Chatting with ChatGPT

I have been reading about AI and ChatGPT for a few weeks about how cheating will be rampant (see https://www.washingtonpost.com/education/2022/12/28/chatbot-cheating-ai-chatbotgpt-teachers/), and no reliable way to detect AI produced work: https://www.washingtonpost.com/technology/2023/06/02/turnitin-ai-cheating-detector-accuracy/. This new technology is not going away, unless something newer and fancier comes along, but believe me, it won’t be a different narrative (and the technology that supersedes it will be far more obnoxious). How can we embrace this technology rather than try to find ways to safeguard against it?

First, I want to admit that when I was teaching, I was really irritated with students’ rampant use of cellphones during class. A colleague was like, “find a way to use the technology, it’s already there.” I grumbled because I didn’t have a smartphone at the time and really did not understand what they were capable of. Looking back, I did not do enough to understand the new technology that was coming out. Somewhat related, for my dissertation, I interviewed experienced math teachers about how prepared or unprepared they felt at various points in their career and the pandemic seemed to impact the experienced teachers' preparedness more than any other disruption. I supposed this was because they could no longer rely on their tried and true techniques for all classroom issues. The technology may have existed to overcome these issues, however, these teachers seemed to not be aware of such technologies that would be successful in bridging these changes (not for a lack of trying though, these were dedicated and skilled teachers). I also did not have many ideas for how teachers could have been resourced to handle the challenges, short of districts, counties, and curriculum developers mobilizing to bring useful technology to teachers' attention (there is only so much teachers can do to keep up). Now ChatGPT exists, and I am still not sure what it is capable of, but I have the time to find out some of its adequacies and inadequacies :).

I decided to visit https://chat.openai.com/ and take on the persona of a geometry student who does not feel confident in their math ability. I chose geometry because I was working on another project about triangles and I figured that would give me an edge (no pun intended) as I was already thinking about triangles in weird ways. The following is part of my exchange with the ChatGPT program, I have edited it to keep the conversation brief (because ChatGPT seems to be overly verbose at times).

I started by briefly sharing some concerns and worries I had as a student. ChatGPT was very kind and provided a long list of things I could do to better prepare myself for class, examples of questions to ask when I am confused, and other fairly reasonable solutions to various challenges. And then I asked about proving triangles (my words and thoughts are in red).

ME: Why doesn't SSA prove triangles are congruent?

ChatGPT

The Side-Side-Angle (SSA) condition, also known as the Side-Side-Angle Congruence Criterion or the Side-Side-Angle (SSA) Theorem, does not guarantee triangle congruence. The reason is that with just the lengths of two sides and the measurement of an angle opposite one of those sides, multiple triangles can be formed with the same SSA measurements but different shapes.

[So far ChatGPT has successfully described SSA, let's read on]

To understand why SSA is not sufficient for proving triangle congruence, consider the following scenario:

Suppose we have two triangles, Triangle ABC and Triangle A'B'C', where:

• The lengths of sides AB and AC are congruent to the lengths of sides A'B' and A'C', respectively (SS).

• The measure of angle A is congruent to the measure of angle A' (A). [Note that ChatGPT is explaining SSA, and defines two sides first, and then an angle, possibly leading to the following mistake]

• However, there is no other known relationship between the remaining sides or angles.

In this case, even though two sides and the included angle are congruent, the triangles may not be congruent. It is possible to have different triangles with the same SSA measurements, as shown in the diagram below:

css

A A'

/ \ / \

/ \ / \

/ \ / \

/ \ / \

/_________\ /_________\

B C B' C'

As you can see, the two triangles have the same SSA measurements (AB ≅ A'B', AC ≅ A'C', ∠A ≅ ∠A'). However, the triangles have different shapes and sizes, indicating that SSA alone is not sufficient to prove triangle congruence.

It's important to note that other criteria, such as Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Side-Angle (ASA), can be used to establish triangle congruence with greater certainty. These criteria ensure that all corresponding sides and angles are congruent, providing a more conclusive proof of triangle congruence.

ME: Isn't the example you just gave a SAS example?

ChatGPT

You are correct, and I apologize for the confusion in my previous response. The example I provided indeed represents a situation where the Side-Angle-Side (SAS) criterion can be used to prove triangle congruence.

In the scenario described, if we have two triangles, Triangle ABC and Triangle A'B'C', where:

• The lengths of sides AB and AC are congruent to the lengths of sides A'B' and A'C', respectively (SS).

• The measure of angle A is congruent to the measure of angle A' (A).

• No other known relationship is given. [This is essentially the same explanation provided above]

In this case, the SAS criterion can be applied to prove the congruence of the triangles. By showing that two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of the other triangle, we can establish their congruence.

The corrected [note: it's the same] diagram for the SAS example would be as follows:

css

A A'

/ \ / \

/ \ / \

/ \ / \

/ \ / \

/_________\ /_________\

B C B' C'

Thank you for pointing out the error, and I apologize for any confusion caused. The Side-Side-Angle (SSA) criterion, as mentioned earlier, does not guarantee triangle congruence, while the Side-Angle-Side (SAS) criterion is a valid method for proving triangle congruence.

ME: I am still confused about SSA criterion

ChatGPT

I apologize for the confusion. You are correct, and I want to clarify the issue with the SSA (Side-Side-Angle) criterion.

The SSA criterion, also known as the "ambiguous case" or "assumed case," does not guarantee triangle congruence in all situations. In some cases, the SSA criterion may result in more than one possible triangle, leading to ambiguity.

To understand this better, let's consider the SSA criterion in detail:

If two sides and the angle opposite one [there it is!] of those sides in one triangle are congruent to two sides and the angle opposite one of those sides in another triangle, it does not necessarily imply congruence.

The SSA criterion leads to ambiguity because there can be multiple triangles formed with the given side lengths and angle measurement. Depending on the relationship between the given side lengths and angle measurement, different triangles can be formed.

Here are the two possible scenarios that arise with the SSA criterion:

1. No Triangle: If the length of the given side is shorter than the length of the other side, the side cannot reach the angle formed by the other two sides. In this case, no triangle can be formed.

2. Two Possible Triangles: If the length of the given side is equal to or longer than the length of the other side, it is possible to form two different triangles. One triangle may have the given angle as an acute angle, while the other triangle may have the given angle as an obtuse angle.

Therefore, the SSA criterion alone is not sufficient to prove triangle congruence because it can result in different possible triangles.

It's important to note that other congruence criteria, such as the Side-Side-Side (SSS), Side-Angle-Side (SAS), and Angle-Side-Angle (ASA) criteria, provide more conclusive evidence for triangle congruence. These criteria establish congruence by considering combinations of sides and angles that lead to a unique triangle.

-----------------------END

My takeaway from this interaction is that AI does not get it right all the time and students need to see that, and ChatGPT provided some pretty good responses regarding feeling vulnerable and regurgitating definitions (and then finally providing a reasonable explanation for why SSA does not prove congruence). One homework or class assignment (perhaps over the course of a unit or something) that comes to mind is to have students sign on and ask ChatGPT math questions relevant to their topic until they spot a mistake, record this mistake and bring it to class to share. This helps students realize that it is a tool, but the crafts-person has to know how to use the tool.