teaching methodologies, still attempted to make math interesting, logical and realistic. I learnt more about area, time, angles and space from him than I ever did from all the other mathematics teachers both in primary as well as secondary school.

Marilyn Frankenstein has summarised several other people’s and my perception of mathematics in her article ‘Mathematics Anxiety: Misconceptions about Learning Mathematics’. She said that the features of current maths curricula are as follows:
- rote calculations
- memory dependence
- unmotivated exercises
- spurious applications of calculation strategies
- authoritarianism in mathematics education
- tests which assume mathematics can be divided into tiny water-tight compartments
(Frankenstein, M., (1989). pp. 183 – 186)

There are certain questions teachers will have to answer before understanding the teaching methodologies for mathematics and they are:
- Who are our students?
- What are their requirements?
- What are their skill sets?
- What are the learning tools available to them?
- How do they like to study?
- How are they influenced?
- What are the methodologies in which they can be taught?
- How would these methodologies help them?

Before we focus on the current teaching methodologies and resources available to pre-service mathematics teachers, it is of primary importance that we focus on who our students are. Times have changed and it does not make sense to live in the past. We as teachers need to understand whom we are teaching if we have to improve our methodologies.

Though most teachers would not like what I am about to say, I will have to emphasise that we need to look at teachers as ‘knowledge service providers’ and our students as ‘end-users’ or ‘customers’. I am well aware of education being a noble profession but I prefer to think of it as any other industry for quality management purposes. Like all industries governed by the 6 Sigma and Lean Toyota project for service and quality assurance, we will have to make changes to service levels so as to assure quality levels are effectively managed.

Our students are our customers and their world is very different from the one that we live in or grew up in. I liked what Sir Ken Robinson said in a Youtube video played during an Issues lecture, “what are we educating them for?” Their world is changing faster than our own. It is more multi-cultural, multi-lingual and global than ours. Beare predicts the future world of our students in his paper on “From an old world-view to a new” by looking at various factors such as demographics, economic growth, environment, consumer patterns, impacts of consumerism, resource availability, etc in the future. (Beare, 2001, pp. 11-17).

Based on Beare’s statement it is possible to understand that ‘the nature of what is to be studied has also changed’. (Booker, G., Bond, Denise., Sparrow, Len., Swan, Paul.,. (2010). p. 7). There are several reasons why students will have to be proficient in numeracy. These external reasons are as follows:
- intellectual advancement in several arenas requiring an understanding of numbers and their functions
- technological growth both in infrastructural and cyber development
- economical instability requiring better understanding of market figures and functions
- environmental degradation that requires numerical knowledge of replantation and natural resources management systems to circumvent the negative impacts
These are just a few areas where mathematics will play a major role in the future lives of our students. However, as teachers, we will have to ensure that students understand the pervasiveness of mathematics in all walks of life, i.e. ‘an innumerate citizen today is as vulnerable as the illiterate peasant of Gutenberg’s time’ (Steen, 1997)

Students learn differently from before. They have access to better learning management systems and the world of knowledge is literally at their finger tips. Everything that has been discovered or is in the process of being discovered is available to them at the touch of a finger tip. As teachers, we will have to improve our technical skills so that we can use the same tools as our students to teach numeracy.

There is a process to follow when teaching mathematics as stated by Alistair McIntosh in ‘When will they ever learn?’ (McIntosh, A., (1977).) These can be enumerated on as follows:
- Don’t start formal work too early: “It is the experience of many good teachers…it is found to be unnecessary before the sixth year has passed…to do any formal Arithmetic on slates” (Reports, 1895/6)
- Use learning materials and start from practical activities: “…examples of any kind upon practical numbers are of very little use, until the learner has discovered the principle from practical examples.” (Margaret Brown, 1977, p. 10)
- Give children problems and freedom initially to find their own methods of solutions: “If a child be requested to divide a number of apples among a certain number of persons, he will contrive a way to do it, and will tell how many each must have.” (Margaret Brown, 1977, p. 10)
- Children must have particular examples from which to generalise: “When the pupil learns by means of abstract examples, it very seldom happens that he understands a practical example the better for it; because he does not discover the connexion until he has performed several practical examples, and begins to generalise them.” (By a teacher of youth, 1840, p. IV)
- Go for relevance and the involvement of the child: “When children explore for themselves they make discoveries…” (Curriculum Bulletin No.1, 1965, p. 1)
- Go for reasons and understanding of processes. Never give mechanical rules: “…when children obtain answers to sums and problems by mere mechanical routine, without knowing why they use the rule…they cannot be said to have been…well versed in arithmetic.” (Margaret Brown, 1976, p. 16)
- Emphasize and encourage discussion by children: “When children explore for themselves, they make discoveries which they want to communicate to their teacher and to other children and this results in frequent discussion. It is this changed relationship which is the most important development of all.” (Curriculum Bulletin No. 1, 1965, p. 1)
- Follow understanding with practice and applications: “When the pupil learns…he does not discover the connexion until he has performed several practical examples, and begins to generalise them.” (By a teacher of youth, 1840, p. IV)
There are various methodologies for teaching mathematics and they depend on what is being taught. I would prefer to use the same teaching process as the one I used when teaching literacy to young adults/mature age students in India. I will start each lesson with a ‘Lead In’ activity that gives students an idea/clue of what the day’s lesson is all about. Based on this clue/hint, there is the ‘Mathematics for Gist’ activity that focuses on a broad over-view of the topic. The next ‘Mathematics for Specifics’ activity focuses on the appropriate area of learning for that particular day. The learning acquired is then determined on a daily basis through ‘Post specifics-activity tasks’. The assessment/work-sheet will help me to gauge the learning curve of each student in the class.

The one lesson that comes to my mind when thinking about the above process is the one regarding an introduction to algebraic expressions using the ‘Leap Frog’ activity. That was the first time I realised the potential behind teaching mathematics. I had so much fun coming up with the algebraic expression that I realised I would like my students to enjoy learning mathematics too. I had re-discovered the joy of learning mathematics as when I was taught by my favourite mathematics teacher. I believe that it is important for students to move from a general idea to a more specific understanding of the mathematical discipline that is being explored for the day.

Based on what I have just said, I can say that my beliefs are a combination of both ‘Content and clarity’ and ‘Content and understanding’ as stated by Beswick (Beswick, K., 2006, p. 19). I believe it is important to increase my knowledge about the subject. Also, it is important that I should constantly update myself about changes in teaching principles and methodologies. I believe that sequencing of information is important since there is a specific reason why we have been teaching basic algorithms first before moving onto fractions or algebraic expressions. However, I would be glad to hear about alternative solutions from students as well as teachers. I do know that students might not know everything and there will be times when the answers will have to be given to them.

I believe that it is mandatory that ‘pupils are aware of different methods of calculation and are able to choose methods in relation to their effectiveness and efficiency in solving a problem’ (Askew, M. et al, n.d). It is important that I emphasise ‘the complementary nature of teaching and learning and valued classroom activity, which involved pupils working together with other pupils and teachers to overcome difficulties and to reach shared understanding’ (Askew, M. et al, n.d).

The road to being an efficient and effective Mathematics teacher involves self-discovery, self-learning and self-proficiency. If this will enable even one student to love Mathematics and discover maybe the greatest ‘Primary Number’ or even a new method for teaching ‘Mathematics’ then I am willing to make the required changes. Hopefully no student of mine will ever dream about “the final test involves escaping from a nightmare, which is depicted as a lifetime of mathematics problems written on a blackboard” (Swan, Paul., 2004, p. 501) and instead will learn to love this subject as much as I have come to love it.

What power do I use as a teacher?

This morning I was speaking to my sister and she told me about an incident in a school at Geelong. An Indian boy, now aged 6 years, had been enrolled in a school at Geelong last year. The boy’s completed his kindergarten at an International school in Bangalore, India. His communication skills, numeracy understanding and literary skills are commendable for a child of his age. However, his parents have seen a marked decline in his social skills a year after joining this school. Though he was commended for his ability to take on leadership roles at his former school, they have been made aware of his unwillingness to interact with his peers or with his teachers. It was only after several sessions with the parents and a teacher from his current grade that they got to know about an incident that had happened last year in the school. The boy not knowing the exact words for excusing himself had wet his pants while in class. Not only did the teacher pull him up before the entire class for this but also asked him “Is this what you do in your culture in your country?”

I am placing my indignation aside and wondering what would I have done if I had been the teacher? How would I have handled the situation? Which power that I have would I have used? This is a situation that requires tact and skill at drawing attention away from the boy’s embarrassment. Do I have the knowledge and the expertise required to do this?

I am aware that the most relevant of all the powers to be used here would be referent power. However, how would one use the right words and actions to make this boy feel less awkward or guilty? How would one draw the other students’ attention away from a situation like this? And how much time did his former grade teacher have to build rapport and understanding with the boy?

I really would like input on this since I am desperately trying to help this boy get back his confidence and love of learning that he had before. I am trying to do what Jaime Escalante would have done, i.e. help a student believe in himself once again though the odds seem stacked against him.

An imaginative approach to learning

What would a person do if he/she were to win the lottery? Or, what would a child do if he/she were to get a box with the most fantastic toys imaginable in it? I guess that’s how I feel right now.

I have been reading ‘An Imaginative Approach to Learning’ by Egan and I feel like I have discovered a treasure trove. In this article are tools that are extremely useful for teaching English to Speakers of Other Languages.

It was an eye-opener to see how I could use tools such as stories, metaphors, binary opposites, rhyme, rhythm and pattern, jokes and humour, mental imagery, gossip (yes ‘gossip’), play and mystery to teach not only the usage of the language but also the use of the language.

I hope to use these tools in my country since it is of primary importance that students are comfortable with ‘oral skills’ before they can move onto learning the ‘theory’ behind the language. Egan says in this article “No one learns to speak alone – or to read and write or to think using theoretic abstraction”. I have always believed that students should be involved in group activities since two minds or more are better than just one. Egan says that learning occurs through shared knowledge or activities.

Also, he says “Be suspicious of the simple claim that young children and people in oral cultures are ‘concrete thinkers.” I have believed that it is important for students to use their imagination as much as possible so that they can improve their communication skills.

There’s so much that a student can learn by narrating a story or discovering the joy of writing a poem, or even creating a play that is a spoof of the latest movie that he/she has seen.

Where would we be if people hadn’t questioned the phenomena that occur in our planet everyday? I am looking forward to asking my students to write out their own story about ‘their’, ‘there’ and ‘they’re’. Who knows, maybe, I will have the next George Lucas or Luciano Pavarotti in my class?

Monday, April 18, 2011

Theories as a guideline for understanding cognitive development of children in an acadmic environment



Lev Vygotsky believes that it is important for the teacher to facilitate the child from the level of independent performance to the level of assisted performance through the zone of proximal development. Vygotsky’s Developmental Theory varies from that of Piaget in this one aspect. He gives a more important role for society’s influence in a child’s development than Piaget did. Indeed, Vygotsky believes that peer learning and teacher facilitation will enable a child to develop better.

Marsh states that it is important that the teacher gives tasks that are sufficiently difficult for the child to perform so that the child is motivated to achieve learning either by asking his/her peer or the teacher. The teacher’s instructions should motivate and not de-motivate the child from learning. Also, these tasks will be easier to perform if the child is helped through the zone of proximal development by the teacher.




Keeping in mind Vygotsky’s theory and Gardner’s Multiple Intelligences theory; it is possible to involve children with learning activities that will enable them to develop their individual intelligences through cumulative effort. If one were to include activities that will be challenging, students can learn to strengthen those intelligences that are relevant to themselves.



In order to ensure that the activities within the classroom will allow for multiple intelligence progression, it is imperative that the teacher use Bloom’s taxonomy so as to monitor whether he/she has a healthy mix of activities that will cater to the needs of each child’s intelligence quotient in the classroom.

Piaget’s theory of Cognitive Development lacks only one aspect and that is – it does not address the issue of society’s role in the development of a child’s knowledge and mental aptitude. Neither did his theory state the role of culture in the learning acquired by a child.

These theories are excellent as a guideline to understanding cognitive development of a child within the academic environment. However, as Barrow (1984) stated ‘these theories and generalisations have not been empirically validated’. So, it is wrong to take them to be the Gospel when it comes to understanding Cognitive Development amongst children. Instead they can be used as guidelines to improving the learning environment in our schools so that children can improve their knowledge and develop their personality so that they can handle their future better.

Saturday, April 16, 2011

Making a difference

I have learnt three very important facts this week, and they are related to the statements/questions:
- “Think about”
- “I would correct my mistakes”
- “So, you want to be a teacher?”

When I walked out of the lecture theatre on Monday, the words that were ringing in my mind were Dr. Howard Nicholas’ “THINK ABOUT”. The emphatic tone that he used when he said those words, made me realise that there are a lot more questions that I need to ask myself so as to improve my skills as a teacher.

Not only did Dr. Nicholas ask us to think about our students and the schools that we will be working in but also he asked us to think about our role as teachers. How prepared am I to take on the responsibility of laying the foundation for future learning in the minds of children who are not only eager to learn in a creative manner but also very impressionable and vulnerable - emotionally and mentally?

This is a role that I cannot take in a flippant manner. It involves constant self-monitoring and constant raising of the bar for myself. Like Jaime Escalante said, “Every year I would correct my mistakes.” His statement reminded me of what my Assistant Manager said when I was working as a Corporate Communications Trainer, “You stop growing when you stop listening to feedback”.



The learning or rather the gaining of knowledge never stops. I would once again have to refer to last week’s learning. If I have to prepare my students for a future that I’m unaware of, then I’ll have to constantly learn and improve myself. There’s no room for mediocrity of teaching nor is there room for ‘taking a break’ from learning for me.

Listening to “Poison Berries” by Penny Ikinger made me realise that we do live in a world where we have lost the optimism and romanticism of the past. But the pessimism and realism of the words, “You’d be better off dead” can be changed if I as a teacher work on realising and improving myself in my areas of concern.



During the workshop discussion, I listened to my colleagues talk about their experiences working in schools in Australia. It was very thought-provoking and I was glad that I got an opportunity to listen to them. Their statements helped me to understand what I needed to do so as to answer the question that was raised in my mind by Dr. Howard Nicholas and the workshop’s questions.

Am I sure that I want to be a trainer? Oh, yes! I would love to work with adults who are attempting to re-start their lives and their careers by improving their communication skills in English. This week’s session and discussion has only helped to re-enforce this thought in my mind.

Why innovate?



I totally agree with Sir Ken Robinson’s statement that “education will take us into a future we cannot grasp”. This one statement made me wonder how different will the world be from what I grew up in and what I know of it. I guess I got my answer when I read what Beare has said through a 5 year old child, Angelica (Beare 2001). It is a truly ‘multi-cultural, multi-national, multi-faith world’ that future students will be living in.

If that is the case then we will have to use innovative teaching methodologies that will concentrate on the creative capacity of a student. These should help students to imbibe and hone skills that will help them to face their future. These skill sets cannot be learnt using the conventional methodologies that curb student creativity. So, that brings us to the question of what are the innovative teaching methodologies that foster creativity.

I guess I got my answer through Miller who talks about ‘timeless learning’ and the ‘characteristics of timeless learning’. However, I am unsure of how effective ‘timeless learning’ will be; especially since the examples given by him were from abstract forms of learning that are not easily quantifiable.

So, that brings me back to how can a student creatively learn skills that will help him/her in the future. Maybe, there are certain fields where it is not possible to be creative with the basics. How can one explain to a child that 1+1=2 always since that is how it has been taught us? Why can it not be 1+1=3? Or, how does one explain to a child that it is always ‘they are’ and not ‘they is’ in the English language?
I guess there will always be certain rules that cannot be taught creatively. But for all the others not only the student but also the teacher can don on his/her thinking cap and become innovative once again.

Training and Learning Activities

Imagine this: On completion of your graduation in Engineering, XYZ Company has recruited you as IT Support Executive. XYZ Company is an IT Business Process Out-sourcing firm providing services to a UK-based company. It’s your first day at work. There are 19 other trainees who either are like you or have worked for 5-6 years in another company.
Even as you interact with your colleagues, a lady walks in and says, “Welcome to XYZ Company. My name is Rachel James and I will be your ‘Corporate Communications’ facilitator for the next two weeks.” Then she goes on to say,
Scenario 1: We will begin with basic communication skills. As stated in the first slide of this power point presentation, basic communication skills are… (Expository teaching)
Scenario 2: We will be learning more about basic communication skills. You can stop me at any moment to clarify doubts. We know that basic communication skills are reading, writing, listening and speaking. Can any one tell me which skills you will be using when interacting on-line with customers from Norwich… (Interactive teaching)
Scenario 3: The first thing we will do is split into 5 groups of 4. We’ll work in our groups for the next two weeks to learn more about basic communication skills and to understand the communication methods used within the company… (Small group teaching/discussion)
Scenario 4: You might or might not know about XYZ Company. In the next 45 minutes we’ll find out more about XYZ Company through company advertising videos, search engines, IT magazines, etc. Once we research the topic, we’ll split into 4 groups of 5 to co-relate the data and create presentations… (Inquiry teaching/problem solving)
Scenario 5: Based on diagnostic tests done during your recruitment, we have created individualised training programmes. Log onto the computer system using your username and password. Double click the ‘Day One’ folder in ‘E-drive’. The folder has a series of tasks that you will complete within a stipulated time… (Individualisation)
Scenario 6: Please log onto the computer system using your username and password. On the desktop, there’s an icon that says ‘Virtual Learning Environment’. Double-click on this icon. You will be paired with a virtual customer… (Models of reality)
Print (1993) states, “In the cycle of the curriculum process, learning activities are integrally related to content and curriculum intent…The selection of appropriate teaching-learning strategies reflects the curriculum developer’s professional understanding of the task at hand and the needs of students…” Print’s (1993) definition forms the selection criteria for my imaginary training programme, i.e.
Content: Communication Skills Use and Usage by Indian IT agents amongst British customers.
Curriculum Intent: Facilitate and improve practical knowledge in the use of communication skills, i.e. reading, writing, speaking and listening, so that trainees can work as a team in the future to handle customer queries effectively.
Teaching Learning Strategies: ‘Small group teaching/discussion’ and ‘Inquiry teaching/problem solving’ are the two strategies to be used.
Learner: A group of tenured and un-tenured IT agents who are not first-language users with limited exposure to British culture.
Learner Needs: Improvement in the use and usage of English language so as to be effective IT support agents.
Now that we know the background to my imaginary training programme, let us look at the learning activities used. Print (1993) defines learning activities as “those activities offered to learners in the teaching-learning situation which are designed to enable them to acquire the designated content and thereby achieve the stated objectives and more broadly, the curriculum’s intent.” Based on Print’s (1993) definition, I have decided to use the above stated two teaching learning strategies. The advantages and areas of concern for using ‘Small group teaching/discussion’ and ‘Inquiry teaching/problem solving’ are as follows:
Small group teaching/discussion: allows me to divide the classroom into small groups. Each group works on the same task but they learn to work independently. They achieve the learning objective by using interactive formats such as group discussion, brain-storming, mind-mapping, etc. The interaction between trainees will result in communication skills improvement. The use of technology for topic research will improve their receptive skills. The discussions will improve their productive skills.
Print (1993) states “small groups are the only…opportunity for (trainees)…to acquire these skills effectively.” Group discussions will result in trainees working together in teams to achieve a common goal. This suits the training purpose of ensuring that in the future they will work as a team to sort out consumer problems.
An important area of concern is that trainees can lose focus. I can circumvent this by placing ‘off-task behaviour’ checks at points where trainees might deviate from learning objectives. Since the learning objective is to improve communication skills, it’s irrelevant how loud the discussions are as long as trainees are not discourteous to each other. It’s to be noted that the advantages of using this learning strategy to improve communication skills far outweighs its ‘areas of concern’.
Inquiry teaching/problem solving: Print (1993) states ‘Inquiry teaching/problem solving’ “ensures…learners are actively engaged in determining answers to questions or resolving problems.” This method will involve trainees in four stages of activity. Later, this will help them to determine a consumer’s area of concern. They’re as follows:
1) Problem awareness: Trainees determine the problem by asking questions and listening to answers provided.
2) Forming tentative hypotheses or possible solutions to the problem or issue: Based on the answers given, trainees will create possible solutions.
3) Researching and collecting data to test those hypotheses: Trainees will ask leading questions that will enable them to test the feasibility of their solution to the problem.
4) Forming conclusions based on the evidence collected and accepting or rejecting the hypotheses/possible solution: Trainees work on determining plausible solutions that can be substantiated with data.
The learning objective here is to ensure that trainees receive enough opportunities to improve their communication skills before they begin interacting with customers. The expository and interactive teaching strategies do not provide these opportunities. Since we require trainees to work in teams once they begin taking calls, ‘individualisation’ and ‘models of reality’ strategies do not work. Both these strategies require trainees’ to work as individuals and they defeat the training programme’s purpose.
Print (1993) suggests four criteria for choosing learning activities and they are diagrammatically sequenced as follows:

I have based my criteria on all four. When you work in the BPO industry, you will realise that there are little or no ‘resources crunch’ or ‘constraints’ based on finance, technology, etc. This multi-billion dollar industry generates enough finance to fund it-self and other industries too. The basis’ here are the first two. If I had a resource crunch or constraints then I would have a contingency plan in place. For example, if I had no access to online systems then I would have given each trainee printed sheets, magazines and other media examples to access information. As a facilitator I do know how to work with what is given to me so that I can achieve the objectives of the learning modules keeping in mind the learners’ styles of learning.
Appropriate content teaching with effective teaching learning strategies that achieve learning standards is an extensive field. Print (1993) provides us with the stepping stones to accomplish those objectives along with information about possible pit falls in disregarding them. Since I am convinced by what Print (1993) has stated, I’ll follow the above process in my training career in the future.

Bibliography:
Print, M. (1993) Curriculum Development and Design (2nd Edition). St. Leonards: Allen & Unwin