skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.


Title: Empirical re-conceptualization: From empirical generalizations to insight and understanding
Award ID(s):
1920538
PAR ID:
10356120
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
The Journal of Mathematical Behavior
Volume:
65
Issue:
C
ISSN:
0732-3123
Page Range / eLocation ID:
100928
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Identifying patterns is an important part of mathematical investigation, but many students struggle to explain or justify their pattern-based generalizations or conjectures. These findings have led some researchers to argue for a de-emphasis on pattern-based activities, but others argue that empirical investigation can support the discovery of insight into a problem’s structure. We introduce a phenomenon we call empirical re-conceptualization, in which learners identify a conjecture based on an empirical pattern, and then re-interpret that conjecture from a structural perspective. We elaborate this construct by drawing on interview data from undergraduate calculus students and research mathematicians, providing a representative example of empirical re-conceptualization from each participant group. Our findings indicate that developing empirical results can foster subsequent insights, which can in turn lead to justification and proof. 
    more » « less
  2. null (Ed.)
  3. null (Ed.)
    We propose a new simple and natural algorithm for learning the optimal Q-value function of a discounted-cost Markov decision process (MDP) when the transition kernels are unknown. Unlike the classical learning algorithms for MDPs, such as Q-learning and actor-critic algorithms, this algorithm does not depend on a stochastic approximation-based method. We show that our algorithm, which we call the empirical Q-value iteration algorithm, converges to the optimal Q-value function. We also give a rate of convergence or a nonasymptotic sample complexity bound and show that an asynchronous (or online) version of the algorithm will also work. Preliminary experimental results suggest a faster rate of convergence to a ballpark estimate for our algorithm compared with stochastic approximation-based algorithms. 
    more » « less
  4. We construct robust empirical Bayes confidence intervals (EBCIs) in a normal means problem. The intervals are centered at the usual linear empirical Bayes estimator, but use a critical value accounting for shrinkage. Parametric EBCIs that assume a normal distribution for the means (Morris (1983b)) may substantially undercover when this assumption is violated. In contrast, our EBCIs control coverage regardless of the means distribution, while remaining close in length to the parametric EBCIs when the means are indeed Gaussian. If the means are treated as fixed, our EBCIs have an average coverage guarantee: the coverage probability is at least 1 −  α on average across the n EBCIs for each of the means. Our empirical application considers the effects of U.S. neighborhoods on intergenerational mobility. 
    more » « less