This book stems from the 13th International Conference on the Teaching of Mathematical Modeling and Applications (ICTMA 13). The first part focuses on research into what it means for students to understand models and modeling processes (6 sections, comprising 23 chapters), while the second part considers what is needed for modeling activities to be productive in classrooms (5 sections, comprising 30 chapters). The authors come from many countries: South Africa, Taiwan, Japan, Mexico, Germany, Israel, Brazil, Argentina, Sweden, Italy, Australia, Canada, Cyprus, U.K., Spain, Switzerland, Denmark, and the U.S. This book is a veritable fount of information about modeling in classrooms at all levels from elementary through university. In addition, it has chapters dealing with how teachers develop models of modeling (Chapters 30–46) and how new technologies influence modeling in classrooms (Chapters 47–51).
If you are not familiar with the difference between mathematical modeling and mathematical problem solving, Section 6 (Chapters 22–30) will help clarify that for you. While they do overlap, they are quite different. Mathematics problems, even open-ended ones, are already formulated mathematically. They are often artificial, puzzle-like tasks that do not encourage students to rely on their common sense or make realistic connections to the problem context. Modeling problems, however, have an undetermined nature, and do often involve using common sense, making decisions about assumptions, and collecting information, before mathematics can be used. In Chapter 28, one finds that models are “more like powerful/shareable/reusable artifacts and tools” than simple answers to traditional questions and word problems. The difference is between “making mathematics practical” and “making practice mathematical”.
Then there’s mathematization. According to Chapter 4, mathematization is the part of the modeling process that occurs after the situation has been structured. It consists of translating the objects and relations of the structured situation into appropriate mathematical terms. Chapter 22 provides twenty-one sample tasks for Grades 9–12 that distinguish between modeling tasks (of long and short duration) and mathematization tasks (of short duration). Sample modeling tasks include: “Which means of transportation is the best?” (2–4 weeks) and “How many times can one brush one’s teeth with a tube of toothpaste?” (one lesson). A sample mathematization task is: A liqueur glass is cone-shaped. What height of liqueur served in the glass makes it half-way full? This, however, seems more like a mathematical problem-solving task to me.
Much more can be learned from this book. In Chapter 6, Noss and Hoyles report their three-year investigation of companies, such as automobile manufacturing and financial services, where the technology in use often hides the underlying mathematical concepts and procedures. While calculation and basic arithmetic are necessary for workers, these are of less importance than a “conceptual grasp of, for example, variables and relationships in the work flow, how graphs and spreadsheets highlight relationships and trends, how systematics data may be used with powerful, predictive tools to control and improve [workplace] processes.” (p. 84). The authors introduce the idea of “Techno-mathematical Literacies” (TmL) — new skills needed in technology-rich workplaces — which can seldom be picked up on the job. The authors discuss industry specific interactive software that they have developed for use with employees to promote effective learning of TmL.
Chapter 11 considers how one can turn general ideas into modeling problems. The authors give five principles for doing this, with Principle 1 being that there should be a genuine link with the real world of the students to provide relevance and motivation (p. 135). They provide sample questions that Australian Year 10 and 11 students investigated in a modeling challenge that allowed them to pose questions themselves. (p. 139). Some of these are: “When will the Aral Sea dry up completely?” and “How much water do we really have left in the Hinze Dam? Will it cater for the current and future population of the Gold Coast of Australia?” The authors note that problems having a social context, rather than a scientific context, are often preferable because they allow more critical analyses of the models produced.
In addition, some chapters discuss model-eliciting activities (MEAs), a concept introduced by Lesh and Doerr in their 2003 book. In Chapter 9, Larsen describes a study of modeling, using an MEA called the “Summer Jobs” problem, which she conducted with college students who were given data on employees’ performance at summer jobs. The question was: “Whom to rehire for next summer?” The students had to structure the problem and pick out the important quantities and relationships. They divided the data into busy, slow, and steady times; considered hours, dollars, dollars per hour; and compared the money made by employees during those times. As with most such modeling problems, at the end they had to write a report justifying their recommendations.
In Chapter 21, Carmona and Greenstein report what happened when they gave the same modeling problem to both 3rd graders and to post-baccalaureates enrolled in a summer workshop. The modeling problem was to find the best way to rank (1st to 5th) twelve soccer teams based on win-loss data, provided as labeled points plotted in a quadrant determined by two unit-less axes simply labeled “wins” and “losses”. While the 3rd graders were given two 1-hour sessions to do this, the post-baccalaureates worked on the problem in one 2-hour session. The somewhat surprising result was that both groups came up with essentially the same ranking and similar reasoning, but the post-baccalaureates used much more sophisticated mathematics.
Several chapters deal with projects and courses for upper level engineering students. Chapter 7 details how a team of industrial engineering undergraduates engaged in a long-term engineering design project for an industry partner. Their assignment was to provide a recommendation on whether the partner should establish a satellite center at a certain location. The students determined that they needed to do detailed cost analyses for several different options. Chapter 15 describes a pilot course offered to ten junior and senior industrial engineering students at the University of Pittsburgh. Part of a four-year six-institution research project funded by NSF, the course consisted of nine model-integrating activities (MIAs) briefly described in the chapter. The authors also provide a “first cut at assessment” in the form of a two-page rubric (pp. 186–187). This chapter might provide a “head start” for others interested in developing, and assessing, engineering modeling courses.
I got the impression that despite the vast amount of information coming out ICTMA 13 and previous conferences, there are many questions about modeling and teaching modeling that could still be investigated. A question that I had that was not answered in the book was: What’s the difference between project-based learning and learning through modeling activities? According to one website I found,
In Project Based Learning (PBL), students go through an extended process of inquiry in response to a complex question, problem, or challenge. While allowing for some degree of student “voice and choice,” rigorous projects are carefully planned, managed, and assessed to help students learn key academic content, practice 21st Century Skills (such as collaboration, communication & critical thinking), and create high-quality, authentic products & presentations.(http://www.bie.org/about/what_is_pbl/)
It would seem that modeling activities are much less structured and more loosely connected to specific curriculum objectives than PBL.
Who might be interested in this book? While not a “how to” book, teachers at all levels can glean information about modeling — its implementation and assessment — as well as find some interesting modeling tasks to adapt. Also, mathematics education researchers can discover the kind of modeling research that has been done and gain insight into what remains to be done.
I have a few quibbles. There are some insignificant typos, such as “close” when “closed” was surely intended (p. 125) and the inclusion of “Simon Fraser” twice — once as part of Stephen Campbell’s last name and again to designate his university (p. xi). Also, the index is very skimpy — several times I tried to find topics that I had read about, but not jotted down the page number of, and could not do so using the index. That said, I found the book a worthwhile read. A plus is the fact that one can skip around and read just what interests one, as the chapters are essentially independent.
Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education.
Modeling and Design Research and Assessment Methodologies.- Modeling in High School and College.- Modeling in Middle Schools.- Modeling in the Primary Grades.- Modeling and Teacher Development.- Modeling vs. Traditional Problem Solving.- Modeling in Engineering and in Other 21st Century Design Sciences.- Technological Tools and Data Modeling.- Modeling and Socio-Cultural Perspectives.- Directions for the Future.