Flow Chart : Compulsory Part with Module 2 (Algebra and Calculus)
Chapter 4 Learning and Teaching
4.3 Choosing Learning and Teaching Approaches and Strategies
The extent to which teachers are successful in facilitating learning among students depends, to some degree, on the teaching approaches and strategies they use. A
variety of learning and teaching approaches can be interwoven and deployed to suit students with different needs, interests, abilities and prior knowledge. Familiarity with the characteristics of Hong Kong students in Mathematics can help teachers adopt the most suitable learning and teaching strategies. In general, Hong Kong students:
have a high regard for Mathematics;
believe that diligence can help them to learn Mathematics;
are strong in basic skills and computation, but are weaker in dealing with non-routine problems;
value drill and practice;
are more used to exposition than activity-based ways of learning such as the use of games or group discussion;
seldom interrupt the flow of teaching by asking questions; and
are usually motivated by extrinsic factors.
Three common pedagogical approaches that can be employed for the effective delivery of the Mathematics Curriculum (S4 – 6) are discussed below.
Teaching as direct instruction
Direct instruction, a very frequently used approach in the Mathematics classroom, can make a positive contribution to the learning of mathematics if it is dynamic, and well planned and organised.
A direct instruction approach is most relevant to contexts which involve explanation, demonstration or modelling to enable learners to gain knowledge and understanding of particular concepts. In this approach, the teacher focuses on directing the learning of the students through providing information or explanations. Direct instruction, in which teachers present factual materials in a logical manner, can be employed effectively for teaching definitions and notations of mathematical terms, rigorous proofs of mathematics theorems and procedures for sketching curves. This approach can also be used with a large group of students to stimulate thinking at the start of an open discussion. Teaching Mathematics through direct instruction can produce a mathematically neat and complete lesson which contains presentations and explanation leading to an intended conclusion. Although it can be interactive, it is a more teacher-centred approach: the teacher asks questions, chooses students to answer them and evaluates the responses; and then he/she probes for more information and may ask the students to justify their answers.
Another major goal of direct instruction is to maximise student learning time. In a typical direct instruction lesson on Mathematics, the teacher usually spends some time lecturing, guides the students through a concept, provides examples and breaks complex ideas into simpler steps. The students are then given simple problems to try on their own; and the simple problems are then combined so that students can solve more complex problems. Finally, the teacher summarises what the students should have learned, and gives them assignments through which he/she can assess the amount of the content the students have learned in that lesson. Direct instruction lessons can also be facilitated by the use of audio-visual materials, when appropriate.
Teaching the concept and notation of combination using the direct instruction approach
Students are introduced to the concept of combination by their teacher. The teacher explains the concept and notation of combination and puts the related formulae on the board. The teacher then derives the formulae and demonstrates with some examples.
The difference between the concepts of combination and permutation is also clearly explained. The teacher then alerts the students to the fact that the terminology of combination in mathematics is different from its common use in daily life – for example, the word “combination” in combination lock does not refer to combination.
After that, the teacher introduces other notations for combination used in various parts of the world. At the end of the lesson, a brief note on what has been taught and some simple exercises on the topic are distributed to the students for consolidation.
The “direct instruction” approach is suitable for teaching the concept and notation of combination for the following reasons:
The concept and the notation are difficult for students to understand and the proofs of the formulae are mainly procedural.
It can be ensured that students have acquired an accurate understanding of the concept and notation.
The notations used in the books from different countries may be different. For instance, the notations of combination n and
Cn are usually found in the books written by British and Chinese scholars respectively. Students will be confused by the notations if this is not explained properly.
The concept and notation of combination can be introduced efficiently in a short time.
Teaching as inquiry
The Mathematics Curriculum (S4 – 6) sets out various learning activities for students to carry out investigations and includes “Inquiry and Investigation” learning units in the Compulsory Part, Module 1 and Module 2. In the inquiry approach, the emphasis is on the process and the action undertaken by the learner – inquiry tasks often focus on students’ thinking and inquiring processing. Students are encouraged to help each other raise questions and develop understanding, with their experiences being seen more as a resource for learning than as a product. Students engage in complex cognitive processes requiring thoughtful discourse. They are invited, for example, to make connections between facts, make predictions, and debate alternatives. This relies heavily upon extensive dialogue among students, and can take place during interactive whole-class teaching or during peer interaction in pairs or groups. This approach promotes critical thinking and problem-solving skills. The topic(s) under investigation should be placed in a meaningful context, and students should be encouraged to explore and discover information through the use of open-ended questions, group discussion, investigations, experiments and hands-on exercises. However, there should be sufficient “wait time”, so that students can explain their thinking adequately and elaborate their answers.
Teaching properties of a cyclic quadrilateral using the inquiry approach
The teacher starts the topic “properties of a cyclic quadrilateral” in the Compulsory Part by asking students to use computers with dynamic geometry software installed and an application file with a circle and a cyclic quadrilateral drawn as in the figure below.
Students are free to:
move the vertices of A, B, C and D;
add lines such as diagonals AC and BD;
extend line segments AB, CD, etc.;
measure sizes of angles; or
measure lengths of line segments such as AB, AC, etc.
Students are asked to record their measurements and make conjectures about the properties such as angle relations or line relations. They are encouraged to discuss their observations and conjectures with each other, and may use the software to provide evidence for them. After that, students are asked to make a presentation on their observations and conjectures, during which the teacher focuses on asking students to explain and justify their conjectures and gives them time to do so. For example, if students observe that the opposite angles of a cyclic quadrilateral are supplementary, the teacher may ask them whether they have verified their conjecture by referring to different quadrilaterals. It should be pointed out that students’
observations and conjectures may differ and the teacher should provide different responses to students who have arrived at different conclusions.
As students have often encountered circles and quadrilaterals in daily life and in their previous learning experiences, they can integrate their prior knowledge and IT skills to explore new aspects of plane geometry. This topic allows students to explore a rich collection of relations, including the basic properties listed in the Compulsory Part and also aspects such as Ptolemy’s Theorem* that is beyond the curriculum. Through engaging in this exploratory activity, they may come to recognise the richness of geometrical knowledge. They can construct knowledge through observing, making conjectures and testifying. Sketching a diagram using IT tools or drawing instruments are good ways for students to justify their hypotheses and conjectures.
This approach is different from the familiar deductive approach to study geometric knowledge.
* Ptolemy’s Theorem: For a cyclic quadrilateral, the sum of the products of the two pairs of opposite sides equals the product of the diagonals.
Teaching as co-construction
This approach puts the focus on the class as a community of learners. It is based on the view that mathematics is a creative human activity and that social interaction in the classroom is crucial to the devgelopment of mathematical knowledge. The
teacher plays a central role in developing a problem-solving environment in which students feel free to talk about mathematics. Tasks for this approach usually involve creating knowledge and developing criteria for judging that knowledge collectively.
A wide variety of linkages are made between members of the class as they contribute to the construction of knowledge. In this approach, the teacher helps students learn how to think for themselves, which may require temporary frameworks or scaffolds such as model answers, demonstrations, prompts and feedback, opportunities for students to explain their arguments to the class, or worksheets for students to work on.
The teacher also requires students to check the process through which they reached a conclusion and indicate how it might be improved.
This approach relies heavily on discussion and sharing among participants. Students learn through student-student and teacher-student interactions. Through co-construction of knowledge, students develop social skills, organise their thinking and develop rational arguments.
Teaching the quadratic formula using the co-construction approach
In teaching “solving quadratic equations by the quadratic formula”, a Mathematics teacher may employ a co-construction approach in which, instead of passively accepting knowledge, students are able to construct concepts and to understand how the formula is deduced. After learning how to solve quadratic equations by the factor method, students are asked to solve quadratic equations with irrational roots.
They will discover that not all quadratic expressions can be factorised into linear expressions with rational coefficients. They will realise that the factor method is limited in its usefulness. By being actively involved in well-constructed tasks – for instance, from solvingx2 −5=0, x2 − x2 −5=0 to x2 −2x−c=0 – and class discussion, students should become responsible for explaining their reasoning orally to the class, developing their own analogical thinking and discovering how the method of completing the square can be extended to solving general quadratic equations. The teacher has to sum up the discussion and refine the constraints (e.g.
the coefficient of the term involving x2must be non-zero) in collaboration with students.
In the co-construction approach, the teacher focuses on making connections between facts and fostering new understanding in students. The teacher encourages students to analyse, interpret and predict information. He/She also
promotes extensive dialogue among students, which is a very important part of learning mathematics. Communication is a way of sharing ideas and clarifying understanding. When students think, discuss and inquire about mathematical concepts, they reap a dual benefit: they communicate to learn mathematics, and they learn to communicate mathematically.
Teaching is developmental rather than merely directive or presentational.
Although algorithms provide an efficient route to getting correct answers, students do not understand how or why these procedures work. Merely directing them to follow the traditional rules or algorithms in a fixed fashion is like expecting them to arrive without having made the journey. The co-construction approach is student-oriented: the teacher serves as an expert providing guidance and does not dictate the working strategies entirely.
Lessons should begin from where the students are, not from where the teacher is.
Students are expected to bring the necessary skills and background knowledge to the task and to be self-motivated. Individual differences are expected, welcomed and supported.
The classroom climate is collegial, supportive and spontaneous. Discussion, group work and active participation are encouraged and expected. Students are active creators of knowledge, not passive receivers of information. The emphasis is on understanding rather than on memorisation and repetition.
Activities are problem-centred and student-driven. Lessons are built on solving problems at the appropriate level for the students. Some easier problems are dealt with early in the lesson and are used to provide paradigms. Busy work and unnecessary repetition are minimised.
As mentioned earlier, in the learning and teaching of mathematics, a single strategy is seldom adopted: an effective teacher of Mathematics very often integrates various strategies when teaching a topic. Strategies should always be used flexibly and be suited to the abilities, interests and needs of students, and the context and development of the lesson. Some common classroom practices which are effective for learning and teaching mathematics include:
expanding concepts or definitions;
analysing a problem in more than one way;
using an idea in more than one setting;
providing models that enhance understanding;
giving examples of applications;
providing examples and counter-examples when explaining concepts;
requesting students to rephrase problems;
practising and drilling;
using scaffolding to demonstrate how to tackle a problem by, for example, induction, deduction, analogy, looking for equivalence and identities, classification, conjecturing, generalising, exhausting, formulating hypotheses, looking for patterns and regularities, charting and graphing, abstracting, intuition, analysing and synthesising, and breaking down a complicated problem into smaller parts;
posing problems by students through changing the knowns, unknowns or constraints;
providing opportunities for discussion and sharing of ideas; and
brainstorming among students.
The example below illustrates how some of the approaches and strategies can be used in the Mathematics classroom:
Teaching one of the properties of the scalar product of vectors using the direct instruction, the inquiry and the co-construction approaches Teachers may integrate various teaching approaches and classroom practices to introduce the properties of the scalar product of vectors so that the lessons can be more vivid and pleasurable. In this example, teaching one of the properties of the scalar product of vectors, |a − b|2 = |a|2 + |b|2 −2(a⋅b), is used as an illustration.
In previous lessons, the teacher has taught the concepts of magnitudes of vectors and the scalar product of vectors using direct instruction. In this lesson, the students are divided into small groups to promote discussion, and the groups are asked to explore the geometrical meaning of the property. Here, the inquiry approach is adopted, with students having to carry out investigations with the newly acquired knowledge related to vectors. During the exploration, the groups may interpret the geometrical meaning differently. Some may consider one of the vectors to be a zero vector and get the above property; but others may relate it to the Pythagoras’ Theorem by constructing two perpendicular vectors a and b with the same initial point.
Hence, the hypotenuse is |a − b| and a⋅b = 0 and the result is then immediate. If some groups arrive at this conclusion, the teacher should guide them to discover that
their interpretation is only valid for special cases. However, the geometrical meaning of this property is related to the cosine formula learned in the Compulsory Part. If some groups can find that the property is the vector version of the cosine formula, they can be invited to explain how they arrived at this geometrical meaning.
If none of the groups can arrive at the actual meaning, the teacher may guide them to find it out by giving prompts. Some well-constructed prompts (or scaffolds), such as asking them to draw various types of triangles and find clues to connect |a−b|, a⋅b, |a| and |b| with the triangles drawn, may be provided. The co-construction approach is adopted here.
After understanding the geometrical meaning, the result can be derived by applying the cosine formula learned in the Compulsory Part. The groups are further asked to explore alternative proofs. Here, the inquiry approach is employed. The groups may not think of proving this property with |x|2 = x⋅x directly. The teacher may give some hints to guide them. In this case, the teacher and the students are co-constructing knowledge. If the students still cannot prove this property, the teacher can demonstrate the proof on the board using the direct instruction approach. Whatever methods the students use, they are invited to explain their proofs to the class. During the explanation, the teacher and students may raise questions and query the reasoning.