This particular seminar has several goals. The most specific 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 how computer simulations can be used to help understand mathematical thinking 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 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 and other general abilities.

A third goal of the course is to illustrate how knowledge in an area can be conveyed at varying levels of detail and from varying perspectives. In the beginning of the course, we will read an award winning trade book about mathematical thinking. Trade books summarize findings from research that are written for an audience of educated people who are not necessarily specialists in the area but who are interested in understanding some of the main findings in it. Next we will read several chapters from a governmental report that is aimed at professionals that presumes somewhat more background in the field, but that also is relatively accessible, and a practice guide aimed at teachers, principals, and others who deal with the pratical challenges of teaching children mathematics (in this case, fractions). The third part of the course will expose you to a yet more specific level of detail, that of journal articles. These focus on very specific topics and research studies, but provide more in-depth information on the topic than any general book or report to the government could. Many journal articles are very complex and require a lot of background; the ones that we will read have been chosen for being relatively accessible even without extensive background in the field.

As mentioned above, the fact that this is a small 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 just taking in 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 somewhat different than most. First, attendance is obligatory; I will expect everyone to be at each class meeting. I realize that on rare occasions, it is impossible to be at a particular class (illness, job interviews), 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.

When it is your turn to lead the discussion, you will be responsible to post discussion questions to the google document that will be shared by the class. In order to obtain access to our shared google document (SGD), send your email address to my research coordinator, Theresa Treasure, at tt2p@andrew.cmu.edu, and she will add you to the access group. You will receive an email notification when you have been added. 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. If you have any difficulties with the SGD, let Theresa know as soon as possible.

The key criteria for grading class participation will be 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 midterms 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. The midterms will include 10 short essay questions, each worth 10 points; most, but not all, will be taken from the questions posed in the class, both by me and by you. The final exam will be similar to the midterm, but it will be cumulative.

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

1. Describe commonalities and differences between the mathematical thinking of humans and other animals;

2. Understand and be able to explain theories of the development of mathematical knowledge;

3. Describe and evaluate research methods for investigating mathematical cognition and development;

4. 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;

5. Critique individual journal articles on such dimensions as whether the conclusions follow from the results, whether the experimental techniques were directly relevant to the issues central to the study, whether confounds were present that call the researchers' conclusions into questions, and whether the researchers ignored relevant evidence favoring a different conclusion;

6. Lead discussions of research studies and the broader issues that motivated the studies; and

7. Explain to others well thought out ideas regarding how mathematics learning could be improved.