How geometric thinking develops
· 2 min read
Geometry asks students to reason in a way that earlier math often doesn’t. A long-standing theory describes how that reasoning develops.
What the research found
- Levels of geometric thinking. The van Hiele theory describes levels of geometric thinking that run from recognizing shapes by their overall appearance, through reasoning about their properties, to deductive proof.
- Levels and proof. Senk found that secondary school geometry students’ van Hiele levels were associated with their achievement in writing geometry proofs.
This research is about how students learn geometry, not about SubjectPath.
Why it matters for diagnosis
A student who struggles with proof may not be missing proof itself. They may still be reasoning about shapes by how they look, rather than by their properties. A diagnostic that only marks a proof right or wrong can miss where that student’s thinking actually is.
How SubjectPath will apply this idea
The Geometry Path is coming soon. It will be built on the Common Core high school math standards and will use the same method as every Path: a map of which skills build on which, answer choices written to reveal specific misconceptions, and practice that starts where understanding breaks down.
What the levels look like
A student reasoning by appearance might say a square isn’t a rectangle because it doesn’t look like one. A student reasoning about properties might say a square is a rectangle, because it has every property a rectangle needs. Proof asks for something further again: a chain of reasoning from definitions and properties to a conclusion. A student working at the first level isn’t ready for proof yet.
What this means for a geometry diagnostic
In principle, a diagnostic built around these ideas could ask which kind of reasoning a student is using, not just whether a final answer is correct. Answer choices that match reasoning by appearance could be told apart from choices that match reasoning by properties.
Sources
- van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Academic Press.
- Senk, S. L. (1989). Van Hiele levels and achievement in writing geometry proofs. Journal for Research in Mathematics Education, 20(3), 309–321.