From: mk_thisisit
Mathematical understanding varies significantly among individuals, with a notable division between those who prefer visual images and those who favor a more abstract or algebraic approach to understanding mathematical concepts [00:02:45].
Personal Experience with Visual Thinking
The speaker notes that they personally find it easier to understand things visually [00:03:27]. This visual thinking process is central to their approach when contemplating mathematics [00:03:41]. While at school, the speaker observed that their way of thinking was different from most peers, as they understood concepts others did not [00:15:05]. At university, studying mathematics, they initially assumed others thought similarly but discovered a wide array of thinking styles [00:15:19]. A key distinction was between visual and non-visual thinkers [00:15:37]. The speaker, a visual thinker, found themselves in the minority, with only three out of fifteen students in their year thinking visually [00:15:40]. Despite being a minority, this visual way of thinking was advantageous for the speaker [00:15:52].
Systemic Bias Against Geometric Thinking
There is a perceived systemic prejudice against mathematicians who excel at geometric thinking [00:00:45]. The speaker argues that the examination process is harder for geometric thinkers compared to those with non-geometric thinking styles [00:19:27]. This means individuals who can think in a more direct, verbal way, rather than visually, tend to perform better in exams [00:19:33].
Examination Challenges for Visual Thinkers
When the speaker specialized in subjects like projective geometry and algebraic geometry, which are highly visual [00:16:43], they achieved the highest grades [00:17:01]. However, their best performance was unexpectedly in algebra, not geometry [00:17:07].
The speaker hypothesizes that this is due to the translation process required for geometric thinking [00:17:12]:
- Algebra: Problems can be solved and written down directly, seemingly engaging a single mental process [00:17:12].
- Geometry: While understanding might be immediate for a geometric thinker, translating the solution into written words for an exam introduces an additional step [00:17:52]. This translation process slows down geometric thinkers, putting them at a disadvantage in timed exams [00:18:06]. In contrast, algebraic solutions could be written down immediately [00:18:34].
Teaching and Student Preferences
As a lecturer, the speaker observed that students did not enthusiastically receive explanations presented visually [00:19:44]. Students preferred formulas that could be written down and manipulated, reflecting their preferred mode of thinking [00:19:57]. This suggests that the entrance tests for universities might be structured in a way that disproportionately affects visual thinkers, leading to a student body that thinks more verbally [00:20:06].
Complexity of Dimensions
The idea that human understanding is limited by the three-dimensionality of the brain is addressed [00:11:16]. However, mathematics used in quantum mechanics often deals with schemes that are not three-dimensional, extending to n-dimensional or even infinite dimensions [00:11:40]. While visual thinking can be limiting in these contexts, it is still possible to think in ways that are not purely visual [00:12:10].
For instance, special relativity, described geometrically by Minkowski, is a four-dimensional geometry [00:12:37]. While the speaker personally prefers to use visual tricks to conceptualize four-dimensional concepts [00:13:11], it is understood that such images are often imprecise when applied to non-Euclidean geometry [00:13:22]. Spacetime (four-dimensional) cannot be conceived as Euclidean three-dimensional space due to the differing character of the time dimension compared to spatial dimensions [00:14:00].
Ultimately, while the brain’s structure may influence how individuals grasp mathematical truths, it doesn’t preclude understanding higher dimensions or abstract concepts [00:14:27].