## Key Ideas > [!abstract] Core Concepts > > - **Engagement doesn't equal learning**: Students can be highly engaged while thinking about irrelevant details rather than mathematical concepts > - **Memory is residue of thought**: Students remember what they think about, so engagement must focus on mathematical thinking > - **Surface vs deep structure focus**: Activities can engage students in cutting, arranging, or visual appeal rather than core concepts ## Definition **Engagement**: Student involvement and interest in learning activities, which may or may not contribute to actual learning depending on what students are thinking about (Willingham, 2009). ## Connected To [[Activity-Based Curriculum]] | [[Motivation]] | [[Surface and Deep Structure]] | [[Memory]] | [[Schema]] --- Dylan Willingham coined the term: _Memory is the residue of thought_. Before we start worrying about things like [[Memory|working memory]] capacity, relevant [[Schema|schemas]], transfer, and problem-solving, we need to ensure that the items being processed in working memory are the items that need to be processed. Students remember what they think about, not what the teacher intends them to think about (Willingham, 2009). When people say they want students to be engaged in their learning, we need to consider what they are actually engaged in: the activity's surface features or the underlying concepts (Chi et al., 1981). In the example above, are students thinking about experimental probability or the best technique to flip a bottle? When students create a poster or PowerPoint on a topic, are they synthesising the information they have learnt or thinking of the design aspects of the poster? When students complete a Tarsia puzzle, are they focused on answering the questions or on neatly cutting out and finding the matching pieces? Students need to be engaged in mathematical thought, not the superficial [[Surface and Deep Structure|surface structures]] of the experience. An [[Activity-Based Curriculum]] engages students in cutting, arranging, decorating, or competing rather than the mathematical thinking that produces learning. > [!cite] TES Resource: Experimental Probability – Bottle Flipping > The author says: I created this PowerPoint as an engaging end of term activity for my low attaining year ten class to investigate the relationship between amount of water in a bottle and the chances of it being flipped successfully. I used the resource to revise experimental probability but the wider investigation covers collecting data, a bit of FDP etc. I know bottle flipping is a bit of a sore point for some but my class were fully engaged and it helped give them an appreciation of experimental probability. ## Curiosity and engagement Curiosity emerges when people notice information gaps in their knowledge (Willingham, 2009). The knowledge base must be sufficient to notice gaps but not so complete that nothing seems new. This creates an optimal difficulty level for engagement. Problems must be solvable but require mental effort. Work that is too easy fails to provide the pleasure of successful problem-solving. Work that is too difficult causes students to mentally disengage. Successful problem-solving provides a pleasurable response that reinforces engagement. However, working memory has severe limitations. People avoid thinking when working memory becomes overloaded. When confusion sets in, mental checking out occurs. Teachers must set appropriate difficulty levels to maintain engagement, assign work at the right level of difficulty for each student's current competence, and adjust when students show signs of confusion by slowing pace or simplifying content. ## Implications for teaching Observing engagement alone does not imply learning. Students can be highly engaged in irrelevant thinking (Willingham, 2009). Engagement can harm learning if students focus on the [[Surface and Deep Structure|surface structures]] of a problem, leaving them less working memory capacity for the deep structures (Sweller et al., 2019). Teachers should review each lesson by considering what students are likely to think about, not what teachers hope they will think about. Begin lessons with the problem that content is meant to solve. Students often do not understand or appreciate the question the lesson addresses. Presenting problems before teaching the solution creates cognitive need. Ensure students grasp why the content matters before teaching it. This approach leverages curiosity by creating information gaps that instruction then fills. ## References Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. *Cognitive Science*, 5(2), 121-152. https://doi.org/10.1207/s15516709cog0502_2 Sweller, J., van Merriënboer, J. J. G., & Paas, F. (2019). Cognitive architecture and instructional design: 20 years later. *Educational Psychology Review*, 31(2), 261-292. https://doi.org/10.1007/s10648-019-09465-5 Willingham, D. T. (2009). *Why don't students like school? A cognitive scientist answers questions about how the mind works and what it means for the classroom*. Jossey-Bass.