## Key Ideas > [!abstract] Core Concepts > > - **Purpose First**: Establish why students should care before diving into content > - **Foundation Security**: Assess and secure prior knowledge before building new learning > - **Clear Communication**: Use bullet-proof definitions and explicit vocabulary instruction ## Definition **Introduction**: The opening phase of instruction that establishes purpose, secures foundations, and sets clear learning expectations for the lesson. ## Overview The lesson introduction affects whether the instruction that follows succeeds or fails. Effective introductions accomplish three essential tasks: they establish why students should care (purpose), verify that foundational knowledge is secure (prior knowledge assessment), and clarify what students will learn (clear communication). These functions are essential - proceeding without student motivation wastes their attention (Ryan & Deci, 2000), building on insecure foundations creates unsustainable [[Cognitive Load Theory|cognitive load]] (Sweller et al., 2019), and unclear vocabulary or definitions causes confusion (Beck et al., 2002). Yet introductions must remain brief (5-10 minutes maximum), avoiding the trap of elaborate contexts or extended motivational speeches that consume time better spent on teaching and practice (Rosenshine, 2012). The strategies that follow provide targeted, efficient approaches to each introduction component, ensuring lessons launch with purpose, clarity, and secure foundations. ## Connected To [[Explicit Teaching Learning Episode]] | [[Cognitive Load Theory]] | [[Retrieval Practice]] --- ## Purpose: Why Should Students Care? Student motivation depends on perceived relevance. When students understand why learning matters, they invest attention and effort (Hulleman & Harackiewicz, 2009). Establishing purpose can follow several paths. One approach is to show how today's learning builds on prior knowledge and opens doors to future topics through a flow diagram or curriculum mapping overview. Rather than treating mathematics as isolated topics, this creates coherence and demonstrates that each skill connects to what students already know. Another approach is to share historical context. Explaining who developed mathematical ideas and why they were needed gives students the human story behind the concepts. This often proves more memorable than abstract definitions. Real-world connections establish authentic relevance when genuine. Examples like prime numbers in cybersecurity or visual images illustrating Pythagoras' theorem (such as desire paths) create connections students recognise. However, elaborate or unfamiliar contexts increase cognitive load (Sweller et al., 2019). Real-world connections should remain brief and genuine; most real-life applications prove messier than mathematical problems. Another effective strategy is the "headache problem" approach used by Dan Meyer (2010). Present a problem that highlights the need for today's learning: a calculation that becomes tedious without the target skill. For example: Common headache problems include calculating repeated addition (693 + 693 + 693 + 693 + 693 + 693 + 693) to motivate multiplication, working out $2^{3651} ÷ 2^{3648}$ to show why index laws matter, writing Earth's mass without standard form notation, calculating $\frac{42}{55} × \frac{35}{6} × \frac{11}{14} × \frac{2}{7}$ to demonstrate fraction cancelling, or finding solutions to $x^2 + 3x - 28 = 0$ to establish the need for factorising quadratics. The headache problem should create struggle for only 2 minutes at most. The goal is motivation, not frustration. Students with a foundation of success associate struggle with eventual breakthrough (Bandura, 1997). ## Learning Intentions and Success Criteria Learning intentions should be communicated explicitly at the start of the lesson so students understand what they will learn. Success criteria become more meaningful when students understand the content, making them most useful at the lesson or topic conclusion rather than the beginning. Throughout the lesson, students must know both what they are learning and why it matters. ## Assess and Secure Prior Knowledge If prerequisite skills are not secure in long-term memory, learning new skills alongside applying them creates unsustainably high cognitive load (Sweller et al., 2019). Assessment must verify that foundations are in place before proceeding. Daily review should include prerequisite checks in Do Now activities, using retrieval practice to maintain prior knowledge (Rosenshine, 2012). Targeted questions before teaching new skills ensure specific competencies are secure. For example, checking that students can create equivalent fractions before teaching fraction simplification. Mini-whiteboard checks gather reliable data from all students rather than relying on volunteers, providing whole-class assessment data (Black & Wiliam, 1998). Decision-making should follow the 80% rule: if fewer than 80% of students demonstrate understanding of prerequisites, address gaps before proceeding (Wilson et al., 2019). This prevents building new learning on insecure foundations. Proofs and derivation showing how concepts develop can be appropriate when teaching conceptual development. For instance, showing how function transformations emerge through coordinate changes helps students understand the reasoning. However, procedural teaching often comes appropriately before conceptual understanding, and starting with concrete methods makes more sense than beginning with abstract reasoning. ## Clear Communication Strategies Definitions should be delivered as single, clear sentences that summarise key ideas. State definitions explicitly and have students repeat them to ensure accurate recall. Detailed explanations can overwhelm working memory in early learning stages, so definitions must remain concise. Vocabulary development requires intentional planning. Mathematical vocabulary should be taught before introducing content that uses it, for instance, teaching "discrete" before starting data collection. Research shows students need to understand nearly all words in a text for comprehension to occur (Nagy & Scott, 2000), making vocabulary checks essential barriers to identify. Discussing word origins aids memory; for example, explaining that "cent" means hundred helps students remember percentages and centuries. Polysemous words, those with different meanings in mathematics versus everyday use, require explicit attention. "Similar" means something specific in geometry but has different connotations in common speech. The mathematical register uses language with greater precision than colloquial language (Pimm, 1987), and students need explicit instruction in how this differs from everyday communication. ## Implementation Sequence Effective lessons follow a structured introduction lasting 5–10 minutes total. The sequence begins with a hook or purpose statement (2 minutes maximum) that establishes why students should care about today's learning. This is followed by a prior knowledge check lasting a variable amount of time, which assesses and secures foundations. The learning intention should be stated clearly in one minute so students understand what they will learn. Vocabulary pre-teaching takes 2–3 minutes to ensure students know essential terms before encountering them in complex contexts. Finally, provide a clear, memorable definition in 1–2 minutes. The entire introduction should remain brief, consuming at most 5–10 minutes. This timing preserves time for substantial teaching and practice. Purpose matters for motivation, but it should not dominate the lesson. ## Quality Indicators Effective introductions show consistent characteristics. Students can articulate why learning matters, prior knowledge gaps have been identified and addressed, key vocabulary is understood before use, clear learning direction has been established, and students are engaged and ready to learn. Conversely, warning signs indicate problems. Students confused about lesson purpose, vocabulary barriers preventing comprehension, prior knowledge assumptions proving incorrect, introductions consuming excessive time, or students remaining passive and disengaged all suggest the introduction needs revision. The introduction creates the foundation for effective learning by establishing purpose, securing prerequisites, and creating clear expectations. When executed well, it maximises the effectiveness of subsequent instruction. ## References Bandura, A. (1997). *Self-efficacy: The exercise of control*. W. H. Freeman. Beck, I. L., McKeown, M. G., & Kucan, L. (2002). *Bringing words to life: Robust vocabulary instruction*. Guilford Press. Black, P., & Wiliam, D. (1998). Assessment and classroom learning. *Assessment in Education: Principles, Policy & Practice*, *5*(1), 7-74. https://doi.org/10.1080/0969595980050102 Boaler, J. (2002). 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