# The „Dismal” State of Mathematics Textbooks and How to Improve It

# BLOG: Heidelberg Laureate Forum

For most students, mathematics and computer science can be difficult, daunting subjects. That may be hard to grasp if you have dedicated your career to one of the two, but surveys from several countries show that it is common for students to struggle with maths. Research shows that textbooks are often a part of the problem. But when textbooks are good, they can make a real difference – and there is an important lesson here.

### Teaching math

Many math textbooks are hard to digest. You do not necessarily need an expert to tell you that. However, it does help when researchers also highlight it.

In 2020, experts from Project 2061, a long-term math and science education reform initiative of the American Association for the Advancement of Science (AAAS), did just that.

They surveyed 12 popular middle school mathematics textbooks from the United States and found that only four of them are well-suited to help the learning process. In fact, none of the most popular ones received good reviews.

“States and school districts are bombarded with information from textbook publishers claiming their materials are aligned with benchmarks and standards. The Project 2061 analysis gives busy educators the solid information they need to make informed choices about which textbooks will help their students improve their understanding of and skills in mathematics,” stated Dr. Gerald Kulm, who led Project 2061’s evaluation.

There was good news in the review. There were some excellent textbooks, the researchers found – and the best ones were not necessarily simpler. In fact, the best series contained plenty of in-depth mathematics, but it had the support and guidance to enable students to make the most of it.

Most textbooks suffered in their explanation of concepts in mathematics, however. They also offered weak instructional support and did not promote student thinking – which is arguably one of the most important things a textbook should do.

Granted, this could be a US-specific problem. The data on how well textbooks fare is scarce for most countries, and there could be big differences from one country to the other. But it seems unlikely that the US alone suffers from this isssue – particularly if you look at the international literature.

### Textbooks could do with some improvements

A study led by University of Michigan researchers looked at math textbooks from 19 countries and found that in the vast majority of cases, the textbooks fell short of supporting and engaging students. The researchers reported a “dismal picture” of mathematics textbooks across the globe.

“The world has failed in terms of providing teachers with the textbooks they need in order to better develop students’ quantitative literacy, mathematical reasoning and to help them learn to solve demanding, real-world problems,” said MSU Distinguished Professor William H. Schmidt, who led the 19-country study, in a press release accompanying the work.

“Across the [mathematics] textbooks we analyzed, there were more than 50,000 exercises. Of those, less than 1% dealt with demanding, real-world applications. In the U.S., only 0.5% of problems met this standard. How can we expect kids to have a sense of if they’ll ever use math in the real world, or gain any experience in doing it if we don’t give them any opportunities to learn?”

Schmidt has spent decades looking at standards in mathematical education. Some 25 years ago, he found that the US covered more mathematical subjects than other countries during the first eight grades – but this strategy just did not work. Other countries that fared well in mathematical education covered fewer topics but had more success.

Comparisons like this one helped improve textbooks across the world, but there is still a lag between what these books should be like and what they are actually like, and reform is relatively slow.

“Every country we studied has a statement supporting increasing math literacy, including a focus on real-world problem-solving. Our policies call for it, but we’re not giving teachers the tools they need to accomplish these goals,” says Schmidt.

There is also a lack of adherence to recommendations from psychology and educational science. For instance, research suggests that most students benefit from interleaved practice – where students are learning two or more related concepts or skills, instead of focusing exclusively on one concept.

“In brief, strong evidence suggests that students benefit from heavy doses of interleaved practice, yet most mathematics texts provide scarcely any,” one recent study concluded.

Schmidt says he hopes clearer and more incisive policy can help bring textbooks up to par, but ultimately, the impact of textbooks on students’ achievements is not fully understood. However, what is clear is that when textbooks are good, they can really make a difference.

Just look at some fields of computer science.

### Textbooks that work

When ACM named Alfred Vaino Aho and Jeffrey David Ullman as the recipients of the 2020 ACM A.M. Turing Award, it was a bit of a surprising announcement. Not because of the scientists’ achievements – the two are well-known as pioneers of algorithms and programming languages. But rather, because of something in the motivation of the prize. The two were celebrated “for fundamental algorithms and theory underlying programming language implementation,” and for “synthesizing these results and those of others in their highly influential books, which educated generations of computer scientists.”

It is not very common for leading researchers in any field to also be successful educators. There are a few notable exceptions (the Feynman Lectures on Physics are still a fantastic resource, for instance) but in general, great textbooks (and their authors) rarely receive the credit they deserve.

But perhaps they should. Textbooks have a massive impact on current and future generations of students, and especially since so many of them fall short, it would perhaps be wise to start cherishing the authors that do a great job.

In their first seminal work, *The Design and Analysis of Computer Algorithms* (1974) Aho and Ullman worked with co-author John Hopcroft, another A.M. Turing Award Laureate. The book became an instant classic in the field. It introduced algorithm design techniques that are now an integral part in almost any computer science curriculum. Their next book, *Principles of Compiler Design* (1977) was even better.

The book became known as the “Dragon Book,” because of its cover that featured a dragon. Of course, it is not easy to write books on computer science, especially at a time when the field was still in its nascent years.

So textbooks can also be a part of positive change. Somehow, great textbooks find a way to shine. They pass the test of time, even when the field itself changes so much. It would obviously be fantastic if we had great textbooks for all fields of mathematics and for all fields, period. It could be a missing puzzle piece for getting more students interested in mathematics. Maybe then, the “dismal picture” would become a bit brighter.

### Books with equity and inclusiveness

There’s another area where textbooks have historically fallen short: equity.

“The images portrayed in textbooks have the ability to influence students’ beliefs about self, ethnicity, social class, and gender. Several studies in the 1970s and 1990s documented the lack of equity in mathematics textbooks with respect to gender; however, few studies have been conducted recently,” a recent study noted.

The paper found that despite some progress, males and Whites are “portrayed as being more mathematical.” This is yet another hurdle that textbooks have to overcome.

Dealing with disability is also something that textbooks have largely overlooked. Mathematics and science textbooks in Braille, for instance, are more expensive and inaccessible because they’re so difficult. But on this side, technical progress can help. A team of researchers from Towson University in Maryland has developed a “universal translator” that can make the production of Braille textbooks much easier and less expensive.

“This project is about equity and equal access to knowledge,” said Martha Siegel, a Professor Emerita from Towson University in Maryland. Siegel met a blind student who, upon ordering a required statistics textbook, had to wait for six months (and pay thousands of dollars) for the book to be delivered. This even caused the student to delay her studies.

“Given the amazing technology available today, we thought it would be easy to piece together existing tools into an automated process,” said Alexei Kolesnikov. This proved to be true, and it is excellent news – but without the empathy and awareness that kickstarted the study in the first place, this translator would have not seen the light of day.

Perhaps herein lies an important key to improving textbooks. It seems we should strive to move past the age of cold, dry textbooks that are torn from the real world and into more genuine, helpful, and relatable books that actually speak to students. The future of mathematics will probably be brighter if we do so.

Andrei Mihai wrote (31. May 2023):

>

The „Dismal” State of Mathematics Textbooks […]>

[…] https://scilogs.spektrum.de/hlf/files/grafik.pngThis graphic poignantly illustrates the dismal state of (certain) Mathematics textbooks by showing the arguably hard-to-digest use of (natural) numbers as index symbols, apparently in a Mathematics textbook.

>

[…] and How to Improve ItIn the case illustrated, an improvement would be brought about by the use of non-numerical (but perhaps otherwise relevant, suggestive, or at least distinctive and memorable) tokens as index symbols;

or by abandoning indices altogether in favor of plain distinctive labels, as appropriate here.

Math apps and even personal math helpers based on ChatGPT are the future.

These future mathematics teachers will monitor the progress of the students and adapt the further teaching material to it.

At some point, math lessons will say goodbye to technical details such as solving systems of equations and reorient themselves to the student’s ability to recognize the mathematical core of practical problems. Because ChatGPT-like programs or other AI programs will do the rest.

Textbooks in the United States typically present mathematical ideas as facts to memorize rather than as meaningful relationships. Not all U.S. mathematics textbooks are alike, however. Sparked by U.S. students’ lower-than-expected performance on national and international assessments, the National Science Foundation launched a major initiative to create new mathematics textbooks based on challenging mathematics curriculum standards and instructional strategies aligned with current research on learning (NCTM, 1989, 2000). This initiative has produced textbooks at the elementary, middle, and high school levels that break the mold of traditional instruction (Reys, Robinson, Sconiers, & Mark, 1999; Trafton, Reys, & Wasman, 2001).

Skills remain a central theme in the new textbooks, but teachers introduce these skills in real-world contexts that enable students to investigate and solve problems. At present, approximately 10–15 percent of U.S. classrooms use these “standards-based” textbooks. They differ from traditional mathematics textbooks in that they present mathematical ideas in various contexts and engage students in exploring ideas, solving problems, sharing strategies, and building new knowledge based on solid conceptual understanding. Teachers no longer simply “cover” material. Rather, they facilitate a classroom learning environment that encourages questioning, conjecturing, and problem formulation and values student thinking and multiple strategies.

Textbooks in the United States typically present mathematical ideas as facts to memorize rather than as meaningful relationships. Not all U.S. mathematics textbooks are alike, however. Sparked by U.S. students’ lower-than-expected performance on national and international assessments, the National Science Foundation launched a major initiative to create new mathematics textbooks based on challenging mathematics curriculum standards and instructional strategies aligned with current research on learning (NCTM, 1989, 2000). This initiative has produced textbooks at the elementary, middle, and high school levels that break the mold of traditional instruction (Reys, Robinson, Sconiers, & Mark, 1999; Trafton, Reys, & Wasman, 2001).

One of the core academic subjects in the United States and throughout the world, mathematics is key to many of our most common daily activities and routines, things like shopping, banking, cooking, and home repair. Further, in today’s increasingly technologically reliant society, more and more jobs require some level of proficiency in mathematics and science. Students who lack sufficient mathematical knowledge and skills are more likely to experience negative outcomes as adults, including fewer opportunities for meaningful employment and a reduced likelihood of economic independence. Because of these factors, it is critical that students begin to develop essential math concepts and skills at an early age. Unfortunately, studies conducted since the 1970s have shown that students in the United States do not perform as well on mathematics assessments as do their peers in other countries.

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