Enabling Students in Mathematics: A Three-Dimensional Perspective for Teaching Mathematics in Grades 6‒12

In Enabling Students in Mathematics, Marshall Gordon addresses the cognitive, social, and psychological dimensions that shape students’ mathematical learning. Throughout the book, ideas and effective teaching skills are presented for the teacher to apply when teaching students for 21st century competiveness. Gordon gives considerations on how 6–12 grade teachers are continually under pressure to cover material in set curricula and the adverse effects it has on students. As he points out, “the imperative to cover the curriculum compels mathematics teachers and textbooks to emphasize presentations of mathematics algorithms and problem-solving techniques as this is the most direct approach to transmit all the content.”

Given the recent NCTM standards and other drastic changes to curricula, one would think that students would gain a fulfilling educational experience in mathematics. With time constraints, however, teachers have trouble finding enough time to have meaningful conversations with students and to have discovery projects on important concepts, such as the Fundamental Theorem of Calculus. The author gives several examples of how to promote rich discussion among students.

More than any other subject and from the start of our student teaching days to our last day of teaching before retirement, we are always faced with the challenge of figuring out how to motivate students to learn mathematics and to enjoy it. Gordon points out that too often we are asking students in grades 6–12 to solve problems, recognize patterns, and memorize procedures. Of course there are patterns to solving mathematical problems. Solely promoting calculations, however, without considering why \(m\) is the letter of choice to represent slopes or why 90^o degrees represent a right angle, not using logic to derive important results, we are doing a great disservice to our students. As Gordon points out, it is as if the calculations teachers are presenting are merely magic tricks. The teachers and students know they work but have little to no understanding as to why they work. For example, on pages 15–17, the authors asks why the Long-Division Algorithm works when dividing \(231\) by \(17\) and why \(\frac23+\frac45 \neq \frac68\) and even how to simplify \(8\div\frac34\). As pointed out on page 26, 96% of U.S. students’ time during seatwork is spent on practicing procedures and 78% of U.S. teachers were about as likely to simply state concepts as opposed to developing them. In addition, in a survey for employers, 59% of students said they were well prepared to analyze and solve complex problems but 24% of employers found fresh college graduates ready to take on the complex problems.

Gordon goes on to talk about mathematical textbooks and discusses a comparison by Lianghuo Fan and Yan Zhu (2007) between China, the U.S.A., and Singapore. In Singapore, textbook series in grades 6–12 devotes time to specific heuristics including teaching the students how to draw diagrams to solve complex problems, change their point of view, and use an equation they themselves derived to solve the problem. In the United States, the majority of the problems in the textbooks surveyed were routine, traditional, and often single-step problems with limited heuristic approaches.

Although there are efforts being made that include “habits of mind” further research is needed. Gordon points out, “engaging more complex problems for a vast population of mathematics students and continues to keep most students limited to learning algorithms and solving one- or two-step problems.” The question now becomes, how can we as teachers motivate students think outside the box? A problem as simple as finding \(y\) if \(80 = 5(y - 79)\) may be difficult for students if they forget the distributive property or don’t realize they can divide both sides by 5 first. As Gordon points out, however, if we “create an easier simpler problem,” for example, \(80 = 5x\), students may have a better idea of where to start. One interesting example using graph theory is presented on pages 34–35. A class of 10th grade students was presented with graphs in which the nodes and edges were expressed as the ordered pairs \(\{(n, e): (3, 3), (4, 6), (5, 10), (6, 15), (7, 21), (8, 28)\}\). The students were asked about the relationship between \(n\) and \(e\). After some clever manipulations, one sees that the number of edges is \(e=\frac{n(n-1)}{2}\).

Gordon stresses that curricula are at most a guide and give the teacher a path on what to cover. Following a curriculum is not enough, however, to ensure that the 21st century student is ready for the next mathematics course, graduation, and the workforce. As Dewey (1916/1944) points out, there is a “failure to take into account the instinctive or native powers of the young; secondly, failure to develop initiative in coping with novel situations; [and] thirdly, an undue emphasis upon drill and other devices which secure automatic skill at the expense of personal perception.”

We must not be naïve, simply giving our students the slope-intercept form for the equation of a line without giving a explanation of why \(m\) was picked to represent slope or why 360^o is used to represent the degrees in a circle instead of, say, 400^o. We must inspire students to think for themselves and with others. If we don’t, we will be robbing our students and not making them ready for the future.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362^nd Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He can be reached at pto2@psu.edu. Webpage: www.personal.psu.edu/pto2. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.