Discrete Geometry: Assessment

What this is for

This rubric is meant to help the teacher understand how students are engaging with the process of extending geometry to new worlds. It is not designed for grading. It can be applied to observations during class discussion, to the Student Handout, or to the worlds and diagrams students produce.

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. Definition extension

The question here is: can the student borrow a concept from Euclidean geometry and make it meaningful in a new context?

At the most basic level, the student cannot separate the concept from its Euclidean appearance. When asked "what is a straight line in this world?" they say things like "you cannot have straight lines with only 6 points" or draw continuous lines between dots. They are treating the Euclidean version as the only version.

A step up is when the student can state a definition that works in the new context but only by copying the Euclidean definition verbatim without adapting it. For example, saying "a circle is round" does not help in a discrete world — "round" has no meaning when there are only finitely many points. The student has the right instinct (borrow from Euclidean geometry) but has not identified which part of the definition transfers.

More sophisticated is when the student identifies the core of the Euclidean definition — the part that generalises — and discards the parts that are specific to Euclidean geometry. For example, recognising that "equidistant from a centre" is the core of the circle definition, while "round" and "continuous curve" are consequences of having infinitely many points. This student can state a working definition and apply it.

The most impressive level is when the student recognises that there may be multiple reasonable ways to extend a concept and can articulate the tradeoffs. For example, a student who says "we could define bisection as finding a midpoint vertex or as splitting at an edge, and these give different answers" is operating at this level. They see the extension as a choice, not a discovery.

2. Navigating multiple definitions

The question here is: can the student hold two definitions simultaneously and trace different conclusions from each?

At the most basic level, the student cannot keep two definitions separate. They switch between A-bisection and B-bisection without noticing, or they forget which definition they are using. When asked "can this line be bisected?" they answer without specifying which kind of bisection.

A step up is when the student can work with one definition at a time but struggles to compare. They can answer "under A-bisection, these lines can be bisected" but cannot then say "under B-bisection, these additional lines can also be bisected" without starting over.

More sophisticated is when the student can hold both definitions in mind, apply them to the same world, and compare the results. They can say: "Under A-bisection, only lines of even length can be bisected. Under B-bisection, all lines can be bisected. So the answer to the original question depends on which definition we choose."

The most impressive level is when the student uses the comparison between definitions to make a substantive mathematical observation. For example: "A-bisection is more restrictive than B-bisection — anything that can be A-bisected can also be B-bisected, but not the other way around. So if we use A-bisection, the question is harder to answer 'yes' to." This student is reasoning about the definitions themselves, not just applying them.

3. Conjecturing and generalising

The question here is: can the student move from specific worlds to general claims about classes of worlds?

At the most basic level, the student works only with the specific world in front of them. They can answer "in this world, the line A-B-C can be bisected" but do not ask whether this is true in other worlds or for other lines.

A step up is when the student checks multiple cases within a world and identifies patterns. "All the even-length lines can be A-bisected and all the odd-length lines cannot." This is pattern recognition within a single world.

More sophisticated is when the student conjectures across worlds. "In the chain world, odd-length lines cannot be A-bisected. I wonder if that is true in all worlds, or just chains." They may test the conjecture by examining the necklace world and finding that the pattern changes.

The most impressive level is when the student formulates a general claim about a class of worlds and attempts to justify it. "In any necklace world with an odd number of points, every straight line can be A-bisected, because the maximum distance is (n-1)/2, which means all distances are less than n/2 and every shortest path has a midpoint vertex." This student is doing genuine mathematical reasoning.

In my implementation, generalisation was the hardest skill. Most students stayed at the level of individual worlds. The students at Ganga were more willing to generalise than those at Indus, partly because the Ganga students had more time to explore and partly because the teacher at Ganga more explicitly encouraged it by asking "what if the number of points were different?"

4. Representation

The question here is: can the student work with graph representations and not confuse representational features with world features?

At the most basic level, the student treats the graph drawing as the world. They measure distances on the paper (physical distance between dots) rather than counting edges. They try to place points halfway along edges. They think the angle between two edges in the drawing matters. One student, Vivaan, drew additional points on edges when trying to bisect a line — he was treating the drawing as a continuous space.

A step up is when the student understands that distance means "number of hops" but still sometimes slips into visual reasoning. They might say "A and E are far apart" looking at the drawing, even if A and E are 2 hops apart in the graph. They are mostly working with the graph structure but occasionally confused by the picture.

More sophisticated is when the student consistently works with the graph structure. They count hops rather than measuring. They recognise that two drawings that look different can represent the same world (if they have the same connections). They are not thrown off by the visual arrangement of the dots.

The most impressive level is when the student reflects on the representation itself. "We are drawing these as dots and lines, but the same world could be represented as a table of distances. The table might make some things easier to see." Or: "The way I drew my world made it hard to see the shortest paths, but when I redrew it differently, the structure was clearer." This student understands that the representation is a tool and that choosing a good representation is part of doing mathematics.

Common representation errors worth discussing

Treating edges as divisible. Students try to place points at the midpoint of an edge. The correction: remind them of the teleportation analogy. You cannot stop halfway through a teleportation.

Confusing drawing distance with graph distance. Students think two points that are drawn close together are nearby in the world. The correction: "Distance in this world means number of hops. These two points are drawn close together, but how many hops does it take to get from one to the other?"

Thinking angles matter. Students think the angle between two edges in the drawing has geometric meaning. The correction: "The drawing is just a picture. You can redraw the same world with different angles and it is still the same world. What matters is which points are connected."

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	Definition extension	Multiple definitions	Generalisation	Representation	Notes
Group 1
Group 2
Group 3
Group 4
Group 5

What to look for across sessions

Session 1 (Bisection)

Can students propose a definition of "straight line" that works in the discrete world, or do they get stuck on Euclidean intuitions?
When the bisection ambiguity arises, do students recognise that the disagreement is about definitions, not about facts?
Can students determine which lines can be A-bisected in the simple world?
Do any students ask about other worlds beyond the ones given?

Session 2 (Circles)

Can students apply the circle definition (equidistant from centre) without being distracted by the non-round appearance?
Do students notice differences between circles in the chain world and circles in the necklace world?
Can students make connections between bisection and circles, or between circle properties and the structure of the world?
Do students conjecture about patterns (e.g., "circles in the necklace always have 2 points or 1 point")?

Session 3 (Triangles and generalisation)

How do students react to collinear triangles? Do they reject them outright, accept them reluctantly, or see the need to refine the definition?
Can students find NC-triangles in the necklace world?
Do students attempt to generalise from specific worlds to classes of worlds?
Can students articulate what structural feature of a world determines whether NC-triangles exist?

End-of-module reflection prompts

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

We started with the question "Can every straight line be bisected?" What is your answer now? (Hint: there is no single answer.)
Pick a concept we explored — straight lines, bisection, circles, or triangles — and describe how it is different in a discrete world compared to Euclidean geometry. What stays the same and what changes?
In one of our worlds, the statement "every straight line can be bisected" was true because there were no straight lines at all. Is this a satisfying answer? Why or why not?
We had to make choices about definitions (like A-bisection vs B-bisection). Did the choice matter? How?
What is one question about discrete worlds that you do not know the answer to but would like to investigate?

Student feedback as evidence

In the implementation, students were asked for feedback at the end of the module. Here are some responses that indicate engagement with the learning outcomes:

On extending definitions:

"We had to think about what words actually mean when the normal picture doesn't work anymore." — Vivaan
"I liked that we got to decide what a circle is. Normally the teacher just tells you the definition." — Karan

On multiple definitions:

"It was surprising that the same question could have different answers depending on the definition." — Devika
"I didn't realise that bisection could mean two different things until we disagreed about it." — Tarini

On the difficulty:

"This was harder than the triangle module because there was no right answer — you had to build everything yourself." — Tanya
"I found it confusing at first but once we had the definitions it became more like a puzzle." — Uday

On frustration (from Indus):

"I did not understand what we were supposed to do. The question did not make sense." — (anonymous)
"It was too open. I would have preferred more structure." — (anonymous)

The negative feedback from Indus is important. This module is the most open-ended in the course, and some students will find that frustrating. The assessment should account for this: a student who is struggling with the openness of the task is not necessarily failing to learn — they may need more scaffolding. The teacher's job is to provide just enough structure to keep the student engaged without closing down the exploration.