## Key Ideas > [!abstract] Core Concepts > > - **Vary one element against constant background**: Hold everything same except single critical feature to direct attention to its effect > - **Reveals mathematical structure**: Students see patterns and relationships by observing how one change affects outcomes > - **Requires fluency prerequisite**: Students need automated procedures before having working memory capacity to notice structural relationships ## Definition **Minimally Different Questions**: Question sequences where only one critical element varies while everything else remains constant, allowing students to observe specific effects and see mathematical structure (Watson & Mason, 2006). ## Connected To [[Surface and Deep Structure]] | [[Fluency]] | [[Memory]] | [[Cognitive Load Theory]] | [[Problem-Solving]] | [[Schema]] --- ## Variation Theory Foundation [Variation Theory](https://variationtheory.com/) provides the theoretical basis for minimally different questions (Marton & Booth, 1997; Watson & Mason, 2006): **Core Principle**: "Meanings are acquired from experiencing differences against a background of sameness, rather than from experiencing sameness against a background of difference." By holding everything else the same and varying one element, students' attention focuses on that element and its effect on the answer. This supports understanding of mathematical [[Surface and Deep Structure|structure]]. This resembles scientific experimentation: one variable changes, the effect on the outcome is observed, and that change can be attributed to that variable. Random variation across multiple elements obscures patterns; systematic variation reveals them. ## Visual Example > [!example] Variation Theory in Action ![[VariationTheory.png]] > > This sequence shows how changing one element (the operation) while keeping numbers constant helps students see the structural differences between addition and subtraction with negative numbers. ## Types of variation ### Parameter variation Parameter variation modifies one number, operation, or structural feature whilst maintaining problem context and format. For example, the sequence $2x + 3 = 11$, $2x + 5 = 11$, $2x + 7 = 11$ holds the coefficient and target constant whilst varying only the added term. Students observe how changing this constant affects the solution method and outcome. ### Strategy selection variation Questions can be designed such that students need to identify which strategy to use. Triangle problems exemplify this approach: students determine whether to use Pythagoras or trigonometry, with minimal changes in given information requiring different approaches. ## Implementation requirements ### Fluency prerequisite Students need spare [[Memory|working memory]] capacity to notice relationships. They must be [[Fluency|fluent]] or close to fluent in the relevant procedure (Cowan, 2001; Sweller et al., 2019). Without automation, working memory is consumed by executing the procedure itself, leaving nothing for pattern recognition (Ericsson & Kintsch, 1995; Logan, 1988). [[Fluency Practice]] must come before minimally different questions. The sequence should build procedural fluency through repetitive practice until responses become automatic, then apply minimally different questions to reveal underlying structure, and progress to more complex variation patterns as understanding deepens. Reversing this sequence wastes the power of minimally different questions because students cannot see patterns whilst struggling with execution. Students notice patterns when basic procedures are automated, which frees cognitive resources for pattern recognition. Variation must be obvious enough that students notice the change and its consequences. ## Design principles Question sequences should change only one element per question, keeping other features constant to make the variation obvious. For example, keeping numbers the same whilst varying the operation directs attention to operational differences. The change must produce noticeable outcome differences that students can compare. Start with obvious variations and move to subtle ones. Include questions that complete the pattern so students can predict the next in sequence, testing their understanding of the underlying relationship. ## Mathematical structure benefits Students develop the ability to see underlying mathematical relationships rather than surface features (Chase & Simon, 1973). Experience with variation helps students apply principles to new contexts (Watson & Mason, 2006) and distinguish between different problem types and required strategies (Chi, Feltovich, & Glaser, 1981). Systematic pattern exposure builds organised knowledge networks (Sweller et al., 2019). ## Key warnings Using variation questions before students have procedural fluency wastes their potential because working memory cannot simultaneously execute procedures and notice patterns. Changing multiple elements simultaneously hides the effect of each change. Variations must produce meaningful differences students can observe, or the sequence becomes arbitrary practice. Students may memorise surface patterns without understanding deep structure if fluency is insufficient. ## Practical examples Fraction addition sequences can vary denominators: $\frac{1}{4} + \frac{1}{4}$, $\frac{1}{4} + \frac{1}{8}$, $\frac{1}{4} + \frac{1}{12}$. Students observe how denominator changes affect solution complexity. Linear graphing sequences can vary slope: $y = 2x + 3$, $y = 3x + 3$, $y = 4x + 3$. Students see how slope changes affect graph steepness whilst y-intercept remains constant. Quadratic factorisation sequences can vary coefficients: $(x + 2)(x + 3)$, $(2x + 2)(x + 3)$, $(3x + 2)(x + 3)$. Students observe how coefficient changes affect expansion patterns. ## Integration with other strategies Use variation once students understand basic procedure from [[Worked Examples]]. Build pattern recognition before attempting unfamiliar [[Problem-Solving]]. Eventually mix different question types after structural understanding develops, applying the [[Interleaving Effect]]. ## References Chase, W. G., & Simon, H. A. (1973). Perception in chess. *Cognitive Psychology*, 4(1), 55-81. https://doi.org/10.1016/0010-0285(73)90004-2 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 Cowan, N. (2001). The magical number 4 in short-term memory: A reconsideration of mental storage capacity. *Behavioral and Brain Sciences*, 24(1), 87-114. https://doi.org/10.1017/S0140525X01003922 Ericsson, K. A., & Kintsch, W. (1995). Long-term working memory. *Psychological Review*, 102(2), 211-245. https://doi.org/10.1037/0033-295X.102.2.211 Logan, G. D. (1988). Toward an instance theory of automatization. *Psychological Review*, 95(4), 492-527. https://doi.org/10.1037/0033-295X.95.4.492 Marton, F., & Booth, S. (1997). *Learning and awareness*. Lawrence Erlbaum Associates. 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 Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. *Mathematical Thinking and Learning*, 8(2), 91-111. https://doi.org/10.1207/s15327833mtl0802_1