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We consider learning mathematics through action research, hacking, discovery, inquiry, learning-by-doing as opposed to the instruct and perform, industrial model of the 19th century. A learning model based on self-awareness, types, functions, structured drawing and formal diagrams addresses the weaknesses of drill and practice and the pitfalls of statistical prediction with Large Language Models. In other words, we build mathematics/informatics education on the activity of a professional mathematician in mathematical modelling and designing programs. This tradition emphasises the role of dialogue and doing mathematics. In the Language/Action approach the teacher designs mathematising situations that scaffold previously encountered, or not-known-how-to-solve problems for the learner while teachers and teacher/interlocutors supervise the process. A critical feature is the written-oral dialogue between the learner and the teacher. As a rule, this is 1 to 1 communication. The role of the teacher/interlocutor, a more knowledgeable other, is mostly performed by a more senior student, 1 per 5 to 7 pupils. After Doug Engelbart we propose the metaphor of human intellect augmented by digital

Motivated by a question posed by Freeman, Oikhberg, Pineau and Taylor, we prove that if $K$ is a compact Hausdorff space with $K^{(α)} eq\varnothing$, where $2<α<ω$, then $C[1,ω^α]$ isometrically embeds into $C(K)$ doing stable phase retrieval (SPR). We also show that the latter cannot be extended to the case $α=2$.

Do LLM agents act on the reasoning they state? This question of process fidelity is central to using LLMs in social simulation, yet it is hard to measure where no reference for correct behavior exists. We study it in acontrolled setting, a Texas Poker simulator with a verifiable reference action for every decision by decomposing the faithfulness gap into two steps: reasoning-conclusion and conclusion-action. The two steps behave oppositely.

Can AI solve all math? What do we actually mean by doing mathematics? How do we communicate mathematics? What is mathematics beyond problem solving? This essay is my attempt to answer these questions.

We give a model of how to infer natural language rules by doing experiments. The model integrates Large Language Models (LLMs) with Monte Carlo algorithms for probabilistic inference, interleaving online belief updates with experiment design under information-theoretic criteria. We conduct a human-model comparison on a Zendo-style task, finding that a critical ingredient for modeling the human data is to assume that humans also consider fuzzy, probabilistic rules, in addition to assuming that humans perform approximately-Bayesian belief updates. We also compare with recent algorithms for using LLMs to generate and revise hypotheses, finding that our online inference method yields higher accuracy at recovering the true underlying rule, and provides better support for designing optimal experiments.

Online lending, a phenomenon which is becoming mainstream due to the migration of consumer finance to the Internet and the adoption of AI based lending models, is an example of learning by doing. This paper studies optimal policies for a direct online lender. This is an instance of a more general problem: how should a decision-maker experiment sequentially in the face of unknown customer (or other) information? Conventional wisdom suggests the decision-maker should take advantage of sequential learning opportunities by conducting multiple small, lean experiments, each building incrementally on the results of earlier ones. Can a single grand experiment, uninformed by earlier experiments, do as well? We find that lean incremental experiments are optimal when the interest rate is exogenous. However, when we extend the lender's action space to setting both the interest rate and the loan amount, we find conditions under which a single grand experiment is optimal. In both cases, income variability can benefit the lender by enabling more effective experimentation. We also study the consumer segmentation associated with each strategy and show that the lender cannot achieve more than half t

Machine learning and statistical modeling methods were used to analyze the impact of climate change on financial wellbeing of fruit farmers in Tunisia and Chile. The analysis was based on face to face interviews with 801 farmers. Three research questions were investigated. First, whether climate change impacts had an effect on how well the farm was doing financially. Second, if climate change was not influential, what factors were important for predicting financial wellbeing of the farm. And third, ascertain whether observed effects on the financial wellbeing of the farm were a result of interactions between predictor variables. This is the first report directly comparing climate change with other factors potentially impacting financial wellbeing of farms. Certain climate change factors, namely increases in temperature and reductions in precipitation, can regionally impact self-perceived financial wellbeing of fruit farmers. Specifically, increases in temperature and reduction in precipitation can have a measurable negative impact on the financial wellbeing of farms in Chile. This effect is less pronounced in Tunisia. Climate impact differences were observed within Chile but not in

In this article, we identify the didactic conditions for learning mathematics in kindergarten. To do so, we rely on the framework of the theory of didactic situations (Brousseau, 1998) and the notion of problem-situation (Douady, 1984). We first explain what constitutes for us the stakes of teaching mathematics in kindergarten and then, based on examples, we highlight the conditions related to the stakes of the pupils' activity, to the characteristics of the situations proposed to the pupils and to the teacher's interventions.

In 1952 Bing astonished the mathematical world with his wild involution on $S^3$. It has been among the most seminal examples in topology. The example depends on finding shrinking homeomorphisms of Bing's decomposition of $S^3$ into points and arcs. If Bing's original homeomorphisms are varied, Bing's original wild involution changes by conjugation, which preserves some analytic properties \cite{fs22} while altering others. In 1988, Bing published a second paper "Shrinking Without Lengthening," answering a question that one of the present authors posed to him in an effort to understand the geometry of the entire conjugacy class. In this paper we produce a counterintuitive construction, namely, a method to shrink the Bing decomposition doing almost nothing at all--neither lengthening much nor rotating much.

Deep learning has achieved a great success in many areas, from computer vision to natural language processing, to game playing, and much more. Yet, what deep learning is really doing is still an open question. There are a lot of works in this direction. For example, [5] tried to explain deep learning by group renormalization, and [6] tried to explain deep learning from the view of functional approximation. In order to address this very crucial question, here we see deep learning from perspective of mechanical learning and learning machine (see [1], [2]). From this particular angle, we can see deep learning much better and answer with confidence: What deep learning is really doing? why it works well, how it works, and how much data is necessary for learning. We also will discuss advantages and disadvantages of deep learning at the end of this work.

Despite a rise in the use of "learning by doing" pedagogical methods in praxis, little is known as to how these methods improve learning outcomes. Here we show that visual association cortex causally contributes to performance benefits of a learning by doing method. This finding derives from transcranial magnetic stimulation (TMS) and a gesture-enriched foreign language (L2) vocabulary learning paradigm performed by 22 young adults. Inhibitory TMS of visual motion cortex reduced learning outcomes for abstract and concrete gesture-enriched words in comparison to sham stimulation. There were no TMS effects on words learned with pictures. These results adjudicate between opposing predictions of two neuroscientific learning theories: While reactivation-based theories predict no functional role of visual motion cortex in vocabulary learning outcomes, the current study supports the predictive coding theory view that specialized sensory cortices precipitate sensorimotor-based learning benefits.

Prosociality is fundamental to human social life, and, accordingly, much research has attempted to explain human prosocial behavior. Capraro and Rand (Judgment and Decision Making, 13, 99-111, 2018) recently provided experimental evidence that prosociality in anonymous, one-shot interactions (such as Prisoner's Dilemma and Dictator Game experiments) is not driven by outcome-based social preferences - as classically assumed - but by a generalized morality preference for "doing the right thing". Here we argue that the key experiments reported in Capraro and Rand (2018) comprise prominent methodological confounds and open questions that bear on influential psychological theory. Specifically, their design confounds: (i) preferences for efficiency with self-interest; and (ii) preferences for action with preferences for morality. Furthermore, their design fails to dissociate the preference to do "good" from the preference to avoid doing "bad". We thus designed and conducted a preregistered, refined and extended test of the morality preference hypothesis (N=801). Consistent with this hypothesis, our findings indicate that prosociality in the anonymous, one-shot Dictator Game is driven by

We develop an approach to finding upper bounds for the number of arithmetic operations necessary for doing harmonic analysis on permutation modules of finite groups. The approach takes advantage of the intrinsic orbital structure of permutation modules, and it uses the multiplicities of irreducible submodules within individual orbital spaces to express the resulting computational bounds. We conclude by showing that these bounds are surprisingly small when dealing with certain permutation modules arising from the action of the symmetric group on tabloids.