articles about mathematics education

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• Published: 19 December 2019

Problematizing teaching and learning mathematics as “given” in STEM education

• Yeping Li 1 &
• Alan H. Schoenfeld 2  

International Journal of STEM Education volume  6 , Article number:  44 ( 2019 ) Cite this article

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Mathematics is fundamental for many professions, especially science, technology, and engineering. Yet, mathematics is often perceived as difficult and many students leave disciplines in science, technology, engineering, and mathematics (STEM) as a result, closing doors to scientific, engineering, and technological careers. In this editorial, we argue that how mathematics is traditionally viewed as “given” or “fixed” for students’ expected acquisition alienates many students and needs to be problematized. We propose an alternative approach to changes in mathematics education and show how the alternative also applies to STEM education.

Introduction

Mathematics is commonly perceived to be difficult (e.g., Fritz et al. 2019 ). Moreover, many believe “it is ok—not everyone can be good at math” (Rattan et al. 2012 ). With such perceptions, many students stop studying mathematics soon after it is no longer required of them. Giving up learning mathematics may seem acceptable to those who see mathematics as “optional,” but it is deeply problematic for society as a whole. Mathematics is a gateway to many scientific and technological fields. Leaving it limits students’ opportunities to learn a range of important subjects, thus limiting their future job opportunities and depriving society of a potential pool of quantitatively literate citizens. This situation needs to be changed, especially as we prepare students for the continuously increasing demand for quantitative and computational literacy over the twenty-first century (e.g., Committee on STEM Education 2018 ).

The goal of this editorial is to re-frame issues of change in mathematics education, with connections to science, technology, engineering, and mathematics (STEM) education. We are hardly the first to call for such changes; the history of mathematics and philosophy has seen ongoing changes in conceptualization of the discipline, and there have been numerous changes in the past century alone (Schoenfeld 2001 ). Yet changes in practice of how mathematics is viewed, taught, and learned have fallen far short of espoused aspirations. While there has been an increased focus on the processes and practices of mathematics (e.g., problem solving) over the past half century, the vast majority of the emphasis is still on what content should be presented to students. It is thus not surprising that significant progress has not been made.

We propose a two-fold reframing. The first shift is to re-emphasize the nature of mathematics—indeed, all of STEM—as a sense-making activity. Mathematics is typically conceptualized and presented as a body of content to be learned and processes to be engaged in, which can be seen in both the NCTM Standards volumes and the Common Core Standards. Alternatively, we believe that all of the mathematics studied in K-12 can be viewed as the codification of experiences of both making sense and sense making through various practices including problem solving, reasoning, communicating, and mathematical modeling, and that students can and should experience it that way. Indeed, much of the inductive part of mathematics has been lost, and the deductive part is often presented as rote procedures rather than a form of sense making. If we arrange for students to have the right experiences, the formal mathematics can serve to organize and systematize those experiences.

The second shift is suggested by the first, with specific attention to classroom instruction. Whether mathematics or STEM, the main focus of most instruction has been on the content and practices of the discipline, and what the teacher should do in order to make it accessible to students. Instead, we urge that the main focus should be on the student’s experience of the discipline – on the affordances the environment provides the student for disciplinary sense making. We will introduce the Teaching for Robust Understanding (TRU) Framework, which can be used to problematize instruction and guide needed reframing. The first dimension of TRU (The Discipline) focuses on the re-framing discussed above: is the content conceptualized as something rich and connected that can be experienced and codified in meaningful ways? The second dimension (Cognitive Demand) examines opportunities students have to do that kind of sense-making and codification. The third (Equitable Access to Content) examines who has such opportunities: is there equitable access to the core ideas? Dimension 4 (Agency, Ownership, and Identity) asks, do students encounter the discipline in ways that enable them to see themselves as sense makers, building both agency and positive disciplinary identities? Finally, dimension 5 (Formative Assessment) asks, does instruction routinely use formative assessment, allowing student thinking to become public so that instruction can be adjusted accordingly?

We begin with a historical background, briefly discussing different views regarding the nature of mathematics. We then problematize traditional approaches to mathematics teaching and learning. Finally, we discuss possible changes in the context of STEM education.

Knowing the background: the development of conceptions about the nature of mathematics

The scholarly understanding of the nature of mathematics has evolved over its long history (e.g., Devlin 2012 ; Dossey 1992 ). Explicit discussions regarding the nature of mathematics took place among Greek mathematicians from 500 BC to 300 AD (see, https://en.wikipedia.org/wiki/Greek_mathematics ). In contrast to the primarily utilitarian approaches that preceded them, the Greeks pioneered the study of mathematics for its own sake and pursued the development and use of generalized mathematical theories and proofs, especially in geometry and measurement (Boyer 1991 ). Different perspectives about the nature of mathematics were gradually developed during that time. Plato perceived the study of mathematics as pursuing the truth that exists in external world beyond people’s mind. Mathematics was treated as a body of knowledge, in the ideal forms, that exists on its own, which human’s mind may or may not sense. Aristotle, Plato’s student, believed that mathematicians constructed mathematical ideas as a result of the idealization of their experience with objects (Dossey 1992 ). In this perspective, Aristotle emphasized logical reasoning and empirical realization of mathematical objects that are accessible to the human senses. The two schools of thought that evolved from Plato’s and Aristotle’s contrasting conceptions of the nature of mathematics have had important implications for the ensuing development of mathematics as a discipline, and for mathematics education.

Several more schools of thought were developed as mathematicians tackled new problems in mathematics (Dossey 1992 ). Davis and Hersh ( 1980 ) provides an entertaining and informative account of these developments. Three major schools of thought in the early 1900s dealt with paradoxes in the real number system and the theory of sets: (1) logicism, as an outgrowth of the Platonic school, accepts the external existence of mathematics and emphasizes the form rather than the interpretation in a specific setting; (2) intuitionism, as influenced by Aristotle’s ideas, only accepts the mathematics to be developed from the natural numbers forward via “valid” patterns of mental reasoning (not empirical realization in Aristotle’s thought); and (3) formalism, also aligned with Aristotle’s ideas, builds mathematics upon the formal axiomatic structures to free mathematics from contradictions. These three schools of thought are similar in that they view the contents of mathematics as products , but they differ in whether products are viewed as pre-existing or created through experience. The development of these three schools of thought illustrates that the view of mathematics as products has its long history in mathematics.

With the gradual development of school mathematics since 1900s (Stanic and Kilpatrick 1992 ), the conception of the nature of mathematics has increasingly received attention from mathematics educators. Which notion of mathematics mathematics education adopts and uses has a direct and strong impact on the way of school mathematics being presented and approached in education. Although the history of school mathematics is relatively short in comparison with mathematics itself, we can find ample examples about the influence of different views of mathematics on curriculum and classroom instruction in the USA and other education systems (e.g., Dossey et al. 2016 ; Li and Lappan 2014 ; Li, Silver, and Li 2014 ; Stanic and Kilpatrick 1992 ). For instance, the “New Math” movement of 1950s and 1960s used the formalism school of thought as the core of reform efforts. The content was presented in a structural format, using the set theoretic language and conceptions. But the result was not a successful progression toward a school mathematics that is best for students and teachers (e.g., Kline 1973 ). Alternatively, Dossey ( 1992 ), in his review of the nature of mathematics, identified and selected scholars’ works and ideas applicable to both professional mathematicians and mathematics educators (e.g., Davis and Hersh 1980 ; Hersh 1986 ; Tymoczko 1986 ). Those scholars' ideas rested on what professional mathematicians do, not what mathematicians think about what mathematics is. Dossey ( 1992 ) specifically cited Hersh ( 1986 ) to emphasize mathematics is about ideas and should be accepted as a human activity, not strictly governed by any one school of thought.

Devlin ( 2000 ) argued that mathematics is not a single entity but has four different faces: (1) computation, formal reasoning, and problem solving; (2) a way of knowing; (3) a creative medium; and (4) applications. Further, he contended school mathematics typically focuses on the first face, makes some reference to the fourth face, but pays almost no attention to the other two faces. His conception of mathematics assembles ideas from the history of mathematics and observes mathematical activities occurring across different settings.

Our brief review shows that the nature of mathematics can be understood as having different faces, rather than being governed by any single school of thought. At the same time, the ideas of Plato and Aristotle continue to influence the ways that mathematicians, mathematics educators, and the general public perceive mathematics. Despite nearly a half century of process-oriented research (see below), let alone Pólya’s work on problem solving, mathematics is still perceived of largely as products —a body of knowledge as highlighted in the three schools (logicist, intuitionist, formalist) of thought, rather than ideas that call for active thinking and creation. The evolving conceptions about the nature of mathematics in history suggests there is room for us to decide how mathematics can be perceived, rather than being bounded by a pre-occupied notion of mathematics as “given” or “fixed.” Each and every learner can experience mathematics through different practices and “own” mathematics as a human activity.

Problematizing what is important for students to learn in and through mathematics

The evolving conceptions about the nature of mathematics suggest that choices exist when deciding what and how to teach and learn mathematics but they do not specify what and how to make the choice. Decisions require articulating options for conceptions of what is important for students to learn in and through mathematics and evaluating the advantages and drawbacks for the students for each option.

According to Stanic and Kilpatrick ( 1992 ), the history of school mathematics curricula presents two important and real changes over the years: one is at the turn of the twentieth century when school mathematics was reformed as a unified and applied curriculum to accommodate dramatically increased student populations from diverse backgrounds, and the other is the “New Math” movement of the 1950s and 1960s, intended to integrate modern mathematics into school curriculum. The perceived failure of the “New Math” movement led to the “Back to Basics” movement in the 1970s, followed by “Problem Solving” in the 1980s, and then the Curriculum Standards movement in the 1990s and after. The history shows school mathematics curricula have emphasized teaching and learning mathematical knowledge and skills, together with problem solving and some applications of mathematics, a picture that is consistent with what Devlin ( 2000 ) refers to as the 1st face and some reference to the 4th face of mathematics.

Therefore, although there have been reforms in mathematics curriculum and instruction, there are hardly real changes in how mathematics is conceptualized and presented in school education in the USA (Stanic and Kilpatrick 1992 ) and other education systems (e.g., Leung and Li 2010 ; Li and Lappan 2014 ). The dominant conception remains mathematics as products , frequently referring to a body of static knowledge and skills that need to be learned and acquired (Fisher 1990 ). This continues to be largely the case in practice, despite advances in conceptualization (see below).

It should be noted that conceptualizing mathematics as “a body of knowledge and skills” is not wrong, especially with such a long history of knowledge creation and accumulation in mathematics, but it is not adequate for school mathematics nowadays. The set of concepts and procedures, after years of development, exceeds what could be covered in any school curricula. Moreover, this body of knowledge and skills keeps growing, as the product of human intelligence and scholarship in mathematics. Devlin ( 2012 ) pointed out that school mathematics mainly covers what was developed in the Greek mathematics, plus just two further advances from the seventh century: calculus and probability theory. It is no wonder if someone questions the value of learning such a small set of knowledge and skills developed more than a thousand years ago. Meanwhile, this body of knowledge and skills are often abstract, static, and “foreign” to many students and teachers who learned to perceive mathematics as an external entity in existence (Plato’s notion) rather than Aristotelian emphasis on experimentation (Cooney 1987 ). It is thus not surprising for so many students and teachers to claim that mathematics is difficult (e.g., Fritz et al. 2019 ) and “it is ok—not everyone can be good at math” (Rattan et al. 2012 ).

What can be made meaningful should be critically important to those who want to (or need to) learn and teach mathematics. In fact, there is significant evidence that students often try to make sense of mathematics that is “presented” or “given” to them, although they made numerous errors that can be decoded to study their thinking (e.g., Ashlock 2010 ). Indeed, misconceptions are best thought of not as errors that need to be “fixed,” but as plausible abstractions on the basis of what students have learned—i.e., attempts at sense-making (Smith et al. 1993 ). Conceiving mathematics as about “ideas,” we can help students to play, own, experience, and think about some key ideas just like what they do in many other activities, such as game play (Gee 2005 ). Definitions of concepts and formal languages and procedures can be postponed until students are ready to consider why and how they are needed. Mathematics should be taken and accepted as a human activity (Dossey 1992 ), and developing students’ mathematical thinking (about ideas) should be emphasized in learning mathematics itself (Devlin 2012 ) and in STEM (Li et al. 2019a ).

Along with the shift from products to ideas in mathematics, scholars have already focused on how people work with ideas in mathematics. Elaborated in detail by Schoenfeld ( in press ), the revolution began with George Pólya (1887–1985) who had a fundamental interest in having students learn and understand content via problem solving. For Pólya, mathematics was about inquiry, sense making, and understanding how and why mathematical ideas (instead of content as products) fit together the way they do. The call for problem solving in the 1980s in the USA was (at least partially) inspired by Pólya’s ideas after a decade of “back to basics” in the 1970s. It has been recognized since that the practices of mathematics (including problem solving) are every bit as important as the content itself, and the two shouldn’t be separated. In the follow-up standards movement, the content and practices have been the “warp and weave” of the fabric doing mathematics, as articulated in Principles and Standards for School Standards (NCTM 2000 ). There were five content standards and five process standards (i.e., problem solving, reasoning, connecting, communicating, representing). It is widely acknowledged, also in the Common Core State Standards in the USA (CCSSI 2010 ), that both content and processes/practices are essential and they form the base for next steps.

Problematizing how mathematics is taught and learned, with connections to STEM education

How the ways that mathematics is often taught cause concerns.

Conceiving mathematics as a body of facts and procedures to be “mastered” has been long-standing in mathematics education practice, and it often results in students’ learning by rote memorization. For example, Schoenfeld ( 1988 ) provided a detailed account of the disasters of a “well-taught” mathematics course, documenting a 10th-grade geometry class taught by a confident and experienced teacher. The teacher taught and managed his class well, and his students also did well on standardized examinations, which focused on content and procedures. At the same time, however, Schoenfeld pointed out that the students developed counterproductive views of mathematics. Although the students developed some level of proficiency in content and procedures, they gained (or were reinforced in) the kinds of beliefs about mathematics as being fragmented and disconnected. Schoenfeld argued that the course led students to develop a robust and counterproductive set of beliefs about the nature of geometry.

Seeking possible origins about students’ counterproductive beliefs about mathematics from mathematics instruction motivated Schoenfeld’s study (Schoenfeld 1988 ). Such an intuitive motivation is also evident in other studies. Keitel ( 2006 ) compared the lessons of two teachers (T1 and T2) in Germany who taught their classes very differently. T1 regularly taught the class emphasizing routine individual practice and memorization of specific algebraic rules. T1 emphasized the importance of such practices for test taking, and the students followed his instruction. Even when T1 one day introduced a non-routine problem that connects algebra and geometry, the overwhelming emphasis on mastering routines and algorithms seemed to overshadow in dealing such a non-routine problem. In contrast, T2’s teaching emphasized students’ initiatives and collaboration, although T2 also used formal routine tasks. At the end, students in T2’s class reported positively about their experience, enjoyed working together, and appreciated the opportunities of thinking mathematically. Studies by Schoenfeld ( 1988 ) and Keitel ( 2006 ) indicate how students’ experience in mathematics classes influences their perceptions of mathematics and also imply the importance of learning about teachers’ perceptions of mathematics that likely guide their instructional practice (Cooney 1987 ).

Rattan et al. ( 2012 ) found that teachers with different perceptions of mathematics teach differently. Specifically, Rattan et al. looked at these teachers holding an entity (fixed) theory of mathematics intelligence (G1) versus incremental theory (G2). Through their studies, Rattan and colleagues found that G1 teachers more readily judged students to have low ability, comforted students for low mathematical ability, and used “kind” strategies (e.g., assigning less homework) unlikely to promote their engagement with the field than G2 teachers. Students who received comfort-oriented feedback perceived their teachers’ entity theory and low expectations and reported lowered motivation and expectations for their own performance. The results suggest how teachers’ inadequate perceptions of mathematics and beliefs about the nature of students’ mathematical intelligence contributed to low achievement, diminished self-esteem and viewed mathematics is only a set of static facts and procedures. Further, the results suggest that how mathematics is taught influences more than students’ proficiency with mathematics content in a class. Sun ( 2018 ) made a similar argument after synthesizing existing literature and analyzing classroom observation data.

Based on the 2012 US national survey of science and mathematics education conducted by Horizon Research, Banilower et al. ( 2013 ) reported that a vast majority of mathematics teachers, from 81% at the high school level to 90% at the elementary level, believe that students should be given definitions of new vocabulary at the beginning of instruction on a mathematical idea. Also, many teachers believe that they should explain an idea to students before having them consider evidence for it and that hands-on activities should be used primarily to reinforce ideas students have already learned. The report suggests many teachers emphasized pedagogical practices of “give” and “present,” perhaps influenced by conceptions of mathematics that are more Platonic than Aristotelian, similar to what was reported about teachers’ practices more than two decades ago (Cooney 1987 ).

How to change?

Given that the evidence demonstrates a compelling case for changing how mathematics is taught, we turn our attention to suggesting how to realize this transformation. Changing how mathematics is taught and learned is not a new endeavor for both mathematics educators and mathematicians (e.g., Li, Silver, and Li 2014 ; Schoenfeld in press ). For example, the “Moore Method,” developed and used by Robert Lee Moore (a famous topologist) in the early twentieth century, shifted instruction from teacher-centered lecturing to student-centered mathematical development (Coppin et al. 2009 ). In its purest form, students were presented with mathematical definitions and asked to develop and/or prove theorems from them after class, without reading mathematics books or using other resources. When students returned to the class, they were asked to prove a theorem. As a result, students developed the mathematics themselves, instead of the instructor presenting the proofs and doing the mathematics for students. The method has had its own success in producing many great mathematicians; however, the high-pressure environment also drowned many students who might have been successful otherwise (Schoenfeld in press ).

Although the “Moore Method” was used primarily in advanced mathematics courses at the post-secondary level, it illustrates how a different conception of mathematics led to a different instructional approach in which students developed mathematics. However, it might be the opposite end of a spectrum, in comparison to approaches that present mathematics to students in accommodating and easy-to-digest ways that can be as much easy as possible. Neither extreme is a good option for K-12 students. Again, it becomes important for us to consider options that can support the value of learning mathematics.

Our discussion in the previous section highlights the importance of taking mathematics as a human activity, ensuring it is meaningful to students, and developing students’ mathematical thinking about ideas, rather than simply absorbing a set of static and disconnected knowledge and skills. We call for a shift in teaching mathematics based on Platonic conceptions to approaches based on more of Aristotelian conceptions. In essence, Plato emphasized ideal forms of mathematical objects, perhaps inaccessible through people’s sense making efforts. As a result, learners lack ownership of the ideal forms of mathematical objects, because mathematical objects cannot and should not be created by human reasoning. In contrast, Aristotle emphasized that mathematical objects are developed through logic reasoning and empirical realization. In other words, mathematical objects exist only when they can be sensed and verified by people's efforts. This differs from Plato’s passive perspective, highlights human ownership of mathematical ideas and encourages people to make mathematics make sense, termed as making sense by McCallum ( 2018 ). Aristotelian conceptions view mathematics as objects that learners can actively develop and structure as mathematically meaningful, which is more in line with what research mathematicians do. McCallum ( 2018 ) argued that both sense-making and making-sense stances are needed for a complete view of mathematics and learning, recognizing that not attending to both stances carries risks. “Just as it is a risk of the sense-making stance that the mathematics gets ignored, it is a risk of the making-sense stance that the sense-maker gets ignored.” (McCallum 2018 ).

In addition, there is the issue of personal identity: if students come to avoid mathematics because they are uncomfortable with it (in fact, mathematics anxiety has become a widespread problem for all ages across the globe, see Luttenberger et al. 2018 ) then mathematics instruction has failed them, regardless of test scores.

In the following, we discuss sense-making and making-sense stances first with specific examples from mathematics. Then, we discuss connections to STEM education.

Sense making is much more than the acquisition of knowledge and skills

Sense making has long been emphasized in mathematics education community. William A. Brownell is a well-known, early 20 th century scholar who advocated the value of sense making in the learning of mathematics. For example, Brownell ( 1945 ) discussed how arithmetic can and should be taught and learned not only as procedures, but also as a meaningful system of thinking. He shared many examples like the following division,

Brownell suggested to ask questions: what does the 5 of 576 mean? Why must 57 be the first partial dividend? Do you actually divide 8 into 57, or into 57…’s? etc., instead of simply letting students memorize how to carry out the procedure. What Brownell advocated has been commonly accepted and emphasized in mathematics education nowadays as sense making (e.g., Schoenfeld 1992 ).

There can be different ways of sense making of the same computation. As an example, the sense making process for the above long division can come out with mental math as: I am looking to see how close I can get to 570 with multiples of 80; 7 multiples of 80 gives me 560, which is close. Of course, given base 10 notation, that’s the same as 8 multiples of 70, which is why the 7 goes over the 57. And when I subtract 560, there are 16 left over, so that’s another 2 8 s. Such a sense-making process also works, as finding the answer (quotient, k ) of 576 ÷ 8 is the same operation as to find k that satisfies 576 = k × 8. In mathematics, division and multiplication are alternate but equivalent ways of doing the same operation.

To help students build numerical reasoning and make sense of computations, many teachers use number talks in their classrooms for students to practice and share these mental math and computation strategies (e.g., Parrish 2011 ). In fact, new terms are being created and used in mathematics education about sense making, such as number sense (e.g., Sowder 1992 ) and symbol sense (Arcavi, 1994 ). Some instructional programs, such as Cognitively Guided Instruction (see, e.g., Carpenter et al., 1997 , 1998 ), make sense making the core of instructional activities. We argue that such activities should be more widely adopted.

Making sense makes the other side of mathematical practice visible, and values idea development and ownership

The making-sense stance, as termed by McCallum ( 2018 ), is not commonly practiced as it is pertinent to expert mathematician’s practices. Where sense making (as discussed previously) emphasizes the process of making sense of what is being learned, making sense emphasizes the process of making mathematics make sense. Making sense highlights the importance for students to experience mathematics through creating, designing, developing, and connecting mathematical ideas. As an example, for the above division computation, 8 \( \overline{\Big)576\ } \) , students may wonder why the division procedure is performed from left to right, which is different from the other operations (addition, subtraction, and multiplication) that are all performed from right to left. In fact, students can be encouraged to explore if the division can also be performed from right to left (i.e., starting from the one’s place). They may discover, with possible support from the teacher, that the division can be done in this way. However, once the division is moved to the high-value places, it will require the process to go back down to the low-value places for completion. In other words, the division process starting from the low-value place would require repeated processes of returning to the low-value places; as a result, it is inefficient. As mathematical procedure is designed to improve efficiency, the division procedure is thus set to be carried out from the high-value place to low-value place (i.e., from left to right). Students who work this out experience mathematics more deeply than the sense-making described by Brownell ( 1945 ).

There are plenty of making-sense opportunities in classroom instruction. For example, kindergarten children are often given opportunities to play with manipulatives like cube trains and snub cubes, to explore and learn about patterns, numbers, and measurement through various connections. The recording of such activities typically results in numerical expressions or operations of these connections. In addition, such activities can also serve as a context to encourage students to design and create a way of “recording” these connections directly with a drawing line next to the connected train cubes. Such a design activity will help students to develop the concept of a number line that includes the original/starting point, unit, and direction (i.e., making mathematics make sense), instead of introducing the number line to students as a mathematical concept being “given” years later.

Learning how to provide students with opportunities to develop mathematics may occur with experience. Huang et al. ( 2010 ) found that expert and novice teachers in China both valued students’ mastering of mathematical knowledge and skills and their development in mathematical thinking methods and abilities. However, novice teachers were particularly concerned about the effectiveness of their guidance, in contrast to expert teachers who emphasized the development of students’ mathematical thinking and higher-order thinking abilities and properly dealing with important and difficult content points. The results suggest that teachers’ perceptions and pedagogical practices can change and improve over time. However, it may be worth asking if support for teacher development would accelerate the process.

Connecting changes in mathematics and STEM education

Although it is commonly acknowledged that mathematics is foundational to STEM, mathematics is being related to STEM education at a distance in practice and also in scholarship development (English 2016 , see additional notes at the end of this editorial). Holding the conception of mathematics as products does not support integrating mathematics with other STEM disciplines, as mathematics can be perceived simply as a set of tools for these disciplines. At the same time, mathematics and science have often proceeded along parallel tracks, with mathematics focused on “problem solving” while science has focused on “inquiry.” To better connect mathematics and other disciplines in STEM, we should focus on ideas and thinking development in mathematics (Li et al. 2019a ), unifying instruction from the student perspective (the Teaching for Robust Understanding framework, discussed below).

Emphasizing both sense making and making sense in mathematics education opens opportunities for connections with similar practices in other STEM disciplines. For example, sense making is very much emphasized in science education (Hogan 2019 ; Kapon 2017 ; Odden and Russ 2019 ), often combined with reflections in engineering (Kilgore et al. 2013 ; Turns et al. 2014 ), and also in the context of using technology (e.g., Antonietti and Cantoia 2000 ; Dick and Hollebrands 2011 ). Science is fundamentally about discovery and understanding of the natural world. This notion provides a natural link to mathematical modeling (e.g., Burkhardt 1981 ). Beyond that, in science education, sense making places a heavy focus on the construction and evaluation of explanation (Kapon 2017 ), and can even be defined as a process of constructing an explanation to resolve a perceived gap or conflict in knowledge (Odden and Russ 2019 ). Design and making play vital roles in engineering and technology education (Dym et al., 2005 ), as is student reflection on these experiences (e.g., Turns et al. 2014 ). Indeed, STEM disciplines share the same conceptual process of sense making as learners, individually or in a group, actively engage with the natural or man-made world, explore it, and then develop, test, refine, and use ideas together with specific explanation. If mathematics was conceived as an “empirical” discipline, connections with other STEM disciplines would be strengthened. In philosophical terms, Lakatos ( 1976 ) made similar claims Footnote 1 .

Similar to the emphasis on sense making placed in the Mathematics Curriculum Standards (e.g., NCTM, 1989 , 2000 ), Next Generation Science Standards (NGSS Lead States 2013 ) prompted a shift in science education away from simply knowing science content and procedures to practicing and using science, together with engineering, to make sense of the world and create the future. In a review, Fitzgerald and Palincsar ( 2019 ) concluded sense making is a productive lens for investigating and characterizing great teaching across multiple disciplines.

Mathematics has stronger linkages to creation and design than traditionally imagined. Therefore, its connections to engineering and technology could be much stronger. However, the deep-rooted conception of mathematics as products has traditionally discouraged students and teachers from considering and valuing design and design thinking (Li et al. 2019b ). Conceiving mathematics as making sense should help promote conceptual changes in mathematical practice to value idea generation and design activity. Connections generated from such a shift will support teaching and learning not only in individual STEM disciplines, but also in integrated STEM education.

At the same time, although STEM education as a commonly recognized field does not have a long history (Li 2014 , 2018a ), its rapid development can help introduce ideas for exploring how mathematics can be taught and learned. For example, the concept of projects is common in engineering professional practice, and the project-based learning (PjBL) as an instructional approach is a key component in some engineering programs (e.g., Berger 2016 ; de los Ríos et al. 2010 ; Mills and Treagust 2003 ). de los Ríos et al. ( 2010 ) highlighted three main advantages of PjBL: (1) development in technical, personal, and contextual competences; (2) students’ engagement with real problems from professional contexts; and (3) collaborative learning facilitated through the integration of teaching and research. Such advantages are important for students’ learning of mathematics and are aligned well with efforts to develop 21 st century skills, including problem solving, communication, collaboration, and critical thinking.

Design-based learning (DBL) is another instructional approach commonly used in engineering and technology fields. Gómez Puente et al. ( 2013 ) conducted a sampled review and concluded that DBL projects consist of open-ended, hands-on, authentic, and multidisciplinary design tasks. Teachers using DBL facilitate both the process for students to gain domain-specific knowledge and thinking activities to generate innovative solutions. Such features could be adapted for mathematics education, especially integrated STEM education, in concert with design and design thinking. In addition to a few examples discussed above about making sense in mathematics, there is a growing body of publications developed by and for mathematics teachers with specific examples of investigations, design projects, and instructional activities associated with STEM (Li et al. 2019b ).

A framework for helping students to gain important experiences in and through mathematics, as connected to other disciplines in STEM

For observing and evaluating classroom instruction in general and mathematics classroom instruction in specific, there are several widely used frameworks and rubrics available. However, a trial use of selected frameworks with sampled mathematics classroom instruction episodes suggested their disagreements on what counts as high-quality instruction, especially with aspects on disciplinary thinking being valued and relevant classroom practices (Schoenfeld et al. 2018 ). The results suggest the importance of choice making, when we consider a framework in discussing and evaluating teaching practices.

Our discussion above highlights the importance of shifting away from viewing mathematics simply as a set of static knowledge and skills, to focusing on ideas and thinking development in teaching and learning mathematics. Further discussion of several aspects of changes specifies the needs of developing and using practices associated with sense making, making sense, and connecting mathematics and STEM education for changes.

To support effective mathematics instruction, we propose the use of the Teaching for Robust Understanding (TRU) framework to help characterize powerful learning environments. With the conception of mathematics as “empirical” and a focus on students’ experience, then the focus of instruction should also be changed. We argue that shift is from instruction conceived as “what should the teacher do” to instruction conceived as “what mathematical experiences should students have in order for them to develop into powerful thinkers?” It is the shift in the frame of TRU that makes it so powerful and pertinent for all these proposed changes. Moreover, TRU only uses a small number of actionable dimensions after distilling the literature on teaching for robust or powerful understanding. That makes TRU a practical mechanism for problematizing instruction.

Figure 1 presents the TRU Math framework that identifies five key dimensions along which powerful classroom environments can be characterized: the mathematics; cognitive demand; equitable access; agency, ownership, and identity; and formative assessment. These five dimensions were distilled from an extensive literature review, thus capturing what the literature considers to be essential. They were tested against classroom videotapes and data on student performance, and the results indicated that classrooms that did well on the TRU dimensions produced students who did correspondingly well on tests of mathematical knowledge, thinking, and problem solving (e.g., Schoenfeld 2014 , 2019 ). In brief, the argument regarding the importance of the five dimensions of TRU Math is as follows. First, the quality of the mathematics discussed (dimension 1) is critical. What individual students learn is unlikely to be richer than what they experience in the classroom. Whether individual students’ understanding rises to the level of what is discussed/presented in the classroom depends on other factors, which are captured in the remaining four dimensions. For example, you surely have had the experience, at a lecture, of hearing beautiful content presented, and then not being able to do any of the assigned problems! The remaining four dimensions capture aspects needed to support the development of all students with respect to sense making, making sense, ownership, and feedback loop. Dimension 2: Cognitive Demand. Are students engaged in sense making and making sense? Are they engaged in “productive struggle”? Dimension 3: Equitable Access. Are all students fully engaged with the central content and practices of the domain so that every student can profit from it? Dimension 4: Agency, Ownership, and Identity. Do all students have opportunities to develop idea ownership and mathematical agency? Dimension 5: Formative Assessment. Are students encouraged and supported to share their thinking with a meaningful feedback loop for instructional adjustment and improvement?

The TRU Mathematics Framework: The five dimensions of powerful mathematics classrooms

The first key point about TRU is that students learn more in classrooms that are powerful along the five TRU dimensions. Second, the shift of attention from the teacher to the environment is fundamentally important. The key question is not “Is the teacher doing particular things to support learning?”; instead, it is, “Are students experiencing instruction so that it is conducive to their growth as mathematical thinkers and learners?” Third, the framework is not prescriptive; it respects teacher autonomy. There are many ways to be an excellent teacher. The question is, Does the learning environment created by the teacher provide each student rich opportunities along the five dimensions of the framework? Specifically, in describing the dimensions of powerful instruction, the framework serves to problematize instruction. Asking “how am I doing along each dimension; how can I improve?” can lead to richer instruction without prescribing or imposing a particular style or particular norms on teachers.

Extending to STEM education

Now, we suggest the following. If you teach biology, chemistry, physics, engineering, or any other STEM field, replace “mathematics” in Fig. 1 with your discipline. The first dimension is about rich content and practices in your field. And the remaining four dimensions are about necessary aspects of your students’ classroom engagement with the discipline. Practices associated with sense making, making sense, and STEM education are all be reflected in these five dimensions, with central attention on students’ experience in such classroom environments. Although the TRU framework was originally developed for characterizing effective mathematics classroom environments, it has been carefully framed in a way that is applicable to many different disciplines (Schoenfeld 2014 ). Our discussion above already specified why sense making, making sense, and specific instructional approaches like PjBL and DBL are shared across disciplines in STEM education. Thus, the TRU framework is applicable to other STEM disciplines. The natural analogue of the TRU framework for any field is given in Fig. 2 .

The domain-general version of the TRU framework

Both the San Francisco Unified School District and the Chicago Public Schools adopted the TRU Math framework and found results within mathematics sufficiently promising that they expanded their efforts to all subject areas for professional development and instruction, using the domain-general TRU framework. Work is still in its early stages. Current efforts might be best conceptualized as a laboratory for exploration rather than a promissory note for improvement across all different disciplines. It will take time to accumulate data to show effectiveness. For further information about the domain-general TRU framework and tools for professional development are available at the TRU framework website, https://truframework.org/

Finally, as a framework, TRU is not a set of specific tools or guidelines, although it can be used to guide their development. To help lead our discussion to something more practical, we can use the framework to check and identify aspects that are typically under-emphasized and move them to center stage in order to improve classroom instruction. Specifically, the following is a list of sample under-emphasized norms and practices that can be identified (Schoenfeld in press ).

Establishing a climate of inquiry, in which mathematics is experienced as a discipline of exploration and sense making.

Developing students’ ownership of ideas through the process of developing, sharing, refining, and using ideas; concepts and language can come later.

Focusing on big ideas, and not losing the forest for the trees.

Making student thinking central to classroom discourse.

Ensuring that classroom discourse is respectful and inviting.

Where to start? Begin by problematizing teaching and the nature of learning environments

Here we start by stipulating that STEM disciplines as practiced, are living, breathing fields of inquiry. Knowledge is important; ideas are important; practices are important. The list given above applied to all STEM disciplines, not just mathematics.

The issue, then, is developing teacher capacity to craft environments that have the properties described immediately above. Here we share some thoughts, and the topic itself can well be discussed extensively in another paper. To make changes in teaching, it should start with assessing and changing teaching practice itself (Hiebert and Morris 2012 ). Opening up teachers’ perceptions of teaching practices should not be done by telling teachers what to do!—the same rules of learning apply to teachers as they apply to students. Learning environments for teachers should offer teachers the same opportunities for rich engagement, challenge, equitable access, and ownership as we hope students will experience (Schoenfeld 2015 ). Working together with teachers to study and reflect on their teaching practices in light of the TRU framework, we can help teachers to find out what their students are experiencing and why changes are needed. The framework can also help guide teachers to learn what changes would be needed, and to try out changes to learn how their students’ learning may differ. It is this iterative and concrete process that can hopefully help shift participating teachers’ perceptions of mathematics. Many tools for problematizing teaching are available at the TRU web site (see https://truframework.org/ ). If teachers can work together with a focus on selected lessons like what teachers often do in China, the process would help form a school-based learning community that can contribute to not only participating teachers’ practice change but also their expertise improvement (Huang et al. 2011 ; Li and Huang 2013 ).

As reported before (Li 2018b ), publications in the International Journal of STEM Education show a mix of individual-disciplinary and multidisciplinary education in STEM over the past several years. Although one journal’s publications are limited in its scope of providing a picture about the scholarship development related to mathematics and STEM education, it can allow us to get a sense of related development.

If taking a closer look at the journal’s publications over the past three years from 2016 to 2018, we found that the number of articles published with a clear focus on mathematics is relatively small: three (out of 21) in 2016, six (out of 34) in 2017, and five (out of 56) in 2018. At the same time, we should point out that these publications from 2016 to 2018 seem to reflect a trend, over these three years, of moving toward issues that can go beyond mathematics itself, as what was noted before (Li 2018b ). Specifically, for these three articles published in 2016, they are all about mathematics education at either elementary school (Ding 2016 ; Zhao et al. 2016 ) or university levels (Schoenfeld et al. 2016 ). Out of the six published in 2017, three are on mathematics education (Hagman et al. 2017 ; Keller et al. 2017 ; Ulrich and Wilkins 2017 ) and the other three on either teacher professional development (Borko et al. 2017 ; Jacobs et al. 2017 ) or connection with engineering (Jehopio and Wesonga 2017 ). For the five published in 2018, two are on mathematics education (Beumann and Wegner 2018 ; Wilkins and Norton 2018 ) and the other three have close association with other disciplines in STEM (Blotnicky et al. 2018 ; Hayward and Laursen 2018 ; Nye et al. 2018 ). This trend likely reflects a growing interest, with close connection to mathematics, in both mathematics education community and a broader STEM education community of developing and sharing multidisciplinary and interdisciplinary scholarship.

Availability of data and materials

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The strategy we developed asks that teachers compare two ways for solving a problem, side by side, and that they follow an instructional routine to lead a discussion to help students understand the difference between the two methods. That discussion is really the heart of this routine because it is fundamentally about sharing reasoning: Teachers ask students to explain why a strategy works, and students must dig into their heads and try to say what they understand. And listening to other people’s reasoning reinforces the process of learning.

GAZETTE: Why is this strategy an improvement over just learning a single method?

STAR: We think that learning multiple strategies for solving problems deepens students’ understanding of the content. There is a direct benefit to learning through comparing multiple methods, but there are also other types of benefits to students’ motivation. In this process, students come to see math a little differently — not just as a set of problems, each of which has exactly one way to solve it that you must memorize, but rather, as a terrain where there are always decisions to be made and multiple strategies that one might need to justify or debate. Because that is what math is.

For teachers, this can also be empowering because they are interested in increasing their students’ understanding, and we’ve given them a set of tools that can help them do that and potentially make the class more interesting as well. It’s important to note, too, that this approach is not something that we invented. In this case, what we’re asking teachers to do is something that they do a little bit of already. Every high school math teacher, for certain topics, is teaching students multiple strategies. It’s built into the curriculum. All that we’re saying is, first, you should do it more because it’s a good thing, and second, when you do it, this is a certain way that we found to be especially effective, both in terms of the visual materials and the pedagogy. It’s not a big stretch for most teachers. Conversations around ways to teach math for the past 30 or 40 years, and perhaps longer, have been emphasizing the use of multiple strategies.

GAZETTE: What are the potential challenges for math teachers to put this in practice?

STAR: If we want teachers to introduce students to multiple ways to solve problems, we must recognize that that is a lot of information for students and teachers. There is a concern that there could be information overload, and that’s very legitimate. Also, a well-intentioned teacher might take our strategy too far. A teacher might say something like, “Well, if comparing two strategies is good, then why don’t I compare three or four or five?” Not that that’s impossible to do well. But the visual materials you would have to design to help students manage that information overload are quite challenging. We don’t recommend that.

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Articles on Mathematics education

Displaying 1 - 20 of 39 articles.

How counting by 10 helps children learn about the meaning of numbers

Helena Osana , Concordia University ; Jairo A. Navarrete-Ulloa , Universidad de O’Higgins (Chile) , and Vera Wagner , Concordia University

Heritage algorithms combine the rigors of science with the infinite possibilities of art and design

Audrey G. Bennett , University of Michigan and Ron Eglash , University of Michigan

The simple reason a viral math equation stumped the internet

Egan J Chernoff , University of Saskatchewan and Rina Zazkis , Simon Fraser University

4 things we’ve learned about math success that might surprise parents

Tina Rapke , York University, Canada and Cristina De Simone , York University, Canada

The key to fixing the gender gap in math and science: Boost women’s confidence

Lara Perez-Felkner , Florida State University

Celebrating Marion Walter – and other unsung female mathematicians

Jennifer Ruef , University of Oregon

These four easy steps can make you a math whiz

20% maths decree sets a dangerous precedent for schooling in South Africa

Clive Kronenberg , Cape Peninsula University of Technology

Why it doesn’t help – and may harm – to fail pupils with poor maths marks

Elizabeth Walton , University of the Witwatersrand

Boredom, alienation and anxiety in the maths classroom? Here’s why

Brian Hudson , University of Sussex

Mastery over mindset: the cost of rolling out a Chinese way of teaching maths

Alexei Vernitski , University of Essex and Sherria Hoskins , University of Portsmouth

What makes a mathematical genius?

David Pearson , Anglia Ruskin University

The rush to calculus is bad for students and their futures in STEM

Kevin Knudson , University of Florida

The Common Core is today’s New Math – which is actually a good thing

How to get children to want to do maths outside the classroom

Steve Humble , Newcastle University

Don’t freak if you can’t solve a math problem that’s gone viral

It’s often the puzzles that baffle that go viral

Jonathan Borwein (Jon) , University of Newcastle

More high school science and maths linked to more dropouts

Washington University in St. Louis

Better at reading than maths? Don’t blame it all on your genes

Kathryn Asbury , University of York

What is 7x8? You’ll need confidence to answer correctly

Adam Boddison , University of Warwick

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Journal for Research in Mathematics Education

An official journal of the National Council of Teachers of Mathematics (NCTM), JRME is the premier research journal in mathematics education and is devoted to the interests of teachers and researchers at all levels--preschool through college.

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Working Interinstitutionally to Apprentice Doctoral Students in Mathematics Education Research

Identity, power, and dignity: a positional analysis of gisela in her high school mathematics classroom.

Multiply minoritized learners face racialized, gendered, and ableist hierarchies of mathematical ability that shape the organization of schools and classrooms and can significantly challenge access to identities as mathematical learners and practitioners as well as to fundamental human dignity. Classrooms and everyday interactions can perpetuate or interrupt these conditions. Contributing to questions about the relationships among identity, power, and dignity in mathematics learning, this article presents a positional interaction analysis of Gisela, a Disabled 10th-grade Latina student, as she took up, challenged, and renegotiated identities of mathematical thinker, learner, and community member over the course of one school year.

Attending to Coherence Among Research Questions, Methods, and Claims in Coding Studies

We consider a kind of study common in mathematics education research: one that allocates qualitative data to categories in a theoretical or conceptual framework. These studies sometimes lack coherence among research questions, sampling and analysis methods, and claims, which can be attributed to tensions in how these aspects are framed. We ground our discussion in examples from five published studies, focusing on the methodological and reporting decisions that increase coherence: answering research questions from the same perspective they are asked (using a variance or a process lens), using (relative) frequencies properly to warrant claims, employing a coherent sampling strategy, and making appropriate generalizations. We argue that attending to coherence can increase the quality and contribution of coding studies.

The Journal for Research in Mathematics Education is published online five times a year—January, March, May, July, and November—at 1906 Association Dr., Reston, VA 20191-1502. Each volume’s index is in the November issue. JRME is indexed in Contents Pages in Education, Current Index to Journals in Education, Education Index, Psychological Abstracts, Social Sciences Citation Index, and MathEduc.

An official journal of the National Council of Teachers of Mathematics (NCTM), JRME is the premier research journal in mathematics education and is devoted to the interests of teachers and researchers at all levels--preschool through college. JRME presents a variety of viewpoints. The views expressed or implied in JRME are not the official position of the Council unless otherwise noted.

JRME is a forum for disciplined inquiry into the teaching and learning of mathematics. The editors encourage submissions including:

• Research reports, addressing important research questions and issues in mathematics education,
• Brief reports of research,
• Research commentaries on issues pertaining to mathematics education research.

More information about each type of submission is available here . If you have questions about the types of manuscripts JRME publishes, please contact [email protected].

Editorial Board

The  JRME  Editorial Board consists of the Editorial Team and Editorial Panel.  The Editorial team, led by JRME Editor Patricio Herbst, leads the review, decision and editorial/publication process for manuscripts.  The Editorial Panel reviews manuscripts, sets policy for the journal, and continually seeks feedback from readers. The following are members of the current JRME Editorial Board.

Editorial Staff   

Editorial Panel  

International Advisory Board   

Headquarters Journal Staff  

The editors of the  Journal for Research in Mathematics Education (JRME)  encourage the submission of a variety of manuscripts.

Manuscripts must be submitted through the JRME Online Submission and Review System . 

Research Reports

JRME publishes a wide variety of research reports that move the field of mathematics education forward. These include, but are not limited to, various genres and designs of empirical research; philosophical, methodological, and historical studies in mathematics education; and literature reviews, syntheses, and theoretical analyses of research in mathematics education. Papers that review well for JRME generally include these Characteristics of a High-Quality Manuscript . The editors strongly encourage all authors to consider these characteristics when preparing a submission to JRME. 

The maximum length for Research Reports is 13,000 words including abstract, references, tables, and figures.

Brief Reports

Brief reports of research are appropriate when a fuller report is available elsewhere or when a more comprehensive follow-up study is planned.

• A brief report of a first study on some topic might stress the rationale, hypotheses, and plans for further work.
• A brief report of a replication or extension of a previously reported study might contrast the results of the two studies, referring to the earlier study for methodological details.
• A brief report of a monograph or other lengthy nonjournal publication might summarize the key findings and implications or might highlight an unusual observation or methodological approach.
• A brief report might provide an executive summary of a large study.

The maximum length for Brief Reports is 5,000 words including abstract, references, tables, and figures. If source materials are needed to evaluate a brief report manuscript, a copy should be included.

Other correspondence regarding manuscripts for Research Reports or Brief Reports should be sent to

Patricio Herbst, JRME Editor, [email protected] .

Research Commentaries

The journal publishes brief (5,000 word), peer-reviewed commentaries on issues that reflect on mathematics education research as a field and steward its development. Research Commentaries differ from Research Reports in that their focus is not to present new findings or empirical results, but rather to comment on issues of interest to the broader research community. 

Research Commentaries are intended to engage the community and increase the breadth of topics addressed in  JRME . Typically, Research Commentaries —

• address mathematics education research as a field and endeavor to move the field forward;
• speak to the readers of the journal as an audience of researchers; and
• speak in ways that have relevance to all mathematics education researchers, even when addressing a particular point or a particular subgroup.

Authors of Research Commentaries should share their perspectives while seeking to invite conversation and dialogue, rather than close off opportunities to learn from others, especially those whose work they might be critiquing. 

Foci of Research Commentaries vary widely. They may include, but are not restricted to the following:

• Discussion of connections between research and NCTM-produced documents
• Advances in research methods
• Discussions of connections among research, policy, and practice
• Analyses of trends in policies for funding research
• Examinations of evaluation studies
• Critical essays on research publications that have implications for the mathematics education research community
• Interpretations of previously published research in JRME that bring insights from an equity lens
• Exchanges among scholars holding contrasting views about research-related issues

Read more about Research Commentaries in our May 2023 editorial . 

The maximum length for Research Commentaries is 5,000 words, including abstract, references, tables, and figures.

Other correspondence regarding Research Commentary manuscripts should be sent to: 

Daniel Chazan, JRME Research Commentary Editor, [email protected] .

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The forms below provide information to authors and help ensure that NCTM complies with all copyright laws: 

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Find more information in this flyer  about how to become a reviewer for JRME . 

The  Journal for Research in Mathematics Education  is available to individuals as part of an  NCTM membership  or may be accessible through an  institutional subscription .

The  Journal for Research in Mathematics Education  ( JRME ), an official journal of the National Council of Teachers of Mathematics (NCTM), is the premier research journal in math education and devoted to the interests of teachers and researchers at all levels--preschool through college.

JRME is published five times a year—January, March, May, July, and November—and presents a variety of viewpoints.  Learn more about   JRME .

© 2024 National Council of Teachers of Mathematics (NCTM)

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Rick Hess Straight Up

Education policy maven Rick Hess of the American Enterprise Institute think tank offers straight talk on matters of policy, politics, research, and reform. Read more from this blog.

What It Takes to Actually Improve Math Education

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Barry Garelick, a veteran math teacher in California and respected observer of math instruction, recently reached out after seeing my Q&A with ST Math’s Andrew Coulson on using visualization to teach math. Garelick is a cogent thinker, clear writer, and author of books including Out on Good Behavior: Teaching math while looking over your shoulder and Math Education in the U.S.: Still Crazy After All These Years . Given all that, I thought his reflections well worth sharing—see what you think.

Rick, I thought your recent interview with Andrew Coulson of ST Math was a fascinating look at how educational products—particularly those that address math—are promoted. In the interview, Coulson states that the “innate ability of visualizing math was not being leveraged to solve a serious education problem: a lack of deep conceptual understanding of mathematics.”

As someone who has been teaching math for the past 10 years and written several books on key issues in math education, this struck a chord for me. I’ve seen the three-decade-long obsession with “deeper understanding” cause more problems than it solves—including overlooking other factors contributing to problems in math education, such as the disdain for memorization, the difference between understanding and procedure, and the issue with trying to teach problem solving solely by teaching generic skills. Undoing these would be a long-overdue step in the right direction to reverse the trends we are seeing in math education.

For starters, many math reformers seem to disdain memorization in favor of cultivating “deeper understanding.” The prevailing belief in current math-reform circles is that drilling kills the soul and makes students hate math and that memorizing the facts obscures understanding. Memorization of multiplication facts and the drills to get there, for example, are thought to obscure the meaning of what multiplication is. Instead of memorizing, students are encouraged to reason their way to “fluently derive” answers. For example, students who do not know that 8×7 is 56 may find the answer by reasoning that if 8×6 is 48, then 8×7 is eight more than 48, or 56. (Ironically, the same people who believe no student should be made to memorize have no problem with students using calculators for multiplication facts.)

Unfortunately, this approach ignores the fact that there are some things in math that need to be memorized and drilled, such as addition and multiplication facts. Repetitive practice lies at the heart of mastery of almost every discipline, and mathematics is no exception. No sensible person would suggest eliminating drills from sports, music, or dance. De-emphasize skill and memorization and you take away the child’s primary scaffold for understanding.

Teaching procedures and standard algorithms is similarly shunned as “rote memorization” that gets in the way of “deeper understanding” in math. But educators who believe this fail to see that using procedures to solve problems actually requires reasoning with such methods—which in itself is a form of understanding. Indeed, iterative practice is key to attaining procedural fluency and conceptual understanding. Understanding, critical thinking, and problem solving come when students can draw on a strong foundation of relevant domain content, which is built through the “rote memorization” of procedure. Whether understanding or procedure is taught first ought to be driven by subject matter and student need—not educational ideology. In short, of course we should teach for understanding. But don’t sacrifice the proficiency gained by learning procedures in the name of understanding by obsessing over it and holding students up when they are ready to move forward.

Finally, although it’s been shown that solving math problems cannot be taught by teaching generic problem-solving skills, math reformers believe that such skills can be taught independent of specific problems. Traditional word problems such as “Two trains traveling toward each other at different speeds. When will they meet?” are held to be inauthentic and not relevant to students’ lives.

Instead, the reformers advocate an approach that presents students “challenging open-ended problems” (sometimes called “rich problems”) for which little or no prior instruction is given and which do not develop any identifiable or transferable skills. For example, “How many boxes would be needed to pack and ship 1 million books collected in a school-based book drive?” In this problem, the size of the books is unknown and varied and the size of the boxes is not stated. While some teachers consider the open-ended nature of the problem to be deep, rich, and unique, students will generally lack the skills required to solve such a problem, such as knowledge of proper experimental approaches, systematic and random errors, organizational skills, and validation and verification. Students are given generic problem-solving techniques (e.g., look for a simpler but similar problem), in the belief that they will develop a “problem-solving habit of mind.” But in the case of the above problem, such techniques simply will not work, leaving students frustrated, confused, and feeling as if they are not good at math.

Instead of having students struggle with little or no prior knowledge of how to approach a problem, students need to be given explicit instruction on solving various types of problems, via worked examples and initial practice problems. After that, they should be given problems that vary in difficulty, forcing students to stretch beyond the examples. Students build up a repertoire of problem-solving techniques as they progress from novice to expert. In my experience, students who are left to struggle with minimal guidance tend to ask, “Why do I need to know this?,” whereas students given proper instruction do not—nor do they care whether the problems are “relevant” to their everyday lives.

At the end of the day, finding a cure for a system that refuses to recognize its ills has proved futile. Parents confronting school administrators are patronized and placated or told that they don’t like the way math is taught because it’s not how they were taught.

Change will not come about by battling school administrations. There must be a recognition that the above approaches to teaching math are not working, as is currently happening with reading, thanks to the efforts of people like Emily Hanford, Natalie Wexler, and others, who have shown that teaching reading via phonics is effective , whereas memorizing words by sight or guessing the word by the context or a picture is not. Until then, only people with the means and access to tutors, learning centers, and private schools will be able to ensure that their students learn the math they need. The rest will be left to the “equitable solutions” of the last three decades that have proved disastrous.

Barry Garelick is a 7 th and 8 th grade math teacher and author of several books on math education, including his most recent, Out on Good Behavior: Teaching math while looking over your shoulder . Garelick, who worked in environmental protection for the federal government before entering the classroom, has also written articles on math education for publications including The Atlantic, Education Next, Nonpartisan Education Review, and Education News.

The opinions expressed in Rick Hess Straight Up are strictly those of the author(s) and do not reflect the opinions or endorsement of Editorial Projects in Education, or any of its publications.

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What’s going on in secondary mathematics education? A conversation

Ted Coe and Catherine A. Roberts

Catherine:   After seven years as the executive director at the American Mathematical Society and a short stint at the Consortium for Mathematics and Its Applications , I am now getting back into collegiate teaching. As an applied mathematician keenly interested in mathematics education, I want to reacquaint myself with the broader education landscape. There are lots of voices clamoring to be heard. I thought it might be helpful to hear from an expert on what is going on in secondary math education, so I invited Ted Coe to address a few of my questions. First, I’ll ask Ted to introduce himself.

Ted : Thank you for the invitation, Catherine. Throughout my career I’ve had the opportunity to serve in many roles across different sectors – it’s been an adventure! I’ve been a high school mathematics teacher, a community college professor and chair, an assistant dean, the mathematics director at Achieve (a DC-based independent, nonpartisan, nonprofit education reform organization ) , and I’m currently the VP of Academic Advocacy for Mathematics at NWEA . I also have the privilege of serving as the treasurer of the Conference Board of the Mathematical Sciences and I support the work of Charles A. Dana Center on the Launch Years project to improve transitions between high school and college.

These days, I am often involved in working with state teams to examine and rethink the mathematics students encounter in the transition space from high school to college. We bring together representatives from all education sectors to try to figure out how we can better meet our students’ needs. It’s wonderful yet challenging work.

Catherine : Thanks, and welcome! Perhaps we could start with the work going on in various U.S. states regarding high school math pathways? What is this? And what do you think faculty in higher education ought to be paying attention to?

Ted: Sure! One of the most exciting things is an attempt to better align secondary mathematical experiences to students’ future careers. It’s typically not an easy discussion. There is a desire at all levels to connect students to mathematics that is engaging and relevant, yet there is also a healthy concern that modifications to high school programs may leave students unprepared as they transition to postsecondary endeavors. Will they be ready for the demands of college classes? When should students be allowed to choose which classes they take? What happens if students change their minds after they have started down a specific pathway?

Many colleges and universities used to require that all students satisfy a math requirement that was generally some version of College Algebra. However, a shift that has happened in many higher education institutions is the recognition that College Algebra is not necessarily seen as the best default math requirement. There are better options that better match mathematics requirements to student programs and interests and serve as more appropriate terminal math course experiences.

Catherine: It seems to me that this shift has been going on for decades!

Ted: Yes, precisely. This has been a change that has taken place over many years, and there have been some wonderful curricular examples over the years that connected content with context in meaningful ways. The 2015 CBMS Survey , for example, highlighted how 58% of two-year colleges had implemented some forms of math pathways. I can only imagine that number has grown significantly since then. But the tricky issue now is to consider how these evolving changes in higher education create issues and opportunities for high school. If college students no longer need College Algebra, we need to have conversations about how much algebra every student needs to see in high school. What algebra should all students experience before digging into the algebra that is more specialized for students heading to Calculus? What different mathematics might make better use of that time?  These conversations get messy, as you can imagine. CBMS helped pull state teams together a few years ago. The Launch Years work of the Dana Center continues to provide support to states as they navigate these waters. Some states have made a lot of progress, although many are still in the early stages. Each state, though, is heading in a different-ish direction, so at some point the field needs to get together and find points of consensus.

Catherine : Back in the early 1990s during the calculus reform movement, I remember when I was searching for a faculty position that my potential future colleagues would ask me which calculus textbook I preferred. I viewed that question as a way to determine if I would fit in as a “traditionalist” or a “reformist”. Now, it seems like controversies are swirling around what our gen ed math courses ought to be – and where College Algebra lives in light of statistics, data science, math modeling, and other approaches to quantitative thinking and learning. And that this is really boiling at the interface between high schools and colleges.

Ted : I find the conversations I’m involved with to be less about College Algebra or not, and more about determining what options should students have. I remember those calculus reform years as well, as it was when I was starting out as a high school teacher, but I’d say the work on pathways is less about polarization and taking sides than it is about how to navigate the mess of issues that arise when systems try to work together. There are challenges that arise when high schools want to make changes to curriculum. Their students go to many different higher education institutions, and so there are mixed signals on what matters. Some colleges recognize different courses, while others do not. When higher education institutions don’t agree, it can cause challenges for high schools.

Calculus surely doesn’t have a lock on rigor in mathematics, but it is still unclear on the high school side to say when some other non-calculus pathway course is or is not rigorous. With the support of higher education faculty, we can answer these questions. Ohio did a pretty remarkable thing on this issue when they brought together representatives from K-12 and higher education to create an Ohio definition of rigor . Ideally, we need state teams made up of members across the education sectors working together to clarify what makes for a respectable and valuable math experience.

Much of the work in states comes back to the role of Algebra 2. It’s tempting to think of that course in terms of our own historical experiences, but it’s always eye-opening when those who do not teach the course examine the course standards and textbooks. It’s important that we focus more on actual content than course names. Washington State is doing some creative work rethinking Algebra 2, which will be interesting to watch. It is in a pilot phase right now.

Readers might take just a moment to reflect on the general notion of college readiness. What did it mean to be college ready in 2010? What will it mean to be college ready in 2030? If those two are different, do you know if your state has made progress in bridging the gap? Everyone who cares about mathematics education, all the different segments, need to work more intently and directly with each other.

Catherine : Something that has fascinated me is seeing such a transformation in the ways we teach and learn. I remember thinking as a college math professor, I ought to be aware of the evidence-based research that could make my teaching more effective. Can you say a little bit about this?

Ted : While many may be looking for some silver bullet, a one-size-fits-all perfect pedagogy, there are too many factors to ever be able to say one thing will always work best. Teaching is both a science and an art, after all. Some pedagogies are better in some situations than others. That said, the learning sciences (in math education and other areas) have helped us to understand the importance of things we might overlook. I like to remember the claim in the national academies’ How People Learn II that “Quite literally, it is neurobiologically impossible to think deeply about or remember information about which one has had no emotion because the healthy brain does not waste energy processing information that does not matter to the individual.” In the higher education space, we know that active learning is effective [see CBMS statement ]. NCTM’s Effective Mathematics Teaching Practices apply to all levels of teaching. There’s one noteworthy shift happening in K-12 focused on Peter Liljedahl’s “ Building T hinking C lassrooms ” At CBMS we are preparing to issue a call for the National Academies to create a Grades 9-14 math framework to address issues around content, pedagogy, and technology. It’s an exciting time.

Catherine : Well, clearly there is a lot for me to be paying attention to, Ted. I really appreciate this primer on the secondary school math education landscape. What is your wish for anyone reading this column?

Ted : There’s a lot going on out there, and it’s messy, complicated work. After all, it isn’t easy to change things that are deeply entrenched across different systems. I hope our readers keep an open mind, regularly talk to people outside of their regular circles, and surface the wide areas of agreement that exist. I also hope they check out the work that is happening (or has happened) in their states, look for ways to get involved, and always keep the best interests of students at the forefront.

Catherine: Thank you so much, Ted. We all really appreciate your time and insights!

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21 Essential Strategies in Teaching Math

Even veteran teachers need to read these.

We all want our kids to succeed in math. In most districts, standardized tests measure students’ understanding, yet nobody wants to teach to the test. Over-reliance on test prep materials and “drill and kill” worksheets steal instructional time while also harming learning and motivation. But sound instruction and good test scores aren’t mutually exclusive. Being intentional and using creative approaches to your instruction can get students excited about math. These essential strategies in teaching mathematics can make this your class’s best math year ever!

1. Raise the bar for all

WeAreTeachers

For math strategies to be effective, teachers must first get students to believe that they can be great mathematicians. Holding high expectations for all students encourages growth. As early as second grade, girls have internalized the idea that math is not for them . It can be a challenge to overcome the socially acceptable thought, I’m not good at math , says Sarah Bax, a math teacher at Hardy Middle School in Washington, D.C.

Rather than success being a function of how much math talent they’re born with, kids need to hear from teachers that anyone who works hard can succeed. “It’s about helping kids have a growth mindset ,” says Bax. “Practice and persistence make you good at math.” Build math equity and tell students about the power and importance of math with enthusiasm and high expectations.

(Psst … you can snag our growth mindset posters for your math classroom here. )

2. Don’t wait—act now!

Look ahead to the specific concepts students need to master for annual end-of-year tests, and pace instruction accordingly. Think about foundational skills they will need in the year ahead.

“You don’t want to be caught off guard come March thinking that students need to know X for the tests the next month,” says Skip Fennell, project director of Elementary Mathematics Specialists and Teacher Leaders Project and professor emeritus at McDaniel College in Westminster, Maryland. Know the specific standards and back-map your teaching from the fall so students are ready, and plan to use effective math strategies accordingly.

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3. Create a testing pathway

You may not even see the results of standardized tests until next school year, but you have to prepare students for it now. Use formative assessments to ensure that students understand the concepts. What you learn can guide your instruction and determine the next steps, says Fennell. “I changed the wording because I didn’t want to suggest that we are in favor of ‘teaching to the test.'”

Testing is not something separate from your instruction. It should be integrated into your planning. Instead of a quick exit question or card, give a five-minute quiz, an open-ended question, or a meaningful homework assignment to confirm students have mastered the math skill covered in the day’s lesson. Additionally, asking students to explain their thinking orally or in writing is a great way to determine their level of understanding. A capable digital resource, designed to monitor your students in real-time, can also be an invaluable tool, providing actionable data to inform your instruction along the way.

4. Observe, modify, and reevaluate

Sometimes we get stuck in a mindset of “a lesson a day” in order to get through the content. However, we should keep our pacing flexible, or kids can fall behind. Walk through your classroom as students work on problems and observe the dynamics. Talk with students individually and include “hinge questions” in your lesson plans to gauge understanding before continuing, suggests Fennell. In response, make decisions to go faster or slower or put students in groups.

5. Read, read, read!

Although we don’t often think of reading as a math strategy, there’s almost nothing better to get students ready to learn a new concept than a great read-aloud. Kids love to be read to, and the more we show students how math is connected to the world around us, the more invested they become. Reading books with math connections helps children see how abstract concepts connect to their lives.

6. Personalize and offer choice

When students are given the opportunity to choose how they learn and demonstrate their understanding of a concept, their buy-in and motivation increase. It gives them the chance to understand their preferred learning style, provides agency over their own learning, and allows for the space to practice different strategies to solve math problems. Give students a variety of options, such as timed exercises, projects, or different materials , to show that they’ve mastered foundational skills. As students show what they’ve learned, teachers can track understanding, figure out where students need additional scaffolding or other assistance, and tailor lessons accordingly.

7. Plant the seeds!

Leave no child inside! A school garden is a great way to apply math concepts in a fun way while instilling a sense of purpose in your students. Measurement, geometry, and data analysis are obvious topics that can be addressed through garden activities, but also consider using the garden to teach operations, fractions, and decimals. Additionally, garden activities can help promote character education goals like cooperation, respect for the earth, and, if the crops are donated to organizations that serve those in need, the value of giving to others.

8. Add apps appropriately

The number of apps (interactive software used on touch-screen devices) available to support math instruction has increased rapidly in recent years. Kids who are reluctant to practice math facts with traditional pencil-and-paper resources will gladly do essentially the same work as long as it’s done on a touch screen. Many apps focus on practice via games, but there are some that encourage children to explore the content at a conceptual level.

9. Encourage math talk

Communicating about math helps students process new learning and build on their thinking. Engage students during conversations and have them describe why they solved a problem in a certain way. “My goal is to get information about what students are thinking and use that to guide my instruction, as opposed to just telling them information and asking them to parrot things back,” says Delise Andrews, who taught math (K–8) and is now a grade 3–5 math coordinator in the Lincoln Public Schools in Nebraska.

Instead of seeking a specific answer, Andrews wants to have deeper discussions to figure out what a student knows and understands. “True learning happens a lot around talking and doing math—not just drilling,” she says. Of course, this math strategy not only requires students to feel comfortable expressing their mathematical thinking, but also assumes that they have been trained to listen respectfully to the reasoning of their classmates.

Learn more: Free Let’s Talk Math Poster

10. The art of math

Almost all kids love art, and visual learners need a math strategy that works for them too, so consider integrating art and math instruction for one of the easiest strategies in teaching mathematics. Many concepts in geometry, such as shapes, symmetry, and transformations (slides, flips, and turns), can be applied in a fun art project. Also consider using art projects to teach concepts like measurement, ratios, and arrays (multiplication/division).

11. Seek to develop understanding

Meaningful math education goes beyond memorizing formulas and procedures. Memorization does not foster understanding. Set high goals, create space for exploration, and work with the students to develop a strong foundation. “Treat the kids like mathematicians,” says Andrews. Present a broad topic, review various strategies for solving a problem, and then elicit a formula or idea from the kids rather than starting with the formula. This creates a stronger conceptual understanding and mental connections with the material for the student.

12. Give students time to reflect

Sometimes teachers get so caught up in meeting the demands of the curriculum and the pressure to “get it all done” that they don’t give students the time to reflect on their learning. Students can be asked to reflect in writing at the end of an assignment or lesson, via class or small group discussion, or in interviews with the teacher. It’s important to give students the time to think about and articulate the meaning of what they’ve learned, what they still don’t understand, and what they want to learn more about. This provides useful information for the teacher and helps the student monitor their own progress and think strategically about how they approach mathematics.

13. Allow for productive struggle

When giving students an authentic problem, ask a big question and let them struggle to figure out several ways to solve it, suggests Andrews. “Your job as a teacher is to make it engaging by asking the right questions at the right time. So you don’t take away their thinking, but you help them move forward to a solution,” she says.

Provide as little information as possible but enough so students can be productive. Effective math teaching supports students as they grapple with mathematical ideas and relationships. Allow them to discover what works and experience setbacks along the way as they adopt a growth mindset about mathematics.

14. Emphasize hands-on learning

WeAreTeachers; Teacher Created Resources

In math, there’s so much that’s abstract. Hands-on learning is a strategy that helps make the conceptual concrete. Consider incorporating math manipulatives whenever possible. For example, you can use LEGO bricks to teach a variety of math skills, including finding area and perimeter and understanding multiplication.

15. Build excitement by rewarding progress

Students—especially those who haven’t experienced success—can have negative attitudes about math. Consider having students earn points and receive certificates, stickers, badges, or trophies as they progress. Weekly announcements and assemblies that celebrate the top players and teams can be really inspiring for students. “Having that recognition and moment is powerful,” says Bax. “Through repeated practice, they get better, and they are motivated.” Through building excitement, this allows for one of the best strategies in teaching mathematics to come to fruition.

16. Choose meaningful tasks

Kids get excited about math when they have to  solve real-life problems. For instance, when teaching sixth graders how to determine area, present tasks related to a house redesign, suggests Fennell. Provide them with the dimensions of the walls and the size of the windows and have them determine how much space is left for the wallpaper. Or ask them to consider how many tiles they would need to fill a deck. You can absolutely introduce problem-based learning, even in a virtual world.

17. Play math games

Life Between Summers/Shape Guess Who via lifebetweensummers.com; 123 Homeschool 4 Me/Tic-Tac-Toe Math Game via 123homeschool4me.com; WeAreTeachers

Student engagement and participation can be a challenge, especially if you’re relying heavily on worksheets. Games, like these first grade math games , are an excellent way to make the learning more fun while simultaneously promoting strategic mathematical thinking, computational fluency , and understanding of operations. Games are especially good for kinesthetic learners and foster a home-school connection when they’re sent home for extra practice.

18. Set up effective math routines

Students generally feel confident and competent in the classroom when they know what to do and why they’re doing it. Establishing routines in your math class and training kids to use them can make math class efficient, effective, and fun! For example, consider starting your class with a number sense routine . Rich, productive small group math discussions don’t happen by themselves, so make sure your students know the “rules of the road” for contributing their ideas and respectfully critiquing the ideas of others.

19. Encourage teacher teamwork and reflection

You can’t teach in a vacuum. Collaborate with other teachers to improve your math instruction skills. Start by discussing the goal for the math lesson and what it will look like, and plan as a team to use the most effective math strategies. “Together, think through the tasks and possible student responses you might encounter,” says Andrews. Reflect on what did and didn’t work to improve your practice.

Learn With Play at Home/Plastic Bottle Number Bowling via learnwithplayathome.com; Math Geek Mama/Skip-Counting Hopscotch via mathgeekmama.com; WeAreTeachers

Adding movement and physical activity to your instruction might seem counterintuitive as a math strategy, but asking kids to get out of their seats can increase their motivation and interest. For example, you could ask students to:

• Make angles with their arms
• Create a square dance that demonstrates different types of patterns
• Complete a shape scavenger hunt in the classroom
• Run or complete other exercises periodically and graph the results

The possibilities of these strategies in teaching mathematics are limited only by your imagination and the math concepts you need to cover. Check out these active math games .

21. Be a lifelong learner

Generally, students will become excited about a subject if their teacher is excited about it. However, it’s hard to be excited about teaching math if your understanding hasn’t changed since you learned it in elementary school. For example, if you teach how to divide fractions by fractions and your understanding is limited to following the “invert and multiply” rule, take the time to understand why the rule works and how it applies to the real world. When you have confidence in your own mathematical expertise, then you can teach math confidently and joyfully to best apply strategies in teaching mathematics.

What do you feel are the most important strategies in teaching mathematics? Share in the comments below.

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Opinion | Teaching and Learning

When teaching students math, concepts matter more than process, by nicola hodkowski     jun 5, 2024.

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As a mathematics education researcher, I study how math instruction impacts students' learning, from following standard math procedures to understanding mathematical concepts. Focusing on the latter, conceptual understanding often involves understanding the “why” of a mathematical concept ; it’s the reasoning behind the math rather than the how or the steps it takes to get to an answer.

So often, in mathematics classrooms, students are shown steps and procedures for solving math problems and then required to demonstrate their rote memorization of these steps independently.

As a result, students' agency, knowledge and ability to transfer the concepts of mathematics suffer. Specifically, students experience diminished confidence in tackling mathematical problems and a decreased ability to apply mathematical reasoning in real-world situations. In addition, students may struggle with more advanced mathematical concepts and problem-solving tasks as they progress in their education.

While procedural fluency is important, conceptual understanding provides a framework for students to build mental relationships between math concepts. It allows students to connect new ideas to what they already know , creating increasing connections toward more advanced mathematics.

If we want mathematics achievement to improve, we need instruction to begin focusing on concepts instead of procedures.

Why Concept Matters More Than Procedure

Conceptual understanding builds on existing understanding to advance knowledge and focuses on the student’s ability to justify and explain. Procedural fluency, on the other hand, is about following steps to arrive at an answer and accuracy.

When considering how students will learn more advanced mathematics concepts, it is important to consider how they will engage with the problems presented to them in class and how those problems will contribute either to their greater conceptual understanding or greater procedural fluency. For example, consider these two math questions and ask yourself: What knowledge is needed to solve each problem?

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High school students’ math scores are still lagging, STAAR results show

Algebra scores have not recovered since the pandemic, raising worries about students’ readiness for STEM-related jobs.

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Partial scores from the state’s standardized test released Friday show high school students are still struggling with algebra, once again raising concerns about young Texans’ readiness to enter high-paying careers in STEM-related fields.

The State of Texas Assessments of Academic Readiness end-of-course tests evaluate high-schoolers in five subjects: Algebra I, Biology, English I, English II and U.S. History. The exams gauge whether students’ grasp of a subject is appropriate for their grade level and if they need additional help to catch up.

The percentage of students who took the test this spring and met grade level for Algebra I was 45%, the same as last year. Since the pandemic, students’ academic performance in the subject has remained mostly unchanged. The latest results are still 17 percentage points below students' scores in spring 2019.

“The data is clear, Texas students continue to struggle with math recovery,” said Gabe Grantham, policy advisor at public policy think tank Texas 2036. “We run the risk of leaving students ill-equipped to enter the future workforce without the basic math skills needed to be successful.”

Education policy analysts closely observe Algebra I results because a wealth of research links the subject to students’ future success in their careers after high school. Kate Greer, the managing director of policy at The Commit Partnership, said STAAR test scores allow researchers to delve into districts that performed better than the state average and form concrete policy proposals to help improve math scores.

“We are still underperforming compared to where we were pre-pandemic, so it is incumbent on us as a state to collectively focus on what we know works,” Greer said. “The value of assessments is it can focus adult behavior, shine a flashlight on opportunities where we can improve more and highlight best practices when we’re seeing impressive growth.”

However, in the past few years, high schoolers have consistently scored better on their English tests since the pandemic. Emergent bilingual students, or students who are learning English as a second language, have steadily performed better on the English I and II tests. The percentage of emergent bilingual students who met grade level went from 12% in 2019 to 30% this spring.

Test results for U.S. history and biology still lag behind pre-pandemic levels, but they are much closer to catching up than in math.

Across all five subjects, low-income students graded lower than students who were not economically disadvantaged. For example, 35% of low-income students met grade level in Algebra I, compared to 61% of all other students.<br> In a push to improve math skills, the Texas Legislature last year passed Senate Bill 2124 , which automatically puts middle schoolers into a higher math class if they do well in previous courses. Lt. Gov. Dan Patrick included reading and math readiness on his list of interim priorities , suggesting that lawmakers will revisit the issue during next year’s legislative session.

Disclosure: Commit Partnership and Texas 2036 have been financial supporters of The Texas Tribune, a nonprofit, nonpartisan news organization that is funded in part by donations from members, foundations and corporate sponsors. Financial supporters play no role in the Tribune's journalism. Find a complete list of them here .

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On Tech: A.I.

How teachers and students feel about a.i..

As the school year begins, their thinking has evolved.

By Natasha Singer

I sat in on a ChatGPT workshop this month for teachers at Walla Walla High School, about 270 miles southeast of Seattle. As a reporter who covers education technology, I have closely followed how generative artificial intelligence has upended education .

Now that the first full school year of the A.I. chatbot era is beginning, I wanted to ask administrators and educators how their thinking had evolved since last spring. Walla Walla, a district that serves some 5,500 students, seemed like a timely location to begin the conversation. After blocking student access to ChatGPT in February, Walla Walla administrators told me they unblocked it last month and are now embracing A.I. tools.

So I jumped at the chance to learn more about how teachers there are planning to use chatbots with their students this academic year. You can read more in my story today about how school districts across the country are repealing their ChatGPT bans.

My colleague Kevin Roose has some great suggestions in his column today on how schools can survive, “and maybe even thrive,” with A.I. tools this fall. Step one, Kevin says: “Assume all students are going to use the technology.”

We recently asked educators, professors, and high school and college students to tell us about their experiences using A.I. chatbots for teaching and learning. We got a massive response — more than 350 submissions. Here are some highlights:

Teaching with A.I.

I love A.I. chatbots! I use them to make variations on quiz questions. I have them check my instructions for clarity. I have them brainstorm activity and assignment ideas. I’ve tried using them to evaluate student essays, but it isn’t great at that.

— Katy Pearce, associate professor, University of Washington

Before they even use ChatGPT, I help students discern what is worth knowing, figuring out how to look it up, and what information or research is worth “outsourcing” to A.I. I also teach students how to think critically about the data collected from the chatbot — what might be missing, what can be improved and how they can expand the “conversation” to get richer feedback.

— Nicole Haddad, Southern Methodist University

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Most US students are recovering from pandemic-era setbacks, but millions are making up little ground

Fifth grade students attend a math lesson with teacher Jana Lamontagne, right, during class at Mount Vernon Community School, in Alexandria, Va., Wednesday, May 1, 2024. (AP Photo/Jacquelyn Martin)

A fifth grade student explains a math answer to his classmate during a math lesson at Mount Vernon Community School, in Alexandria, Va., Wednesday, May 1, 2024. (AP Photo/Jacquelyn Martin)

Fifth grade students attend a math lesson with teacher Alex Ventresca, right, during class at Mount Vernon Community School, in Alexandria, Va., Wednesday, May 1, 2024. (AP Photo/Jacquelyn Martin)

A fifth grade student attends a math lesson during class at Mount Vernon Community School, in Alexandria, Va., Wednesday, May 1, 2024. (AP Photo/Jacquelyn Martin)

Fifth grade students work on computers during a math class at Mount Vernon Community School, in Alexandria, Va., Wednesday, May 1, 2024. (AP Photo/Jacquelyn Martin)

Fifth grade teacher Jana Lamontagne, center, teaches a math lesson during class at Mount Vernon Community School, in Alexandria, Va., Wednesday, May 1, 2024. (AP Photo/Jacquelyn Martin)

Osbell, 9, works on the Ignite program with a live tutor, during a third grade English language arts class at Mount Vernon Community School, in Alexandria, Va., Wednesday, May 1, 2024. (AP Photo/Jacquelyn Martin)

Bridget, 9, attends a third grade English language arts class at Mount Vernon Community School, in Alexandria, Va., Wednesday, May 1, 2024. (AP Photo/Jacquelyn Martin)

Jaelene, 9, works on a computer during a third grade English language arts class at Mount Vernon Community School, in Alexandria, Va., Wednesday, May 1, 2024. (AP Photo/Jacquelyn Martin)

Students work on a writing assignment during a third grade English language arts class at Mount Vernon Community School, in Alexandria, Va., Wednesday, May 1, 2024. (AP Photo/Jacquelyn Martin)

Wells Preparatory Elementary School principal Vincent Izuegbu talks about the school’s mission to overcome the effects of remote pandemic learning Friday, March 8, 2024, in Chicago. In the classroom, the school put a sharper focus on collaboration. Along with academic setbacks, students came back from school closures with lower maturity levels, said Izuegbu. “We do not let 10 minutes go by without a teacher giving students the opportunity to engage with the subject,” Izuegbu said. “That’s very, very important in terms of the growth that we’ve seen.” (AP Photo/Charles Rex Arbogast)

The Wells Preparatory Elementary School Student Creed hangs on the wall behind principal Vincent Izuegbu on Friday, March 8, 2024, in Chicago. At Wells Preparatory Elementary on the South Side, just 3% of students met state reading standards in 2021. Last year, 30% hit the mark. Federal relief allowed the school to hire an interventionist for the first time, and teachers get paid to team up on recovery outside working hours. (AP Photo/Charles Rex Arbogast)

Wells Preparatory Elementary School student Olorunkemi Atoyebi, responds to a question during an interview with The Associated Press on Friday, March 8, 2024, in Chicago. Atoyebi was an A student before the pandemic, but after spending fifth grade behind a computer screen, she fell behind. During remote learning, she was nervous about stopping class to ask questions. Before long, math lessons stopped making sense. When she returned to in classroom learning other students worked in groups, her math teacher helped her one-on-one. Atoyebi learned a rhyming song to help memorize multiplication tables. Over time, it began to click. “They made me feel more confident in everything,” said Atoyebi, now 14. “My confidence started going up, my grades started going up, my scores started going up. Everything has felt like I understand it better."(AP Photo/Charles Rex Arbogast)

Students at the Wells Preparatory Elementary School make their way to the cafeteria past reminders of the education and subjects they are receiving on Friday, March 8, 2024, in Chicago. In Chicago Public Schools, the average reading score went up by the equivalent of 70% of a grade level from 2022 to 2023. Math gains were less dramatic, with students still behind almost half a grade level compared to 2019. Chicago officials credit the improvement to changes made possible with nearly $3 billion in federal relief. (AP Photo/Charles Rex Arbogast)

The desktop of a student at the Wells Preparatory Elementary School reflects the literature they are studying in Charlotte Owens’ fifth grade class on Friday, March 8, 2024, in Chicago. In Chicago Public Schools, the average reading score went up by the equivalent of 70% of a grade level from 2022 to 2023. Math gains were less dramatic, with students still behind almost half a grade level compared to 2019. Chicago officials credit the improvement to changes made possible with nearly $3 billion in federal relief. (AP Photo/Charles Rex Arbogast)

Wells Preparatory Elementary School teacher Charlotte Owens, left, works with her fifth grade students during the literature segment of their day, Friday, March 8, 2024, in Chicago. In Chicago Public Schools, the average reading score went up by the equivalent of 70% of a grade level from 2022 to 2023. Math gains were less dramatic, with students still behind almost half a grade level compared to 2019. Chicago officials credit the improvement to changes made possible with nearly $3 billion in federal relief. (AP Photo/Charles Rex Arbogast)

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ALEXANDRIA, Va. (AP) — On one side of the classroom, students circled teacher Maria Fletcher and practiced vowel sounds. In another corner, children read together from a book. Scattered elsewhere, students sat at laptop computers and got reading help from online tutors.

For the third graders at Mount Vernon Community School in Virginia, it was an ordinary school day. But educators were racing to get students learning more, faster, and to overcome setbacks that have persisted since schools closed for the COVID-19 pandemic four years ago.

America’s schools have started to make progress toward getting students back on track. But improvement has been slow and uneven across geography and economic status, with millions of students — often those from marginalized groups — making up little or no ground.

Nationally, students made up one-third of their pandemic losses in math during the past school year and one-quarter of the losses in reading, according to the Education Recovery Scorecard , an analysis of state and national test scores by researchers at Harvard and Stanford.

But in nine states, including Virginia, reading scores continued to fall during the 2022-23 school year after previous decreases during the pandemic.

Clouding the recovery is a looming financial crisis. States have used some money from the historic $190 billion in federal pandemic relief to help students catch up , but that money runs out later this year.

“The recovery is not finished, and it won’t be finished without state action,” said Thomas Kane, a Harvard economist behind the scorecard. “States need to start planning for what they’re going to do when the federal money runs out in September. And I think few states have actually started that discussion.”

Virginia lawmakers approved an extra $418 million last year to accelerate recovery. Massachusetts officials set aside $3.2 million to provide math tutoring for fourth and eighth grade students who are behind grade level, along with $8 million for literacy tutoring.

But among other states with lagging progress, few said they were changing their strategies or spending more to speed up improvement.

Virginia hired online tutoring companies and gave schools a “playbook” showing how to build effective tutoring programs. Lisa Coons, Virginia’s superintendent of public instruction, said last year’s state test scores were a wake-up call.

“We weren’t recovering as fast as we needed,” Coons said in an interview.

U.S. Education Secretary Miguel Cardona has called for states to continue funding extra academic help for students as the federal money expires.

“We just can’t stop now,” he said at a May 30 conference for education journalists. “The states need to recognize these interventions work. Funding public education does make a difference.”

In Virginia, the Alexandria district received $2.3 million in additional state money to expand tutoring.

At Mount Vernon, where classes are taught in English and Spanish, students are divided into groups and rotate through stations customized to their skill level. Those who need the most help get online tutoring. In Fletcher’s classroom, a handful of students wore headsets and worked with tutors through Ignite Learning, one of the companies hired by the state.

With tutors in high demand, the online option has been a big help, Mount Vernon principal Jennifer Hamilton said.

“That’s something that we just could not provide here,” she said.

Ana Marisela Ventura Moreno said her 9-year-old daughter, Sabrina, benefited significantly from extra reading help last year during second grade, but she’s still catching up.

“She needs to get better. She’s not at the level she should be,” the mother said in Spanish. She noted the school did not offer the tutoring help this year, but she did not know why.

Alexandria education officials say students scoring below proficient or close to that cutoff receive high-intensity tutoring help and they have to prioritize students with the greatest needs. Alexandria trailed the state average on math and reading exams in 2023, but it’s slowly improving.

More worrying to officials are the gaps: Among poorer students at Mount Vernon, just 24% scored proficient in math and 28% hit the mark in reading. That’s far lower than the rates among wealthier students, and the divide is growing wider.

Failing to get students back on track could have serious consequences. The researchers at Harvard and Stanford found communities with higher test scores have higher incomes and lower rates of arrest and incarceration. If pandemic setbacks become permanent, it could follow students for life.

The Education Recovery Scorecard tracks about 30 states, all of which made at least some improvement in math from 2022 to 2023. The states whose reading scores fell in that span, in addition to Virginia, were Nevada, California, South Dakota, Wyoming, Indiana, Oklahoma, Connecticut and Washington.

Only a few states have rebounded to pre-pandemic testing levels. Alabama was the only state where math achievement increased past 2019 levels, while Illinois, Mississippi and Louisiana accomplished that in reading.

In Chicago Public Schools, the average reading score went up by the equivalent of 70% of a grade level from 2022 to 2023. Math gains were less dramatic, with students still behind almost half a grade level compared with 2019. Chicago officials credit the improvement to changes made possible with nearly $3 billion in federal relief.

The district trained hundreds of Chicago residents to work as tutors. Every school building got an interventionist, an educator who focuses on helping struggling students.

The district also used federal money for home visits and expanded arts education in an effort to re-engage students.

“Academic recovery in isolation, just through ‘drill and kill,’ either tutoring or interventions, is not effective,” said Bogdana Chkoumbova, the district’s chief education officer. “Students need to feel engaged.”

At Wells Preparatory Elementary on the city’s South Side, just 3% of students met state reading standards in 2021. Last year, 30% hit the mark. Federal relief allowed the school to hire an interventionist for the first time, and teachers get paid to team up on recovery outside working hours.

In the classroom, the school put a sharper focus on collaboration. Along with academic setbacks, students came back from school closures with lower maturity levels, principal Vincent Izuegbu said. By building lessons around discussion, officials found students took more interest in learning.

“We do not let 10 minutes go by without a teacher giving students the opportunity to engage with the subject,” Izuegbu said. “That’s very, very important in terms of the growth that we’ve seen.”

Olorunkemi Atoyebi was an A student before the pandemic, but after spending fifth grade learning at home, she fell behind. During remote learning, she was nervous about stopping class to ask questions. Before long, math lessons stopped making sense.

When she returned to school, she struggled with multiplication and terms such as “dividend” and “divisor” confused her.

While other students worked in groups, her math teacher took her aside for individual help. Atoyebi learned a rhyming song to help memorize multiplication tables. Over time, it began to click.

“They made me feel more confident in everything,” said Atoyebi, now 14. “My grades started going up. My scores started going up. Everything has felt like I understand it better.”

Associated Press writers Michael Melia in Hartford, Connecticut, and Chrissie Thompson in Las Vegas contributed to this report.

The Associated Press’ education coverage receives financial support from multiple private foundations. AP is solely responsible for all content. Find AP’s standards for working with philanthropies, a list of supporters and funded coverage areas at AP.org .

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Mathematics identity instrument development for fifth through twelfth grade students

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• Juliana Utley

The influence of students’ prior numeracy achievement on later numeracy achievement as a function of gender and year levels

Inquiry-based mathematics education and attitudes towards mathematics: tracking profiles for teaching

• Inés M. Gómez-Chacón
• Adrián Bacelo
• Andrés Palacios

Influences on student decisions to enrol in higher-level mathematics courses

• Gregory Hine
• Chris Forlin
• Paola Chivers

Exploring spherical units as perceptual clues in enumerating 3D arrays

• Edgar Alstad
• Maren Berre
• Per Nilsson

Children engaging with partitive quotient tasks: elucidating qualitative heterogeneity within the Image Having layer of the Pirie–Kieren model

• Lois George
• Chronoula Voutsina

Cognitive tuning in the STEM classroom: communication processes supporting children’s changing conceptions about data

• Lyn English
• Katie Makar

Generalization: strategies and representations used by sixth to eighth graders in a functional context

• M. C. Cañadas

Approaches to investigating young children’s mathematical activity

• Jennifer Way

Unpacking mathematics preservice teachers’ conceptions of equity

• Rebecca McGraw
• Anthony Fernandes
• Becca Jarnutowski

How current perspectives on algorithmic thinking can be applied to students’ engagement in algorithmatizing tasks

• Timothy H. Lehmann

The relationship between prospective teachers’ mathematics knowledge for teaching and their ability to notice student thinking

• Sandy M. Spitzer
• Christine M. Phelps-Gregory

Implementing a pedagogical cycle to support data modelling and statistical reasoning in years 1 and 2 through the Interdisciplinary Mathematics and Science (IMS) project

• Joanne Mulligan
• Russell Tytler
• Melinda Kirk

The stability of mathematics students’ beliefs about working with CAS

• Scott Cameron
• Vicki Steinle

How tools mediate elementary students’ algebraic reasoning about evens and odds

• Susanne Strachota
• Ana Stephens

Shifts in students’ predictive reasoning from data tables in years 3 and 4

• Gabrielle Oslington
• Penny Van Bergen

Assessing students’ understanding of time concepts in Years 3 and 4: insights from the development and use of a one-to-one task-based interview

• Margaret Thomas
• Doug M. Clarke
• Philip C. Clarkson

Interpreting young children’s multiplicative strategies through their drawn representations

• Katherin Cartwright

Beyond counting accurately: a longitudinal study of preschoolers’ emerging understandings of the structure of the number sequence

• Brandon G. McMillan
• Nicholas C. Johnson
• Jennifer Ricketts Schexnayder

Examining fifth graders’ conceptual understanding of numbers and operations using an online three-tier test

• Iwan A. J. Sianturi
• Zaleha Ismail
• Der-Ching Yang

The development of high school students’ statistical literacy across grade level

• Achmad Badrun Kurnia
• Sitti Maesuri Patahuddin

Correction to: Teachers as designers of instructional tasks

• Berinderjeet Kaur
• Yew Hoong Leong
• Catherine Attard

The role of movement in young children’s spatial experiences: a review of early childhood mathematics education research

• Catherine McCluskey
• Anna Kilderry
• Virginia Kinnear

“If you’re a dude, you’re a chick, whatever the hell in between, you need to know about maths”: the Australian and Canadian general public’s views of gender and mathematics

• Jennifer Hall
• Cinzia Di Placido

The university mathematics lecture: to record, or not to record, that is the question

• Maria Meehan
• Emma Howard

Exploring children’s negotiation of meanings about “D” in 2D and 3D shapes in a year 5/6 New Zealand primary classroom

• Shweta Sharma

Methodologies to reveal young Australian Indigenous students’ mathematical proficiency

• Bronwyn Reid O’Connor

Conceptualising critical mathematical thinking in young students

• Chrissy Monteleone
• Jodie Miller
• Elizabeth Warren

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CSULB PREM Students Travel During Summer 2024

The PREM summer has started with CSULB undergrad Josh Luna and CSULB M.S. student Tyler Hadsell attending the High Energy X-ray Techniques (HEXT) workshop at the Cornell High Energy Synchrotron Source (CHESS) at Cornell University in Ithaca, NY. Josh and Tyler are members of Prof. Ojeda-Aristizabal's group at CSULB.

Josh then went straight to Ohio State to start his 10-week research experience for undergraduates (REU) at the Center for Emergent Materials (MRSEC) hosted by PREM affiliated OSU faculty Prof. Jeannie Lau. Josh met his CSULB classmate, undergraduate Movindu Dissanayake, who is also spending a couple weeks learning various sample making and characterization techniques to bring back to CSULB.

CSULB M.S. student Ivan Pelayo, Prof. Ojeda-Aristizabal, and Prof. Michael Peterson were visiting the Center for Emergent Materials.