Feedback influences children's reasoning about math equivalence: A meta-analytic review

ABSTRACT

Decades of research have focused on children's reasoning about math equivalence problems for both practical and theoretical insights. Not only are math equivalence problems foundational in arithmetic and algebra, they also represent a class of problems on which children's thinking is resistant to change. Feedback is one instructional tool that can serve as a key trigger of cognitive change. In this paper, we review all experimental studies (N = 8) on the effects of feedback on children's (ages 6–11) understanding of math equivalence. Meta-analytic results indicate that feedback has positive effects for low-knowledge learners and negative effects for high-knowledge learners, and these effects are stronger for procedural outcomes than conceptual outcomes. Findings highlight the variable influences of feedback on math equivalence understanding and suggest that models of thinking and reasoning need to consider learner characteristics, learning outcomes, and learning materials, as well as the dynamic interactions among them.     

Self-knowledge,belief, ability (and agency?)

Matthew Boyle [(2011). “Transparent Self-Knowledge.” Aristotelian Society Supplementary Volume 85 (1): 223–241. doi:10.1111/j.1467-8349.2011.00204.x] has defended an account of doxastic self-knowledge which he calls “Reflectivism”. I distinguish two claims within Reflectivism: (A) that believing that p and knowing oneself to believe that p are not two distinct cognitive states, but two aspects of the same cognitive state, and (B) that this is because we are in some sense agents in relation to our beliefs. I find claim (A) compelling, but argue that its tenability depends on how we view the metaphysics of knowledge, something Boyle does not consider. I argue that in the context of the standard account of knowledge as a kind of true belief – what I call the Belief Account of knowledge – the claim faces serious problems, and that these simply disappear if we instead adopt an Ability Account of knowledge, along the lines of that defended by John Hyman [(1999). “How knowledge Works.” The Philosophical Quarterly 49 (197): 433–451; John Hyman (2015). Action, Knowledge, and the Will. Oxford: Oxford University Press]. I find claim (B) less compelling, and a secondary aim of the paper is to suggest that once we reject the Belief Account of knowledge, and move over to an Ability Account, there is no explanatory role for (B) left to play.     

Knowing Versus Caring: The Role of Affect and Cognition in Political Perceptions

This paper examines the importance of political knowledge in shaping accurate perceptions of the political world—specifically, how levels of general political knowledge influence the accuracy of specific political judgments, how those judgments might also be shaped by "wishful thinking," and how political knowledge attenuates the impact of wishful thinking on political judgments. Predictions of who would win the U.S. presidential election in 1984, 1988, 1992, and 1996, as surveyed in the National Election Studies conducted in those years, were used as a measure of the accuracy of political perceptions. Analysis of these data reveals that both political knowledge and wishful thinking are important determinants of the accuracy of people's perceptions; in addition, the impact of wishful thinking on perceptions is attenuated by political knowledge.     

“Intrinsically” or just “Instrumentally” Valuable? On Structural Types of Values of Scientific Knowledge

Debates about scientific (though rarely about otherforms of) knowledge, research policies or academic trainingoften involve a controversy about whether scientificknowledge possesses just “instrumental” value or also “intrinsic” value. Questioning this common simpleopposition, I scrutinize the issues involved in terms of agreater variety of structural types of values attributableto (scientific) knowledge. (Intermittently, I address thepuzzling habit of attributing “intrinsic” value to quitedifferent things, e.g. also to nature, in environmentalethics.) After some remarks on relevant broader philosophicaldebates about scientific knowledge, I pave a path throughthe (terminological) thicket of structural types of values. Our initial simple opposition is shown to conflate thedistinctions intrinsic/extrinsic and instrumental (or justuseful)/final. Next, I consider the value(s) of knowledgeand knowing in general and their possible value components(like the values of truth and justifiedness). After havingdiscussed the types of value of everyday knowledge,especially its functional and constitutive value (notionsintroduced earlier), I argue that these can or should alsobe attributed to scientific knowledge, thus departing fromboth objectivist and sociological views of science. One could say that I offer a certain defense of theintrinsic value of scientific knowing (and the inherentvalue of scientific knowledge) and some importantdifferentiations of its “instrumental values”. I alsocaution (in relation with my puzzle) against drawing hastymoral conclusions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Comparing the effects of generating questions,testing, and restudying on students' long-term recall in university learning

We compared the long-term effects of generating questions by learners with answering questions (i.e., testing) and restudying in the context of a university lecture. In contrast to previous studies, students were not prepared for the learning strategies, learning content was experimentally controlled, and effects on factual and transfer knowledge were examined. Students' overall recall performance after one week profited from generating questions and testing but not from restudying. When analyzing the effects on both knowledge types separately, traditional analyses revealed that only factual knowledge appeared to benefit from testing. However, additional Bayesian analyses suggested that generating questions and testing similarly benefit factual and transfer knowledge compared with restudying. The generation of questions thus seems to be another powerful learning strategy, yielding similar effects as testing on long-term retention of coherent learning content in educational contexts, and these effects emerge for factual and transfer knowledge.     

The role of inhibition and speed of processing in mathematics

The current study highlights the importance of inhibitory ability in facilitating performance in mathematics. To understand the role of inhibition in mathematical knowledge, this study tested 102 college students on a series of standardized complex math exercises. Inhibition tasks varied by task and stimuli (letters, numbers, and arrows). The result showed stronger associations between math abilities and strength of inhibition with more inhibition found for the flanker task (involving inhibition of peripheral stimuli) than for the Navon task (requiring inhibition of a potential response). The results supported the view that Inhibition in general (which is not unique to numerical stimuli) is associated with math abilities. General, rather than specific inhibition was associated with math abilities and finally, speed of processing masked the relationship between math and inhibition. These results confirm the assumption that the relation between inhibition and math performance is modulated by the task that measures the inhibition, and by additional predictors such as processing speed.