# Novice Learner and Expert Tutor:
Evaluating Math Reasoning Abilities of
Large Language Models with Misconceptions

Naiming Liu
Rice University
<EMAIL_ADDRESS>
Shashank Sonkar
Rice University
<EMAIL_ADDRESS>
Zichao Wang
Adobe Research
<EMAIL_ADDRESS>
Simon Woodhead
Eedi
<EMAIL_ADDRESS>
Richard G. Baraniuk
Rice University
<EMAIL_ADDRESS>

###### Abstract

We propose novel evaluations for mathematical reasoning capabilities of Large
Language Models (LLMs) based on mathematical misconceptions. Our primary
approach is to simulate LLMs as a novice learner and an expert tutor, aiming
to identify the _i_ ncorrect answer to math question resulted from a specific
misconception and to recognize the misconception(s) behind an incorrect
answer, respectively. Contrary to traditional LLMs-based mathematical
evaluations that focus on answering math questions correctly, our approach
takes inspirations from principles in educational learning sciences. We
explicitly ask LLMs to mimic a novice learner by answering questions in a
specific incorrect manner based on incomplete knowledge; and to mimic an
expert tutor by identifying misconception(s) corresponding to an incorrect
answer to a question. Using simple grade-school math problems, our experiments
reveal that, while LLMs can easily answer these questions correctly, they
struggle to identify 1) the incorrect answer corresponding to specific
incomplete knowledge (misconceptions); 2) the misconceptions that explain
particular incorrect answers. Our study indicates new opportunities for
enhancing LLMs’ math reasoning capabilities, especially on developing robust
student simulation and expert tutoring models in the educational applications
such as intelligent tutoring systems.

##  1 Introduction

The fast-evolving development in math problem solving capabilities of Large
Language Models (LLMs) demands appropriate evaluations to measure progress and
identify opportunities for further improvements. Most, if not all, existing
evaluation approaches compute the percentage of math questions that LLMs can
correctly solve, given a set of such questions. For instance, the state-of-
the-art LLM, GPT-4 (Peng et al., 2023) has achieved a commendable performance
on benchmark datasets, including GSM8K (grade-school math word problems)
(Cobbe et al., 2021), MATH (competition-level complex problems) (Hendrycks et
al., 2021), and NumGLUE (numerical reasoning) (Mishra et al., 2022).

In this paper, we ask a different question: how do LLMs perform if we instruct
them to complete a math problem incorrectly, and, moreover, not just
incorrectly in any arbitrary way but in a specific manner? On the flip side,
if given a specifically incorrect answer to a math problem, can the LLMs
detect and analyze the specific error in the answer? Analyzing LLMs’
performance in this setting can reveal if LLMs truly understand the
mathematical concepts required to answer a particular math question, enhance
the explainability of their mathematical problem-solving behaviors, and
suggest a way to evaluate their math reasoning robustness in real-world use
cases such as education. Our new approach of evaluating LLMs can also be seen
as a special instance of counterfactual reasoning (Sonkar and Baraniuk, 2023),
in which LLMs will need to abide to new knowledge and discard the information
it has already encoded. Our investigation also has far-reaching potential
implications for intelligent tutoring systems in education (Sonkar et al.,
2023a, b) with an emphasis on correctly understanding student’s error and
misconceptions for effective guidance.

###  1.1 Contributions

In this work, we propose "novice learner and expert tutor" approach to
evaluate the mathematical reasoning capabilities of LLMs based on math
misconceptions. Instead of solely focusing on the LLMs’ success rate in
answering math questions correctly, we measure whether LLMs can identify the
wrong answer to a question corresponding to a specific set of math
misconceptions, simulating a novice learner, and whether LLMs can identify the
set of misconceptions that can explain a particular wrong answer to a
question. Such investigation provides a new way to evaluate LLMs’ capabilities
in mathematical reasoning and has far-reaching implications for education
applications. We conduct experiments on grade-school math problems and
demonstrate that LLMs face challenges in replicating the behaviors of 1)
novice learners who make mistakes, and 2) expert tutors who precisely
understand why learners make certain mistakes.

###  1.2 Problem formulation

We define a natural language representation of a mathematical misconception
$m$ as an element in a set of misconceptions $\mathcal{M}$. For each multiple
choice question $q$, we denote its four possible answer choices as
$\mathcal{A}_{q}$ where the correct answer is represented as $a_{q}$ and the
incorrect answer choice caused by the misconception $m$ as $a_{q,m}$. For
simplicity, we assume each incorrect answer choice is determined by a single
misconception.

For “novice learners” evaluation, we obtain the LLM $f_{\theta}$’s output
answer to question $q$ based on the specific misconceptions:
$\hat{a}_{q,m}=f_{\theta}(p(q,m,\mathcal{A}_{q}))$ where $p(\cdot)$ is the
prompt. We then compute the accuracy
$s_{1}=\frac{1}{N}\sum_{i=1}^{N}{1}\\{a_{q,m}=\hat{a}_{q,m}\\}$.

For the "expert tutors" evaluation, we derive LLM $f_{\theta}$’s output
misconception given a question, a particular incorrect answer and a subset of
misconceptions $\mathcal{M}_{n}\subset\mathcal{M}$:
$\hat{m}=f_{\theta}(p(q,a_{q,m},\mathcal{M}_{n}))$, where $p(\cdot)$ is the
prompt. We compute the accuracy as
$s_{2}=\frac{1}{N}\sum_{i=1}^{N}{1}\\{m_{i}=\hat{m}_{i}\\}$.

###  1.3 Dataset

| w/o exp | with exp
---|---|---
GPT-3.5 | 0.576 $\pm$ 0.025 | 0.805 $\pm$ 0.027
GPT-4 | 0.774 $\pm$ 0.019 | 0.948 $\pm$ 0.009
Table 1: Accuracy of both GPT-3.5 and GPT-4 when answering math questions in
our dataset.

Our research utilizes a dataset from Eedi’s platform111https://eedi.com/home,
which comprises multiple-choice grade-school level math questions. More
details of the dataset and preprocessing steps are in Appendix A. We explore
two prompting strategies to examine the accuracy of LLMs to correctly answer
the questions. The “w/o exp" prompting strategy provides LLMs with a question
and four answer choices, while requesting correct answer as the output. The
“with exp" prompt demands the LLMs to justify the answer with explanation
prior to providing an answer itself, which is similar to the Chain-of-Thoughts
prompting (Wei et al., 2022). Table 1 provides the accuracy of question
answering, where GPT-4 could achieve a near-perfect accuracy of 94.8%. It
clearly demonstrates that LLMs can accurately solving the math questions in
the dataset, when providing with appropriate prompts. It solidifies the
baseline performance of LLMs with our dataset, which offers potential for
further exploration of LLM’s ability to answer math questions incorrectly in a
specific manner.

Figure 1: (Left) An example question from the dataset with question, choices
and the corresponding misconceptions. (Middle) A depiction of experiment 1 in
which we use LLMs to mimic the role of novice learners. (Right) An
illustration of experiment 2 where we use LLMs to simulate expert tutors.

##  2 Experiments

We conduct two experiments to evaluate if LLMs can effectively simulate novice
learners and expert tutors, as shown in Figure 1. We use with exp prompting
strategy mentioned in Section 1.3 unless defined otherwise. We utilize GPT-3.5
and GPT-4 (Peng et al., 2023) by OpenAI, with a temperature of 0.9. To
automate the evaluation process, we prompt the LLMs to deliver structured
responses in a predefined json format: { "Explanation": "…", "Choices":
"A/B/C/D"  }. We ensure statistical significance by reporting all results
based on the mean and standard deviation derived from four random, independent
trials.

###  2.1 Experiment 1: Using LLMs to simulate a novice learner

Our first experiment was designed to explore the potential of LLMs to mimic an
novice learner by answering math questions incorrectly in a particular
pattern. More specifically, we present LLMs with a datapoint
$(q,m,\mathcal{A}_{q})$. Instead of seeking a correct answer, we require the
LLMs to provide the specific incorrect answer $a_{q,m}$ that aligned with the
provided math misconception $m$ because our primary objective is to assess
LLMs’ potential to distinguish predictable error patterns based on incomplete
knowledge (misconception). In order to conduct a comprehensive experiment, we
explore four prompt settings as follows: (1). [zero (w/o exp)]: zero-shot
prompting without asking for explanations; (2). [zero (with exp)]: zero-shot
prompting requires explanations; (3). [few (rand)]: few-shot prompting uses 2
random examples and requires explanations; (4). [few (same)]: few-shot
prompting uses 2 examples of the same misconceptions as the given
misconception and requires explanation (only a subset of 152 datapoint is
suitable)

We provide the prompts for zero (with exp) below and the remaining prompts in
Appendix B.

You are a student with some math misconceptions.
Your task is to answer a multiple-choice math question as a student who
doesn’t understand the concepts completely and has some math misconceptions.
Your answer should be based on the given misconception. DO NOT just give the
correct answer! Also provide some explanations on why you choose your answer.

Table 2 provides the results for each prompting strategy. The results reveal
that an effective zero-shot prompting strategy can yield an accuracy of 61.7%
for GPT-4 model to select the “appropriate" incorrect choice based on a
specific math misconception. While the accuracy significantly exceeds the rate
of random selection, the performance is still not quite satisfactory when
compared to GPT-4’s near-perfect performance at answering the questions
accurately. The few-shot prompting strategy with randomly selected examples
leads to a even worse performance than zero-shot prompting. Our hypothesis is
that the randomly selected examples might not always be relevant or precise,
potentially introducing more distractions and “noise" in the learning
processing. Furthermore, the relatively higher accuracy of few-shot prompting
with same misconceptions suggest that appropriate few-shot examples can
ultimately enhance performance. In the future work, we aim to extend our study
with in-context learning to choose the most suitable examples.

Overall, although LLMs demonstrate impressive accuracy when providing accurate
answers to grade-school math questions, their ability to act as novice
learners and answer the question incorrectly in a specific way remains
limited. There’s still considerable potential for improving the way LLMs
“controllably" generate specified erroneous responses to math problems
following particular instructions.

| zero (w/o exp) | zero (with exp)
---|---|---
GPT-3.5 |  0.371 $\pm$ 0.010 |  0.425 $\pm$ 0.011
GPT-4 |  0.534 $\pm$ 0.007 |  0.617 $\pm$ 0.005
| few (rand) | few (same)
GPT-3.5 |  0.415 $\pm$ 0.023 |  0.497 $\pm$ 0.031
GPT-4 |  0.600 $\pm$ 0.014 |  0.679 $\pm$ 0.020
Table 2: Accuracy of evaluating GPT-3.5 and GPT-4 models’ capability to behave
as a novice learner. We use four prompting strategies. Compare to results in
Table 1, LLMs perform considerably worse in simulating novice learners with
incomplete knowledge.

Figure 2: Performance of GPT models mimicking expert tutors. Increasing the
number of misconceptions results in a decline of accuracy in detecting the
misconception that explains a wrong answer.

###  2.2 Experiment 2: Using LLMs to simulate an expert tutor

We explore the capability of LLMs to simulate an expert tutor, particularly on
pinpointing misconception(s) specific to the incorrect student answer of a
math question. We provide the LLMs with a datapoint
$(q,a_{q,m},\mathcal{M}_{n})$, where $\mathcal{M}_{n}$ contains a sequence of
$n$ numbers of misconceptions. Among these misconceptions, a specific
misconception $m$ corresponds to the incorrect answer $a_{q,m}$ while the rest
of $n-1$ misconceptions are randomly selected from the misconceptions pool
$\mathcal{M}$. We then prompt the LLMs to identify the correct misconception
$m$. Additionally, we incrementally increase the number of misconceptions
presented to the LLMs. We use the following prompt to guide the LLMs:

You are an expert math tutor who knows about all grade-school level math
misconceptions.
Your task is to select the accurate type of misconceptions your student have
based on the (incorrect) answer he/she gives to a multiple-choice math
question. You will be given $n$ misconception types. Your selected
misconception type should correspond to the given question and answer. You
should also provide some explanations on why you choose the misconception.

Our results in Figure 2 reveal that with a starting point of $n=4$ (one real
and three random) misconceptions, GPT-4 achieves an impressive accuracy of
91.9%. It demonstrates that LLMs, when provided with a limited amount of
misconceptions, can effectively mirror a tutor’s ability to identify the
correct misconceptions causing inaccurate answer. However, an inversely
proportional relationship between the number of misconceptions and the
performance of both LLMs is observed as we increase the number of
misconceptions. The performance drops down to an average of 39.8% for GPT-4 as
the misconception number increases to 100. The ideal proficiency level of an
expert tutor would involve the recognition and differentiation of at least 100
types of grade-school level math misconceptions effortlessly. Judging from
this stringent criterion, LLMs still have much room for improvement in
approximating the comprehensive knowledge and adaptive problem-solving acumen
of an expert tutor.

##  3 Conclusion

We propose novel evaluations of the mathematical reasoning capabilities of
LLMs based on math misconceptions. Our experiments indicate that LLMs struggle
to simulate “novice” learners with specific knowledge gaps. We also find that
even though LLMs are capable of emulating a “tutor”, they could not simulate
the proficiency of an “expert” tutor, even on simple grade-school level
questions. Our evaluations reveal opportunities to further improve
mathematical question answering capabilities of LLMs to additionally consider
scenarios where incomplete knowledge is assumed and misconceptions are taken
into account. Doing so will help probe whether LLMs truly understand the
mathematical concepts, improve the LLMs’ explainability when performing math
question answering, and unlock their potential use in improving intelligent
tutoring systems in education use cases. Avenues of future work include 1)
expanding our current evaluation to additional scenarios and LLMs, such as
LLaMA-2 (Touvron et al., 2023) or Vicuna (Chiang et al., 2023); and 2)
improving LLMs’ capabilities in handling and understanding misconceptions.

## References

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##  Appendix A Dataset Statistics

Our Eedi dataset includes multiple-choice grade-school level math questions.
Each question contains four answer choices, of which only one is correct.
Additionally, we have a list of expert-curated student misconceptions
corresponding to the incorrect choices, though not every choice possesses an
associated misconception. We preprocess the dataset using the two following
steps: 1). remove all the questions that contain images or graphs; 2). remove
all the questions whose different choices reveal identical misconceptions.
After the preprocessing steps, the dataset contains 213 questions, paired with
482 choices that revealed a collective total of 319 student misconceptions. An
example of a math question, the corresponding misconceptions and an instance
of GPT-4’s incorrect answer to the question is provided in Figure 3.

Figure 3: An sample question from our dataset along with associated
misconceptions for each incorrect choice. It also includes an erroneous
response given by GPT-4.

##  Appendix B Prompts

Prompt for LLMs to correctly answer the questions (w/o exp):

Your are a helpful assistant for answering simple math questions.
Your task is to give the correct answer of the provided math multiple choice
questions. Put your answer as the letter index (A/B/C/D) in json format as
below.
{"Answer": "A/B/C/D" }

Prompt for LLMs to correctly answer the questions (w/o exp):

Your are a helpful assistant for answering simple math questions.
Your task is to give the correct answer of the provided math multiple choice
questions. Also provide explanations on why you choose the answer. Put your
answer as the letter index (A/B/C/D) in json format as below.
{"Explanation": "...", "Answer": "A/B/C/D" }

Prompt for LLMs to act as novice learners (zero (w/o exp)):

You are a student with some math misconceptions.
Your task is to answer a multiple-choice math questions as a student who
doesn’t understand the concepts completely and has some math misconceptions.
Your answer should be based on the given misconception. Choose a choice from
the four answer choices provided and DO NOT just give the correct answer! Put
your answer (A/B/C/D) in json format with your choice as shown below.
{"Choice": "A/B/C/D" }

Prompt for LLMs to act as novice learners (few (rand / same):

You are a student with some math misconceptions.
Your task is to answer a multiple-choice math questions as a student who
doesn’t understand the concepts completely and has some misconceptions. Some
examples of questions, misconceptions and answer triples are provided below:
{few-shot examples...}
Your answer should be based on the given misconception and DO NOT just give
the correct answer! You should also provide some explanations on why you
choose the choice. Put your answer in json format with your choice and
explanation as shown below.
{"Explanation": "...", "Answer": "A/B/C/D" }