Wednesday, April 18, 2018

The Exercise with No Wrong Answer: Notice and Wonder

Guest post by May Mei, Denison University.

How often have your students said nothing rather than risk saying something wrong? And how often in our own writing are we so paralyzed by the fear of imperfection that we end up writing nothing at all? 



Enter Notice and Wonder, the exercise that has no wrong answers. After all, everything you can observe about a problem is a valid thing to notice and every question you can ask about a problem is a valid thing to wonder.

Notice and Wonder is a way for instructors to create a safe place of exploration by allowing students to brainstorm before attempting to solve a problem. It’s a simple process-students are presented with a problem and before attempting to solve it, they are asked what they notice about it. When all students have contributed, or nothing new is being noticed, students can then answer what they wonder.

This provides opportunities to discuss what is still unknown and puts all students on equal ground-everyone can wonder about something. Because there are no wrong answers for either of these questions, all students can participate in the activity.

This handout describes Notice and Wonder as an in-class activity, but I like to use it to review for an exam.

If my students have an upcoming exam, I will use the class period before as review, and sometime before that review period I ask each of my students to email me one thing they noticed and one thing they’re still wondering about. Just before class on the review day, I put all the students’ Notices and Wonders into one document and distribute it to the class. As I'm making this document I'm able to recognize themes and repeated observations or questions and can focus on those during the in-class review.

The purpose of the exercise is to help me see what my students need, but also for the students to take inventory of what content the course has covered and to self-assess their understanding.

Note that a brief discussion about what makes a fruitful Wonder may be needed. Let’s consider an upper level course, such as an introductory course on proof techniques. To help students develop ‘good’ Wonders, I provide them with the following examples and ask about the different levels of self-reflection that they display.

"I wonder in Chapter 6, Exercise 5 in which it asks us to prove 3 is irrational, which definitions to use. Just like the proof we did for2, here I would say suppose 3 is rational so therefore3can be written as a/b where a and b are in Z. But once I squared both sides and did some algebra it does not come out to show a2 is even. Since we haven't done irrational numbers any other way I am confused as to what to do."

3 reasons to consider incorporating this exercise into your math courses:

  • Notice and Wonder ingrains good habits of mind. This exercise provides a way for students to engage with material after the initial in-class exposure. The Notice component encourages students to draw connections that may not have been apparent in the first read-through while the Wonder component asks students to evaluate their comprehension of the material.
  • Notice and Wonder minimizes instructional prep time. I spend about 20 minutes compiling responses. For a class of 25 students, this generates much more material than I can cover in a 50 minute class. The responses provide students with something to work on after class, when they may feel compelled to study.  The student-generated ideas provide guidance about how and what to study for students who are unsure of how to proceed and need to develop useful study habits.
  • Notice and Wonder allows students to gain insight into the thought processes of their peers. How many times have you heard a student say something to the effect of "everyone gets it but me"? Students gain the benefit of seeing that other people have questions, and maybe the same ones as them. Students can also see questions that they may not have thought to ask, and can’t yet answer. 

I'm always impressed with the wonderful gamut of things students notice and wonder.  Thus the practice makes not only supports student learning, it also makes my own teaching more effective and enjoyable.

Tuesday, April 3, 2018

Design Practices to Maximize Students Learning

By Karen Keene, North Carolina State University, Beth Burroughs, Montana State University, and Hortensia Soto, University of Northern Colorado
This semester Teaching Tidbits continues its posts highlighting the new Instructional Practices Guide (IP Guide) from the Mathematical Association of America (MAA). This evidence-based guide is a complement to the Curriculum Guide published in 2015. The guide provides significant resources for faculty focused on teaching mathematics in evidence-based ways. There are three focus chapters in the guide, Assessment Practices, Design Practices and, Classroom Practices, along with some additional sections that explain the importance of evidence-based instructional practices. Karen Keene and Beth Burroughs served as lead writers for the Design Practices and Hortensia Soto was a project team member and co-editor of the MAA IP Guide.


College professors have been planning for their classroom instruction for as long as universities have existed.  Planning for instruction is one facet of the practice of design. The MAA IP Guide addresses design practices as

“the plans and choices instructors make before they teach and what they do after they teach to modify and revise for the future. Design practices inform the construction of the learning environment and curriculum and support instructors in implementing pedagogies that maximize student learning.”

Design practices include planning for the content, but much more as well.  Designing to maximize student learning requires professors to consider many things as they plan, but also to use the results of teaching to continue to revise and modify teaching in the future. Consideration of what is known about teaching practices and how students learn is necessary for all parts of the design. The design practices chapter of the IP Guide includes questions that instructors could ask themselves while designing instruction (i.e., how can I be sure to be inclusive in my instruction?), as well as many suggestions that focus on designing cognitive and affective learning goals, developing tasks and other ideas for instruction, and creating learning environments. To focus on student learning, instructors need to design the learning environments, the tasks and the homework based on the student learning objectives.

In the design practices chapter, the authors offer design principles and considerations, educational research, and real examples provided by faculty in the field.  Readers can access the chapter for a quick planning idea, or to consider making bigger instructional changes that focus on student learning; both are exciting and possible.

Tuesday, March 20, 2018

Three Ways to Help Teach Growth Mindset

By Deanna HaunspergerCarleton College and MAA President



Every fall I teach a differential calculus course at Carleton College that is five days a week instead of our usual three-days-per-week format. This course is designed to give students a review of algebra and pre-calculus and trigonometry skills just-in-time as I’m teaching the calculus material. It’s the lowest entry point we have for students who want or need to learn calculus, and it is where I introduce students to the idea of a growth mindset.


On their mathematical autobiography cards that students write for me the first day of class, they often admit to feeling unsuccessful in their previous math class, being nervous about the material, and worried they’re not smart enough to succeed this time. What I enjoy most about these students is that they are in my class on the first day, regardless of background or perceived ability, ready to learn.

I know from day one that one of my biggest responsibilities as a mathematician is to give my students the confidence to be successful. They need to come at this material with a fresh start, open their notebooks to a fresh page, and use a new mindset: a growth mindset.

Growth mindset, as defined by psychologist Carol Dweck, is the belief that mathematical (or any) ability is not something you’re born with, but something that can be developed through dedication, hard work, and good strategies. She and her colleagues have shown that students who believe in a fixed mindset – that you’re either born with a certain ability or intelligence or you’re not – are defeated by mistakes because they don’t think they are capable of improving. Growth mindset students, however, take mistakes as a challenge to work harder or dig in more deeply. They believe they can grow their brains to understand more.

Of course we want our math students to have a growth mindset so that when they face problems they don’t know how to solve, they engage with the problem and persevere. But how do we teach growth mindset? Here are my three ways:

  1. Tell them. I was talking to the director of our Learning and Teaching Center a few years ago over coffee. I wasn’t seeking advice at the time, I was just kvetching about my students and the things I thought they should know about being a successful student. “How can they not know that being in class is important? How can they not know that getting enough sleep and eating well helps? How can they not know that if they work at something long and hard and try different strategies they’ll get better at it?” He looked at me and said, “Well, have you told them?” No, I had to admit, I hadn’t. I don’t know why, but it had never occurred to me than in addition to teaching math, I needed to teach my students how to learn math.

    So now on day one I tell them that showing up to class well-rested and well-nourished is important. I tell my students that finding study buddies is important and that keeping up with their homework is important. I also tell them all about growth mindset and how they can be successful if they engage the material and persevere. In fact, I have a handout I give them on “How to be Successful in a College Math Classroom” that contains these and other suggestions.
  2. Remind them. Before the first exam, I bring up these tips for success again. Not everyone is fully listening the first day of class, so it is important to continue to remind students of the expectations I have of them. This time, I tell them a personal story; this is not difficult for me because having a growth mindset helped me survive graduate school. My first year of graduate school, I took graduate abstract algebra without having had undergraduate abstract algebra. It turns out this was not a good idea. I felt defeated after one term, redoubled my efforts the second term, dug in even deeper the third term, and I ended up passing my algebra prelim at the end of the year on my first attempt. The material in that course did not come to me through divine intervention. I worked very hard to learn it, and I put in the hours and the focus to develop a growth mindset.
  3. Use growth mindset-appropriate words throughout the term. I am, sincerely, very proud of the efforts that the students put in throughout the term, and I love being their cheerleader. I don’t commend their talent or intelligence, though. Instead, I write “Great improvement; I can see you studied a long time for this exam!” “Excellent work!” on their exams. I acknowledge the hard work their brains are doing during class and over time, they are building new stronger connections between the neurons in their brains, and that’s why they need adequate rest and nutrition. Exams are not meant to judge students. Exams assess how much students have learned and indicate whether students have put in enough work to master the material.
Of course, not all students are successful in this class that meets every day; it’s a lot of hard work. But a couple years ago, a student who had dropped the course one fall, signed up for it again the next fall. He went from failing one year to earning A’s the next year. “What’s different?,” I asked him. “I learned how to work hard and focus,” he replied. Now I make sure to slip that story into the class each year as well.

Once they understand the growth mindset, students also feel slightly more in control of their own grades in the class, since they are seeing a more direct correlation between their time on task and their grade in the class.

This made such a positive change in my calculus class, that I brought it into all the classes I teach now. I see a difference in my classes, especially in the attitude of some women. If this change in frame of mind improves the classroom experience for even a few students each term, it’s well worth the extra few minutes in class.

Editor’s note: For more on the Growth Mindset in the math classroom, please see the MAA Instructional Practices Guide sections on classroom practices as well as the equity in practice section.

Tuesday, March 6, 2018

Fostering Student Engagement through Enhanced Classroom Practices


By guest writers April Strom Scottsdale Community College and James Álvarez University of Texas at Arlington

This is the third and final installment of Teaching Tidbits highlighting the new Instructional Practices Guide (IP Guide) from the Mathematical Association of America (MAA). This evidence-based guide is a complement to the Curriculum Guide published in 2015. The guide provides significant resources for faculty focused on teaching mathematics using ideas grounded in research. Many thanks to April  Strom and James Álvarez , lead writers for the Classroom Practices section, for providing this post.


Have you wondered how to increase student engagement in your courses or searched for ideas to help curious colleagues enhance their classroom practices? Well, the Mathematical Association of America’s Instructional Practices Guide (MAA IP Guide) may be an answer for you! The MAA IP Guide is purposefully written for all college and university mathematics faculty and graduate students who wish to enhance their instructional practices. A valuable feature of the MAA IP Guide is that it doesn't need to be read from cover to cover. Rather readers can begin, depending on their interests, with any chapter. The MAA IP Guide contains 4 main chapters: Classroom Practices, Assessment, Design Practices, and Cross-cutting Themes (such as Technology and Equity). In this post, we take a deeper dive into the Classroom Practices chapter of the MAA IP Guide, which can be downloaded in full for free here.

The Classroom Practices chapter is partitioned into two primary sections: (1) fostering student engagement and (2) selecting appropriate mathematical tasks. Moreover, the chapter is designed such that quick-to-implement instructional practices are presented upfront in the chapter followed by ideas that require more preparation to fully implement. The message is clear: we want to embrace, encourage, and support active learning strategies in the teaching and learning of collegiate mathematics!

Fostering Student Engagement: Classroom practices aimed to foster student engagement can build from the research-based idea that students learn best when they are engaged in their learning (e.g., Freeman, et al., 2014). Consistent use of active learning strategies in the classroom also provide a pathway for more equitable learning outcomes for students with demographic characteristics who have been historically underrepresented in science, technology, engineering, and mathematics (STEM) fields (e.g., Laursen, Hassi, Kogan, & Weston, 2014). In this section we illustrate what it means to be actively engaged in learning and offer suggestions to foster student engagement.

To foster student engagement, the MAA IP Guide promotes the notion of building a classroom community from the first day of class. Community and sense of belonging are more likely to flourish in classrooms where the instructor incorporates student-centered learning approaches (c.f. Slavin, 1996; Rendon, 1994). Thus, establishing norms for active engagement or taking steps to increase a student’s sense of belonging to the classroom community also impacts the quality of student engagement in the classroom. We begin by providing suggestions on how to build a classroom community and then describe quick-to-implement strategies (e.g., wait time after questioning and one-minute papers), followed by more elaborate strategies that may require more preparation (e.g., collaborative learning strategies, flipped classroom, just-in-time teaching).

Of course, fostering student engagement through active learning strategies requires thoughtful consideration of the mathematical tasks that will support the work you want to accomplish with your students. In the next section, we focus on part 2 of the MAA IP Guide:

Selecting Appropriate Mathematical Tasks: Stein, Grover, and Henningsen (1996) define a mathematical task as a set of problems or a single complex problem that focuses student attention on a particular mathematical idea. Selecting appropriate mathematical tasks is critical for fostering student engagement because the tasks chosen provide the conduit for meaningful discussion and mathematical reasoning. But, how does an instructor know when a mathematical task is appropriate? There does not appear to be one single idea of what constitutes appropriateness in the research literature or in practice. Rather, appropriateness appears to be determined from a linear combination of a number of factors. The successful selection of an appropriate mathematical task seems to involve two related ideas:

  1. The intrinsic appropriateness of the task, by which we mean the aspects of the task itself that lend itself to effective learning; and
  2. The extrinsic appropriateness of the task, by which we mean external factors involving the learning environment that affect how well students will learn from the task.

In this section we elaborate on group-worthy tasks, which provide opportunities for students to develop deeper mathematical meaning for ideas, model and apply their knowledge to new situations, make connections across representations and ideas, and engage in higher-level reasoning where students discuss assumptions, general reasoning strategies, and conclusions. Group-worthy tasks can be characterized in terms of the cognitive demand required and they are often considered as high-level cognitive demand (see Stein et al. (1996) for a discussion on low-level and high-level cognitive demand).

When implementing active learning strategies in the classroom, it is important to keep in mind the notion of communication: reading, writing, presenting, and visualizing of mathematics. The MAA IP Guide leverages the Common Core Standards for Mathematical Practice, specifically the idea of SMP3: constructs viable arguments and critiques the reasoning of others, where students are expected to justify their thinking publicly through verbal and written mathematics. Students construct viable arguments as they engage in mathematical problem solving tasks by articulating their reasoning as they demonstrate their solution to the problem. These arguments can be made for solutions to abstract problems and proofs as well as for mathematical modeling and other problems with real world connections. Students should come to interpret the word “viable” as “possible” so that during presentations students recognize that they are considering a possible solution which requires analysis in order to determine its mathematical worthiness.

Impact on Teaching Evaluations
Whenever new teaching strategies are implemented in the classroom, faculty take a risk that student feedback surveys of their teaching may not accurately reflect the positive changes made or the deep learning achieved. Although the use of student feedback surveys as the only tool for evaluating teaching is highly problematic, in such cases it is important that faculty communicate their efforts. Documenting significant efforts to implement new strategies and collecting evidence of positive change can support this communication. Some possible ways to do this include:

  1. Keep notes; use them to write a brief summary of the changes made and the rationale for the changes to be shared with administrators or tenure and promotion committees.
  2. Document positive student feedback and comments, especially regarding their learning experience.
  3. Put negative comments into perspective. For example, if students make negative comments about working on open-ended problems, provide a rationale and explanation for the implementation of these types of tasks.
  4. Save examples of student work that represent the quality of the mathematical work and learning taking place and include or explain this in the summary of changes to the course delivery.

Although negative student feedback surveys of teaching may be discouraging, it is important to put the survey feedback into context and to remember that implementing new strategies and techniques takes practice (i.e., don’t give up after the first time you try it). Students also need time and practice to acclimate to new ways of learning, so having several courses that require active engagement may also affect the way they react to active engagement in your course. Bringing these issues to the attention of the person or committee that evaluates your teaching and collecting evidence of your efforts to supplement student feedback surveys may help mitigate possible apprehension in trying new strategies to encourage active engagement in the classroom.

To learn more about Classroom Practice, download the MAA IP Guide and use it as a resource to increase student engagement. And don’t forget to share broadly with others!  

References
Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences, 111(23), 8410-8415.

Laursen, S. L., Hassi, M. L., Kogan, M., & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A multi-institution study. Journal for Research in Mathematics Education, 45(4), 406-418.

Rendon, L. I. (1994). Validating culturally diverse students toward a new model of learning and student development. Innovative Higher Education, 19(1), 33.

Slavin, R. W. (1996). Research on cooperative learning and achievement: What we know, what we need to know. Contemporary Educational Psychology, 21, 43-69.

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.

Tuesday, February 20, 2018

4 Ways to Promote Gender Equity in Your Classroom

By Jessica DeshlerWest Virginia University

There is something beautiful about the structure of mathematics that we can all appreciate, but it’s equally beautiful because it can be creative and messy. So is the teaching of mathematics. As mathematicians, we know and understand the complexities involved in our discipline, but sometimes overlook the underlying complexities of our classroom environment when preparing to teach.


You’ve likely heard about the leaky pipeline – the phenomenon that describes the loss of women from STEM fields at various points in the academic pipeline. Because many undergraduate women leave the STEM pipeline after taking a mathematics course, our discipline can especially benefit from classroom practices known to help retain and support these students.

You might wonder whether the gender breakdown in our classes or variation in our students’ cultural and social backgrounds matter. We posit that these do matter, and that they can impact whether students are comfortable contributing to discussions, volunteering to present work on the board, or seeking help during office hours. We have some control, though, over how social interactions affect learning in our classrooms. Below are several ways you can support gender equity in your classroom. These techniques are meant to be inclusive and support all students, but are particularly important and empowering for undergraduate women in our classrooms. Links are included for suggestions that have appeared in previous Teaching Tidbits posts.

  • Don’t be the Authority in the Classroom. Help your students find ways to stop relying on you as the expert, and use the authority inherent in mathematics to become the experts. Through collaborative activities, students can express themselves and their mathematical ideas to their peers, developing self-reliance and focusing on the mathematics, not what the instructor says
  •  Language Matters. Research has shown that even in elementary school, acknowledging the gender of our students reinforces stereotypes. While we might not be saying ‘boys and girls’ in our Calculus classes, we are certainly using language that affects our students. This recent Teaching Tidbits post provides several ways for us to use language inclusively to support our students’ identities as mathematicians including statements like “When a mathematician approaches this problem, she…” or “When you explain it like that, you are really thinking like a mathematician.” 
  •  Don’t Lecture. If you’re reading Teaching Tidbits, chances are you are interested in doing more than lecturing to your students. However, lecturing is still the preferred teaching method of many mathematics instructors. Research has shown us over and over that interactive teaching is one of the best ways to reduce the gender gap in achievement, and a 2014 report told us just how much we were neglecting all students when using only lecture in our classrooms. Moving from ‘sage on the stage’ to ‘guide on the side’ is a powerful way to give all students, especially women, the opportunity to engage in classroom activities and discussions. One technique for providing this type of classroom experience is through Inquiry Based Learning, described in a recent post with some resources here.
  • Know Your Own Biases. One of the most important social interaction factors that can play out in our classroom is implicit bias. Before we can address any bias we see in our students, we need to understand our own biases. These freely accessible Implicit Association Tests allow us to face biases we might not know we’re carrying with us and help us to become more equitable instructors.


Additional related resources:

 Deshler, J. & Burroughs, E., (2013). Teaching Mathematics with Women in Mind, Notices of the American Mathematical Society, http://www.ams.org/notices/201309/rnoti-p1156.pdf.

Tuesday, February 6, 2018

MAA IP Guide – Assessment

By Rick Cleary (guest blogger), Babson College

A note from the Editors: This semester Teaching Tidbits will have several posts highlighting the new Instructional Practices Guide (IP Guide) from the Mathematical Association of America (MAA). The MAA has a long tradition of reporting what content should be taught in the mathematics classroom through its Curriculum Guide; now the new IP Guide addresses how things could be taught in the mathematics classroom, how one could to design that experience, and how one could assess that experience. The suggested practices are well grounded in research on student learning. In our first post about the IP Guide, we dive deeper into the Assessment Practices section of the guide. Thanks to Rick Cleary, a lead writer for this section, for providing this post.
The opening statement of the Assessment chapter of the MAA Instructional Practices guide makes the following claim: Effective assessment occurs when we clearly state high-quality goals for student learning, give students frequent informal feedback about their progress toward these goals, and evaluate student growth and proficiency based on these goals. The chapter details some of the ways that effective assessment can be implemented in various types of courses. Many of the same assessment principles apply, whether you are from a big or small school, whether you teach large or small numbers of students, no matter what your lecture/active learning balance, on or off-line, developmental courses through graduate seminars. This portion of the IP Guide is designed to get colleagues thinking and talking about grounding both formative assessments that take place throughout the course and summative assessments at the end of a course in appropriate learning goals.

There is a fine line between assessments that are challenging and assessments that are discouraging. Once students become discouraged, it is hard to get them back on track. For example, traditional lecture-based instruction methods have been associated with traditional summative assessment procedures such as timed exams with questions in very specific formats. Recent research in mathematics education recommends classroom practices that provide ongoing lower stakes assessment to promote student engagement. New technology such as clickers and online polls or quizzes can help faculty provide these types of opportunities. Through vignettes grounded in the experience of the writers, the IP Guide illustrates these developments, providing instructors the tools they need to be creative as they design appropriate and equitable assessments for their courses.

The IP Guide chapter on Assessment provides both a research framework and practical tips needed to implement effective assessments that encourage, rather than discourage, student learning. It considers ways to make assessment consistent with course design and practice to promote effective learning for all students. Rather than seeing assessment as a mandate from an administration or an accrediting agency, the IP guide shows there is great value in creating a positive culture of assessment for students, faculty and departments.

Download a copy of the MAA Instructional Practices Guide today.

Tuesday, January 23, 2018

The One Question Calculus Final

By Lew Ludwig (Editor-in-Chief), Denison University


As the semester begins and we prep for classes, the practice of backward course design is a powerful way to get the most of the learning experience for our students. With this in mind, I thought I would share one of my favorite exam questions for a first semester calculus course, which appears below. I call it “the one question calculus final.” Now of course this is tongue-in-cheek, as the one question has over 15 questions. Nonetheless, this one question tour de force covers the full range of a first semester calc course. To substantiate this bold claim, I found a comprehensive list of typical topics in such a course at Wolfram Mathworld Classroom. The chart below cross-references each alphabetically listed topic with its specific question. While some questions touch on a range of topics, the cross-referencing refers to the primary reference.


Why I like this question:
I really like this question because it requires students to problem-solve, not just memorize a procedure. For example, instead of providing a typical composite function and asking for the derivative, my students must understand that the function in question (f) is a composition, know how to apply the chain rule, then read the graph to fill in the missing values. While I admit that question (m) may be a stretch for a Riemann sum, question (p) helps students realize the a definite integral is just a question about area. I especially like questions such as (j) and (k) that help students intuitively use important results like the intermediate value theorem or the mean value theorem. Finally, I like this question because no piece of technology can do the work for you. I can safely permit graphing calculators during my final without the fear of some CAS (computer algebra system) making short work of my exam.

How I use this question:
Please do not unleash this question on your students without prior exposure! My students have been working with this type of question for the whole semester. The beginning of the semester would focus more on limit questions like (a)-(e). By the second test of the semester, my students work on questions like (f) and (g) to understand the with mechanics of differentiation. By the third test, we get practice with applications of the derivative with questions such as (h)-(l). And by the end of the semester, questions of the type (m)-(p) test students’ understanding of the definite and indefinite integral. To make sure students do not forget prior material, my tests include questions from previous tests. Cognitive psychologists refer to this technique as interleaving.

How I grade this question:
Since this question has so many parts, I only count each sub-question for one point out of a 100-point final exam. Okay, I do ask other questions beside this one! I grade each question as right or wrong, no partial credit. While this may seem harsh, by giving each question a small point value, a student can miss a few of these questions without serious detriment to the overall grade. Moreover, past tests have shown a student’s exam score tracks fairly closely with performance on this question.

How to modify this question:
Of course there is a myriad of ways this question could be modified. For one, change the graph. When I initially developed this question, I would make sketches of the graph by hand. Now the online graphing program Desmos helps me produce graphs that are easy to read and export into LaTex or word processing programs. Students can contribute by creating their own questions for a graph you provide. Or you can turn that on end and have students provide a graph based on questions you provide. However you use it, you will find this focused cumulative approach will help deepen your students understanding of calculus.