TEACHING MATHEMATICS TO EMPOWER YOUNG PEOPLE AND HENCE ECONOMIC GROWTH

Claude M Packer, CD, JP,President, The Mico University College, Kingston, Jamaica

INTRODUCTION

Research findings consistently showed teacher quality as a significant factor influencing students’ performance in the subject. McKinsey, et. al. (2007) in their report noted that educational jurisdictions which were achieving high literacy and numeracy levels had one common feature — teachers of high quality.

This finding led McKinsey, et. al. (2007) to conclude that:

“...the quality of an education system cannot exceed the quality of its

teachers...” (p. 16).

and Kong (2011):

“Teachers are ultimately what make education

succeed.”

Also Katzenmeyer and Moller (2001) state:

“Student learning depends first, last, and always on the

quality of...teachers” (p. 22)

Good teachers are usually aware of the specific weaknesses in their own practice through reflective thinking and appropriate self evaluation. They often make the effort to gain understanding of specific best practices and are usually self motivated to make the necessary improvements for good teaching. In fact, good teaching has been shown to be the most important determinant of student outcomes (e.g. see McKinsey & Company, 2007, 2009, 2010).

It is very important to know the competencies that a

teacher must have to teach with meaning for

understanding. A rich repertoire of mathematical knowledge and skills that

relate directly to the curriculum, instructions and

student learning are mandatory.

Schmidt et al (2011) indicated that:

“the most important competencies tend to be tacit, like skills involved in playing the concert piano, learned but not necessarily available to consciousness.” (p. 1266)

It is important to know what tacit knowledge includes. According to Davis (2011):

“Many instantiations involved to introduce and elaborate concepts, e.g., analogies, metaphors, and applications.” (p. 1505)

Instantiation of mathematics concepts has not been systematically incorporated in teacher preparation. According to Davis (2011):

“This gap raises interesting issues, which can be highlighted through popular understanding of multiplication, for example, ...; repeated addition and/or a grouping process. This definition works well for natural numbers, but it begins to break down as early as the middle grades. How, for example, does one add 5/8 to itself ¾ times, d to itself π times, or -2 to itself -3 times?” (p. 1506)

It is clear that an assessment of the teaching and learning of mathematics using a variety of data collection methods including observation must be done to determine the extent to which: - teachers understand the content which they are

presenting to students. teachers have the pedagogical content-

knowledge required to teach for understanding. weaknesses in literacy are impacting the students'

ability to understand the concepts being taught and more importantly, their ability to read and understand a question or problem.

Some Emphases for Pedagogical Renewal

Principals and Heads of Departments of schools, must provide effective leadership to ensure that mathematics teachers plan effectively and provide adequate coaching and counselling for weak students.

Frequent departmental seminars should be encouraged for experienced teachers to mentor the younger teachers.

In certain cultures, the learning of mathematics is a priority, for example, the Chinese and Indian cultures.

Most schools in China can be relied on to have a qualified and committed teaching staff and the educational system is geared towards learning; discipline and conformity (Plafker, 2007).

Students from the Chinese culture where the main emphasis is based on pedagogy; Singapore, Korea, Japan and Hong Kong, have outperformed the world on the “every four year” TIMSS tests designed for elementary and junior high school students around the world.

Schoenfield (2008) explained why:

They see success as a function of persistence and doggedness and

willingness to work hard for twenty-two minutes to make sense of something that

people would give up on after thirty seconds (p. 246).

The students of certain countries in Europe, for example, Hungary, are also achieving significantly in the field.

To enhance learning experiences, that may allow students to construct their own mathematical knowledge periodically, many more teachers should always provide more classroom activities and structure innovations that can capture children's sustained interest for prolonged periods of time.

Students need to learn to manipulate and manage databases in order to create and communicate mathematical information.

Techniques must be found to empower students to think creatively and critically and to think how mathematicians think and thus be better problem solvers. Mathematics is problem solving!

Teachers must make an effort to teach mathematics contextually so that students will be able to connect mathematical ideas to relevant real world experiences.

Hexagons

Symmetry

We ought to emphasize the Constructivist’s (Connell, 1998; Novak, 1977; Brooks, 1999) view on learning when we teach mathematics because mathematics is manmade.

One therefore needs to emphasize the idea of students constructing their own knowledge and ensure that complex calculations do not prevent this experience. Students could explore, for example, through an experiment, to ascertain the value of Pi (π).

So π is a constant and approximately 3 to one significant figure. A more accurate result is approximately 3.142 to four significant figures which is universally employed, but π is an irrational number and cannot be expressed as 3 1/7.

It would be fascinating for students to further explore that π can be expressed to many decimal places (non terminating or non repeating) using a calculator, for example, π 3.1415926535897932384626433832795 ...

The primary school students can be guided to construct their own algorithms to perform certain mathematical tasks, for example; the operation of addition:

The sum: 84 + 29 = 113, could also be done

Mathematics educators must utilise technology to: teach via concepts and not by rote, teach contextually, teach to foster creativity, map concepts together to teach high order

concepts, use Piaget’s model—from concrete to

semi-concrete to the abstract,

make sure that the basics are understood and mastered, for example, the real number system,

use spacial concepts and contrived models to eliminate abstraction,

use art and colour to add meaning especially in geometry,

use real world experiences to motivate, and give immediate feedback and encourage

lots of practice to reinforce what is learnt.

Hexagonal PrismSquare-base Pyramid

Cube

Too many students are being discouraged and turned off by frequent tests, thus eliminating the opportunity for them to learn at their own pace, to think creatively and also enjoy the fascination of numbers.

The teachers have no time to give the students these experiences because of the “passing exam syndrome” that we have developed. Glasser (1969) states:

Memory is not education, answers are not knowledge. Certainty and memory are the

enemies of thinking, the destroyers of creativity and originality (p. 38).

Assessing students’ readiness to learn mathematics in a new class or a new topic is critical to the success of teaching in mathematics. Three methods of making these assessments are: examining records, testing, and observing. (Schewinger, 1999).

When a child has learned through extensive repetition, math anxiety may develop when the necessary amount of repetition is no longer possible.

Only a small portion of what is to be learned should be memorized. Basic addition, subtraction and multiplication facts and certain formulas might be memorized.

However, even for these facts, it is best to use them extensively in a variety of contexts so that they are learned with connections and meanings rather than as abstract lists and mnemonics (such as SOHCAHTOA in Trigonometry).

SOH : Sine is opposite over hypotenuse

CAH : Cosine is adjacent over hypotenuse

TOA : Tangent is opposite over adjacent

Mathematical knowledge must be acquired, facts must be learned, mathematical concepts must be developed, and mathematical processes must be practiced.

If lessons are set in a problem solving context, three important benefits accrue. First, students can see what is to be learned as interrelated with other concepts, and thus find it easier to attach meaning to new ideas.

Second, students must see that the mathematical concepts and processes have real uses and applications.

Third, students must be provided with an opportunity to learn problem solving and problem posing skills and strategies.

Problem-solving Survey

A mathematics laboratory is a fundamental tool to support the delivery of mathematics teacher education programmes. The laboratory would provide teacher educators with the resources needed to effectively engage students within courses, using effective methodologies.

They would be engaged in a manner which would support the development of their own conceptual knowledge.

Some Directionsfor Improvement

The new approach puts less emphasis on learning rules (for example, knowing the rules for adding fractions or solving quadratic equations) and being able to use them.

There is a shift away from “rule-based” knowledge of word and formula and algorithm and toward construction of one’s collection of basic metaphors or assimilation paradigms.

And so, Professor Ang Ken Cheng (2011) of the Mathematics and Mathematics Education Academic Group (MME) of the National Institute of Education (NIE), Singapore, suggested a formula for a good mathematics teacher:

gmt should => 

“Love of math and love of teaching”

+

“The ability to make math come alive through passion for both subject and teaching”

+

“Being an exemplary role model of learning”

+

“Being an advocate of using multiple pedagogies”

+

“Being a thinking teacher in order to cope with change, in our time, to optimise

learning”.

+

“Mastery of mathematical content and demonstrate good pedagogical sense”

+

“The ability to assess student’s learning differences since it is difficult for a ‘one size-

fits-all’ computer software to facilitate learning”

+

“Being cognisant that learning can be, at times, a complex and dynamic process”

–

“The thought that mathematics is just functions and formulae”

+

“Not only being able to talk about formulae and algorithms and methods, but to explain

concepts and how they can be utilised in real life experiences”

+

“Understanding the mathematics ‘hidden curriculum’; values such as perseverance

for problem solving and creativity, for example”

+

“The ability to add value to students’ lives”

+

...

Mathematics as communication will have a significant bearing on how mathematics is taught. See Figure 4 on the Components of Mathematics as Communication

MATHas

Communication

MATHas

Communication

talkingclass presentations, discuss strategies, results of surveys, co-operative group

work

writinglogs, letters,

journals, reports

representinggraphs, tables, charts, words,

symbols, manipulative

readingtables, charts,

statistics on sports page, menu,

children’s literature where math is

involved

listeningothers’ solutions,

class reports, directions,

others’ strategies

Student Assessment to Support Meaningful Learning

Assessment should not only involve closed book examinations, but should also include projects, portfolios, oral examinations, video presentations, laboratory work, creation of CDs, etc.

Teaching Math via Projects

Mathematics Teaching

Emphasis must be placed on a clear, precise description of the basic ideas and concepts being taught, with worked examples and applications where appropriate.

add

Correct, precise, orderly, spoken and written mathematics must be used at all times.

Homework should be utilised as a key component of learning.

Individual pupil mistakes should provide teaching points for the whole class and therefore should be utilised.

Teaching and evaluation must be embedded with each other as shown in Figure 7.

Too much taking of notes should not be encouraged in a mathematics class; engage the students in much discussion to enhance their understanding for enjoyment, motivation and independent study for knowledge creation and real world applications.