Related papers: Learning Mathematics through incorrect Problems
Imagine a student -- let's call him `E', and make him a "he" -- that is enrolled in Calculus 2, and who believes that to pass in maths courses he only needs to memorize methods and apply them quickly and without errors. Let's imagine that…
Helping students learn from their own mistakes can help them develop habits of mind while learning physics content. Based upon cognitive apprenticeship model, we asked students to self-diagnose their mistakes and learn from reflecting on…
We propose a semester-long Bayesian statistics course for undergraduate students with calculus and probability background. We cultivate students' Bayesian thinking with Bayesian methods applied to real data problems. We leverage modern…
Parsons problems (PPs) have shown promise in structured problem solving by providing scaffolding that decomposes the problem and requires learners to reconstruct the solution. However, some students face difficulties when first learning…
One of the outstanding problems of philosophy of science and mathematics today is whether there is just "one" unique mathematics or the same can be bifurcated into "pure" and "applied" categories. A novel solution for this problem is…
We outline a new tool that can promote coherence within and across higher education mathematics courses by focusing on problem-solving: the Mathematical Problem-Solving Pipeline or MPSP. The MPSP can be used for teaching mathematics and…
While functional programming is an efficient way to express complex software, functional programming languages have a steep learning curve. Haskell can be challenging to learn for students who were only introduced to imperative programming.…
Recent studies report that many machine reading comprehension (MRC) models can perform closely to or even better than humans on benchmark datasets. However, existing works indicate that many MRC models may learn shortcuts to outwit these…
To precisely evaluate a language model's capability for logical reading comprehension, we present a dataset for testing the understanding of the rationale behind critical reasoning. For questions taken from an existing multiplechoice…
Developing expertise in physics entails learning to use mathematics effectively and efficiently as applied to the context of physical situations. Doing so involves coordinating a variety of concepts and skills including mathematical…
We explore the use of expert iteration in the context of language modeling applied to formal mathematics. We show that at same compute budget, expert iteration, by which we mean proof search interleaved with learning, dramatically…
In Constraint Programming, constraints are usually represented as predicates allowing or forbidding combinations of values. However, some algorithms exploit a finer representation: error functions. Their usage comes with a price though: it…
Real-life conjectures do not come with instructions saying whether they they should be proven or, instead, refuted. Yet, as we now know, in either case the final argument produced had better be not just convincing but actually verifiable in…
One can notice that quite often difference between so-called "standard students" and "gifted" ones is not because that first are less smart, but they have different "orientation", they consider subject as a collections of rules which should…
Language models have demonstrated remarkable performance in solving reasoning tasks; however, even the strongest models still occasionally make reasoning mistakes. Recently, there has been active research aimed at improving reasoning…
There has been an increasing trend of females performing better than males academically across the mathematical engineering courses. To confirm this assumption, final marks of two independent samples of students from Calculus courses across…
We consider learning a predictive model to be subsequently used for a given downstream task (described by an algorithm) that requires access to the model evaluation. This task need not be prediction, and this situation is frequently…
Expanding our knowledge of student difficulties in advanced undergraduate electromagnetism is essential if we are to develop effective instructional interventions. Drawing on an analysis of course materials, in-class observations and…
Machine learning tasks entail the use of complex computational pipelines to reach quantitative and qualitative conclusions. If some of the activities in a pipeline produce erroneous or uninformative outputs, the pipeline may fail or produce…
Understanding of free-format multi-step arithmetic word problems with extraneous information is discussed. A model including a full set of general skills necessary for understanding such problems was developed and computer implemented. The…