Effective
instruction of a unit typically involves three phases of learning: Readiness,
Engagement and Mastery.
Phase
1 - Readiness
Student
readiness to learn is vital to learning success. In the readiness phase of learning,
teachers prepare students so that they are ready to learn. This requires considerations
of prior knowledge, motivating contexts, and learning environment.
[1]
Prior Knowledge
For students
to be ready to learn, teachers need to know students’ prior knowledge in relation
to the new learning. This requires knowing whether students have the pre-requisite
concepts and skills. Some form of diagnostic assessment is necessary to check that
students are ready to learn.
[2]
Motivating Contexts
For students
to be ready to learn, teachers need to provide motivating contexts for learning.
These contexts should be developmentally appropriate. For example, younger students
may like contexts such as stories and songs, and play-based activities such as games,
whereas older students may appreciate contexts related to everyday life so that
they can see the relevance and meaningfulness of mathematics. For the more advanced
students, applications in other disciplines can serve as motivation for learning.
[3] Learning
Environment
Shared rules
help promote respectful and emotionally-safe interactions between teacher and students
and among students that are necessary for productive and purposeful learning. Established
procedures for organising students and managing resources will also facilitate a
smooth start and transitions during lessons.
Phase 2 - Engagement
This is the main
phase of learning where teachers use a repertoire of pedagogies to engage students
in learning new concepts and skills. Three pedagogical approaches form the spine
that supports most of the mathematics instruction in the classroom. They are not
mutually exclusive and could be used in different parts of a lesson or unit. For
example, the lesson or unit could start with an activity, followed by teacher-led
inquiry and end with direct instruction.
[1] Activity-based
learning
This approach
is about learning by doing. It is particularly effective for teaching mathematical
concepts and skills at primary and lower secondary levels, but is also effective
at higher levels. Students engage in activities to explore and learn mathematical
concepts and skills, individually or in groups. They could use manipulatives or
other resources to construct meanings and understandings. From concrete manipulatives
and experiences, students are guided to uncover abstract mathematical concepts or
results.
For example,
to develop problem solving skills, students investigate whether rectangles with
the same perimeter can have different areas. Students are given sheets of 1-cm square
grids to draw and cut out different rectangles of a given perimeter (e.g. 12 cm).
They will record the length, breadth and area of each rectangle that they have cut
out on a record sheet. Questions will be posed for students to discuss during the
activity e.g.
’How do you figure
out the length and
breadth of a
rectangle given its
perimeter’ ‘What assumptions
do you make about
the length/breadth of the
rectangle?’ Students further
explore different strategies
and explain why the
strategies work or
do not work and
finally, derive a
conclusion as a
team. During the discussion,
students are also
encouraged to communicate
their ideas using appropriate
mathematical language. Throughout
the activity, the teacher
will be observing what
the students say and
do and constantly making
the decision on the
appropriate amount of feedback to be provided to them. Teacher ends the activity
by summarising and highlighting some of the strategies that students use.
[2]
Teacher-directed inquiry
This
approach is about learning through guided inquiry. Instead of giving the
answers, teachers lead students to explore, investigate and find answers on
their own. Students learn to focus on specific questions and ideas and are
engaged in communicating, explaining and reflecting on their answers. They also
learn to pose questions, process information and data and seek appropriate
methods and solutions. This enhances the development of mathematical processes
and 21st century competencies.
For
example, in teaching the topic on Symmetry, teacher first shows two groups of
shapes – symmetric and non-symmetric shapes, without introducing the concept of
symmetry. Students are asked how the shapes are classified and how one group is
different from the other. Teacher tests the rule for classification suggested
by students. Eventually, teacher guides students to focus on the attributes of
symmetric shapes and directs them to concept of symmetry. To check whether
students understand the concept of symmetry, teacher gives students paper cut-outs
of symmetric and non-symmetric shapes and asks them to classify them in the
same way. Students can fold the shapes into halves and decide if they are
symmetrical. They also visualise how a line of symmetry divides a symmetric
shape into halves that fit exactly over each other.
[3]
Direct instruction
This
approach is about explicit teaching. Teachers introduce, explain and
demonstrate new concepts and skills. Direct instruction is most effective when
students are told what they will be learning and what they are expected to be
able to do. This helps them focus on the learning goals. Teachers draw
connections, pose questions, emphasise key concepts, and role-model thinking.
Holding students’ attention is critical. Stimuli such as videos, graphic
images, real-world contexts, and even humour, aid in maintaining a high level
of attention.
For
example, in teaching problem solving, the teacher demonstrates how to use
Pólya’s four-step problem-solving strategy and models thinking aloud to make
visible the thinking processes. While explaining and demonstrating, the teacher
also probes students’ understanding of the process by asking questions and
giving feedback to the students who response to the questions. Teachers also
use the students’ response to modify her explanation and demonstration so that
students are better able to follow the process. Teachers use additional
examples if necessary and assign work for students to do on the spot. Teachers
check on students’ work and selectively pick a few responses for further
discussion. During lesson closure, the teacher reviews the key learning points of
the lesson to consolidate the learning.
Phase 3 - Mastery
This is the final
phase of learning where teachers help students consolidate and extend their learning.
The mastery approaches include:
[1] Motivated
Practice
Students need
practice to achieve mastery. Practice can be motivating and fun. Practice must include
repetition and variation to achieve proficiency and flexibility. Structuring practice
in the form of games is one good strategy to make practice motivating and fun, while
allowing for repetition and variation. There should be a range of activities, from
simple recall of facts to application of concepts.
[2] Reflective
Review
It is important
that students consolidate and deepen their learning through tasks that allow them
to reflect on their learning. This is a good habit that needs to be cultivated from
an early age and it supports the development of metacognition. Summarising their
learning using concept maps, writing journals to reflect on their learning and making
connections between mathematical ideas and between mathematics and other subjects
should be encouraged. Sharing such reflections through blogs makes learning social.
[3] Extended
Learning
Students who
are mathematically inclined should have opportunities to extend their learning.
These can be in the form of more challenging tasks that stretch their thinking and
deepen their understanding.