Development of Mathematical Thinking, Spring, 2020

Wed 3:00-4:50 pm – – – – – Room GDH 539

Semester Dates: January 22^{nd} to May 6^{th}

Professor: Dr. Robert S. Siegler

Dr. Siegler’s Office Location and Hours — GDH 465:

Tuesday, 3:00 – 4:00 pm

Wednesday, 2:00 – 3:00 pm

*Note: If you would like to visit Professor Siegler during his office hours, please send Shawn an email at sxl2102@tc.columbia.edu beforehand. Thank you.*

The course utilizes Professor Siegler’s website in conjunction with a shared google doc for readings and class presentations. __Please visit siegler.tc.columbia.edu for all course readings, instructions, and access to the shared google doc.__

**Shared google doc for Spring 2020:**

https://docs.google.com/document/d/15eINlRM5-rjq4CG5B1TKitRN4Q0X4qLQxXuwKNCk8h0

Our textbook will be: Dehaene, S. (2011). *The number sense: How the mind creates mathematics.* NY: Oxford University Press. This can be purchased at the bookstore or on Amazon.

This seminar focuses on the development of mathematical thinking. The seminar format allows a depth of engagement with the material much greater than the experience in lecture classes, where students tend to learn a little bit about a lot of subjects. The experience in a seminar is more personal and more focused. It offers the opportunity to acquire deep understanding of a particular subject, but it’s also more demanding in some ways. In particular, the success of a seminar depends on thoughtful participation by students, not just on whether students can remember what the professor and the textbook told them.

The most specific goal of the course is to provide an opportunity for you to learn how children acquire understanding of mathematics. You will learn about topics ranging from whether infants are born with a basic sense of number to the way that board games can improve low-income preschoolers’ mathematical understanding to why children in East Asia outperform children in the U.S. in mathematics.

A somewhat more general goal is to illustrate how psychologists who study children’s thinking go about investigating an area and the varied types of evidence that they collect to shed light on the issues in that area. For example, this course examines the evolutionary roots of numerical understanding in animals other than humans, the basic understanding of numbers that infants bring to the world, how the culture in which children grow up influences their mathematics learning, whether the brain is specialized for learning mathematics, and how mathematical ability is related to intelligence, information processing, and other general abilities.

As mentioned above, the fact that this is a seminar, rather than a large lecture, offers both opportunities and challenges. The opportunities are for people to express themselves actively on a regular basis, rather than sitting back and absorbing what a lecturer tells them. The challenges are that with no one giving a lecture, the quality of the class depends at least as much on what you do as on what I do. For this reason, the ground rules of the class are different than most. First, attendance is obligatory; I expect everyone to be at each class meeting. I realize that on rare occasions, it is impossible to be at a particular class, but these exceptions should be kept to a minimum. Second, everyone is expected to actively participate in the discussion. This is essential if the class is to be a true seminar, rather than degenerating into a rotating lectureship. Third, everyone is expected to be at class on time.

Grades in the course will be based on a midterm, a final, and class participation. Class participation will include two or three discussions of articles that are led by each of you and also your participation when one of your classmates or I are leading the discussion. The topics where you will lead the discussion will be chosen by you at the beginning of the semester. When leading the discussion, it is important to pose good questions to bring out the main points and different perspectives on the issues raised in the article. It is especially important to participate actively when other people lead the discussion, to insure that the experience is a good one for them and for the entire class.

When it is your turn to lead the discussion, you will be responsible for posting discussion questions to the Shared Google Document for the class. With the exception of the discussion questions already posted in the syllabus, discussion questions for each class should be posted on our SGD at least a week before the relevant class. Once you and your presentation partner generate the questions for your presentation, email them to me for my approval. I’ll provide comments and suggests revisions if necessary, so there might be more than one round of emails. Typically, we aim for 10-20 questions per session; if there are two presentations on a given day, each group would generate 5-10 questions. If you have any difficulties with the SGD, let Shawn know as soon as possible. The SGD is organized into three sections, described below:

* Section 1* is the Class Overview. This explains what the class is about, what you should learn from taking the class, how the class will be conducted, and provides information about the grading system and tests.

*Please be sure to read Section 2 as it contains vital information that will be helpful in understanding how the class is organized. Thank you!*

* Section 2* is the Reading Assignments Table. This table has the topic for each class in the first column (under the date) and the reading assignments for each class in the second column. Once all the reading assignments have been made, the third column will show which members of the class are responsible for writing the study questions for the reading in that row. Except for the textbook chapters and the readings assigned to Dr. Siegler, there should be two students assigned to each reading assignment.

* Section 3* contains the Dates of Each Class, the Topics to be Discussed, the Reading Assignments, and the Study Questions for each reading.

*This is where you should post your study questions for the readings assigned to you.*Section 4 has the same information that is in the Reading Assignments Table, just not in table format. The date of the class is listed first, then the topic for the class, and then each reading assignment is listed individually so you can post your study questions underneath your reading. If you have any questions at all about this, let Shawn know and he will help you.

The key criteria for first-rate class discussion is high quality and reasonable quantity of contributions when you are not leading the discussion and posing important and stimulating questions and leading an interesting discussion when you are. Remember: If you contribute interesting and informed perspectives when others lead the discussion, they are likely to do the same for you.

The midterm and final will be based on the textbook (Dehaene, 2011, The Number Sense: How the Mind Creates Mathematics, Revised & Expanded Edition), the outside readings, and the discussions. Both the midterm and final will include 5 essay questions.

By the end of the course, you should be able to:

- Describe commonalities and differences between the mathematical thinking of humans and other animals;
- Understand and be able to explain theories of the development of mathematical knowledge;
- Describe and evaluate research methods for investigating mathematical cognition and development;
- Understand sources of individual differences in mathematical learning, both mathematics learning difficulties and exceptional mathematical achievement, as well as differences among nations in mathematics achievement;
- Critique journal articles on such dimensions as whether the conclusions follow from the results, whether the experimental techniques were directly relevant to the central issues raised by the authors, whether confounds were present that call into question the researchers’ conclusions, and whether the researchers ignored relevant evidence favoring a different conclusion;
- Lead discussions of research studies and the broader issues that motivated the studies in a way that leads your classmates to participate actively; and
- Explain to others well thought out ideas regarding how mathematics learning could be improved.

**Class Readings and Questions:**

**Instructions: All Class Readings are available below and not on Canvas. Please click on the links to the respective readings to access the content.**

**January 22, 2019: Introduction**

Dehaene, Introduction/Preface: pp ix-xxii (no study questions today!)

**January 29, 2019: Talented and Gifted Animals**

**Readings:**

Dehaene, Preface & Introduction Ch 1: Talented and Gifted Animals, pp 3-29, Dr. Siegler

1. What are the implications of the Clever Hans story for whether researchers should conduct their own studies (as opposed to letting research assistants who do not know the hypotheses of the study conduct them)? How have animal cognition researchers gotten around the problem of subtle, unconscious biases among experimenters?f

2. What are the implications of the Clever Hans story for interpreting difficult to understand modern claims, such as those regarding ESP (extra sensory perception), seers such as Jean Dixon, and 1- and 2-year-olds’ learning to read and do math from being presented flashcards?

3. Are you surprised that animals such as rats and pigeons can do simple numerical operations? Why might evolution have prepared these animals to be able to process numbers in relatively precise ways?

4. Why might animals code approximations of numbers rather than exact numbers? Why might the amount of variation of rats’ number of bar presses be greater for larger numbers than for smaller ones (p. 9)?

5. What, if anything, do you think it means that in Meck and Church (1983), the rats’ midpoint of bar presses was 4 rather than 5, when the original numbers that they needed to generate were 2 and 8 (pp. 10-11)?

6. Does the experiment described on p. 14 convince you that the chimps are adding the fractions? What other possible interpretations would need to be ruled out before you would accept this conclusion?

7. What is the accumulator model? What are its advantages and disadvantages as a model of arithmetic for humans and other animals?

8. What are distance and magnitude effects? Why do you think they’re so widespread among animals (including people)?

9. Does Dehaene’s neural model (pp. 20-23) imply that animals have neurons dedicated to detecting, for example, the numbers 45 and 50? If not, how do animals make this discrimination, which Dehaene earlier indicated that they can make?

10. Dehaene describes the chimpanzee Ai as having learned to add pairs of the first 9 numbers with 95% accuracy (p. 36) and indicates that analyses of the chimp’s response times suggest that he uses serial counting for all except the first few numbers. What are the implications of animals being able to learn to do simple numerical activities but taking a great amount of time and training to do so?

11. Describe each stage of the Boysen experiment with Sheba. At what point, if any, would you say that Sheba demonstrated an understanding of numbers?

12. What were the implications of Sheba understanding immediately that the symbol for representing 2+2 would be 4, as opposed to her requiring training before she showed similar understanding?

1. What are the implications of infant chicks, after they were conditioned to expect 5 dots in the middle of a line, looking to the left when 2 dots were on the left and right sides and looking to the right when 8 dots labeled the left and right locations?

2. Why do you think the investigators then presented chicks with 20 dots in the middle of the line, mainly to see where they looked when 8 dots were shown?

3. What evidence suggested that the typical L-R number line might be produced by educational factors?

4. Why did Rugani et al. try in Experiment 3 to control for total area, total perimeter., and density of the displays on the cover of the choices? Why were they concerned that the displays covered different areas and had different perimeters and densities, given that the labels on the two doors were identical?

1. Nieder distinguishes between non-symbolic and symbolic number systems. What are each of them, and how does Nieder say they are related?

2. What does it mean that a wide variety of species, including close relatives of humans such as chimpanzees and less close ones such as crows and chickens, represent non- symbolic numbers in much the same way as humans? What does it say about the role of culture in numerical development?

3. The large majority of human societies represent numbers on a left to right mental number line, as do chickens, but some societies seem to represent numbers from right to left. What do these findings together mean regarding the roles of biology and culture in numerical development?

4. What is single-cell recording of neurons? How is it done? What has it told us about numerical representations?

5. Which areas of the brain are most consistently involved in representing both non- symbolic and symbolic numbers? When is each most involved?

6. What does it tell us that very few neurons represent both non-symbolic and symbolic numbers, whereas far more neurons represent one or the other?

**February 5, 2019: Babies Who Count**

**Readings:**

Dehaene, Ch 2: Babies Who Count, pp. 30-52, Dr. Siegler

13. 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 older children’s lack of competence in other aspects?

14. Were the procedures used by Mehler and Bever (1967) (pp. 33-34) and by McGarrigle and Donaldson (1974) really conservation problems? Why or why not?

15. If 3- and 4-year-olds interpret the second question on each conservation trial in the Mehler and Bever (1967) study as meaning that the experimenter wants a different answer, why would both 2- and 5-year-olds not interpret it that way?

16. Why would infants attend to three objects when they hear three sounds (p. 40)? Do they have good reason to do so? Can you think of reasons other than their matching the number of sights and sounds why they might look more at three objects when hearing three sounds and two objects when hearing two sounds?

17. 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?

18. Do you find the interpretation in the last paragraph on p. 44 of why babies attend to number but not identity to be convincing? Why or why not?

19. Dehaene suggests that babies assume that a truck can turn into a ball but not that one object could become two objects (p. 48). Is this reasoning sound; why or why not?

20. Dehaene bases many of his interpretations in evolutionary theory. Would it do babies much good for survival to recognize the number of objects in very small sets if they did not know which of two sets was more numerous? Why might evolution provide such an ability if it wouldn’t do babies much good?

**Assigned Readings:**

**February 12, 2019: The Mental Number Line**

**Readings:**

Dehaene, Ch 3: The Adult Number Line, pp. 53-76, Dr. Siegler

21. Do Dehaene’s examples (p. 54) support his claim that “most if not all civilizations stop using this system beyond the number 3?” Does the exact point at which the initial system is abandoned matter for our interpretation of the phenomenon (why does Dehaene think it does)?

24. Does it surprise you that human number abilities resemble those of rats and pigeons in many ways (p. 61)? Can you generate a hypothesis regarding how humans and animals are similar in number processing and how they differ?

25. Why do you think that the extensive training that Dehaene provided on whether 1, 4, 6, and 9 were greater or smaller than 5 failed to change the distance effect for those numbers (p. 63)? What does this tell us about numerical knowledge and learning in general?

26. Why do currencies in all countries have more coins and bills corresponding to small values than large ones? Might this be related to the reasons why people and other animals seem to have a logarithmic ruler for representing quantity?

27. What does it mean to say, “Understanding numbers, then, occurs as a reflex” (p. 66)? Does this claim apply to all numbers?

28. If numerical magnitude is understood as a reflex, and we are upset by discontinuities as in the train platform and calendar examples, why is $3.99 perceived as noticeably less than $4.00 (p. 68)? Doesn’t the general logic of this section argue in the opposite direction?

29. Why do numbers seem to be represented spatially? Is it just because we’ve all seen number lines in school (or do we have number lines because people tend to represent numbers spatially)? What evidence would help you answer the question?

30. What is the SNARC effect? In what way does it provide evidence for spatial representations of numbers?

32. “If we did not already possess some internal non-verbal representation of the quantity ‘eight,’ we would probably be unable to attribute a meaning to the digit 8” (p. 75). Do you agree with this statement? Do we also have a non-verbal representation of the number “53”? How about “553”? If not, are these numbers meaningless?

33. What defines rational numbers? Irrational numbers? Complex numbers?

**Assigned Readings:**

**February 19, 2019: The Language of Numbers**

**Readings:**

Dehaene, Ch 4: The Language of Numbers, pp. 79-103, Dr. Siegler

35. Most languages have systematic, hierarchical terms for representing numbers; for example, to represent “426”, we concatenate a number of hundreds, a number of 10’s, and a number of 1’s. Not all languages do so, however. What do you think determines whether a language does or does not adopt a hierarchical system for representing numbers?

36. Why did base 10 become the most common system in widely dispersed societies? Why not base 2, 5, or 20?

37. Why was the invention of 0 so important?

39. If the East Asian system of representing numbers is more efficient (pp. 89-90), as it seems to be, why don’t we in the U. S. just adopt an English equivalent?

40. If university students could vote, would they choose the Chinese model of number names?

43. Do you find it surprising that the numbers 1-3 appear so much more often than other numbers? Why might this occur?

**Assigned Readings:**

**February 26, 2019: Development of Basic Arithmetic**

**Readings:**

Table Song Link: https://youtu.be/9XzfQUXqiYY (This guy created a set of table song)

Dehaene, Ch 5: Small Heads for Big Calculations, pp. 104-128, Dr. Siegler

46. Why do young children like to count objects, if they do not know the purpose of What is the difference between learning multiplication counting (p. 106-107)?

47. If children possess the accumulator model from birth, as Dehaene suggests, why does it take them until the end of their fourth year to understand the purpose of counting?

48. What role do fingers serve in children’s learning of arithmetic?

50. Do some of you use strategies other than retrieval on problems such as 6X9 and 7X8? What strategies do you use? Does it surprise you that about half of college students do this?

51. Is the comparison to multiplication of the arbitrary facts on p. 112-113 a persuasive one? What is the difference between learning multiplication facts and learning the logical statements?

53. Toward the bottom of p. 117, 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?

54. What does it mean to say that on 7X9=20, parity is violated (p. 117)? Are there parity rules for all four arithmetic operations? If so, what are they? Did you know these rules before you thought about them in connection with this question and the comment in the text?

55. Why don’t textbooks typically spell out the long subtraction algorithm (p. 118)? Would arithmetic learning be enhanced if they did so?

56. Does the frequency of subtraction bugs indicate that, “the very occurrence of such absurd errors suggests that the child’s brain registers and executes most calculation algorithms without caring much about their meaning?” (p. 119)? What alternative explanations can you generate?

57. Dehaene argues that relying more on calculators, rather than practice with arithmetic facts, would lead to better understanding of mathematical concepts. What is his logic? What logic would lead to the opposite prediction? Do you think that children today who are good at calculation become adults who are generally good at mathematics? Why?

60. Why does adding the numerators and adding the denominators work in the example of Michael Jordan’s shooting but not in the pie example (p. 126)? How would you teach children the difference between the two cases? Would you follow Dehaene’s advice to tell children that when discussing fractions addition, think of pies rather than scoring averages?

**Assigned Readings:**

**March 4, 2019: Development of Advanced Arithmetic**

**Assigned Readings:**

**March 11, 2019: Midterm!**

**March 18, 2019: No class today! Semester holiday**

**March 25, 2019: Individual Differences in Mathematical Thinking**

Dehaene, Ch 6: Geniuses and Prodigies, pp. 129-157, Dr. Siegler

61. Dehaene emphasizes similarities between mathematical geniuses and idiot savants who are good at identifying prime numbers and related feats. Do you see the unusual abilities of the two groups as being related? What kind of data would be crucial for deciding whether they are in fact related?

62. Why do multiples of 9 have ones digits that descend by one number for each multiple (9, 18, 27, 36…) (P. 133)?

63. Dehaene (p. 136) notes that Einstein, among others, claims that his great insights occurred without the involvement of language. Do you think that this is generally true for the rest of us, who have insights from time to time but are not among the great geniuses of history?

64. Claims were made in the 19th Century that the brains of women were smaller than those of men, and that women were therefore less intelligent. Subsequent research has shown that the brains of women are indeed smaller on average than those of men, but that there are no differences in average intelligence between men and women. Why, then, might women’s brains typically be smaller?

65. As Dehaene notes on p. 141, the larger sizes of certain brain areas relevant to musical performance in musicians is as likely due to their experience playing their instrument as to any innate difference in their brains. Design an experiment to determine if there are innate differences in the brains of musicians independent of their musical experience. Don’t worry about the expense of the experiment.

66. The gender differences in mathematical test scores that Dehaene describes on pp. 143-146 have changed somewhat in the years since the studies that he describes were done. Which phenomena that he describes would you guess have changed (and why), and which would you expect to have stayed the same?

67. Within the testosterone explanation of sex differences in mathematical giftedness that Dehaene advances on p. 146, why might the number of men at the superior level of math achievement be higher than the number of women, even if average math achievement is the same for men and women?

69. Great mental calculators almost always calculate from left to right. Why do you think they do this?

70. It is plausible that children learn through experience with the particular numbers that there are some numbers, such as 12, that can be divided into equal size groups, and others, such as 13, that cannot (p. 153)? But how does anyone learn this (except from instruction) for large and uncommon numbers such as 389 or for roots of 179,859,375?

71. In his conclusion (pp. 155-156), Dehaene says that biology probably plays a part in great mathematical achievement, but then says that biology does “not weigh much compared to the powers of learning, fueled by a passion for numbers.” Is it purely a matter of a supportive environment whether a person develops a passion for numbers? Might biology play a role there too?

Dehaene, Ch 7: Losing Number Sense, pp. 161-190, Dr. Siegler

72. What are the ideas of “dissociation” and “double dissociation”? Why is double dissociation a particularly valued type of evidence in the study of brain-damaged patients? What kinds of hypotheses does it allow us to rule out?

74. What is the logic of split-brain studies; what have they told us about processing of numbers in each hemisphere? Why might findings from them not be generalizable to people with intact brains?

75. Do people need to understand math to answer problems such as 4 minus 3? Is Mr. M’s problem not understanding the magnitudes of the numbers 4 and 3, or does it lie elsewhere (and if so, what might the difficulty be)?

76. Why might the inferior parietal lobe be specialized for mathematical processing, in particular for “the number sense”? What principles of brain organization are implicit in Dehaene’s arguments for the view that the inferior parietal lobe is the prime location of the number sense?

77. If Mr. M doesn’t have a sense of magnitude, how can he estimate “the duration of Columbus’ trip to the New World or the distance from Marseilles to Paris” (p. 178)?

78. Does the specificity of processing pathways for reading and understanding different types of material surprise you (pp. 179-181)? What are the implications of such extreme specificity of content knowledge for the future of cognitive neuroscience?

79. Some authorities have argued that the prefrontal cortex is particularly crucial in making human beings human. Do Dehaene’s descriptions of prefrontal functioning in math support or contradict this view? How so?

80. What is cortical plasticity? Why is it so important for learning mathematics and other “unnatural” activities?

**Assigned Readings:**

**April 1, 2019: Development of Rational Number Knowledge**

**Assigned Readings:**

**April 8, 2019: Role of Input in Math Learning**

**Assigned Readings:**

**April 15, 2019: Interventions for Improving Mathematical Knowledge (I)**

**Assigned Readings:**

**April 22, 2019: Interventions for Improving Mathematical Knowledge (II)**

**Assigned Readings:**

**April 29, 2019:**

**Assigned Readings:**

Policy Implications of Math Development Research

Rittle-Johnson, B., and Jordan, N. C. (2016). Synthesis of IES-Funded Research on Mathematics: 2002–2013 (NCER 2016-2003) Washington, DC: National Center for Education Research, Institute of Education Sciences, U.S. Department of Education. This report is available on the Institute website at ies.ed.gov/ncer/pubs/20162003/