Transitioning to
Online Introductory Math

Defining Quality and Efficiency in Online Math Instruction

Defining Quality

The variety, diversity, and incongruity among available course evaluation instruments speak to the disagreement in higher education about elements fundamental to course quality.[1] Given that lack of consensus, it was necessary to assemble a series of quality dimensions in a novel way with a specific focus on introductory math instruction in online instructional modalities.

Using existing frameworks and input from experienced math faculty from TPSE Math, the following constellation of quality dimensions and indicators were identified as being the foundation of quality instruction for any introductory math courses offered online. These dimensions include equity, quantitative reasoning, alignment of learning outcomes, and intellectual challenge.


Not all students come to higher education with the same academic and life experiences, making it incumbent on their colleges and universities, as well as individual instructors to design learning experiences that minimize or mitigate these differences. When learning moves to an online environment, these considerations are thrown into higher relief; the inequities that are part of higher education more broadly continue to influence education online. The rapid shift to online learning during the COVID-19 pandemic highlighted many of the structural and infrastructural inequalities that may remain hidden during in-person instruction, including access to high speed internet and the hardware to use it and access to quiet places to learn and study online.[2]

These inequities constitute one of the many dimensions to consider in the development of equitable online learning environments. Every Learner Everywhere (ELE), a consortium of higher education and technology organizations, also suggest attending to the following dimensions:[3]

  • Academic: students’ preparation for online learning
  • Pedagogical: course design and quality of instruction
  • Psychological: students’ perceptions about the course and instructor
  • Social: students’ connectedness with peers

There are a variety of frameworks, including validation theory, culturally responsive teaching, social presence, and universal design in learning, that can ameliorate the challenges that can arise within each of these dimensions.[4] When embodied in the learning environment, they can increase equity in the learning environment.

However asymptotic, making progress towards more equitable learning environments is possible when instructional and institutional choices are aligned and made from a place of equity-mindedness…

There is not a one-to-one relationship between any given aspect of inequality and an instructional solution. However asymptotic, making progress towards more equitable learning environments is possible when instructional and institutional choices are aligned and made from a place of equity-mindedness, meaning that the academic and administrative decision-makers “reflect on their own and their colleagues’ role in and responsibility for student success” rather than attributing inequitable outcomes to students’ unpreparedness.[5]

Indicators of Equity

Issues of equity, or more pointedly inequity, abound in all aspects of higher education. It is incumbent on instructors and institutions moving introductory math courses online to acutely attend to how such a change may impact different student communities differently and working diligently to mitigate those differences.[6] Presumed in this imperative is that student success outcomes, such as the rates at which students are earning a D or F, or Withdrawing (DFW rates), are actively being monitored and acted upon when differences emerge.

DFW rates, along with course grades, persistence rates, and enrollments in subsequent courses, are all lagging indicators, meaning that these metrics reveal how equitable the completed course may have been.

DFW rates, along with course grades, persistence rates, and enrollments in subsequent courses, are all lagging indicators, meaning that these metrics reveal how equitable the completed course may have been. Formative assessment elements should be built into the online course and monitored in real-time to provide the opportunity to intervene individually or comprehensively before the end of the course. Such elements could include frequent, low-stakes assessments as well as tracking usage and consumption of online course resources. These early indicators can alert instructors to the potential for inequitable outcomes before the end of the course when there is still the opportunity to take corrective action.

Conceptual Understanding & Quantitative Reasoning

The goal of any introductory math course must be, to some degree, the development of conceptual understanding and quantitative reasoning. Conceptual understanding is “the ability to explain why something happens mathematically, using logical reasoning as opposed to empirical evidence”[7] while quantitative reasoning is the constellation of skills, competencies, and habits of mind that add up to the “ability to reason and solve quantitative problems from a wide array of authentic contexts and everyday life situations.”[8] Although higher education organizations and associations posit somewhat different specifications of which knowledge, skills, and abilities constitute conceptual understanding and quantitative reasoning, they all involve the recognition of, communication about, and application of numeric data to some specific purpose.[9]

Centering conceptual understanding and quantitative reasoning in introductory math courses, as opposed to only procedural fluency, is essential because of the foundational role these courses play in so many academic disciplines beyond majoring in math.[10] The numeracy embodied in conceptual understanding and quantitative reasoning is essential throughout a traditional undergraduate curriculum as well as in supporting success after college. This emphasis is in contrast to how many introductory math courses are delivered, with a focus on mechanistic problem solving, time-based mental calculations, or rote memorization of formulas and procedures.[11]

Indicators of Quantitative Reasoning
Evidence of an emphasis on quantitative reasoning can be found in the activities, assignments, and assessments used in the course. These course elements should focus on the real-world application and utility of the underlying quantitative skills, not simply mechanistic or algorithmic problem solving. The complexity and sophistication of the tasks asked of students can be an indicator of this emphasis, such as more narrative activities that call on students to articulate their problem solving process.

Institutions should develop the student data infrastructure to track student progress in the courses to which introductory math gateways provide access.

As a general education requirement and prerequisite for courses in other disciplines, introductory math courses contribute directly to the success of students in their later studies, making that success an essential metric. Institutions should develop the student data infrastructure to track student progress in the courses to which introductory math gateways provide access. Tracking back student success metrics (grades, persistence, etc.) and institutional measures (DFW rates) can demonstrate which introductory math courses, pedagogies, and/or instructors are best preparing students for the later application of their conceptual understanding of mathematics in diverse academic settings.

Alignment of Learning Goals and Outcomes

Well-articulated, appropriately calibrated, and clearly communicated learning goals and outcomes are the foundation of most effective course design initiatives.[12] For introductory math courses, these principles are further enhanced when the course in question aligns with other aspects of students’ mathematical instruction. Students’ prior knowledge has an outsized influence on their learning experiences, so it is essential that the learning outcomes for introductory math courses are a fluid connection between the mathematics preparation that students already have and the mathematics they will need in the future.[13]

Knowing what students know coming in can help instructors and course designers ensure that ‘baton’ of student learning is successfully handed off during the transition from high school to college. Ensuring this smooth transition into introductory math courses keeps students making progress in the academic career without unnecessarily predicating course activities on mistaken assumptions about what students know and can do. Formative assessments early in introductory courses can help students and instructors better understand what foundational content is already well understood and what content should be reinforced.

It is also important to note that introductory math courses do not exist as discrete entities, separate from the remainder of the students’ education. The entire undergraduate experience, or at least the general education portion of which introductory math is a part, should exist within a coherent framework of learning goals and outcomes. The specifics of this academic super-structure will look different for each institution, but introductory math courses should advance students toward some cohesive learning goal that aligns with other requirements and transcends any individual course. These alignments can be made manifest in programs like Math Pathways, wherein specific introductory math courses are tailored to the mathematical knowledge and skills required for particular paths through the curriculum.[14]

Indicators of Alignment
There are two places in the curriculum to look for evidence that the learning goals and outcomes for an online introductory math course are well-aligned to the general education or institutional learning goals and outcomes. Each of these necessary, but not sufficient, components must clearly describe the knowledge, skills, and abilities students should be able to demonstrate at the end of the learning experience. Ultimately it is the interlocking relationship between the two sets of goals and outcomes that ensure proper alignment between any required introductory math course and the ‘bigger picture’ of student learning.

One way to ensure the appropriate alignment of courses is to engage in curriculum mapping, or the process of visualizing the connections between elements of the curriculum and the outcomes those elements support. A change in the curriculum as impactful as moving introductory math online is an important opportunity to establish – or reestablish – the explicit and direct connection between the introductory math courses and the learning required for success in later courses.

Intellectual Challenge

Related to the theme of purposeful learning that runs through the quantitative reasoning dimension of quality, providing introductory math students with intellectual challenge is an essential aspect of quality in any learning experience, but especially introductory math courses. Embodying an intellectual challenge is not about calibrating the difficulty of course assignments or exam questions. It is about both designing course tasks “just past students’ current achievement level, but well within their reach” as well as connecting those activities to “authentic, real-world tasks relevant to students’ academic life.”[15] This approach builds on Sanford’s classic model of challenge and support, in which the learning experiences provide the necessary scaffolding and content for students to successfully and safely take intellectual risks thereby building their own understanding.[16]

An important part of providing appropriate intellectual challenge is maintaining student motivation for the learning. The DNR (duality, necessity, and repeated reasoning) framework for math instruction posits that “[f]or students to learn what we intend to teach them, they must have a need for it, where ‘need’ means intellectual need, not social or economic need.”[17] This approach suggests several different strategies for cultivating intellectual need, in which questions raised in the students mind naturally compel them to pursue the answer in order to satisfy that need.

Indicators of Intellectual Challenge
One of the key elements of challenge and support is the reciprocal relationship between the two. For every challenge presented to students, there should be appropriate support built into the course and the online learning environment. One effective way to provide such support is to ensure timely feedback on student work, a task made easier and faster with adaptive learning technologies. Prompt responses to student work keep students from moving too far forward armed with misunderstandings or insufficient skills.

In order to maintain intellectual challenge in a course, the majority of the activities and assignments should require students to engage in more cognitively complex tasks, as defined by Bloom’s Taxonomy of Learning.[18] These higher level categories are creating, evaluating, and analyzing, as opposed to remembering or applying course concepts. An indicator that a course will provide motivating and meaningful intellectual challenge is the specific nature of the experiences students will have and the degree to which those experiences are cognitively complex.

Quality in Online Learning Environments

While online learning environments are certainly different from face-to-face, on ground classrooms, the differences in modalities do not change the fundamental elements of quality for introductory math courses. The basic aim of involving students in the experiences that will best prepare them to achieve the course’s learning outcomes does not change, regardless of the teaching modality. It is only the nature of those learning experiences that may be different, and in many cases, should be. Simply replicating the traditional classroom experience in an online environment is not sufficient.

Simply replicating the traditional classroom experience in an online environment is not sufficient.

There are innumerable ‘best practices,’ how-to guides, and assorted frameworks for the design and delivery of effective online learning.[19] One of the most widely adopted sets of standards are those from Quality Matters (QM), a non-profit organization supporting online learning in multiple educational sectors. In addition to a general purpose set of standards for online courses, QM developed an Emergency Remote Instruction Checklist for higher education to support instructors during the sudden shift to online learning resulting from the COVID-19 pandemic.[20] While that emergency modality shift is different from a more deliberate and thought-out move to online learning, the general guidance on transitions remains sound.

QM suggests starting by orienting students to the new learning environment and teaching them first how to learn online before attending to course content. The suggestions also remind instructors to attend to social presence, or the relational, human component of the teaching and learning process. In traditional classrooms, social presence usually results from physical proximity, but must be specifically cultivated in an online learning environment. Students in online classes want to develop a rapport with their instructors, to feel that a person is teaching them and not a screen.[21]

It is also essential to orient students to the online learning environment, especially if they have limited experience learning in a virtual classroom. Educational experts note that prior knowledge can help or hinder learning, a truism that applies equally to content as it does to teaching modality.[22] Most faculty assume, rightly or wrongly, that students come into their traditional classrooms knowing how to successfully navigate that environment and use the artifacts of learning, such as the syllabus, office hours, and the learning management system (LMS). Following the transition to online learning, faculty must forgo all assumptions about what students may already know about the learning environment and teach them how to access learning materials, the norms and venues for communication, and how to self-assess their own learning.

The quality of the online “classroom” and students’ experience of online learning is predicated on the successful implementation of evidence-based course design and teaching practices…

The quality of the online “classroom” and students’ experience of online learning is predicated on the successful implementation of evidence-based course design and teaching practices by those designing and teaching the course. As online learning is a new modality for many introductory math instructors, institutions undertaking a transition to online learning for these courses should provide substantial faculty development and course design support. Institutional faculty development staff and resource centers should be leveraged to this purpose, if available. If such opportunities are not available, the acquisition and integration of such resources from external sources, through paid consultants or additional training for existing instructional staff, should be part of the transition planning process.


  1. Goutam Kumar Kundu, “Quality in Higher Education from Different Perspectives: A Literature Review,” International Journal for Quality Research, 11(1), p. 17-34, 2017, DOI: 10.18421/IJQR11.01-02.
  2. Daniel Rossman and Emily Schwartz, “Online Learning During COVID-19: Digital and Educational Divides Have Similar Boundaries,” Ithaka S+R, April 27, 2020,
  3. L. O’Keefe, J. Rafferty, A. Gunder, and K. Vignare, “Delivering High-Quality Instruction Online in Response to COVID-19: Faculty Playbook,” Every Learner Everywhere, May 18, 2020,
  4. Michelle Pacansky-Brock, Michael Smedshammer, and Kim Vincent-Layton, “Humanizing Online Teaching to Equitize Higher Education.” Current Issues in Education, 21 (2), June 18, 2020,
  5. Estela Mara Bensimon, “The Underestimated Significance of Practitioner Knowledge in the Scholarship of Student Success,” The Review of Higher Education, Volume 30, No. 4, p. 446, 2007,
  6. “Position Statement: Distance Education in College Mathematics in the First Two Years,” American Mathematical Association of Two-Year Colleges (AMATYC), 2019,
  7. Martha L. Abell, Linda Braddy, Doug Ensley, Lewis Ludwig, and Hortensia Soto, “Instructional Practices Guide,” Mathematical Association of America, 2018,
  8. Association of American Colleges and Universities, “Quantitative Literacy VALUE Rubric,”
  9. See, for example AAC&U, 2014; Mathematical Association of America, 1994; Roohr et al., 2014.
  10. Ed Aboufadel, Linda Braddy, Jenna Carpenter, Lloyd Douglas, and Rick Gillman, “The Importance of Mathematical Sciences at Colleges and Universities in the 21st Century.” Mathematical Association of America, October 2018,
  11. Martha L. Abell, Linda Braddy, Doug Ensley, Lewis Ludwig, and Hortensia Soto, “Instructional Practices Guide,” Mathematical Association of America, 2018,
  12. See, for example: Cornell University Center for Teaching Innovation,
  13. Susan A. Ambrose, Michael W. Bridges, Michele DiPietro, Marsha C. Lovett, and Marie K. Norman, “How Learning Works: Seven Research-Based Principles for Smart Teaching.” John Wiley & Sons, 2010.
  14. Pamela Burdman, Kathy Booth, Chris Thorn, Peter Riley Bahr, Jon McNaughtan, and Grant Jackson, “Multiple Paths Forward: Diversifying Mathematics as a Strategy for College Success,” 2018,
  15. IDEA Teaching Methods, 2012, p. 1,
  16. Nevitt Stanford, “Self and Society,” New York: Atherton Press, 1966.
  17. Quoted in Evan Fuller, Jeffrey M. Rabin, and Guershon Harel, Intellectual need and problem-free activity in the mathematics classroom, International Journal for Studies in Mathematics Education 4(1) 80-114, 2011.
  18. David R. Krathwohl, “A Revision of Bloom’s Taxonomy: An Overview.” Theory into Practice, 41, no. 4, 2002.
  19. See TPSE Math’s top ten list of recommended practices for every online instructor developed over the summer 2020 in collaboration with American Mathematical Association of Two-Year Colleges, American Mathematical Society, Mathematical Association of America, Society for Industrial and Applied Mathematics, and Charles A. Dana Center at the University of Texas at Austin:
  20. To learn more about Quality Matters’ rubric standards for online courses, visit QM’s Emergency Remote Instruction Checklist can be found here,
  21. Rebecca A. Glazier, “A Shift to Online Classes this Fall Could Lead to a Retention Crisis,” EdSurge, 2020,
  22. Susan A. Ambrose et. al., 2010.