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.

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.

Tuesday, December 12, 2017

Use the MAA Instructional Practices Guide to Maximize Student Engagement with Math

By guest bloggers Martha Abell, Georgia Southern University, and Linda Braddy, Tarrant County College

If you enjoy Teaching Tidbits, be sure to get an electronic copy of the Mathematical Association of America Instructional Practices Guide, which is chock full of great ideas to help you and your students. The guide is available as a free download on the MAA website. The MAA IP Guide is intended for all instructors of mathematics:
  • New graduate teaching assistants 
  • Experienced senior instructors 
  • Contingent faculty member at a two-year institution 
  • New faculty member at a doctoral-granting institution 
  • Instructors who want to transform their own teaching 
  • Mathematicians delivering professional development to colleagues
The guide boldly responds to challenges articulated in the 2014 INGenIOuS report and the 2016 Common Vision report, which both call for transformation within the mathematical sciences community toward deeper, more meaningful learning experiences. The MAA IP Guide was written by faculty from a wide range of institutions and professional associations. It supports the use of evidence-based instructional strategies to actively engage students in the learning process by providing “how to” suggestions. The supporting research focuses on effective teaching, deep student learning and student engagement with the mathematics both inside and outside the classroom.

The content organization is based on three interconnected foundations of effective teaching: classroom practices, assessment practices, and course design practices. It also addresses two key cross-cutting themes, technology and equity, that permeate all three practice areas.

The Classroom Practices chapter provides multiple entry points to help instructors implement practices that foster student engagement. Topics include building community within the classroom, using collaborative learning strategies, developing persistence in problem-solving, and selecting appropriate tasks.

The Assessment Practices chapter offers guiding principles to assess student learning through both summative and formative assessments. Most instructors routinely employ summative assessments such as quizzes and exams, but some may not be as familiar with formative assessment practices which can inform their decisions during the course regarding “next steps” in instruction based on students’ current needs.

The Design Practices chapter guides instruction planning and revision to maximize student learning. Each chapter includes vignettes, practical tips, and references to research-based studies that support the effectiveness of the practices. Each chapter also offers strategies for trying new instructional methods and avoiding common pitfalls.

The MAA IP Guide will be a topic of discussion at the Joint Mathematics Meetings in San Diego in January 2018, most notably at the MAA Invited Paper Session on this topic, Thursday from 8:00-10:50. The writing team is excited to present this resource to the community in support of deep, meaningful experiences for all instructors and students. We promote the use of engaging instructional practices in your own department by sharing this resource with your colleagues!

Tuesday, November 28, 2017

“I’m Worried About My Grade.” How to Pre-empt the End of Semester Panic

By Julie M. Phelps, Contributing Editor, Valencia College

As the end of the semester nears, educators brace for the inevitable student questions about their final grades. If you are anything like me, this can be distracting from my goal of the class: to teach mathematics and make a difference in student lives. Sometimes I find myself dreading the last couple weeks because many students are stressed and solely focused on the final grade, not on learning ways to utilize mathematics in their major.

This year I decided to try something new and pre-empt student questions about their grades by having them reflect on their class work and engagement. To do so, I showed a YouTube video called “I am worried about my grade” to my College Algebra students at the beginning of the semester (just before the first test). While very basic, the video goes over the many ways we educators evaluate student performance, and how we make time for students to approach us outside of class.

After showing the video, I asked the student to reflect on it and write a brief essay about what they saw in the video. Here are the themes from their essays that you can use to set course expectations early in the semester (and pre-empt the end of semester panic):

1. I need to be a responsible college student/adult.
  • Ask for help/ask questions/make sure I understand/go to tutoring. 
  • Always try to improve your work/use the resources. 
  • Turn in assignments on time/don’t procrastinate. 
  • Come to class/participate. 
  • Passing the class is up to the student. 
  • Take your education seriously/have good work ethic/don’t be lazy/take the initiative/have a good attitude. 
  • Learn from your mistakes. 
  • Manage your time. 
  • Taking this class is not about a grade but about learning mathematics.
2. There are plenty of opportunities to get help.
  • Go to office hours.
  • When the professor takes time to be flexible within your schedule, make sure to show up.
  • Go to tutoring.
3. The professor is there to help!
  • Establish a connection with the professor.
  • "Professors have a lot of students, so if they offer to help...accept it!"

Three notable quotes from student essays on being responsible and engaged students:
  • “OMG… I’m the bear… you’re talking directly to me aren’t you? Don’t drop me… I get it! I need to make a schedule, study, and ask for help immediately. Thanks for the wake-up call.”
  • “By showing me this example, Dr. Phelps showed me that grades aren’t given, they’re earned. After realizing this, I’m going to utilize every resource that is available to me to show that I have the potential to get passing grade and higher, and that I have the capacity to be good at math and enjoy the work that I do in the class and in the other math classes.”
  • “The video that we watched in class shows in a very droll way a conversation between a student and a teacher. I identify his [the students’] attitude as a student from high school or a middle school, since they only want to pass the year, not to learn things for the future, they can’t see the utility of their learning.”
Because of this activity, I am now able to focus on what the end-of-term should be about: teaching mathematics for long-term use and retention. At the end of the semester, we educators only have a few more classes to make an impact on our students, and our students only have a few more class meetings to get the most out of the semester-long experience. I encourage you to have this conversation with your students before the end of the semester so everyone can have productive class time.

Tuesday, November 7, 2017

The Role of Failure and Struggle in the Mathematics Classroom

By Dana Ernst, Contributing Editor, Northern Arizona University

The purpose of this post is to generate discussion and to get us thinking deeply about our teaching practices. This post introduces teaching concepts that researchers and educators developed to promote student success.

In an attempt to understand who is successful and why, psychologist Angela Duckworth has spent years studying groups of people in a variety of challenging situations. Over and over again, one characteristic surfaced as a significant predictor of success: grit. According to Duckworth, grit is passion and perseverance for long-term goals. One of the big open questions in cognitive psychology is how to develop grit.

Growth Mindset
Research suggests that one of the key ingredients to fostering grit is adopting a growth mindset. Psychologist Carol Dweck defines growth mindset as the view that intelligence and abilities can be developed with effort. In contrast, a fixed mindset is the belief that one's talent, intelligence, and abilities are fixed traits with little room for improvement. Dweck has found that:
 "People’s theories about their own intelligence had a significant impact on their motivation, effort, and approach to challenges. Those who believe their abilities are malleable are more likely to embrace challenges and persist despite failure." 
 The claim is that those with a fixed mindset will tend to avoid challenges, while those with a growth mindset will embrace challenges. Research has shown that effort has the potential to physically alter our brains, strengthening neural pathways and essentially making one smarter.

A growth mindset increases potential. Teachers who have a growth mindset about their students can help share this perspective and reap its benefits. Views like "some people just aren't good at mathematics," or "not everyone is cut out to be a math major" exhibit a fixed mindset and may severely limit the potential of our students.

Productive Failure
At the core of the growth mindset paradigm is the notion of productive failure. While mistakes and failure are part of the learning process, productive failures provide an opportunity to learn and grow. According to Manu Kapur, productive failure activates parts of the brain that trigger deeper learning. Unfortunately, failure and mistakes are stigmatized in our culture, especially in many mathematics classrooms.

One of the key ingredients to a successful active learning classroom is getting the students on board. One approach is to explain the concept of productive failure to our students and go out of our way to point out when we see it happen in class (this is a part of being more transparent in the classroom). Some teachers even make productive failure part of the course grade.

If we want students to feel comfortable taking risks, making mistakes, and failing, then we need to provide an environment where this type of behavior is encouraged. I'm not suggesting that we lower standards (or allow wrong or inadequate answers to be sufficient), but give space for tinkering and failure along the way. Our assessment and grading practices need to jive with our philosophy of productive failure. I think that it is important to build opportunities for failure into our everyday structure.

Productive Struggle
Due to the negative connotation of the word "failure,” I started referring to productive failure as productive struggle, thinking that these two phrases meant the same thing. However, over time, I have come to view these two concepts as related but not identical. Let's do a little experiment. Take a minute to look at the following list of word pairs, but do not write anything down.

Now, without looking at the list of words, write down as many pairs as you can. You do not need to remember where any missing letters were nor which column a pair was in. Next, looking at the table below, count how many pairs you found in column A versus column B.

This list of words is from The Talent Code by Daniel Coyle (although I've rearranged the order a bit). According to Coyle, studies show that on average people remember three times as many pairs in column B, the one with missing letters. The claim is that a moment of struggle (cognitive demand) makes all the difference. I regularly utilize this exercise on the first day of class as part of my Setting the Stage activity, the purpose of which is to get students on board with an active learning approach. The point of the exercise is to have students experience in a simple, yet profound way, the value of productive struggle.

This exercise does an excellent job of distinguishing the difference between productive failure and productive struggle. The key to success here is the struggle, not failure. This exercise also helps pinpoint the "productive" part. Imagine we removed most of the letters from the words in column B. As more letters are removed, the less productive one will be in figuring out what the pairs are. It is important to keep this in mind when designing tasks for our students to engage in. There is a "Goldilocks zone" (related to the zone of proximal development), where struggle will be most productive. How much scaffolding we should provide our students will depend greatly on who our students are.

The Goldilocks Zone
The following images, taken from Patrick Rault, provide an analogy for productive struggle and productive failure.

All three images are meant to represent a challenge or task. In Figure A, all the obstacles have been removed; struggle is minimized, and failure is unlikely. There is no opportunity to develop grit, and we have provided too much scaffolding for our students. This is what happens when teachers demonstrate a certain type of problem and then ask students to do problems using the demonstrated method. Certainly a useful skill, but we want more.

In Figure B the rocks are slippery, and the path is not clear; the risk of failure is high, and most attempts to cross the river will involve a tremendous amount of struggle. Whether the failure and struggle are productive depends on how agile the person is crossing the river. For many, the risks may be too high. Perhaps only a few make it across. This is what happens when we only provide students with tasks that are mostly beyond their current reach.

Figure C is meant to represent the Goldilocks zone. The path across the river isn't trivial, but hopefully the risk of failure and amount of struggle isn't too great. As students become more proficient at crossing the river, the rocks in Figure C should be moved further apart, and potentially introduce a maze of rocks for students to navigate. Ultimately, we want students to have developed enough grit to strive for crossing the river in Figure B. It's possible that eventually the stepping stones in Figure B become the new Goldilocks zone.

Stan Yoshinobu says that the right question to ask is "how are my students intelligent," as opposed to "how intelligent are my students?" What the Goldilocks zone looks like for one group of students may be wildly different for another group of students. In fact, the ideal set of stepping stones is different for each student. This may make the task of providing this experience for our students sound too daunting, but we would be lying to ourselves if we thought the problem went away by not providing the opportunity. Being forced to confront these issues is a feature, not a bug.

Here are some questions to ponder. We welcome respectful dialogue in the comments section.

  • To what extent is grit necessary or sufficient for a student (or even a research mathematician) to be successful in mathematics? 
  • How do we go about fostering a growth mindset and altering a fixed mindset? 
  • How do we provide a classroom environment where risk-taking is encouraged, mistakes and failures are valued as part of the learning process, and high standards are maintained? 
  • How can we identify productive versus nonproductive failure and struggle? In particular, how can we locate the Goldilocks zone for a given set of students?