Missing Input: How Imbalanced Distributions of Textbook Problems Affect Mathematics Learning

Abstract

Understanding how environments influence learning requires attending not only to what is present but also to what is absent. In the context of mathematics learning, this means attending not only to problems that children encounter frequently in textbooks but also to ones that appear rarely. We present research in this article showing that students perform surprisingly poorly on seemingly simple fraction and decimal arithmetic problems that are seldom seen in textbooks. Next, we describe imbalanced distributions in textbooks of mixed notation arithmetic and comparison problems, and we hypothesize similar relations between the frequency of those types of problems and student accuracy on those tasks. Finally, we review findings about relations between textbook input and student performance in whole number arithmetic and mathematical equality, and we propose a hypothesis regarding when imbalanced distributions of problems are most detrimental. We conclude that presenting more balanced distributions of problems and helping children understand mathematical principles that differentiate legitimate from flawed solution strategies offer promising ways of improving mathematics education.

Arithmetic word problem solving. Analysis of Singaporean and Spanish textbooks

“…In fact, topics that are not included in textbooks are not usually taught and learned in class (Schmidt et al, 1997). For instance, students are often less proficient in solving certain fraction and decimal mathematical problems, which are rarely found in books (Siegler & Oppenzato, 2021).…”

“…In this vein, it is feasible that textbooks from high-performing countries such as Singapore may present more opportunities for students to solve a wider variety of AWPs than those from low-to mid-performing countries. Indeed, lack of experience with some types of problems and/or certain types of schematic representations may hinder children's learning (Siegler & Oppenzato, 2021). There is evidence that textbooks in highperforming countries contain a more diversified and balanced distribution of both additive and multiplicative AWPs across different problem types than textbooks in countries such as the U.S.…”

“…Second, given that experience with different types of problems may enhance children's learning (Siegler & Oppenzato, 2021), we expected that Singaporean textbooks would include a richer or wider variety of AWPs according to the semantic-mathematical structure (Hypothesis 2a). Furthermore, we expected that the distribution of AWPs by type of structure (additive and multiplicative) in Singaporean textbooks would be more balanced.…”

“…The current study is focused on primary school math textbooks, as they are widely used by teachers to support student mastery of math problems (Depaepe et al, 2009;Hiebert et al, 2003). Furthermore, there is evidence that the content of textbooks may influence the level of competence that students develop (e.g., Fagginger Auer et al, 2016;Heinze et al, 2009;Törnroos, 2005;Siegler & Oppenzato, 2021;Sievert et al, 2019Sievert et al, , 2021. Specifically, we looked at arithmetic word problems (AWPs) since they are considered prime tools for promoting the development of student problem-solving competence in primary school (Verschaffel et al, 2020).…”

Textbooks and Students’ Knowledge

“…Comparisons within Slovenian mathematics education have shown that teachers in Slovenia lag behind other countries in teaching decimals and fractions, as these topics are covered later (Mullis et al, 2016). Another very important message emerges from recent research on the relationships between textbook content and mathematics learning, focusing on how children learn about rational numbers (i.e., fractions, decimals, percentages) (Siegler & Oppenzato, 2021). It is known from previous research that fraction knowledge in the fifth grade uniquely predicts mathematics achievement five years later in secondary school, even after controlling for socioeconomic status, IQ, reading comprehension and whole number arithmetic knowledge.…”