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Formal mathematical reasoning is unique in its precision: any valid conclusion can be justified by a sequence of base axioms. But human-written proofs or solutions rarely operate at that level. Instead, obvious steps are skipped to provide a simple, lucid argument. This is especially important in an educational setting, where too many details in an example solution, or too few, can confuse a student. What are the key steps for humans in a given formal solution? We investigate several computational hypotheses in the context of equation solving. Specifically, we take a reinforcement learning agent that solves equations using low-level axioms, and propose a series of methods for abstracting its solutions by selecting key steps. We consider methods based on the semantic distance between subsequent steps, based on the steps with the highest uncertainty for the agent, and based on transitions between latent &quot;high-level skills&quot; learned from a large number of agent-produced solutions. In a human evaluation we find that skill-base simplifications were judged most useful. These results suggest new directions for understanding human mathematical reasoning.

CogSci 2022

Le to the Reader: Abstracting Solutions in Mathematical Reasoning

Formal mathematical reasoning is unique in its precision: any valid conclusion can be justified by a sequence of base axioms. But human-written proofs or solutions rarely operate at that level. Instead, obvious steps are skipped to provide a simple, lucid argument. This is especially important in an educational setting, where too many details in an example solution, or too few, can confuse a student. What are the key steps for humans in a given formal solution? We investigate several computational hypotheses in the context of equation solving. Specifically, we take a reinforcement learning agent that solves equations using low-level axioms, and propose a series of methods for abstracting its solutions by selecting key steps. We consider methods based on the semantic distance between subsequent steps, based on the steps with the highest uncertainty for the agent, and based on transitions between latent "high-level skills" learned from a large number of agent-produced solutions. In a human evaluation we find that skill-base simplifications were judged most useful. These results suggest new directions for understanding human mathematical reasoning.

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We investigated individuals’ representation of composite numbers (i.e., those with factors other than 1 and themselves). Some numbers are “more composite” (have more factors) than others (e.g., 72 vs. 74). We hypothesized that people mentally represent numbers in factor neighborhoods. This should result in a compositeness penalty: performance on factoring problems (e.g., x * 9 = 72) should be slower for more composite products, because retrieving the corresponding factor requires inhibiting neighboring factor pairs. We measured 50 undergraduates’ time to correctly solve factoring problems for composite products (6 to 100). We found that for well-practiced facts (i.e., the 12x12 multiplication table), performance slowed as compositeness increased, even after controlling for problem size and factor size. However, there was no compositeness penalty for “non-table” facts. These findings support our factor neighborhood hypothesis for well-practiced facts and suggest that people's representations of composite numbers are influenced by mathematics instruction (i.e., “multiplication tables”).

Neighborhood effects for composite (i.e., non-prime) numbers

Burns & Krygier (2015) demonstrated that people could exhibit a strong bias towards the smaller first digits, in a way similar to that described by Benford’s law. This paper sought to expand the scope of this phenomenon and to test a possible explanation, the Recognition Hypothesis that a Benford bias is due to life-long environmental exposure to this statistical relationship. Participants completed three numerical tasks: A Generation Task requiring answering trivia questions; a Selection Task requiring selecting between two numerical responses; and an Estimation task requiring estimating the number of jelly beans in a jar. The results found no evidence of any first digit effect in the Recognition Task, some evidence of Benford bias in the Generation Task and strong evidence in the Estimation Task. Future research should focus on alternatives to the Recognition Hypothesis and investigate the parameters of Benford bias in generation tasks

Why do People fit to Benford’s Law? – A Test of the Recognition Hypothesis

The ability to calculate fluently is one of the basic academic skills, but certain students have difficulty acquiring it, such as those with developmental dyscalculia. In this study, we focus on the visual perception of numbers as a factor related to computational fluency, and examine whether enlarging the letters of the numbers or widening the line spacing in a calculation task would increase computational fluency. Four hundred and thirty-six junior high school students were asked to perform a two-digit addition or subtraction task in which font size and line width were manipulated, and the number of correct answers per minute was used as a measure of computational fluency. The results showed that enlarging the font size increased the computational fluency of all participants, and that widening the line spacing contributed to the fluency of participants with low computational skills, while it may decrease that of high computational skills.

How can visual perception of numbers affect computational fluency?

Images are a powerful medium known to induce empathy and emotional response in people. In political protests it has the power for a people-initiated policy change and signifies the deep symbolism of a political system. In this study, we aim to quantify the range of emotional connection a person experiences for photographs of a farmers' protest.The protest was the headlines in all media at the time this experiment was conducted and had polarized public opinion. Each photograph is identified to have a set of physical and semantic features. The three selected features were presence of police, gender and close-up (vs.long-shot) in the frame. The intensity on a range of emotions (fear, disgust, anger, sadness, optimism, pessimism, surprise, shock, happiness, and respect) experienced by the viewer for each feature was collected. By statistical and dimensionality analyses, we isolate and identify influencing factors in an image. We found that the presence of police in aggressive actions and close-up shots of had the highest variation in the emotional responses of participants. Interestingly, the gender of the protesters did not show statistically significant effects. The findings from the exploratory investigation highlights the powerful role photographic features have on emotional responses of people, an understudied but critical factor in a world immersed in social media.

Exploring Empathy and a Range of Emotions Towards Protest Photographs

Threatening stimuli capture visual attention more rapidly than benign stimuli. The canonical interpretation of this robust finding is that the brain evolved a “fear module” enabling rapid detection of threats common at the time of mammalian evolution, such as snakes and spiders. This rapid attentional capture is thought to enable prioritized processing of threatening stimuli, providing a survival advantage, and is assumed to be universal. However, these findings have been documented almost entirely in WEIRD (white, educated, industrialized, rich, and democratic) populations. Here, we address this gap by examining threat detection in a remote African culture, the Himba. Using a touch screen visual search task, we found that both evolutionary-relevant (snakes and spiders) and modern threats (knives and syringes) captured attention more rapidly than benign stimuli. To our knowledge, this is the first study showing that the same kind of threats that rapidly capture visual attention in the West also rapidly capture visual attention in the Himba.

Visual attention to threat in the Himba, a remote people of Namibia

How possessions gain personal meaning has been a topic of interest to diverse fields studying the mind. Traditionally, the personal value of possessions is hypothesized to be driven by their special histories (Belk, 1988). We provide evidence that people value possessions for their ability to regulate distress. During the pandemic, people (N=333) reported having coping-possessions that helped them to regulate emotions (49%), solve problems (26%), or both (25%). Coping-possessions had high personal and monetary values (M=4.82 and 4.45, respectively; one-sample t-tests against midpoint, ps < .001) with problem-focused possessions having higher monetary values than emotion-focused ones (p < .001). The majority of coping-possessions, however, had mundane histories (e.g., bought at a store) rather than noteworthy ones (p < .001), suggesting that the emotional functions of objects contribute to their perceived personal value. This work also has implications for how adults use objects to regulate their emotions (Timpano & Port, 2021).

The emotional value in objects: Insights from coping with the COVID-19 pandemic

Research in event cognition has focused on how people perceive
and remember events under experimental conditions. This
research study aims to explore the temporal duration of
self-reported events from daily life (Sreekumar, et al., 2018;
Zhuang, et al., 2012). The small amount of prior work that
exists suggests that daily event durations have a Gaussian
distribution and that people have prior beliefs that reflect this
reality (Griffiths & Tenenbaum, 2006). Forty-eight participants
provided activity duration data as they went about their
everyday lives for 14 days. Descriptive analyses and activity
duration modeling (mixture models of gaussian, gamma, normal
and exponential distributions) were used to characterize event
durations within activity types. Results show that most of the
events present an exponential pattern of durations, while others
show a bimodal pattern. Although some preplanned events have
a characteristic time, many daily events have a substantial
exponential component.

How long are real-life events?

Transformer-based language representation models, such as GPT-3 and DeBERTa, have yielded state-of-the-art results on several benchmarks in Natural Language Processing (NLP), including question answering, story generation and information extraction. However, little is currently understood about their ability to make (approximately rational) decisions, even when simple bets with clear net-positive, zero-risk choices are presented. We design a suite of multi-option decision-making experiments in the same vein as other classic studies (Kahneman & Tversky, 1979). Applying over 20 structural variants of these ‘benchmarks’ (to test for generalization and robustness) to three established language representation models, we use detailed experiments to show that the models lack fundamental decision-making ability, even after several controls. Techniques such as selective fine-tuning can help improve performance partially, but more fundamental NLP research may be needed before such models can be robustly and generally applied in real-world problem domains where decision making is essential.

Evaluating Language Representation Models on Approximately Rational Decision Making Problems

Compositionality has been argued to a necessary component of interpreting language, yet there appear to be many linguistic phenomena that do not overtly exhibit semantic compositional behavior. One of the challenges involves the phenomena of contextual modulations referred to collectively as semantic coercion or logical metonymy. In this paper, we present a computational model that provides the “compositional flexibility” in the interpretation of a verb with its arguments, for such coercive contexts in English. Specifically, we argue that such constructions typically have surface structural correlates in the form of dense paraphrases, and that these forms can be used to model the masked content in the coerced compositional context. We present preliminary results using a transformer architecture on a masked completion task. Our results show that modeling logical metonymy is a challenging task but can be substantially improved by fine-tuning through dense paraphrasing.

Interpreting Logical Metonymy through Dense Paraphrasing

Limits and risks of GPT-3 applications
Recent GPT-3 applications have fed new optimism in AI research. To explore limits and risks, we investigated how far we can get in developing a so-called digital replica of Daniel Dennett, generating text outputs from a large language model trained on Dennett’s philosophical writing. In consultation with Dennett himself, we compare several fine-tuning strategies and evaluate outputs. Analyzing the failures and the successes will allow us to address technical and ethical issues such as:
1. How accurate can such models be with current technology?
2. To what extent (if at all) might it be acceptable to present a model’s outputs as representing an author’s view?
3. Should copyright holders have control over such models?
4. How one can address risks of making such models public (including possible over-reliance on their accuracy)?
5. How good are experts and non-experts in distinguishing humans from replicas?

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Le to the Reader: Abstracting Solutions in Mathematical Reasoning

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