Triangle Theory Building: Assessment

What this is for

This rubric is meant to help the teacher understand how students are engaging with the assumption digging process. It is not designed for grading. It can be applied to observations during class discussion, to the Student Handout, or to the proof trees students build on the board.

There are four things worth paying attention to, and I have tried to describe what each looks like at different levels of sophistication. The levels are rough — students will be at different levels on different dimensions, and a given student might shift between levels within a single session.

1. Quality of claims listed

The question here is: can the student articulate things they believe to be true about triangles, and are those claims well-formed?

At the most basic level, a student lists vague or poorly formed claims ("triangles are shapes") or lists very few. This is fine in the first few minutes — the inventory will grow.

A step up is when the student lists standard textbook facts (angle sum is 180, Pythagoras' theorem, area formula) but only well-known results. Most groups will be here at the start.

More sophisticated is when the student lists claims that are both well-known and "obvious" — things like "triangles have three sides" or "triangles have three angles." In both schools where I ran this, students initially omitted these obvious facts. The moment a student realises these need to be stated is a sign of growing rigour.

The most impressive thing to see is a student who lists claims and can already identify relationships between them — "I think this one might follow from that one" — before any proof work has been done. This is rare in the first session but can happen.

2. Quality of proof attempts

The question here is: can the student construct a chain of reasoning from premises to conclusion?

At the most basic level, the student restates the claim or gives an example rather than an argument. "The angles of an equilateral triangle are 60 degrees each because I measured them." This is an empirical approach, not a deductive one — and it is a useful thing to flag.

A step up is when the student gives a correct argument but skips steps or relies on implicit assumptions without noticing. For example, proving the angles of an equilateral triangle are equal by saying "angles opposite equal sides are equal, and all the sides are equal" without mentioning transitivity of equality. This was the most common level across both schools.

More sophisticated is when the student writes out a step-by-step proof and can identify what each step assumes. They might say: "This step uses the fact that angles opposite equal sides are equal, and this step uses transitivity." They are aware of the proof tree they are building.

The most impressive level is when the student spots missing assumptions or hidden dependencies without being prompted. Vivaan's observation about transitivity ("if we work like that, then this follows") is an example — he noticed that the proof relied on a rule that had not been stated.

3. Reasoning about evidence: distinguishing proof from example

The question here is: does the student understand the difference between a proof and an example?

At the most basic level, the student treats examples as proofs. "I drew a triangle and measured the angles — they add up to 180." This is a common starting point and is not wrong in everyday reasoning, but it is not what we are doing here.

A step up is when the student understands that a proof is needed but confuses a statement with its converse. This is one of the four common errors from the implementation: a group claimed that "sides opposite greater angles are greater" was the same as "angles opposite greater sides are greater." When this happens, ask: "Are these really the same claim? Can you think of a situation where one is true but the other is not?"

More sophisticated is when the student can identify which claims in a proof are being used as premises and which are being proved, and can recognise when a proof is circular. In the implementation, Uday used a cyclic quadrilateral to show two angles were equal, then tried to use those equal angles to show the quadrilateral was cyclic. When caught, he said "we are using the same thing" — showing he could recognise the circularity once it was pointed out.

The most sophisticated level is when the student can evaluate whether a proof is complete — whether all the premises have been stated, whether the conclusion really follows, and whether any of the premises themselves need justification. This is essentially the assumption digging skill in full.

Common errors worth discussing

These four types of flawed reasoning came up repeatedly across both schools. They are not marks against the student — they are teaching opportunities.

Applying a theorem outside its scope. Anya applied a claim about sides and angles of the same triangle to sides of two different triangles. She was comparing the diameter of a circle to a chord and concluded the angle must be larger. The response: "That claim is about one triangle. You are comparing two different triangles. Why would the claim apply across triangles?"

Confusing a statement with its converse. A group presented "angles opposite greater sides are greater" as equivalent to "sides opposite greater angles are greater." The response: "If it rains, the ground is wet. If the ground is wet, did it rain? Are these the same?"

Circular reasoning. Students defined straight lines as collinear points and collinear as points on a straight line. Also, Uday assumed a quadrilateral was cyclic to prove two angles were equal, then used the equal angles to justify that it was cyclic. The response: draw the loop on the board. Make the circularity visible.

Over-generalising from examples. Less common in this module than in the Podgon Game, but it does happen. "I measured the angles of this triangle and they add up to 180, so it is always 180." The response: "You have shown it for one triangle. What about all the triangles you did not measure?"

4. Communication and precision of mathematical language

The question here is: can the student articulate their reasoning precisely enough that someone else could follow it?

At the most basic level, the student uses vague language or gestures at the board rather than stating claims precisely. "This equals that because of the thing we said before." In the implementation, one student said a proof was "obvious" when asked to explain a step — pointing at a picture rather than stating a reason.

A step up is when the student uses mathematical terms but imprecisely. For example, saying "the sides are equal" without specifying which sides, or "by congruence" without specifying which congruence test. In the implementation, a student said "CPCT" (corresponding parts of congruent triangles) without being able to state which parts correspond to which.

More sophisticated is when the student states claims with enough precision for another person to evaluate them. Quantifiers are used correctly: "all the angles" vs "two of the angles." Assumptions are stated explicitly rather than left implicit. In the implementation, Devika's precise statement — "Equilateral triangles are a type of isosceles triangle but every isosceles triangle is not an equilateral triangle" — showed this level of precision.

The most sophisticated level is when the student notices imprecision in others' statements and can suggest corrections. In the implementation, Vivaan caught the circularity in the straight-line definition and Pankaj specified the missing claim in Arnav's proof. Students who do this are modelling the rigour the module aims to develop.

Observation sheet

You can use something like the following during sessions to record quick notes on each group. Focus on one or two dimensions per session — you cannot track everything at once.

Group	Claims	Proof attempts	Reasoning	Communication	Notes
Group 1
Group 2
Group 3
Group 4
Group 5

What to look for across sessions

Session 1 (Equilateral triangles)

Are students able to construct even a rough proof, or do they rely on "it is obvious" or examples?
Do they notice when a new claim enters the proof that was not on the board?
Do they understand the distinction between the two directions of the equilateral triangle proof (equal sides implies equal angles vs equal angles implies equal sides)?

Session 2 (Classification)

Can students articulate definitions precisely enough to resolve disagreements (like whether equilateral is a type of isosceles)?
Do they understand what makes a classification complete? Can they say why the angle-based classification requires the angle sum property?
Can they explain why some classifications are useful and others are not (the 47.5-degree challenge)?

Session 3 (Deeper digging)

Can students identify when a proof chain has become circular?
Do they understand the concept of an undefined entity — that not everything can be defined?
Can they distinguish between a definition, a theorem, and an axiom?

End-of-module reflection prompts

These can be given as written tasks at the end of the last session, or as group discussion prompts.

Pick a claim from the board and trace it back as far as you can. What does it ultimately rest on? Draw the chain.
Did you find any circular reasoning during the module? Describe the circle and how you resolved it (or how you would resolve it).
What is the difference between a definition and a theorem? Give an example of each from our work on triangles.
We said that axioms are where the digging stops. Why can't we just keep digging? What goes wrong if we try to prove everything?
One student from a previous class said this module "intrigued us to prove things we normally would not question." Another said "we kept going endlessly." What do you think? Is there value in questioning things you already believe to be true?

Student feedback as evidence

In the implementation, students were asked "What did you learn?" and "What did you not understand?" at the end of the first and last sessions. Here are some of the responses that indicate genuine engagement with the learning outcomes:

On relationships between claims:

"We learned relationships between statements or definitions of properties of triangle. It is that there is an interrelationship between them." — Vivek
"I got to know how each property of triangle are interconnected." — Deepa
"I liked the way we figured out the interdependency between properties." — Tushar

On definitions and axioms:

"We learned about definitions and axioms and the difference between them." — Anya
"One thing can be defined with other things that we know. Not all things can be defined." — Sana
"During the triangle session we were asked to define everything... and I enjoyed that." — Meghna

On questioning assumptions:

"It intrigued us to prove things we normally would not question." — Uday
"Not to take anything given in the textbooks for granted." — Tarini and Tanya
"In a classroom we only use these theorems blindly." — Vivek

These responses do not prove that students achieved the learning outcomes, but they do indicate that students were sensitised to the right ideas. The strongest evidence comes from observing whether students can actually do the reasoning — which is what the observation rubric above is for.