What is Early Math and Why Should We Care?

Posted on November 10, 2015 by Ben Braun

By Jennifer S. McCray, Assistant Research Scientist and Director of the Early Math Collaborative at Erikson Institute

Effective early childhood math teaching is much more challenging than most people anticipate.  Because the math is foundational, many people assume it takes little understanding to teach it, and unfortunately this is distinctly not the case.  In fact, the most foundational math ideas — about what quantity is, about how hierarchical inclusion makes our number system work, about the things that all different shapes and sizes of triangles have in common — are highly abstract ones.  While we should not expect or encourage young children to formally recite these ideas, they are perfectly capable of grappling with them.  Further, they need to do so to develop the kind of robust understanding that will not crumble under the necessary memorization of number words and symbols that is to come in kindergarten.  In preschool, before there is really any opportunity for “procedural” math, it is important that we give children ample opportunity to think about math conceptually.  In this essay I will discuss several profound ideas from early childhood mathematics, including examples of effective early math classrooms.  Along the way I will share some of the resources that my colleagues and I have developed to help early childhood educators develop as skillful teachers of early mathematics.

Erikson.ChristopherHouse.178.scaled Continue reading →

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Active Learning in Mathematics, Part VI: Mathematicians’ Training as Teachers

Posted on November 1, 2015 by Priscilla Bremser

By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.

Editor’s note: This is the sixth and final article in a series devoted to active learning in mathematics courses. The other articles in the series can be found here.

How are mathematicians trained as teachers, what are the effects of this training, and what can we do to improve the quality of this training? We feel these questions are particularly important at this time, as a clamor of recent calls for the dramatic improvement of postsecondary education, made from both inside and outside of the mathematical community, has not abated. From the outside, we hear this call in venues ranging from opinion pieces in major newspapers [1,2,3] to federal advisory reports to the President of the United States [4] and beyond. The message has also been clearly conveyed by leadership from professional societies in the mathematical community: in early 2014, an article titled “Meeting the Challenges of Improved Post-Secondary Education in the Mathematical Sciences” was published in the AMS Notices, MAA Focus, SIAM News, and AMSTAT News through a coordinated effort by the professional societies — we urge any readers who have not already done so to read this statement.

Yet in order to be effective and achieve meaningful change, any actions taken by our professional societies and other leadership in the mathematics community must get buy-in from individual mathematicians who are in the classroom daily, working face-to-face with students. From our training in both mathematics and the teaching of mathematics, we each carry disciplinary habits, ways of thinking, biases, and strengths, many of which occur subconsciously as part of our mathematical culture, and all of which impact our teaching. In order to improve mathematical teaching and learning on a large scale, we must all work to better understand how mathematicians grow and develop as teachers, so that we may more thoughtfully respond to the educational challenges of our time. In this article, we focus our discussion on the topic of pedagogical training and development for graduate students and early-career faculty, with a view toward active learning. Continue reading →

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Active Learning in Mathematics, Part V: The Role of “Telling” in Active Learning

Posted on October 20, 2015 by Elise Lockwood

By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.

Editor’s note: This is the fifth article in a series devoted to active learning in mathematics courses. The other articles in the series can be found here.

Facts, methods, and insights all are essential to all of us, all enter all our subjects, and our principal job as teachers is to sort out the what, the how, and the why, point the student in the right direction, and then, especially when it comes to the why, stay out of his way so that he may proceed full steam ahead.
— Paul Halmos (Halmos, pp. 848-854)

Because of our passion and love for our subject, mathematicians want to share with students the joy, excitement, and beauty of doing mathematics. Our natural human impulse is to do so by telling students about the ways we have come to understand our discipline, to shed light on the subtleties that surround most mathematical ideas, and to explain the fundamental insights of our field. As we have discussed in our previous articles in this series, there is strong evidence that these goals of inspiring students and helping them deeply learn mathematics are often most effectively reached through the use of active learning techniques. Yet there are some good reasons why we might choose to tell students about mathematics when the time is right. In this article we will explore the act of instructor “telling” and discuss some of the roles that telling can play in active learning environments. We seek to balance our inclination to tell students about math, which is inherently passive for the students, with our desire to foster students’ active engagement with mathematical ideas. By doing so, we can simultaneously acknowledge the value of telling while challenging the idea that traditional telling is the best or only way to communicate mathematics with students. Continue reading →

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Active Learning in Mathematics, Part IV: Personal Reflections

Posted on October 10, 2015 by Ben Braun

By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.

Editor’s note: This is the fourth article in a series devoted to active learning in mathematics courses.  The other articles in the series can be found here.

In contrast to our first three articles in this series on active learning, in this article we take a more personal approach to the subject.  Below, the contributing editors for this blog share aspects of our journeys into active learning, including the fundamental reasons we began using active learning methods, why we have persisted in using them, and some of our most visceral responses to our own experiences with these methods, both positive and negative.  As is clear from these reflections, mathematicians begin using active learning techniques for many different reasons, from personal experiences as students (both good and bad) to the influence of colleagues, conferences, and workshops.  The path to active learning is not always a smooth one, and is almost always a winding road.

Because of this, we believe it is important for mathematics teachers to share their own experiences, both positive and negative, in the search for more meaningful student engagement and learning.  We invite all our readers to share their own stories in the comments at the end of this post.  We also recognize that many other mathematicians have shared their experiences in other venues, so at the end of this article we provide a collection of links to essays, blog posts, and book chapters that we have found inspirational. Continue reading →

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Active Learning in Mathematics, Part III: Teaching Techniques and Environments

Posted on October 1, 2015 by Ben Braun

By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.

Editor’s note: This is the third article in a series devoted to active learning in mathematics courses.  The other articles in the series can be found here.

It is common in the mathematical community for the phrases “active learning” and “inquiry-based learning” (IBL) to be associated with a particular teaching technique that emphasizes having students independently work and present to their peers in a classroom environment with little-to-no lecturing done on the part of the instructor.  Yet it is counterproductive for this method to be a dominant cultural interpretation of “active learning,” as it does not represent the range of teaching styles and techniques that fall along the active learning and IBL spectrums as considered by mathematicians who use these pedagogies, mathematics education researchers, federal and private funding agencies, and professional societies such as the AMS, MAA, SIAM, ASA, AMATYC, and NCTM.  In this article we will provide multiple examples of active learning techniques and environments that arise at institutions with different needs and constraints.  We begin by reflecting on general qualities of classroom environments that support student learning.  Continue reading →

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Active Learning in Mathematics, Part II: Levels of Cognitive Demand

Posted on September 20, 2015 by Art Duval

By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.

Editor’s note: This is the second article in a series devoted to active learning in mathematics courses.  The other articles in the series can be found here.

Mathematics faculty are well-aware that students face challenges when encountering difficult problems, and it is common to hear instructors remark that successful students have high levels of “mathematical maturity,” or are particularly “creative,” or write “elegant” solutions to problems.  To appreciate research results regarding active learning, it is useful to make these ideas more precise.  Motivated by research in education, psychology, and sociology, language has been developed that can help mathematicians clarify what we mean when we talk about difficulty levels of problems, and the types of difficulty levels problems can have. This expanded vocabulary is in large part motivated by…

…the “cognitive revolution” [of the 1970’s and 1980’s]… [which] produced a significant reconceptualization of what it means to understand subject matter in different domains. There was a fundamental shift from an exclusive emphasis on knowledge — what does the student know? — to a focus on what students know and can do with their knowledge. The idea was not that knowledge is unimportant. Clearly, the more one knows, the greater the potential for that knowledge to be used. Rather, the idea was that having the knowledge was not enough; being able to use it in the appropriate circumstances is an essential component of proficiency.

— Alan Schoenfeld, Assessing Mathematical Proficiency [17]

In this article, we will explore the concept and language of “level of cognitive demand” for tasks that students encounter.  A primary motivation for our discussion is the important observation in the 2014 Proceedings of the National Academy of Science (PNAS) article “Active learning increases student performance in science, engineering, and mathematics” by Freeman, et al. [8], that active learning has a greater impact on student performance on concept inventories than on instructor-written examinations.  Concept inventories are “tests of the most basic conceptual comprehension of foundations of a subject and not of computation skill” and are “quite different from final exams and make no pretense of testing everything in a course” [5].  The Calculus Concept Inventory is the most well-known inventory in mathematics, though compared to disciplines such as physics these inventories are less robust since they are in relatively early stages of development.  Freeman et al. state:

Although student achievement was higher under active learning for both [instructor-written course examinations and concept inventories], we hypothesize that the difference in gains for examinations versus concept inventories may be due to the two types of assessments testing qualitatively different cognitive skills.  This is consistent with previous research indicating that active learning has a greater impact on student mastery of higher- versus lower-level cognitive skills…

After introducing levels of cognitive demand in this article, our next article in this series will directly connect this topic to active learning techniques that are frequently used and promoted for postsecondary mathematics courses.

Continue reading →

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Active Learning in Mathematics, Part I: The Challenge of Defining Active Learning

Posted on September 10, 2015 by Ben Braun

By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.

Editor’s note: This is the first article in a series devoted to active learning in mathematics courses.  The other articles in the series can be found here.

“…if the experiments analyzed here had been conducted as randomized controlled trials of medical interventions, they may have been stopped for benefit.”

So strong is the evidence supporting the positive effects of active learning techniques in postsecondary mathematics and science courses that Freeman, et.al, made the statement above in their 2014 Proceedings of the National Academy of Science (PNAS) article Active learning increases student performance in science, engineering, and mathematics.  Yet faculty adoption of active learning strategies has become a bottleneck in post-secondary mathematics teaching advancement.  Inspired by the aforementioned PNAS article, a landmark meta-analysis of 225 studies regarding the positive effects of active learning, we will devote a series of posts to the topic of active learning in mathematics courses.  

An immediate challenge that arises when discussing active learning in mathematics is that the phrase “active learning” is not well-defined.  Interpretations by mathematics faculty of this phrase range broadly, from completely unstructured small group work to the occasional use of student response systems (e.g., clicker) in large lectures.  In this article we discuss several descriptions from the literature, including what we will take as our working understanding throughout this series of posts, discuss important considerations in the adaptation of such methods, and highlight some important aspects of the PNAS article. Continue reading →

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The Secret Question (Are We Actually Good at Math?)

Posted on September 1, 2015 by Ben Braun

By Benjamin Braun, Editor-in-Chief, University of Kentucky

“How many of you feel, deep down in your most private thoughts, that you aren’t actually any good at math? That by some miracle, you’ve managed to fake your way to this point, but you’re always at least a little worried that your secret will be revealed? That you’ll be found out?”

Over half of my students’ hands went into the air in response to my question, some shooting up decisively from eagerness, others hesitantly, gingerly, eyes glancing around to check the responses of their peers before fully extending their reach.  Self-conscious chuckling darted through the room from some students, the laughter of relief, while other students whose hands weren’t raised looked around in surprised confusion at the general response.  

“I want you to discuss the following question with your groups,” I said.  “How is it that so many of you have developed negative feelings about your own abilities, despite the fact that you are all in a mathematics course at a well-respected university?”

If this interaction took place in a math course satisfying a general education requirement, I don’t think anyone would be surprised.  Yet this discussion repeats itself semester after semester in my upper-level undergraduate courses, for which the prerequisites are at least two semesters of calculus and in which almost every student is either a mathematics major or minor.  I’ve had similar interactions with students taking first-semester calculus, with experienced elementary school teachers in professional development workshops, with doctoral students in pure mathematics research seminars, and with fellow research mathematicians over drinks after dinner.  These conversations are about a secret we rarely discuss, an invisible undercurrent of embarrassment and self-doubt that flows through American mathematical culture, shared by many but revealed by few.  At every level of achievement, no matter what we’ve done, no matter how much we’ve accomplished, many of us believe that we’re simply not good at math. Continue reading →

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Why High-Impact Educational Practices (Despite Being So Labor–Intensive) Keep Me Coming For More

Posted on August 20, 2015 by Priscilla Bremser

By Maria Mercedes Franco, Coordinator for Undergraduate Research & Associate Professor, Mathematics & Computer Science, Queensborough Community College-The City University of New York (CUNY)

By the time I was finishing graduate school, I had done much soul-searching and had come to realize that I have a passion for teaching and a strong commitment to the mission of public education. With my new awareness came the opportunity to interview for (and soon after accept) a position at Queensborough Community College, where I was encouraged early on to incorporate innovative pedagogies into my teaching. Now on my tenth year at the college, I look back and say without hesitation that High-Impact Educational Practices have brought me closer to larger and more diverse groups of learners – and closer to my ideals for higher education – than any other practice. Continue reading →

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Let Your Students Do Some Grading? Using Peer Assessment to Help Students Understand Key Concepts

Posted on August 10, 2015 by Elise Lockwood

By Elise Lockwood, Contributing Editor, Oregon State University

On many occasions when I grade my students’ proofs, or when I read their solution to a particularly interesting problem, I am surprised by something I read. Sometimes I am surprised because I am disappointed with a given argument or a hand-wavy proof, but often I am surprised because I am impressed by a clever insight or an eloquent way of expressing an argument. Indeed, there have been occasions when I have learned something through the experience of grading my students’ work. Also, seeing the sheer variety of solution strategies that my students offer helps me to appreciate various mathematical approaches and makes me more attuned to their respective mathematical ways of thinking.

In this post I will discuss an activity that I call peer grading, by which I mean having students provide formative, written feedback on their classmates’ assignments. This involves giving students the opportunity to engage with and analyze work that their classmates have done. Peer grading has been used by other teachers (see the references at the end of this post), and my personal reflections on the value of engaging in the process of grading have convinced me that students can similarly benefit from grading other students’ work.

Continue reading →

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