When I was first tasked to take charge of a large-scale mathematics evaluation during the heyday of No Child Left Behind (NCLB) , one of my friends sent me the following message, just to make me chuckle:
Subject: The History of Teaching Math [in the USA]
Teaching Math in 1950:
Teaching Math in 1960:
Teaching Math in 1970:
Teaching Math in 1980:
Teaching Math in 1990:
Teaching Match in 2000:
Despite the cynicism contained in this portrait of the history of teaching mathematics in the USA, it is a fact that improving K-12 students’ opportunities to learn and performance in mathematics and science has been of major concern for several decades (National Council of Teachers of Mathematics 2000; National Mathematics Advisory Panel 2008; National Research Council 2001). Despite waves of education reforms on K-12 mathematics and science teaching and learning, American students’ performance in these subjects remains lackluster when compared to their international peers (Loveless 2013; National Center for Education Statistics 2004; Programme for International Student Assessment 2003). Scholars from different disciplines and with different perspectives have offered a variety of reasons for why this is so. These explanations range from language (e.g., Fuson and Kwon 1991), social beliefs and cultural values (e.g., Chazan 2000; Stevenson et al. 1993), parental involvement (e.g., Stevenson and Stigler 1992), teaching and learning behaviors (e.g., Hiebert et al. 2005; Stevenson et al. 1986; Stevenson and Stigler 1992; Stigler and Hiebert 1999, 2009), to curriculum and textbooks (e.g., Cai et al. 2002; Schmidt et al. 1997). But one area that remains relatively underexplored and underemphasized is teachers’ lack of systematic opportunities to study and develop in-depth mathematical content understanding of topics that they are charged to teach at K-12 levels (Shulman 1986). Though research on mathematics education has done a good job of documenting teachers’ lack of mathematical understanding and its impact on student learning (e.g., Ball 1990, 1993, 2000; Ball et al. 2005; Ball and Rowan 2004; Cohen and Ball 2001; Hill and Ball 2004), we know little about where the deficiency of mathematical understanding came from, or what might be the possible remedies besides the typical argument for improving the pedagogy (e.g., Green 2015; Lampert 2001).
This book shows how several key components of the mathematics education system in the USA work against the goal of improving students’ mathematics learning by failing to provide their teachers with opportunities to learn the mathematics they are charged to teach. These components consist of classroom mathematics learning experiences, undergraduate or graduate training, and in-service professional development. These components coincide with three critical stages where teachers of mathematics gain mathematical knowledge to teach: namely, as K-12 learners themselves, as prospective teachers, and as in-service teachers. This book argues that unless we recognize the importance of teaching future mathematics teachers explicitly and rigorously the topics they are expected to teach, teachers will continue to recycle a body of incoherent and incomprehensible mathematical knowledge to their students, because these are the only types of mathematical knowledge they have at their disposal, both in terms of what they themselves have learned as K-12 students and in terms of the mathematical resources available to them, including the textbooks they have to teach with as mathematics teachers.
In the sections that follow, I will describe when I first noticed the problem with mathematical content understanding through an example of classroom mathematics teaching and learning. My observations have led me to a research path investigating both students’ and teachers’ opportunities to learn mathematics. This research path is the basis for this book. The chapter ends with an overview of the subsequent chapters.
How Did I Become Interested in Studying Mathematical Content Understanding?
The following episode describes one elementary teacher who was reviewing and teaching her 2nd graders the concept of probability through experiment . She first asked one student to read the definition of probability and then did the experiment to show what it meant.
- T1
- Yesterday, we used cubes. Today we are going to use something different. Here is a card, one side is red and one side is blue. I’m going to drop the card on the floor 10 times. John, read the definition of probability.
- John
- [reads the definition]
- T
- What is the probability that it will end up more red or more blue?
- John
- [no response]
- T
- We are going to flip it 10 times to see what comes up. I need a helper, Amanda, to check off on the chart whether it comes up blue or red, the probability or how likely something will happen. Anita, I’d like you to call it when it drops on the floor.
The teacher dropped the card 10 times and the results were as follows:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
|---|---|---|---|---|---|---|---|---|---|---|
Blue | × | × | × | × | × | × | ||||
Red | × | × | × | × |
- T
- How many times did it land on blue?
- Students
- [counted] 6
- T
- How many times did it land on red?
- Students
- [counted] 4
- T
- Did it end up in a tie?
- Students
- No.
- T
- Close to a tie?
- Students
- Yes.
- T
- It’s pretty close to being even.
In fact, this teacher was following the textbook . Although the textbook’s idea of teaching the concept of probability through experiment was a good one, whether students were able to understand the concept or not ultimately depends on the teacher’s understanding of the concept, particularly when students were confused by the seemingly inconsistent experiment results and what they had learned about the concept of probability, as the following interaction shows:
- T
- Let’s try a different object. The next object is a penny. Steve, how many sides to a coin?
- Steve
- 2
- T
- How many sides are heads?
- Steve
- 1
- T
- How many sides are tails?
- Steve
- 1
- T
- Is there a greater chance for it to be heads, tails or equal?
- John
- Equal, because there are 2 sides.
- T
- What about this? [She refers to the experiment results of the card chart.] Why?
- David
- Because there are 2 sides, 1 heads and 1 tails
- T
- Does anyone think any differently?
- Brandi
- Heads more.
- T
- Why? Raise hands if you agree.[Some students did]
- T
- Raise your hands if you think it will land on tails.[Some students did]
In this example, most of the interactions proceeded well—the teacher asked a few scaffolding questions before turning to the first main question, that is, “Is there a greater chance for it to be heads, tails or equal” if she were to drop the coin. John answered the question correctly by saying that it would be “equal”, but he did not just stop by offering only an answer. John also explained that the reason why ...