Traditional algebra word problems have a bad rap and for good reason. Students are hardly enamored with content of the typical word problem, and its relevance to the real world is questionable at best. Amdahl and Loats (1995) captured this sentiment in their amusing tour of beginning algebra:
“Folks who write math books live very different lives from you and me. They seem to spend a lot of time on trains, for example, which leave cities you and I rarely visit, in hopes of meeting their buddies on trains at destinations in-between….They launch rockets across rivers, build bridges, and agonize over how tall various trees are. After a couple of years of math classes, you’ll be uncomfortable hiking through the woods without your calculator handy.” (p. 104)
Much has been said about the state of American manufacturing in the last year, and a series of recent reports present an intricate picture that takes us beyond some of the confusion and common misconceptions. Except for the understandable decline in manufacturing during the recent recession, manufacturing productivity since 2000 has been surprisingly robust. Ball State University’s report even suggests that growth in manufacturing going forward is steady and on an upward path. With all of the news of outsourcing in areas such as textiles, furniture, and apparel, how can this be?
In my previous blog, I argued for a dual topic approach to curriculum design. The framework outlined in that blog is based on a variety of research.
Some of this research is drawn from psychology and studies of human learning. These involve the development of automaticity and controlling cognitive load. Other design elements are associated with what we have learned over the years from international research, particularly the way successful countries focus on fewer topics with greater depth in their math curricula. Still other research is a synthesis of what we believe are best instructional practices in remedial and special education.
Defining a High-Standards Math Curriculum for Struggling Students, Part 2 of 2
I made the case in my previous blog that adjusting the pace of instruction for struggling students in a high-standards curriculum is imperative. We all have different aptitudes for a given endeavor—from music to mathematics—and it is unrealistic to expect that all students can learn the same set of complex ideas in the same, fixed period of time.
It takes time for research to be translated into practice, particularly when it comes to textbooks. For example, it was nearly 20 years ago when U.S. math educators examined the textbooks and instructional practices of highly successful countries around the world, only to determine what we already knew. American math textbooks were “a mile wide and an inch deep.” In contrast, international curricula typically contained fewer topics that were addressed in greater depth.1
The traditional structure of math textbooks as you move across the grade levels has been unfortunately predictable. James Flanders’ analysis of elementary and middle school texts in the late 1980s characterized the typical text as bloated with all kinds of review and extra content.2 Almost 30 years later, we still have the same problem in many of our math textbooks.3 This problem remains in spite of the fact that efforts like the National Council of Teachers of Mathematics (NCTM) standards to infuse more conceptual understanding and problem solving in textbooks occurred in the intervening years.
One of the biggest impacts of the No Child Left Behind (NCLB) legislation has been the infusion of the word “research” into the language and thinking about education. Teachers are encouraged to ask, “Is my classroom curriculum research based?” “What about specialized interventions for students with disabilities?” It is a short step to asking the same question about today’s mathematics standards, and prominent researchers at Vanderbilt University have done just this in a series of randomized control studies.1
What the researchers described as “very low achieving” fourth grade students were randomly assigned to either inclusive or specialized intensive classrooms. Instruction in both conditions was guided by grade-level standards for fractions. It wasn’t a major surprise that the students in the inclusive classrooms performed poorly over the three years of the study as teachers fully implemented the Common Core State Standards. Researchers, however, were dismayed to find that the performance of students in specialized intensive settings also decreased over time, despite the use of their intervention curriculum in these studies.
Debates over math standards, whether they are the NCTM Standards or the Common Core Math Standards, often spill into the question, “What do they mean for struggling students?” There are many issues behind this question, not the least of which is the exceedingly heterogeneous group of students frequently called “struggling.” Unpacking that issue alone is an essay in itself. For our purposes, what standards mean for struggling students can be distilled into at least two basic questions:
Are high standards appropriate for struggling students?
If they are not appropriate, are they even relevant?
We understand the issue more and more every day. For years, we’ve been told that our students don’t stack up in math when compared with their peers in other countries. Our performance isn’t that bad at the fourth grade, but TIMSS and PISA data clearly show significant comparative declines as our students end eighth and tenth grade. One of many interpretations of these data is that math at the intermediate and middle grades is an exceedingly weak link in our educational system.
Were that not enough, the link between mathematical competence and success in the workplace is becoming ever clearer as the economy slowly emerges from a deep recession. A recent and fascinating issue of the Atlantic Monthly (Davidson, 2012) provides a lucid account of the extraordinary gaps in knowledge between highly successful manufacturing workers and their less-skilled counterparts who are employed, at least for now, on the same factory floor. The former possess increasing amounts of quantitative knowledge, while the latter live in fear of automation or outsourcing. Success in math at the middle grades, which is obviously fundamental to success in high school and beyond, is a cornerstone for securing the future for American students.