1. What are cardinality and ordinality and what are different ways of assessing children's understanding of them?
2. What are the differences between learning to count and learning fractions? What is infinite divisibility and how does it relate counting to fractions?
3. Why do you think the scale scores of Hispanics and blacks increased more than the scores of whites from 1978 to 2004?
4. What are Piaget and Vygotsky's explanations for how children acquire mathematical knowledge before school? What information do we have on typically occurring mathematical experiences? What were the researchers' 3 main goals in conducting the study? (Tudge article)
5. What is the intraparietal sulcus' main function in our brain? What are the three characteristics of numerical activation in the intraparietal sulcus? Briefly explain each.
6. Why do you think children fail number conservation tasks, when they have other types of understanding about numbers? Why do they fail class inclusion tasks (six tulips, two roses, more tulips or flowers)? How can we reconcile infants' and toddlers' competence in some aspects of numerical understanding with much older children's lack of competence in other aspects?
7. Is the comparison to multiplication of the arbitrary facts on p. 127 (Dehaene) a persuasive one? What is the difference between learning multiplication facts and learning the logical statements on p. 127?
8. What is distributed practice and how is it related to forgetting curves? Describe the study of longer-term benefits of distributed practice administered by Bahrick and Hall (1991) using algebra and geometry tests.
9. What does single cell recording add to the information that can be gained from PET and other imaging technologies?
10. What does it mean to say, "understanding numbers occurs as a reflex" (Dehaene, p. 78)?
11. What are distance and magnitude effects? Why do you think they're so widespread among animals (including people)?
12. U.S. students lack conceptual understandings of fractions even after several years of practice. What are some facets that lead to such lack of conceptual understanding of fractions?
13. Toward the bottom of p. 131, Dehaene provides an example, and writes, "suggesting that in parallel to calculating the exact result, our brain also computes a coarse estimate of its size." What general lesson does this example, together with his general emphasis on the verbal nature of arithmetic, have for understanding how the brain solves problems?
14. What is the difference between verbatim and gist memory? Which is more important for reasoning and long term retention, and why? What is the gist of numerical information?
15. Explain the role of the dorsolateral prefrontal cortex in the development of problem solving strategies.
16. What do you think are the causes of same-perimeter/same-area misconception and the illusion of linearity?
17. What role do fingers serve in children's learning of arithmetic?
18. Explain Weber's Law and how it relates to neuronal numerical representations.
19. Should the long looking time in Wynn's experiments and other habituation paradigm studies on the dishabituation trial be viewed as indicative of surprise? Can you think of other reasons why babies might look for a long time at certain displays even if they were not surprised?
20. What is stereotype threat and how does it contribute to the variation of mathematical performance among different groups (gender or race)?
21. What is the idea of "double dissociation"? Why is it a particularly valued type of evidence in the study of brain-damaged patients? What kinds of hypotheses does it allow us to rule out?
22. In Siegler, 2009, what were the conclusions regarding the linear number board game and what does playing this game improve?
23. Does the frequency of subtraction bugs indicate that, "the child's brain registers and executes most calculation algorithms without caring much about their meaning?" (p. 133)? What alternative explanations can you generate?
24. What is the SNARC effect? In what way does it provide evidence for spatial representations of numbers?
25. As of now, how can you best explain the invert and multiply procedure used in fractions? For example, how could you explain the computational steps to get the answer to 4 divided by ½ = 8? Why do you think so many teachers are unable to provide an explanation for this method?
