Pedagogical Models in a Mathematical Context:
An effective mathematics program balances many pedagogical tools in response to student needs and relies on the professional judgment of the teacher. In a mathematics classroom, there is a balance between explicit instruction and inquiry. While universal principles of high quality instruction apply to mathematical contexts, there are some principles that are unique to mathematics that are necessary to understand.
- (I do) I explain the procedure or concept. (Explicit instruction)
- (We do) We work examples together.
- (You do) You apply what you just learned to solve a word problem.
- (You do) You tackle a problem you may not know how to solve yet.
- (We do) We talk together about your thinking and your work.
- (I do) I help connect the class discussion to the goal of the lesson. (Explicit instruction)
Explicit instruction is an important component of a balanced mathematical program. Within this document, explicit instruction refers to the following:
Explicit Instruction Is….
Explicitly planned, well organized, and logically presented in keeping with learners’ maturity and personal experiences. Difficulties and misconceptions anticipated; a variety of strategies planned to differentiate.
Explicit Instruction is Not….
Haphazard, unorganized, improvised, and sporadic.
Interactive: (a) requiring learners to think, conjecture, dialog, and experiment; and (b) requiring teachers to deviate from the strategy plan in response to learners’ thinking, conjectures, dialogs, and experiments. Learners contribute to the lesson.
Lecturing (“Chalk and Talk”)/Teacher Centred.
Connecting new ideas to prior learnings, and connects procedures to models. Helping students make sense of strategies and procedures and where the math content is found in the learner’s social or physical world.
Teaching math as “piecemeal”—a series of discrete and disconnected ideas. Teaching procedures as steps in an algorithm to be memorized and duplicated.
Effective when used prior to learners being asked to practise the skill or strategy.
Telling, but not following up with practise. Activities are presented without giving students a firm understanding of the mathematics involved.
Integrating a variety of instructional strategies.
Teaching concepts one way.
Responsive: Using ongoing formative assessments and check-ins to determine learner readiness for more difficult problems or to work more independently.
Brief, engaging, and purposeful.
Teacher talk dominating class time.
Demonstrating the strategy, allowing more repetition to students who need it, and allowing alternative challenges to learners who need easier entry points or enrichment. Differentiated.
“One size fits all”, with limited examples, or too much repetition for students who are ready to try on their own, or not enough explanation or support for students who require more. Offering only one level or type of examples.
Small Group Instruction:
Small group instruction (sometimes referred to as guided math, small group instruction, stations, or side by side conferencing) is a way of either discovering what needs to be taught, or practising and reinforcing what has already been taught well. Practicing in a small group allows for a variety of experiences and opportunities for differentiation. Small group instruction may include a “teacher table”, where the teacher can work intensively with a small group of students to differentiate, reinforce, and assess. This may also include conferring with students and providing strategic support or nudging a learner to move to the next level of understanding.
- Whatever stations or tasks you choose need to reinforce the math concept being taught in a meaningful way.
- It is not enough for the task to be engaging—it must also be purposeful and support learning.
- Groups should be flexible and purposeful.
- Groups should be based on formative assessment.
- It is important that groups are based on the specific, targeted learning needs of students.
Small Group Instruction Is….
Continually changing flexible groupings to meet the learning needs of learners.
Small Group Instruction is Not….
Establishing static groups that remain unchanged.
Varying instructional time and content based on learner need.
Each learner receives the same amount of instruction and same task.
Culturally responsive teaching based on observation.
One size fits all work and activities.
Using a variety of activities and modes of learning to reinforce understanding of math.
Overuse of technology or paper pencil tasks.
Rigorous, accountable engaging learning activities.
Fun games that are chosen because learners like them, not because they provide meaningful practice.
Learners learning independently, interdependently and collaboratively.
Learners wondering what to do with their time.
Learners are clear about expectations. They look for help from peers and have alternative tasks if they are unable to complete tasks at stations.
Learners interrupt the teacher who is at a table with a group, or students sit and wait until their time at a station is over.
Learner work samples, notes on conversations, and observations of ability are collected to create a diverse portfolio of learners’ accomplishments.
Learners are not accountable for the learning they do at stations.
“I hear and I forget. I see and I remember. I do and I understand.” – Confucius
During independent practice a student demonstrates their understanding of the concept by using it independently. During this time, students can be challenging their
own thinking and teachers can be nudging them or “What if-ing?” them to delve deeper into concepts of which they have a basic understanding. Guided instruction and independent practice are interwoven as teachers use this opportunity to gather formative assessment data around the needs of every student at each end of the understanding spectrum. Independent practice allows the teacher to work with small groups for strategic intervention, guided math support, or conferring with individual learners.
Teachers will use their professional judgment to determine what pedagogical practice is needed. Learners who are needing support may require more explicit instruction. Conversely, learners who appear to be independent may still need guided support to nudge them and extend their learning to deeper application of their mathematical thinking. A learner who may have been independent in one area, may require more explicit instruction in another. Teachers will determine the instructional practice that best meets the needs of the learners collectively and individually.
Teachers circle back to previously taught concepts in order to consolidate learning. This ensures that concepts are not taught in isolation, memorized and then forgotten. Spaced practice enables the student to demonstrate their understanding at any point in time. Key outcomes need to be revisited throughout the year. Spaced practice provides multiple opportunities for students to demonstrate understanding over time.
“I never teach my pupils, I only attempt to provide the conditions in which they can learn.” – Einstein
Inquiry learning provides students with opportunities to build knowledge, abilities, and inquiring habits of mind that lead to deeper understanding of their world and human experience.1
Contained in each Curriculum document at www.curriculum.gov.sk.ca.
Saskatchewan curriculum advocates an approach to inquiry that is cyclical, multidirectional, reiterative, and can occur in the moment, with planning for a lesson or a unit, in a subject area, or interdisciplinary approach. Advocates of inquiry provide a variety of models for implementation to structure or explain the work involved in the inquiry. These are not steps, nor are they a research model, but an organizing framework. An approach can be applied to a single question, a lesson, a unit, or even a long-term inquiry.2
Inquiry in Math Is…
An instructional stance posing questions that are open, inviting investigation, mining strategies and reasoning, inviting conjecture and meaning making
Inquiry in Math is Not…
Directing students through prescribed steps to reach a predetermined outcome or utilizing a set of strategies
Engaging, accessible and inclusive. Has multiple entry points: “Low floor high ceiling”
Out of reach for some students – wide open, inaccessible challenges.
Tightly connected to curricular outcome(s), designed purposefully.
“Fun” activities for the sake of variety in the program
Varied investigations from small questions and predictions to open student exploration where students are allowed to engage in productive struggle
Always a big, student-driven exploration without a clear learning intention that leaves students feeling overwhelmed/helpless
Built on student curiosity and draws on diverse backgrounds, interests, and experience, providing multiple paths to understanding in authentic ways.
Based on the teacher’s imposition of a specific process or contrived mathematics
Promoting student ownership of learning/teacher as coach/facilitator
Teacher as presenter where students are passive learners
Embedded within robust and balanced learning
Overwhelming, a complete shift in practice or an add-on.