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Calculus education research on integration is coalescing around the theme that teaching integration based on quantitative reasoning is crucial for robust understanding of and usage of integrals. This paper contributes to this International Journal for Research in Undergraduate Education special issue on the teaching and learning of definite integrals by reviewing the research literature on quantitatively-based approaches to integration, in order to pull together the ideas that are spread across many papers in many outlets. We note that the literature in this area has largely developed along two distinct paradigms, which we call "adding up pieces and accumulation from rate." While they are both based in quantitative reasoning, there are critical differences between them that have important ramifications for teaching and learning integration. We use these two paradigms to organize our review and we use the literature to discuss the meanings, formalizations, foci, reasoning, and modeling in these approaches.

Recent research on integration has shown the importance of quantities-based meanings for integrals. However, this research body is still in need of detailed empirical accounts of "how" students develop such understandings across an entire unit on integration. This paper contributes by providing one such account, based on a quantities-based orientation called "adding up pieces" (AUP). Our study examines how three separate pairs of students learned definite integrals and integral functions within interview settings over four consecutive interview-lessons, meant to correspond to four consecutive in-class sessions. The first interview developed the partition, target quantity, and sum structure for definite integral, which were solidified in the second interview. The third and fourth interviews extended this AUP structure to a variable upper bound, output, and function structure for integral functions in preparation for the Fundamental Theorem of Calculus (FTC).

In this article we introduce a type of teaching scholarship called Lesson Analysis (LA) that helps build instructional knowledge, and an associated genre of writing called Lesson Analysis Manuscript (LAM) that shares this instructional knowledge. We briefly describe the problem LA is intended to solve, as well as key features of a LAM. In LA, instructors develop lessons that overcome specific instructional challenges. LAMs are lesson-specific manuscripts that help others understand details of the lesson that addressed the instructional challenge, and include such information as the instructional context, the rationale behind the instructional decisions, typical student thinking, and guidance on carrying out discussions and implementing tasks. We have illustrated this genre by including an entire example LAM as supplemental material located online. The example LAM focuses on helping students understand the connection between dot products and the connection of dot products to measures of similarity between vectors.

Virtual reality (VR) research in mathematics education has centred largely on geometry content. This paper contributes by describing VR modules developed for another area that heavily involves three-dimensional objects: multivariable calculus. This paper also contributes by describing an empirical study on students' experiences inside the VR modules through student (conceptual) noticing. One finding was that colourful objects had moderate associated conceptual noticing, but that accompanying animation drastically improved the conceptual noticing. Symbolic and textual elements were conceptually noticed much less. While narration was meant to guide the students' noticing, it often did not produce the conceptual noticing intended, though animation again was a key factor. The conceptual noticing was also found to be connected to the students' emerging understandings of the mathematical ideas discussed in the modules. We end by discussing implications for: (1) designing VR modules generally; and for (2) learning specific mathematical content within VR.

The calculus concepts of concavity and inflection points are critical for a complete understanding of quantities' behavior, making them important topics of research for those interested in the intersections of STEM disciplines. This study seeks to provide insight into this area by reporting on trends in students' concept projections of these concepts in a range of real-world contexts, including temperature, economics, human height, and the universe's expansion. While the students mostly thought of concavity and inflection points in pure mathematics as shapes associated with graphs, the projections with the real-world contexts were based much more on changing rates of change. Pedagogically important student confusions were also evident in their attempts at applying concavity and inflection points to the real-world contexts, and this paper uses the concept projection perspective to identify possible sources of these confusions.

The association of thyroid function with essential and non-essential amino acids is understudied, despite their common metabolic roles. Thus, our aim was to evaluate the association of thyroid function with the levels of branched-chain amino acids (BCAAs—leucine, isoleucine, and valine) and of alanine in the general population. We utilized data from the São Paulo research center of ELSA-Brasil, a longitudinal population-based cohort study. Thyroid parameters included thyroid stimulating hormone (TSH), free T4 and free T3 levels, and the FT4:FT3 ratio. BCAAs and alanine were analyzed on a fully automated NMR platform. The current analysis included euthyroid participants and participants with subclinical hyperthyroidism and hypothyroidism. We used Pearson's coefficient to quantify the correlation between thyroid-related parameters and amino acids. Linear regression models were performed to analyze whether thyroid parameters were associated with BCAAs and alanine levels. We included 4098 participants (51.3 ± 9.0 years old, 51.5% women) in this study. In the most adjusted model, higher levels of TSH were associated with higher levels of alanine, FT4 levels were inversely associated with isoleucine levels, FT3 levels were statistically significant and positively associated with valine and leucine, and the T3:T4 ratio was positively associated with all amino acids. We observed that subclinical hypothyroidism was positively associated with isoleucine and alanine levels in all models, even after full adjustment. Our findings highlight the association of subclinical hypothyroidism and thyroid-related parameters (including TSH, free T4, free T3, and FT4:FT3 ratio) with BCAAs and alanine. Further studies are needed to explore the mechanisms underlying this association. These insights contribute to our understanding of the influence of thyroid-related parameters on BCAA and alanine metabolism. [ABSTRACT FROM AUTHOR]

BackgroundThe transition from no coronary artery calcium (CAC) to detectable CAC is important, as even mild CAC is associated with increased cardiovascular events. We sought to characterize the anatomic distribution and burden of newly detectable CAC over 10-year follow-up.MethodsWe evaluated 3112 participants (mean age, 58 years; 64% female) with baseline CAC = 0 from the Multi-Ethnic Study of Atherosclerosis. Participants underwent repeat CAC testing at different time intervals (between 2-10 years after baseline) per the Multi-Ethnic Study of Atherosclerosis protocol. Among participants who developed CAC on a follow-up scan, we used logistic regression and marginal probability modeling to describe the coronary distribution and burden of new CAC by age, sex, and race after adjustment for cardiovascular risk factors and time to detection.ResultsA total of 1125 participants developed detectable CAC during follow-up with a mean time to detection of 6.1 ± 3 years. New CAC was most commonly isolated to 1 vessel (72% of participants), with the left anterior descending artery (44% of total) most commonly affected followed by the right coronary (12%), left circumflex (10%), and left main (6%). These patterns were similar across age, sex, and race. In multivariate models, residual predictors of multivessel CAC (28% of total) included male sex, African American or Hispanic race, hypertension, obesity, and diabetes. At the first detection of CAC >0, burden was usually low with median Agatston CAC score of 7.1 and 100.ConclusionNew-onset CAC most commonly involves just 1 vessel, occurs in the left anterior descending artery, and has low CAC burden. New CAC can be detected at an early stage when aggressive preventive strategies may provide benefit.

The derivative framework described by Zandieh (2000) has been an important tool in calculus education research, and many researchers have revisited the framework to elaborate on it, extend it, or refine certain aspects of it. We continue this process by using the framework to put forward a suggestion on what might constitute a "target understanding" of the derivative concept for first-semester calculus teaching and learning. We draw on empirical results from psychological research into analogical reasoning in order to create a guideline for such a "target understanding," which includes a process-object comprehension of the ratio-limit-function layers in three distinct graphical-symbolic-verbal/rate-physical contexts. Our recommended target understanding creates a testable hypothesis, in that obtaining this target may allow students to apply their derivative understanding to additional contexts, as needed. We provide the results of an empirical study regarding this hypothesis and show that they support our recommendation.

Many mathematical concepts may have prototypical images associated with them. While prototypes can be beneficial for efficient thinking or reasoning, they may also have self-attributes that may impact reasoning about the concept. It is essential that mathematics educators understand these prototype images in order to fully recognize their benefits and limitations. In this paper, I examine prototypes in a context in which they seem to play an important role: graphical representations of the calculus concept of the definite integral. I use student data to empirically describe the makeup of the definite integral prototype image, and I report on the frequency of its appearance among student, instructor, and textbook image data. I end by discussing the possible benefits and drawbacks of this particular prototype, as well as what the results of this study may say about prototypes more generally.