Tuesday, May 15, 2018

Farewell from the Teaching Tidbits Blog

Thank you for visiting the Teaching Tidbits blog, hosted by the Mathematical Association of America, written for mathematics instructors by mathematics instructors. Since 2016, we have been posting regularly during the academic year to help you keep up with the latest educational research and pedagogical practices as the MAA IP Guide was being developed.

With 121,470 total pageviews, and 34 total blog posts, we have helped you find and implement the latest advancements in evidence-based pedagogy for your math courses. We all want to engage our math students and invigorate our math classes, and we recognize the challenges of finding the time to research, plan, and execute new ideas. The short and practical posts can serve as a evidence-based resource for new faculty workshops and ongoing professional development. You might even try using one as food for thought to spark discussion about pedagogy in department meetings. Keep this site bookmarked and connect others to the collective wisdom of the Teaching Tidbits blog.

We hope you will let us know how the Teaching Tidbits blog is helpful to you as you improve your teaching and your students' learning.

Thank you for your support!

Tuesday, May 8, 2018

Teaching Tidbits Blog 5 Most Popular Posts

As the summer approaches, the MAA Teaching Tidbits blog is coming to a close after two successful years of helping to increase student engagement in math courses and helping to build confidence and skills in math faculty.

The blog was designed to be a source for evidence-based teaching practices while the MAA Instructional Practices Guide was in development. Now that the MAA Instructional Practices Guide is available, we encourage our readers to refer to our blog posts for an in-depth treatment of course design, classroom practices, and assessment. Thank you to our readers for their support. We hope you found Teaching Tidbits a useful resource to help improve your math classroom.

Of our 34 posts over the last two years, these are the 5 most popular posts:

We hope you will continue to share the tips on this blog with colleagues and let us know how it has been useful to you. Thank you for your support of the blog!

Tuesday, May 1, 2018

5 Reasons Math Modeling Should have a Place in your Undergraduate Curriculum

by Rachel Levy, Harvey Mudd College

I believe students should see mathematical modeling in school because this way of engaging mathematics can be both inclusive and personally empowering.

Mathematical modeling describes processes used to understand, describe, and predict real-world situations by employing whatever quantitative and computational approaches are useful. Many aspects of applied mathematics, statistics, operations research, algorithms and, data science can fall under this umbrella.

When I say mathematical modeling should have a place in your curriculum, I don’t only mean exposing students to ubiquitous well-known models such as the mass-spring system or predator-prey. I mean the creative process of modeling, like the work students would do in a mathematical modeling competition such as the M3 Challenge or MCM/ICM or in the workplace. I believe this can start as early as kindergarten, though of course here we focus on undergraduate education.

Mathematical modeling is a topic of increased visibility in the education system. First, it appears in the US Common Core Standards for Mathematical Practice (CCSSM-MP4) for students in K-12.  Second, job opportunities in business, industry, and government for those trained in the mathematical sciences are increasing while numbers of tenure-track opportunities are flat or decreasing. As someone on a job panel put it:  I don’t care what flavor of mathematics you studied in school and bring to the job, once you work for my company, you are an applied mathematician.

Here are 5 characteristics of mathematical modeling that can make it inclusive and personally empowering:

1. Mathematical modeling involves genuine choices. Students can be involved in choosing the situation, the topic, the mathematical and computational tools and, the metric for the success of their model. When students make choices, they can feel a different connection to and ownership of mathematics – mathematical tools and ideas are theirs to choose and use. In this way doing mathematics is more like the creative act of a research mathematician, a writer or an artist – the goal of the work is to create something new and of intellectual value.

2. Students learn how objective functions and assumptions/constraints matter. The general public and the media express some skepticism about mathematical modeling because it seems like maybe you can make the data tell any story you want.

Mathematical modeling helps people learn that by clearly communicating your objective function (goal) and assumptions/constraints you can more honestly and clearly convey why the model makes the prediction that it does. Students also learn that aspects of social justice and equity can be brought into play through these aspects of the model. For example, maybe your business goal is to help the people while staying financially viable, rather than just making the most money.

3. Mathematical modeling can draw on interdisciplinary ideas and ways of knowing. Because modeling problems are situated in the real world, they almost always have interdisciplinary aspects.  Maybe the important information is about how a business is run, or how a machine works, or how humans behave. What are the preferences of the end user? What will make a solution useful and usable?  How can visualization help communicate the methods and solutions?

4. Real world problems help students see that mathematics is everywhere. People sometimes think of mathematical modeling as only applied mathematics, but once the problem is extracted and refined to its mathematical form, students can see the structure and beauty of that mathematics.  Mathematical modeling can give students ways to practice using familiar tools as well as reach for new mathematical and computational ideas.

Reflective aspects of the modeling assignment can help teachers see how student perspectives on mathematics are changing. After modeling, students often report seeing things in a new way (such as noticing an elevator bouncing as it reaches the bottom of a building) and wondering how to describe this mathematically. Students can also envision career pathways that will call on these skills.

5. Mathematical Modeling is a team sport that focuses on process as much as product. It may be possible to work on a model alone, but almost always models benefit from multiple perspectives and ways of thinking. Well-structured modeling problems encourage teams to start with a simple common solution and then provide opportunities for teammates to add some complexity and test to see if it improves the model. Teamwork has many challenges, and teams need to be monitored so that students are respectful of all voices.

A rich modeling problem will have multiple solution methodologies and solutions (which depend on the selected objective and constraints). Students can experience mathematics as more than a course with feedback that judges when their answers are right and when they are wrong. This eliminates the phenomenon when every “X” mark on a problem can be like a small cut in a student’s confidence and erode their sense of belonging as a mathematics-doer.  Instead, students can discuss how various aspects of solutions make them useful and what ways of communicating solutions are most effective. Students can iterate and improve their answers. Instructors can find ideas of value in every solution.

Here are three ways to make mathematical modeling available to your students:
  •  Build a final project into a course (could be Calc, Prob/Stats, Lin Al, DEs…) and make sure the project involves some choice. You can still point to a particular tool (such as single value decomposition or compartment modeling) if you need the project to hit a learning objective while leaving the context open to the students.
  • Propose a mathematical modeling course for your department. It doesn’t have to be a full number of units – students can take it as an elective. It could be a once a week seminar and it could be pass-fail. Or it could be a full unit, graded course.
  • Advise a modeling competition (MCM/ICM) team. Provide time and space for students to do some practice problems and read winning papers. Discuss mathematical and computational approaches. This could be a student club activity or a departmental endeavor.
To learn more about the teaching and learning of mathematical modeling, see these free downloadable reports:

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!  

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.