Chuck Kalish | University of Wisconsin-Madison (original) (raw)
Papers by Chuck Kalish
Cognitive Science, 2017
Should we give learners a lot of practice with a few problems, or a little practice with a variet... more Should we give learners a lot of practice with a few problems, or a little practice with a variety of problems? The best practice set depends on the way people are learning. We describe two models people employ when learning arithmetic problems. We show that features of the task environment influence model use. When problems are presented in a purely symbolic format, people learn an item-specific model. When the task format linked problems to representations of magnitudes, people learn a continuous model. We also test the effects of different practice sets on learning. In both formats people learned the practice sets well with a few repeated examples. With a continuous magnitude format people showed better transfer with a wide variety of practice problems. Variety led to poor learning in the symbolic format. In ongoing research we are attempting to identify the optimal practice set for each type of learning model.
Cognitive Science, 2015
In math education the goal is for children not only to master the materials and problems presente... more In math education the goal is for children not only to master the materials and problems presented, but to understand underlying principles and properties that can be applied broadly to new problems and situations. Teachers in the classroom and policy-makers in Washington thus are both faced with what is essentially a cognitive question: What instructional regimes and practices will produce rapid learning, deep understanding, and broad transfer? This question has often been approached without connection to cognitive theories of learning, memory, and representation, but the gap has begun to narrow. On one hand, it is now known that domain-general learning mechanisms can acquire quite abstract and structured representations that go beyond the perceptual structure of the environment—a critical requirement for any theory of mathematical knowledge. Conversely studies in math cognition have revealed counter-intuitive behaviors that find ready explanations in cognitive models of learning i...
Cognitive Science, 2015
In a categorization task involving both labeled and unlabeled data, it has been shown that humans... more In a categorization task involving both labeled and unlabeled data, it has been shown that humans make use of the underlying distribution of the unlabeled examples. It has also been shown that humans are sensitive to shifts in this distribution, and will change predicted classifications based on these shifts. It is not immediately obvious what causes these shifts – what specific properties of these distributions humans are sensitive to. Assuming a parametric model of human categorization learning, we can ask which parameters or sets of parameters humans fix after exposure to labeled data and which are adjustable to fit subsequent unlabeled data. We formulate models to describe different parameter sets which humans may be sensitive to and a dataset which optimally discriminates among these models. Experimental results indicate that humans are sensitive to all parameters, with the closest model fit being an unconstrained version of semi-supervised learning using expectation maximization.
Psychology of Learning and Motivation, 2014
Abstract There are many accounts of how humans make inductive inferences. Two broad classes of ac... more Abstract There are many accounts of how humans make inductive inferences. Two broad classes of accounts are characterized as “theory based” or “similarity based.” This distinction has organized a substantial amount of empirical work in the field, but the exact dimensions of contrast between the accounts are not always clear. Recently, both accounts have used concepts from formal statistics and theories of statistical learning to characterize human inductive inference. We extend these links to provide a unified perspective on induction based on the relation between descriptive and inferential statistics. Most work in Psychology has focused on descriptive problems: Which patterns do people notice or represent in experience? We suggest that it is solutions to the inferential problem of generalizing or applying those patterns that reveals the more fundamental distinction between accounts of human induction. Specifically, similarity-based accounts imply that people make transductive inferences, while theory-based accounts imply that people make evidential inferences. In characterizing claims about descriptive and inferential components of induction, we highlight points of agreement and disagreement between alternative accounts. Adopting the common framework of statistical inference also motivates a set of empirical hypotheses about inductive inference and its development across age and experience. The common perspective of statistical inference reframes debates between theory-based and similarity-based accounts: These are not conflicting theoretical perspectives, but rather different predictions about empirical results.
When the distribution of unlabeled data in feature space lies along a manifold, the information i... more When the distribution of unlabeled data in feature space lies along a manifold, the information it provides may be used by a learner to assist classification in a semi-supervised setting. While manifold learning is well-known in machine learning, the use of manifolds in human learning is largely unstudied. We perform a set of experiments which test a human's ability to use a manifold in a semisupervised learning task, under varying conditions. We show that humans may be encouraged into using the manifold, overcoming the strong preference for a simple, axis-parallel linear boundary.
Journal of Experimental Child Psychology, 2016
What do children learn from biased samples? Most samples people encounter are biased in some way,... more What do children learn from biased samples? Most samples people encounter are biased in some way, and responses to bias can distinguish among different theories of inductive inference. 67 four-to eight-year-old children learned to make conditional predictions about a set of sample items. They then made predictions about the properties of new instances or old instances from the training set. The experiment compared unbiased and biased sampling. Given unbiased samples, participants used what they learned to make predictions about population and sample instances. With biased samples, children were less accurate/confident about inferences about the population than the sample. Children used information in a biased sample to make predictions about items in that sample, but were less likely to generalize to new items than when samples were unbiased.
Cognitive science, Jan 21, 2015
A common practice in textbooks is to introduce concepts or strategies in association with specifi... more A common practice in textbooks is to introduce concepts or strategies in association with specific people. This practice aligns with research suggesting that using "real-world" contexts in textbooks increases students' motivation and engagement. However, other research suggests this practice may interfere with transfer by distracting students or leading them to tie new knowledge too closely to the original learning context. The current study investigates the effects on learning and transfer of connecting mathematics strategies to specific people. A total of 180 college students were presented with an example of a problem-solving strategy that was either linked with a specific person (e.g., "Juan's strategy") or presented without a person. Students who saw the example without a person were more likely to correctly transfer the novel strategy to new problems than students who saw the example presented with a person. These findings are the first evidence tha...
Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2011
Methods We adopted the sort-re-sort procedure used by Medin et al.(1997). In one-on-one interview... more Methods We adopted the sort-re-sort procedure used by Medin et al.(1997). In one-on-one interviews, participants were presented with a set of cards that contained numbers or geometric shapes, the selections of which were based upon the varying of dimensions identified by previous research (Feldman, 2000; Miller & Gelman, 1983; see Figure 1).
Journal of Numerical Cognition
K-12 students often rely on testing examples to explore and determine the truth of mathematical c... more K-12 students often rely on testing examples to explore and determine the truth of mathematical conjectures. However, little is known about how K-12 students choose examples and what elements are important when considering example choice. In other domains, experts give explicit consideration to the typicality of examples – how representative a given item is of a general class. In a pilot study, we interviewed 20 middle school students who classified examples as typical or unusual and justified their classification. We then gave middle school students and mathematicians a survey where they rated the typicality of mathematical objects in two contexts – an everyday context (commonness in everyday life) and a mathematical context (how likely conjectures that hold for the object are to hold for other objects). Mathematicians had distinct notions of everyday and mathematical typicality – they recognized that the objects often seen in everyday life can have mathematical properties that can...
There are many accounts of how humans make inductive inferences. Two broad classes of accounts a... more There are many accounts of how humans make inductive inferences. Two broad classes of accounts are characterized as “theory based” or “similarity based.” This distinction has organized a substantial amount of empirical work in the field, but the exact dimensions of contrast between the accounts are not always clear. Recently, both accounts have used concepts from formal statistics and theories of statistical learning to characterize human inductive inference. We extend these links to provide a unified perspective on induction based on the relation between descriptive and inferential statistics. Most work in Psychology has focused on descriptive problems: Which patterns do people notice or represent in experience? We suggest that it is solutions to the inferential problem of generalizing or applying those patterns that reveals the more fundamental distinction between accounts of human induction. Specifically, similarity-based accounts imply that people make transductive inferences, while theory-based accounts imply that people make evidential inferences. In characterizing claims about descriptive and inferential components of induction, we highlight points of agreement and disagreement between alternative accounts. Adopting the common framework of statistical inference also motivates a set of empirical hypotheses about inductive inference and its development across age and experience. The common perspective of statistical inference reframes debates between theory-based and similarity-based accounts: These are not conflicting theoretical perspectives, but rather different predictions about empirical results.
in The Conceptual Mind: New Directions in the Study of Concepts
How do concepts acquire a normative structure, and how do children acquire the concept of NORMATI... more How do concepts acquire a normative structure, and how do children acquire the concept of NORMATIVITY? Humans clearly have capacities to make judgments of obligations, permissions, and prohibitions. Such capacities would seem to depend on possessing certain kinds of concepts. Theories of concepts must explain and accommodate normative content. Moreover, studying norms may also shed new light on the nature and acquisition of concepts.
Cognitive Science, 2017
Should we give learners a lot of practice with a few problems, or a little practice with a variet... more Should we give learners a lot of practice with a few problems, or a little practice with a variety of problems? The best practice set depends on the way people are learning. We describe two models people employ when learning arithmetic problems. We show that features of the task environment influence model use. When problems are presented in a purely symbolic format, people learn an item-specific model. When the task format linked problems to representations of magnitudes, people learn a continuous model. We also test the effects of different practice sets on learning. In both formats people learned the practice sets well with a few repeated examples. With a continuous magnitude format people showed better transfer with a wide variety of practice problems. Variety led to poor learning in the symbolic format. In ongoing research we are attempting to identify the optimal practice set for each type of learning model.
Cognitive Science, 2015
In math education the goal is for children not only to master the materials and problems presente... more In math education the goal is for children not only to master the materials and problems presented, but to understand underlying principles and properties that can be applied broadly to new problems and situations. Teachers in the classroom and policy-makers in Washington thus are both faced with what is essentially a cognitive question: What instructional regimes and practices will produce rapid learning, deep understanding, and broad transfer? This question has often been approached without connection to cognitive theories of learning, memory, and representation, but the gap has begun to narrow. On one hand, it is now known that domain-general learning mechanisms can acquire quite abstract and structured representations that go beyond the perceptual structure of the environment—a critical requirement for any theory of mathematical knowledge. Conversely studies in math cognition have revealed counter-intuitive behaviors that find ready explanations in cognitive models of learning i...
Cognitive Science, 2015
In a categorization task involving both labeled and unlabeled data, it has been shown that humans... more In a categorization task involving both labeled and unlabeled data, it has been shown that humans make use of the underlying distribution of the unlabeled examples. It has also been shown that humans are sensitive to shifts in this distribution, and will change predicted classifications based on these shifts. It is not immediately obvious what causes these shifts – what specific properties of these distributions humans are sensitive to. Assuming a parametric model of human categorization learning, we can ask which parameters or sets of parameters humans fix after exposure to labeled data and which are adjustable to fit subsequent unlabeled data. We formulate models to describe different parameter sets which humans may be sensitive to and a dataset which optimally discriminates among these models. Experimental results indicate that humans are sensitive to all parameters, with the closest model fit being an unconstrained version of semi-supervised learning using expectation maximization.
Psychology of Learning and Motivation, 2014
Abstract There are many accounts of how humans make inductive inferences. Two broad classes of ac... more Abstract There are many accounts of how humans make inductive inferences. Two broad classes of accounts are characterized as “theory based” or “similarity based.” This distinction has organized a substantial amount of empirical work in the field, but the exact dimensions of contrast between the accounts are not always clear. Recently, both accounts have used concepts from formal statistics and theories of statistical learning to characterize human inductive inference. We extend these links to provide a unified perspective on induction based on the relation between descriptive and inferential statistics. Most work in Psychology has focused on descriptive problems: Which patterns do people notice or represent in experience? We suggest that it is solutions to the inferential problem of generalizing or applying those patterns that reveals the more fundamental distinction between accounts of human induction. Specifically, similarity-based accounts imply that people make transductive inferences, while theory-based accounts imply that people make evidential inferences. In characterizing claims about descriptive and inferential components of induction, we highlight points of agreement and disagreement between alternative accounts. Adopting the common framework of statistical inference also motivates a set of empirical hypotheses about inductive inference and its development across age and experience. The common perspective of statistical inference reframes debates between theory-based and similarity-based accounts: These are not conflicting theoretical perspectives, but rather different predictions about empirical results.
When the distribution of unlabeled data in feature space lies along a manifold, the information i... more When the distribution of unlabeled data in feature space lies along a manifold, the information it provides may be used by a learner to assist classification in a semi-supervised setting. While manifold learning is well-known in machine learning, the use of manifolds in human learning is largely unstudied. We perform a set of experiments which test a human's ability to use a manifold in a semisupervised learning task, under varying conditions. We show that humans may be encouraged into using the manifold, overcoming the strong preference for a simple, axis-parallel linear boundary.
Journal of Experimental Child Psychology, 2016
What do children learn from biased samples? Most samples people encounter are biased in some way,... more What do children learn from biased samples? Most samples people encounter are biased in some way, and responses to bias can distinguish among different theories of inductive inference. 67 four-to eight-year-old children learned to make conditional predictions about a set of sample items. They then made predictions about the properties of new instances or old instances from the training set. The experiment compared unbiased and biased sampling. Given unbiased samples, participants used what they learned to make predictions about population and sample instances. With biased samples, children were less accurate/confident about inferences about the population than the sample. Children used information in a biased sample to make predictions about items in that sample, but were less likely to generalize to new items than when samples were unbiased.
Cognitive science, Jan 21, 2015
A common practice in textbooks is to introduce concepts or strategies in association with specifi... more A common practice in textbooks is to introduce concepts or strategies in association with specific people. This practice aligns with research suggesting that using "real-world" contexts in textbooks increases students' motivation and engagement. However, other research suggests this practice may interfere with transfer by distracting students or leading them to tie new knowledge too closely to the original learning context. The current study investigates the effects on learning and transfer of connecting mathematics strategies to specific people. A total of 180 college students were presented with an example of a problem-solving strategy that was either linked with a specific person (e.g., "Juan's strategy") or presented without a person. Students who saw the example without a person were more likely to correctly transfer the novel strategy to new problems than students who saw the example presented with a person. These findings are the first evidence tha...
Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2011
Methods We adopted the sort-re-sort procedure used by Medin et al.(1997). In one-on-one interview... more Methods We adopted the sort-re-sort procedure used by Medin et al.(1997). In one-on-one interviews, participants were presented with a set of cards that contained numbers or geometric shapes, the selections of which were based upon the varying of dimensions identified by previous research (Feldman, 2000; Miller & Gelman, 1983; see Figure 1).
Journal of Numerical Cognition
K-12 students often rely on testing examples to explore and determine the truth of mathematical c... more K-12 students often rely on testing examples to explore and determine the truth of mathematical conjectures. However, little is known about how K-12 students choose examples and what elements are important when considering example choice. In other domains, experts give explicit consideration to the typicality of examples – how representative a given item is of a general class. In a pilot study, we interviewed 20 middle school students who classified examples as typical or unusual and justified their classification. We then gave middle school students and mathematicians a survey where they rated the typicality of mathematical objects in two contexts – an everyday context (commonness in everyday life) and a mathematical context (how likely conjectures that hold for the object are to hold for other objects). Mathematicians had distinct notions of everyday and mathematical typicality – they recognized that the objects often seen in everyday life can have mathematical properties that can...
There are many accounts of how humans make inductive inferences. Two broad classes of accounts a... more There are many accounts of how humans make inductive inferences. Two broad classes of accounts are characterized as “theory based” or “similarity based.” This distinction has organized a substantial amount of empirical work in the field, but the exact dimensions of contrast between the accounts are not always clear. Recently, both accounts have used concepts from formal statistics and theories of statistical learning to characterize human inductive inference. We extend these links to provide a unified perspective on induction based on the relation between descriptive and inferential statistics. Most work in Psychology has focused on descriptive problems: Which patterns do people notice or represent in experience? We suggest that it is solutions to the inferential problem of generalizing or applying those patterns that reveals the more fundamental distinction between accounts of human induction. Specifically, similarity-based accounts imply that people make transductive inferences, while theory-based accounts imply that people make evidential inferences. In characterizing claims about descriptive and inferential components of induction, we highlight points of agreement and disagreement between alternative accounts. Adopting the common framework of statistical inference also motivates a set of empirical hypotheses about inductive inference and its development across age and experience. The common perspective of statistical inference reframes debates between theory-based and similarity-based accounts: These are not conflicting theoretical perspectives, but rather different predictions about empirical results.
in The Conceptual Mind: New Directions in the Study of Concepts
How do concepts acquire a normative structure, and how do children acquire the concept of NORMATI... more How do concepts acquire a normative structure, and how do children acquire the concept of NORMATIVITY? Humans clearly have capacities to make judgments of obligations, permissions, and prohibitions. Such capacities would seem to depend on possessing certain kinds of concepts. Theories of concepts must explain and accommodate normative content. Moreover, studying norms may also shed new light on the nature and acquisition of concepts.