Three Phases of Mathematics Learning



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.