Study Suggests the Need for an Intergrated Learning Styles Approach to Calculus

By Jessica Knott
Associate Editor
Editor, Twitter

This week, I had the chance to talk with Dr. Daniel McGee (CV), former middle and high school teacher in Francistown, Botswana, Peace Corps volunteer, and researcher/biostatistician for the Centers for Disease Control in Atlanta, Georgia. Dr. McGee’s work in creating a free, public access online learning system for primary and secondary students in Puerto Rico has gained traction in recent years, becoming the basis for online pre-calculus materials that will now be used in schools throughout Puerto Rico.

As his projects began generating larger and larger data sets, he became interested in exploring the insights they might provide on learning styles, learning types, and learning in general. This interview provides an overview of the motivations behind the study as well as a brief discussion of some of his key findings.

Daniel Lee McGee, Professor, Mathematics, University of Puerto Rico

Daniel Lee McGee, Professor, Mathematics, University of Puerto Rico

JK: What did this study find, exactly? Why is this important?

DM: There were two important results of this study.

Result 1: In general, most efforts to define the categories for learning types have been a priori in nature. Researchers will start with a set of learning types and then will categorize students. This study takes an a posteriori approach to student learning types. We gather a great deal of data on a lot of students. The data comes from questionnaires and results on quizzes and exams from an online learning system. Rather than starting with predefined categories for students, we look for natural groups of students based on similar responses to questionnaires and similar results on quizzes and exams. These natural groupings lend insight into the natural learning styles of the students taking the course. The first result of the study was that students were not scattered randomly, they did form natural groupings. So the vast amount of information available with online learning systems does allow us, at least in Puerto Rico, to identify student learning types in an a posteriori manner.

The vast amount of information available with online learning systems does allow us, at least in Puerto Rico, to identify student learning types in an a posteriori manner.

Importance of result 1: Our results indicate that large groups of students seem to organize themselves into distinct clusters. The ability to identify these clusters and their associated strengths and weaknesses will allow professors with large groups of students using online systems to better address the particular needs of the distinct learning types that are in their class at a particular time. 

Result 2: The most important aspect of the natural groupings of the students at the University of Puerto Rico (UPR) was that success with algebraic, verbal and geometric tasks was linked. The natural groupings of students that we found were either very successful, mediocre, or very poor with all three representations. There was no such thing as an “algebraic learner” who was weak in geometry.

Importance of result 2: Puerto Rico tends to be a very traditional and algebraic oriented environment. The clusters that were found at the UPR showed that geometry and algebra seem to mutually reinforce one another with successful students. And without this mutual understanding, students are not successful. Correspondingly, a more balanced approach emphasizing the relationship between geometry and its associated algebra is more consistent with our model for a successful student.

JK: Tell me what you mean by an algebraic environment? Verbal? How do you think identifying these clusters will be helpful?

DM: An algebraic environment means that they tend to emphasize the manipulation of algebraic formulas: Students start with an algebraic expression, perform various actions and end up with another algebraic expression. For example, they might start with 2x+y=3 and change it to y=-2x+3. However, the algebraic expression would seldom be associated with a geometric figure or a real-life situation.

Successful [Calculus I] students appeared to need a unified approach, which emphasized verbal situations, geometric figures, algebraic expressions and the relations between them.

In a practical sense, the clusters were helpful to us in that they identified that successful students appeared to need a unified approach, which emphasized verbal situations, geometric figures, algebraic expressions and the relations between them. Correspondingly, as many professors were overemphasizing algebra, it would suggest that more emphasis on geometry and real life situations would be helpful. Hence, it lets professors address the needs of their own particular students as identified by the learning types in their class. From the perspective of math behaviorists, by selecting the questions in the questionnaires and on the quizzes very carefully, these natural groupings can provide insight into student learning types as they are not based on preconceptions but represent natural groupings.

JK: What would you suggest that professors and practitioners do with this data? How can this help them?

DM: The first thing I’d like to see is whether performance on actual tasks supports behaviorists categories of learning. For example, there are tests to determine whether a student is a “right brain” or “left brain” thinker. And these categories have preconceptions regarding the sort of tasks in which a student should excel. If “left brain” and “right brain” thinkers are equally distributed in each of the clusters we generate, this would indicate that there is little relationship between these categories and the ability to perform different types of tasks. Correspondingly, knowing this propensity in advance would be of little use. However, if each cluster were strongly associated with either a “left brain” or “right brain” propensity, this would indicate that this physiological propensity would be useful to know in advance in order to better design the material to the specific needs of the student.

We found that there was a large cluster of students that learn by rote memorization. An appropriate intervention for this group might be a workshop exploring the value of conceptual understanding vs. rote memorization ….

From the perspective of a professor that simply wishes to teach his class well, this sort of data would allow the professor to address a certain profile associated with a single cluster containing many students. For example, we found that there was a large cluster of students that learn by rote memorization. An appropriate intervention for this group might be a workshop exploring the value of conceptual understanding vs. rote memorization with the goal of convincing them that conceptual understanding is a better investment for their futures in STEM fields.

JK: What made you undertake this research? What was your “lightbulb,” as it were?

DM: I work with research on both informatics and math education. As my projects with online learning systems obtain large quantities of data for many students, it was rather natural to explore what insight this large amount of data might give us on learning types.

10 Responses

  1. I hope that Dr. McGee’s full paper on this topic becomes available soon. The clustering is stark. Although performance differs, there’s an interesting divide alluded to in the interview between those who rely on memory to pass courses and those who attempt to understand.

    Having this a posteriori data and clustering allows us, as Dr. McGee says, to compare various a priori tests and see if any correlation exists. There are too many education theories based on someone’s intuition and followed up with some sort of instrument to classify people.

    Often these classifications are used as inherent personality traits to pigeonhole individuals. Someone may be labeled as unable to learn math, for example. The idea that anyone of reasonable intelligence cannot learn math has been shown to be invalid. Yet, this sort of classification persists.

    In the case of Dr. McGee’s study, he found four clusters based on his numerous testing points. The fact that these four clusters tend to below to two super-clusters of memorizers and concept-seekers should tell us much.

    While today’s education theories focus on learning concepts and being able to reason, those theories are just the verbalization of centuries-old wisdom. It’s not what you know that truly counts; it’s what you know how to do with what you know.

    There’s been a long shift from work that essentially copies a script from previous workers without any necessity for change to work that requires adaptability. Some see a stark divide at the year 2000 and call learning to be adaptable 21st century skills. That, of course, is complete nonsense. We just require more of these people as each decade passes and more automated-style work passes on to automatons.

    The message for teaching and learning is really quite simple and echoes much of what has been written in ETC-J before. Stop creating memorizers. These people, with some coaching, are great at taking tests but will fail at the test of life.

    While we all agree that some things must be learned by rote (e.g. alphabet, digits, addition, spelling), whenever possible, teaching should avoid telling students to memorize. Sometimes, it’s hard to tell the difference. I saw it in math classes where students were told to recognize a particular pattern in a math problem and then apply a specific algorithmic solution. To the untrained mind, this may have seemed like learning to think because students had to use algebra to solve the problem. However, the one-to-one association of pattern and step-by-step algorithm is just memory of higher level constructs than most are used to.

    How can we step away from this disastrous mode of teaching? A great first step would be to eliminate all high-stakes testing, including state-mandated tests and teachers’ own final exams. There’s much more that can be done, but one example must suffice if this note is not to become of book length.

    • I’ve shifted jobs (I’m now Executive Director of the Kentucky Center for Mathematics) so I’ve had to take a bit of a pause in order to move from Puerto Rico to Kentucky and settle into my new world.

      As my new job provides data from p-20, I’m very interested in looking at younger students. Unfortunately, the only state wide data we have available right now comes from “high stakes” testing. In order to succeed with state-wide tests, some school districts have gone to a memorization scheme using what they call “math facts.” An elementary school teacher recently told me that she doesn’t want students taking the time to think concepts through, she wants basic math memorized so students can respond as fast as possible. As the best memorizers are often those with an underlying grasp of the concepts (i.e., it is often easier to memorize when you’re familiar with what you’re memorizing…), it makes it very difficult to pin down the destructive nature of this course. I’m wrestling with the appropriate data analysis to capture the phenomenon. I’d certainly appreciate any thoughts!


      • Kudos to you for recognizing the problem. Wish I could really help you.

        We all have to memorize the symbols for numbers and the multiplication table if we’re to operate well with numbers. Some memorization is unavoidable and even valuable. Number symbols and multiplication tables have little, albeit some, underlying concepts.

        I had to fight teachers over this very issue while my children were in school. I never really “won” in the sense of creating change, but I did save my kids from that terrible fate — for the most part. Parents can’t do everything.

        Somewhere, there should be tests that focus on concepts, especially on combining two concepts in one problem. In my experience, combining concepts that have not been combined during instruction time really separates the memorizers from the learners.

        I’m not familiar with mathematics education instruments and so can be of no more help. Possibly, you could be the one to create this tool.

        • Daniel, I’d love to follow up with you on this interview, talk to you about your new role, and your future plans! Interested?

          • Hi Jessica,

            I’m sorry for the very significant delay getting back to you! I’m adapting to a far larger scale of work and dataand simply didn’t take a pause to follow up for a long time! If you are still interested, I would love to chat about our work at the Kentucky Center for Mathematics. We are working with far larger data sets however the content is less specific so I’m not sure where we’ll end up going!

        • Hi Harry

          Interestingly enough, we are working on what we call “fluency assessments” here at the Kentucky Center for Mathematics. The idea is that assessment is formative and the goal is to assure that numbers are understood in their entirety. One of my favorite questions (while not unique to our system) is “make up a story associated with 24/6 =4” It currently counts on more of an interview format which may make it untenable however we’re looking at the problem!.

          • I certainly am a big fan of formative assessments. They can be woven into the thread of instruction relatively unobtrusively, especially in online learning, and used to adapt the material to the student.

            Even without adaptation, formative assessments can help students to fill in gaps in learning on their own. Often, they don’t know what they don’t know. While not as strong a tool as adaptive learning, continuing self-assessment is better than high-stakes testing and can flag learning problems for teachers to address.

            Post-activity assessment also is valuable to see if the activity created learning, but also to tell students when they have not paid sufficient attention during the activity. These assessments can help students to reflect on what they have been doing. Of course, they can also be used in adaptive learning as well. I try to include a couple of challenging questions in my post-lab assessments (in Smart Science® labs) so that students develop a deep understanding of the material.

            In the pre-lab area, I focus on three primary areas: adequate prior knowledge, preparatory understanding, and high-order thinking.

            Only by moving away from those old-fashioned drill-style questions can we improve the use of assessments — IMHO.

      • Could this be the same Dan McGee who once taught at Mater Spei College as a PCV ? In that case you should remember me as a former colleague. Tony King

      • Interesting to come across your name & face after many years, Dan.
        Tony King (Mater Spei College, FT circa 1980s)

  2. Very interesting work. Is there comparable data for any other subjects?

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