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"Reasoning Mathematical models."
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A conceptual model of mathematical reasoning for school mathematics
The development of students' mathematical reasoning (MR) is a goal of several curricula and an essential element of the culture of the mathematics education research community. But what mathematical reasoning consists of is not always clear; it is generally assumed that everyone has a sense of what it is. Wanting to clarify the elements of MR, this research project aimed to qualify it from a theoretical perspective, with an elaboration that would not only indicate its ways of being thought about and espoused but also serve as a tool for reflection and thereby contribute to the further evolution of the cultures of the teaching and research communities in mathematics education. To achieve such an elaboration, a literature search based on anasynthesis (Legendre, 2005) was undertaken. From the analysis of the mathematics education research literature on MR and taking a commognitive perspective (Sfard, 2008), the synthesis that was carried out led to conceptualizing a model of mathematical reasoning. This model, which is herein described, is constituted of two main aspects: a structural aspect and a process aspect, both of which are needed to capture the central characteristics of MR.
Journal Article
Large language models encode clinical knowledge
Large language models (LLMs) have demonstrated impressive capabilities, but the bar for clinical applications is high. Attempts to assess the clinical knowledge of models typically rely on automated evaluations based on limited benchmarks. Here, to address these limitations, we present MultiMedQA, a benchmark combining six existing medical question answering datasets spanning professional medicine, research and consumer queries and a new dataset of medical questions searched online, HealthSearchQA. We propose a human evaluation framework for model answers along multiple axes including factuality, comprehension, reasoning, possible harm and bias. In addition, we evaluate Pathways Language Model
1
(PaLM, a 540-billion parameter LLM) and its instruction-tuned variant, Flan-PaLM
2
on MultiMedQA. Using a combination of prompting strategies, Flan-PaLM achieves state-of-the-art accuracy on every MultiMedQA multiple-choice dataset (MedQA
3
, MedMCQA
4
, PubMedQA
5
and Measuring Massive Multitask Language Understanding (MMLU) clinical topics
6
), including 67.6% accuracy on MedQA (US Medical Licensing Exam-style questions), surpassing the prior state of the art by more than 17%. However, human evaluation reveals key gaps. To resolve this, we introduce instruction prompt tuning, a parameter-efficient approach for aligning LLMs to new domains using a few exemplars. The resulting model, Med-PaLM, performs encouragingly, but remains inferior to clinicians. We show that comprehension, knowledge recall and reasoning improve with model scale and instruction prompt tuning, suggesting the potential utility of LLMs in medicine. Our human evaluations reveal limitations of today’s models, reinforcing the importance of both evaluation frameworks and method development in creating safe, helpful LLMs for clinical applications.
Med-PaLM, a state-of-the-art large language model for medicine, is introduced and evaluated across several medical question answering tasks, demonstrating the promise of these models in this domain.
Journal Article
The World in the Model
During the last two centuries, the way economic science is done has changed radically: it has become a social science based on mathematical models in place of words. This book describes and analyses that change - both historically and philosophically - using a series of case studies to illuminate the nature and the implications of these changes. It is not a technical book; it is written for the intelligent person who wants to understand how economics works from the inside out. This book will be of interest to economists and science studies scholars (historians, sociologists and philosophers of science). But it also aims at a wider readership in the public intellectual sphere, building on the current interest in all things economic and on the recent failure of the so-called economic model, which has shaped our beliefs and the world we live in.
eBook
Semantic memory: A review of methods, models, and current challenges
Adult semantic memory has been traditionally conceptualized as a relatively static memory system that consists of knowledge about the world, concepts, and symbols. Considerable work in the past few decades has challenged this static view of semantic memory, and instead proposed a more fluid and flexible system that is sensitive to context, task demands, and perceptual and sensorimotor information from the environment. This paper (1) reviews traditional and modern computational models of semantic memory, within the umbrella of network (free association-based), feature (property generation norms-based), and distributional semantic (natural language corpora-based) models, (2) discusses the contribution of these models to important debates in the literature regarding knowledge representation (localist vs. distributed representations) and learning (error-free/Hebbian learning vs. error-driven/predictive learning), and (3) evaluates how modern computational models (neural network, retrieval-based, and topic models) are revisiting the traditional “static” conceptualization of semantic memory and tackling important challenges in semantic modeling such as addressing temporal, contextual, and attentional influences, as well as incorporating grounding and compositionality into semantic representations. The review also identifies new challenges regarding the abundance and availability of data, the generalization of semantic models to other languages, and the role of social interaction and collaboration in language learning and development. The concluding section advocates the need for integrating representational accounts of semantic memory with process-based accounts of cognitive behavior, as well as the need for explicit comparisons of computational models to human baselines in semantic tasks to adequately assess their psychological plausibility as models of human semantic memory.
Journal Article
Combining data and theory for derivable scientific discovery with AI-Descartes
Scientists aim to discover meaningful formulae that accurately describe experimental data. Mathematical models of natural phenomena can be manually created from domain knowledge and fitted to data, or, in contrast, created automatically from large datasets with machine-learning algorithms. The problem of incorporating prior knowledge expressed as constraints on the functional form of a learned model has been studied before, while finding models that are consistent with prior knowledge expressed via general logical axioms is an open problem. We develop a method to enable principled derivations of models of natural phenomena from axiomatic knowledge and experimental data by combining logical reasoning with symbolic regression. We demonstrate these concepts for Kepler’s third law of planetary motion, Einstein’s relativistic time-dilation law, and Langmuir’s theory of adsorption. We show we can discover governing laws from few data points when logical reasoning is used to distinguish between candidate formulae having similar error on the data.
Automatic extraction of consistent governing laws from data is a challenging problem. The authors propose a method that takes as input experimental data and background theory and combines symbolic regression with logical reasoning to obtain scientifically meaningful symbolic formulas.
Journal Article
A generative vision model that trains with high data efficiency and breaks text-based CAPTCHAs
Proving that we are human is now part of many tasks that we do on the internet, such as creating an email account, voting in an online poll, or even downloading a scientific paper. One of the most popular tests is text-based CAPTCHA, where would-be users are asked to decipher letters that may be distorted, partially obscured, or shown against a busy background. This test is used because computers find it tricky, but (most) humans do not. George et al. developed a hierarchical model for computer vision that was able to solve CAPTCHAs with a high accuracy rate using comparatively little training data. The results suggest that moving away from text-based CAPTCHAs, as some online services have done, may be a good idea. Science , this issue p. eaag2612 A hierarchical computer vision model solves CAPTCHAs with a high accuracy rate using relatively little training data. Learning from a few examples and generalizing to markedly different situations are capabilities of human visual intelligence that are yet to be matched by leading machine learning models. By drawing inspiration from systems neuroscience, we introduce a probabilistic generative model for vision in which message-passing–based inference handles recognition, segmentation, and reasoning in a unified way. The model demonstrates excellent generalization and occlusion-reasoning capabilities and outperforms deep neural networks on a challenging scene text recognition benchmark while being 300-fold more data efficient. In addition, the model fundamentally breaks the defense of modern text-based CAPTCHAs (Completely Automated Public Turing test to tell Computers and Humans Apart) by generatively segmenting characters without CAPTCHA-specific heuristics. Our model emphasizes aspects such as data efficiency and compositionality that may be important in the path toward general artificial intelligence.
Journal Article
Styles of reasoning for mathematics education
Although reasoning is a central concept in mathematics education research, the discipline is still in need of a coherent theoretical framework of mathematical reasoning. With respect to epistemological problems in the dominant discourses on proof, mathematical modelling, and post-truth politics in the discipline, and in accordance with trends in the philosophy of mathematics and in mathematics education research in general, it is argued that it is necessary to give a relativist account of mathematical reasoning. Hacking’s framework of styles of reasoning is introduced as a possible solution. This framework distinguished between at least six different styles of reasoning, many of which are closely connected to mathematics, and argues that these frameworks define what we accept as decidable assertions, as justifications for such assertions, and as possible objects of such assertions. The article ends with a discussion of the implications of the framework for chosen fields of mathematics education research, which may motivate more focussed studies in the future.
Journal Article
Five ways to ensure that models serve society: a manifesto
Pandemic politics highlight how predictions need to be transparent and humble to invite insight, not blame.
Pandemic politics highlight how predictions need to be transparent and humble to invite insight, not blame.
Journal Article
Visual Genome: Connecting Language and Vision Using Crowdsourced Dense Image Annotations
Despite progress in perceptual tasks such as image classification, computers still perform poorly on cognitive tasks such as image description and question answering. Cognition is core to tasks that involve not just recognizing, but reasoning about our visual world. However, models used to tackle the rich content in images for cognitive tasks are still being trained using the same datasets designed for perceptual tasks. To achieve success at cognitive tasks, models need to understand the interactions and relationships between objects in an image. When asked “What vehicle is the person riding?”, computers will need to identify the objects in an image as well as the relationships
riding(man, carriage)
and
pulling(horse, carriage)
to answer correctly that “the person is riding a horse-drawn carriage.” In this paper, we present the Visual Genome dataset to enable the modeling of such relationships. We collect dense annotations of objects, attributes, and relationships within each image to learn these models. Specifically, our dataset contains over 108K images where each image has an average of
35
objects,
26
attributes, and
21
pairwise relationships between objects. We canonicalize the objects, attributes, relationships, and noun phrases in region descriptions and questions answer pairs to WordNet synsets. Together, these annotations represent the densest and largest dataset of image descriptions, objects, attributes, relationships, and question answer pairs.
Journal Article