Tuesday, November 12, 2019

Interview: Professor Matthew Boelkins, Grand Valley State University and Active Calculus

Hi everyone! A massive thanks to Professor Matthew Boelkins, Grand Valley State University, for taking the time to share some info about his project, Active Calculus.  -SY

I've had the good fortune to serve on the math faculty at Grand Valley State University in Allendale, MI, for more than 20 years.  I'm also one of the two editors-in-chief of PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies), and have held that position for 5 years, following 5 years as associate editor.  My interests span a wide range of undergraduate teaching and learning of mathematics, but much of my recent creative energy has been focused on developing textbooks that encourage active learning.

2. Briefly describe “Active Calculus”? How much does it cost to the student?
Active Calculus (there are both single and multivariable texts; I'm the lead author of the single variable one, and will focus on that in my responses) is a free, open-source textbook designed for a standard calculus sequence taught from an active learning perspective.  Rather than lots of worked examples, most of the text is structured around activities for students that are designed to be completed before or during class, the latter with encouragement and feedback from peers and instructor.  Interested people can learn more at https://activecalculus.org/.

3. What are some of the reasons why you decided to write “Active Calculus?”
There were two main reasons.  First, I read an article in MAA FOCUS about the \$21M home built by the author of a popular calculus textbook. I figured that if the author had earned that much in royalties, the publisher had likely made \$210M.  It was no longer tenable for me to ask my students to pay \\$150 for a textbook that had ideas in it that had been well understood by humankind for decades, even centuries.  I wanted to write something that would be free for them.

In addition, while at the time I began writing (around 2010) I didn't fully know the scholarship that tells us why active learning is better,  I used a lot of active learning in my own teaching and had found my students to be more successful than when I used to lecture.  Having developed a collection of activities for calculus to use in my own teaching, I had the idea to use those as the foundation for the textbook, and that's what led to Active Calculus.  When the Freeman report came out in 2014 (https://www.pnas.org/content/111/23/8410), that provided added motivation for making the text a good one.

4. What is a typical day like in a course using “Active Calculus” when you teach using it? Can you use it with flipped learning or with IBL methods?
For each class meeting, I prepare a written script with estimated times.  In advance of most classes, students complete a "daily prep assignment" that normally consists of a short reading, a preview activity from the text, and 1-2 additional questions; students spend 30-45 minutes completing these, and their work is graded on effort and completeness for 5\% of their semester mark.  Most classes then begin with a "daily prep debrief & discuss" (6-8 minutes) where students check in with one another and see what questions they might want to discuss as a class.

From there, we usually engage in some brief (5-7 minutes) lecture & discussion to build on daily prep and set stage for an activity.  Students then work in groups of 3-4 on an activity from the text for 15-20 minutes, followed by or including some discussion for closure, transition, and new ideas (adding another 5-10 minutes), and then we are on to the next activity for 15-20 minutes.  Often I teach our 4-credit calculus class on a schedule with two 2-hour meetings a week, so we basically rinse and repeat this schedule for a second hour, but without a daily prep assignment to start the 2nd hour.

I've written a blog post that has some more information and detail, including references to key preface sections of the text for students and instructors, as well as a video for students on how to use the text: https://opencalculus.wordpress.com/2019/08/05/how-to-use-active-calculus/

I think the text is particularly well-suited to a flipped learning setup:  when I have students complete daily prep assignments, they are doing some key basic learning on their own outside of class.  The in-class activity-driven style also fits with engaging students during class in the some of the most important and demanding work of the course.  Some of my GVSU colleagues have created screencasts that accompany the text, and these would work especially well for a flipped experience:

For IBL practitioners who have worked with that approach using more traditional texts like Stewart or Hughes-Hallett, I think you'd find Active Calculus to be a suitable companion for such a course.

5. What are some of the responses by faculty and students after using “Active Calculus”?
For views of some faculty who have reviewed or used the text, see https://open.umn.edu/opentextbooks/textbooks/active-calculus-2-0.  I get a lot of email traffic about the text, and many of those responses express gratitude for the text.  Recently, an instructor who is a first-time user this fall sent me a very kind thank-you note: "I wanted to write to say how much I've appreciated Active Calculus.  I was a bit suspect (I've been using Stewart), but I am so impressed with it.  I like the problems, the students working in class, the organization ... it's really quite good.  I'm excited to be teaching this class again.  And thanks for saving our students some money as well."

This fall, I also surveyed Active Calculus users via my email list and Twitter, and in an open comment part of the survey, I got additional feedback such as:

"I appreciate you and your work on this so much. It's really let me teach calculus in the way I want to without having to create the material from scratch. A million thanks."

"The book is well thought out with great examples. When I teach the class, I really feel like I am actively working through the book with the students. One comment our Quantitative & Symbolic Reasoning Center director told me was that students in Active Calculus sections are asking why a concept is true and being stuck on the theory, while students not using AC are asking questions on the algebra and not thinking about the concepts of the course. "

"I really appreciate the structure and the way that the text works. It has been a game changer for my teaching of Single Variable Calculus! Our pre-calc teacher is now using Active Prelude as one of her core texts and we are excited to see how that influences students being prepared for my class."

"I've been using Active Calculus since my first year of teaching AP Calculus AB and I can't imagine using any other curriculum.  I cannot thank you enough for the time and energy that you have put into your work and my hundreds of students over the years thank you as well."

As to what students say, I think that many students find the book different at first, and thus at times frustrating.  They have been trained to expect a book that has lots of fully-worked examples, followed by exercises that are similar to the examples.  Active Calculus is not that way.

Some of the instructors who responded to my survey commented on what students say:

"It's an excellent book. Students and I both really appreciate its clarity and accessibility. Just yesterday I was recommending to a student the very nice "Summary" bits at the end of each section, and they said, "oh, that's exactly what I was looking for.""

"I really like the text and got mostly positive comments from the students on the course evaluations last year.  In the past, the comments were almost all negative or neutral."

"I asked my students and they expressed appreciation for how easy it is to read. Some students also said they wished there were more examples, which gave me an opportunity to remind them why there aren't more examples. :)"

6. How widespread is “Active Calculus”?
One of the challenges of having a text that's free and available online is knowing exactly where it is being used.

I'm aware of at least 17 4-year universities, 2 2-year colleges, and 5 high schools that have formally adopted Active Calculus as their required textbook.  People from at least another 30 institutions have responded to my recent survey to say they are using it fully in their own course, even though the text hasn't been adopted by all of their peers.

Here's the list of adopters I have at present:
California State University, Monterey Bay
Carroll College
Doane University
Dordt University
Lane Community College
Laurentian University (@ St. Lawrence College)
Lebanon Valley College
Lenoir-Rhyne University
Scottsdale Community College
Sonoma State University
St. Mary's College of Maryland
Texas Lutheran University
The College of Idaho
Vermont Commons School
Vernon Hills High School
Westfield State University
Westminster College (SLC)
Westmont College
Wyoming Seminary

7. Could this book be used in high school?
Absolutely, and several of them have publicly shared their experiences.  For instance, Dave Sabol of St. Ignatius High School has used AC for several years and writes about corresponding activities he has developed on his blog, https://therationalradical.wordpress.com/calculus-resources/.  Jim Pardun and Steve Korney of Vernon Hills High School recently presented on their work teaching AP Calulus using AC at the regional NCTM conference in Nashville.

The overall goal is to keep making both Active Calculus and Active Prelude to Calculus better.

For Active Calculus, the original text was written in LaTeX and later converted to PreTeXt, which allows the HTML output.  I'm realizing that I haven't yet taken advantage of many of the features PreTeXt offers, such as having better cross-referencing, a better index, and more interactive features.  A first goal is to make revisions to take advantage of those features.  I am also considering adding Sage cells to the text to offer some embedded computation and experimentation for students; having some interactive computational opportunities is a second thing I hope to incorporate in the not distant future.  And a third significant goal is to have some additional exercises, ideally with many of them focused on modeling and applications.

For Active Prelude, this is the first year the text has been public, so I'm waiting on some user feedback to see where to focus energies next.

9. Anything else?
People can learn more at https://activecalculus.org and https://opencalculus.wordpress.com/, and I always appreciate hearing directly from users or people interested in the text by email at boelkinm at gvsu dot edu.

Thursday, October 24, 2019

Even Mr. Miyagi had student buy-in issues. If you've seen the movie, The Karate Kid, you know these "wax on, wax off" training scenes, which is clearly active, Daniel-san-centered teaching. You wouldn't just want to only watch videos or watch demos, if you need to face real humans in a karate tournament. It seems like avoiding a kick to the head might require more than factual knowledge of what a kick is and knowing about the existence of a block or an avoidance move.

Let's take a look at a key segment, where Daniel doesn't see the point of all his practice, and has a blow up.

When viewing this scene from a teaching point of view, it comes down to Mr. Miyagi not framing and signposting the activities.  "You're doing this to get stronger at ..."  Instead, he assigned Daniel exercises, and did not tell him what these exercises were for. So naturally Daniel was upset, because he could not see the connection to learning karate. Finally, Mr. Miyagi does some student buy-in work, and has Daniel demonstrate that he actually has strengthened his defense abilities. At that point, Daniel made a big step forward in student buy-in, after getting some positive feedback from this teacher.

(Now of course this a movie, and drama is needed. I wouldn't change the movie, nor is this a film critique.)

If we boil things down to their essence, the situation is (a) instructors know what the connections are and the purpose of the work, however (b) students don't always see (or can't see) these connections, since they lack expert insight.  Hence, it's vital for teachers to signpost, "We are learning in this way, because..."

Returning to The Karate Kid, at some point, there has to be authentic meaning to the work. Daniel fully buys in after the "anniversary scene", where it is revealed that Mr. Miyagi had suffered tragedy in his family life, while earning a congressional medal of honor for his military service to the United States. This experience provides much needed perspective for the young man. The next scene is Daniel practicing with intent on his own - a sign of full buy-in.  Game on!

This suggests that buy-in has multiple levels. It's one thing to see why you are doing an exercise. It's another to be fully committed to the path forward. This is where assignments and activities focused on the meaning and value of the work can be useful. This can include but is not limited to the value of problem solving, growth mindset, and inspirational content, where students have an opportunity to learn universal lessons about what their education is for.

Wax on, wax off,
Wax on, wax off...

Friday, September 27, 2019

Cold Calling Pitfalls

This blog post is about cold calling. Randomly calling on students can go well, but it's not so easy as pulling a name and asking, "What's the answer!?"

The pitfalls of basic random cold calling is that it's not an equitable practice in its barebones form. Some issues include (but are not limited to) the following.

1. Making a mistake in front of the whole class can be stressful. This is a high risk situation.
2. Not everyone thinks fast, so they may be in the process of figuring something out, and then called on the spot to have the goods, right then and there.
3. Thinking fast is not necessarily smart, and thinking slowly is not necessarily not-smart. But having to be quick on your feet biases the definition of smart as having the answer quickly in class on the spot.
4. Stereotype threat can be triggered. Some students are the only one of their identity in your class. If this student makes a mistake, then the stereotypes could be "validated."
What can we do?
Randomizing who contributes does have a positive facet. It spreads the chances around. That's good. The key is to do it in an inclusive way.

Here's one example:
Instead of going with something like "What's the answer!?"you can add think-pair-share into the mix.  First, state the question or task, and ask students to think first and the talk to their partner. Then when you select a person at random, the prompt could be, "Share with us what you discussed with your partner." What students share could be a partial answer, a question, a comment, the whole answer, or perhaps asking for another minute to finish their conversation. If framed as a way to generate discussion and engagement, rather than having to be right, this can reduce anxiety and be a more inviting experience.

Riffs: We can think of think-pair-share as the base layer, and add on or adapt layers above it. Instead of randomly calling on a pair, you could visit a few groups and see what they tried. You can usually very quickly find a pair willing to share. Quiet students who have a useful idea can be encouraged to share, and this specific teacher move is where you enact inclusion! The quiet student is invited in by the instructor and is validated for having something worthy of being shared.

Keeping track of who has shared will then help you spread chances equitably.  This riff works well earlier in the term, as you are developing student buy-in and comfort with talking about and discussing math. It's lower stakes, and doesn't put people on the spot.

Another riff is for classes that include student presentations. Assigning randomly the first chimers is a way to spread who makes the comments. Otherwise, it's the usual people raising their hands again and again. This can be done by selecting two students, then asking everyone to talk to their partner about the presented work. The selected students chime in first. They can ask a question, comment, or give a compliment. Again the contributions are done in a way, where students have time to think, discuss, and prepare remarks. Once the first chimers have made their comments, you can open the floor for further comments.

Cold calling can be warmed up with some additions and tweak. Using a small amount of time, allowing for thinking and talking, and taking perceived judgment off the table, creates an environment where students are less concerned about how they might appear and more focused on the Math.

Monday, September 16, 2019

Mathematics and Social Justice: Interview with Dr. Kyndall Brown, Executive Director, California Math Project

Hi everyone! It's my great pleasure to share with you an interview with Dr. Kyndall Brown, California Math Project. The main theme of this interview is Mathematics and social justice.

Please tell us about yourself and your current position and how you became interested in mathematics education.
I am the executive director of the California Mathematics Project. I oversee a statewide network of professional development organizations whose charge is to provide professional and leadership development for k-12 teachers of mathematics. There are 19 sites of CMP that are housed at UC and CSU campuses. We bring together mathematicians, mathematics educators, and classroom teacher leaders to put on institutes, as well as school-based professional development programs. We are supported by state and federal funds. We are mandated to center our work in low-performing schools and district.

I started my teaching career in 1985 at Manual Arts HS in LAUSD. Because I had a BS in mathematics, I was able to receive an emergency credential to teach secondary mathematics. I immediately discovered that I needed some support if I was going to get the hang of teaching. I think this is where my interest in mathematics education began. During the Summer after my first year of teaching, in order to get the units I needed to renew my emergency credential, I attended two CMP institutes; one at UCLA, and one at CSULA.  In the fall of 1986, I entered a credential program at CSUDH, where I had my first courses in educational psychology and learning theory. I became exposed to Jean Piaget’s theory of constructivism, which had a huge impact on my teaching  practice. I always took advantage of professional development opportunities and learned a lot from them. Fast forward to 1994. I was teaching at a high school when a coworker and colleague of mine invited me to apply to attend a Summer leadership institute at the UCLA Mathematics Project. I attended the four-week institute and ended up becoming a teacher leader for the project. I presented workshops and sessions at other UCLAMP leadership institutes. UCLAMP would use me as a professional development provider in some of their fee-for service contracts with schools in LA County. In 1999, was asked to come on full time at UCLAMP as co-director. Several years later, I took over as the director of UCLAMP. In 2012, I became the executive director of the CMP statewide office.

Math instructors may not be aware of the issues related to Mathematics and social justice. Why does equity and inclusion matter?
Equity and inclusion matters because we live in a multi-racial, multicultural, democratic society. In order for a democracy to function, we need an educated electorate. Technology is becoming more ubiquitous. More industries and jobs are becoming automated. Instead of being taught how to perform rote computations with speed and accuracy, students need to learn how to become critical problem solvers

Education needs to keep up with the trends in the larger society.  The Partnership for 21st Century Learning (P21) has created a framework that describes the skills learners will need to be successful in the 21st century. http://www.battelleforkids.org/networks/p21

P21 states that 21st century  learners must:
• Learn from and working collaboratively with individuals representing diverse cultures, religions, and lifestyles in a spirit of mutual respect and open dialogue in personal, work, and community contexts
• Be open and responsive to new and diverse perspectives; incorporate group input and feedback into the work
• Communicate effectively in diverse environments (including multi-lingual)
• Demonstrate ability to work effectively and respectfully with diverse teams
• Respect cultural differences and work effectively with people from a range of social and cultural backgrounds.
• These skills can and should be incorporated into all mathematics instruction K-12.
If we choose to not educate all of the people in our society, then we will perpetuate a caste system based largely on race and socio-economic status. When people do not have access to a quality education, they are usually relegated to low-income employment, and are more likely to get caught up in cycles of poverty and the criminal justice system.There is a shortage of people qualified to fill some of the most important jobs like nurses, doctors, engineers, architects, as well as skilled trades like electricians, carpenters, machinists, welders, mechanics, and plumbers. We cannot continue to progress and a society if we do not have people to do these important jobs.

Dr. Robert Moses was a voting rights activist during the Civil Right Movement in the 1950’s and 60’s. Dr. Moses’ work in the southern US was focused on increasing the literacy of African-American share croppers so that they could gain the right to vote. In the 1980’s, Dr. Moses recognized that, similar to the civil rights movement in the 60’s, too many African-American students were mathematically illiterate. He recognized that lack of access to algebra would cause African-American students to be left out of the technological careers of the 21st century. For this reason, he created the Algebra Project curriculum. The Algebra Project curriculum is a hands-on, culturally relevant curriculum that uses real-world context to teach Algebra concepts. (Radical Equations: Civil Rights from Mississippi to the Algebra Project, R. P. Moses, 2002, Beacon Press)

Tell us about the work you are doing related to mathematics and social justice.
My work on mathematics and social justice is an outgrowth of my long-term interest in equity and access in mathematics. I have always taught in either low-performing schools, or schools that had large numbers of students who were low-achieving. I have always believed that it is my job to make sure that even the lowest achieving had access to high quality mathematics instruction. When I began associating with UCLAMP, I was pleased to see their emphasis on access and equity. Those early discussions naturally led into culturally relevant and responsive pedagogy. In 2005, Rethinking Mathematics: Social Justice by the Numbers was published. The book had a number of social justice lessons that  I began to use in professional development, as well as mathematics methods courses. Since then, a number of books have been published on the topic of social justice and mathematics education. In 2016, TODOS:Mathematics for All, and NCSM published a joint position statement on Mathematics through the lens of Social Justice. I have been using that document in a number of different professional development workshops and keynotes. For the past two years, I have co-presented a pre-conference workshop on Social Justice and Mathematics  for NCTM. I also co-facilitated a two-day institute on Social Justice and Mathematics in the LA area in 2017. I have also created social justice lessons. The is a bi-annual conference entitled Creating Balance in an Unjust World that focuses on social justice and mathematics.

Are mathematics classrooms inclusive, equitable? Why or why not?
In general, no. Mathematics classes still tend to engage in tracking. Mathematics classes are still very teacher centered. Only students in advanced mathematics classes have opportunities to engage in high level, cognitively demanding mathematics. Too many students are exposed to mathematics instruction that is based upon rote memorization and drill-and-practice. Their are huge achievement/opportunity gaps in mathematics performance that are based upon race and class. Low-income students of color are usually concentrated in the lower-level classes, while affluent white and Asian students are more likely to be found enrolled in advanced mathematics courses. This is exacerbated by a lack of qualified teachers of mathematics that results in low-performing schools facing shortages of qualified mathematics teachers and limited course offerings.

Why are tracking and teacher-centered instruction problematic?
There have been education scholars that have been study the ill-effects of tracking for years (Jeannie Oaks). Tracking labels some students as mathematically capable and others as not capable. These labels are usually based upon standardized test scores or other types of assessments and are not very valid or reliable.

Dr. Lee Stiff worked with a school district in North Carolina that was concerned that their African American students were not enrolled in 8th grade algebra. After learning the admission policy for algebra one, Stiff identified a large number of African American students who met the admission criteria, but had not been admitted to algebra one. After the qualified students were identified, they were properly placed in algebra one, and all of them passed. The situation has become so dire in California, that in 2015 the state legislature passed a law that schools have to make their mathematics placement criteria public and create a plan to inform all parents of the  placement criteria.
San Francisco Unified School District took the bold step a few yeas ago to eliminate all tracking in mathematics. The district eliminated acceleration in middle school and require all students to enroll in algebra in the ninth grade. Acceleration cannot occur until after the students have taken geometry.
For example, in the elementary grades, students are required to complete timed tests, where they have to complete 100 multiplication problems in one minute. These types of assessments send the message that mathematics is about speed and accuracy as opposed to critical thinking and problem solving.
Once a student gets tracked, it is virtually impossible for them to get out of the track they have been placed into. This can impact whether or not students will ever have access to college preparatory mathematics when they get to high school, which will impact whether or not they can get into college, and what they can study when they get there.

Teacher-centered instruction is a philosophy of teaching that dominates mathematics instruction in the US at all levels. A teacher-centered classroom assumes that students come to the learning environment as empty vessels to be filled with the wisdom and knowledge of the teachers. Teachers provide a lecture to students where they passively take notes. The teacher demonstrates how to work mathematical exercises. Students repeat the procedures shown to them by the teacher. Students are given several mathematical exercises to practice on their own. Students are given a few dozen mathematical exercises to complete as homework. Teachers don’t find out what students understand until they assess them with a quiz or test. Education researchers refer to this method of teaching as Initiate-Response-Evaluation (IRE). This method of teaching works for a very few students.
A student centered approach to teaching is one that connects mathematics content to student’s prior knowledge. Students are given high level, cognitively demanding mathematical tasks to solve. Students are allowed to work together in groups to solve problems. Teachers support students by asking probing questions to push their thinking. Teachers provide instruction when students need it in order to solve problems. Students are provided with the tools and technology necessary to engage in complex mathematical tasks. Teachers use a variety of strategies to assess student understanding beyond paper and pencil tests.  This method of teaching opens up opportunities for all students to be successful in the mathematics classroom.

What can math teachers do to make progress?
There are examples of schools and districts that have had success in creating more equitable environments for students.

San Lorenzo HS in northern CA, Core Principles
• All teachers and students are learners
• Working from strengths while making space for vulnerability
• Redefining “smart”
• Redefining what it means to do math in school
• The importance of relationships
• Department Goals
• Detracking 8th Grade Mathematics and  Algebra I
• Creating a reform-based curriculum that supported Complex Instruction pedagogy
• Reducing rates of D/F grades
• Understanding the needs and experiences of African-American students
• Responding to the challenges of  standardized tests
Results
• At the start of 9th grade, San Lorenzo students were scoring at significantly lower levels than the students at the  2  comparison schools
• Within 2 years, Railside students were significantly outperforming students at the other schools
• San Lorenzo students were more positive about mathematics, took more mathematics, and planned to pursue mathematics in college
• Achievement differences among different ethnic groups were reduced or eliminated
• By their senior year, 41% of San Lorenzo students were taking advanced courses compared to 27% of students at the other schools
• A few years ago San Francisco Unified School District detracked their mathematics program as well.

What can parents do?
• Play games that involve mathematics (e.g. card games, dice games, dominoes, puzzles, etc.)
• Share with their children how they use mathematics in their work and home lives. S
• Get involved at school. Meet their children’s mathematics teachers.
• Have their children explain the solutions to their mathematics problems from school.
• Advocate for high quality mathematics instruction at school.
• Read “Mathematical Mindsets” by Jo Boaler, Stanford University

How can administrators support these efforts?
Be knowledgeable of the California Framework and standards. Know what high quality mathematics instruction looks like. Visit mathematics classrooms and provide guidance and feedback to teachers (SERP 5X8 card[b]). Provide time for mathematics teachers to collaborate, visit each others classrooms, evaluate student work together.

How can college math instructors get involved?
College math instructors can get involved in several ways.
1. Focus on your classes first
2. Read the TODOS joint position statement
3. Start or join a Math Teacher Circle or Math Circle
4. Read Eric Gutstein's book, Reading and Writing the World with Mathematics: Toward a Pedagogy for Social Justice
6. The AMS-MAA published a textbook on Mathematics for Social Justice (https://bookstore.ams.org/clrm-60/)
I think college mathematics instructors have a larger distance to travel on this issue. College mathematics instructors have not taken educational psychology courses or mathematics methods courses. They did not have supervised teaching prior to starting their teaching careers. As an undergraduate mathematics major, all of the instruction I experienced was teacher centered.  College mathematics instructors need to first educate themselves about what it means to teach mathematics for understanding. Once mathematics instructors are up to speed on teaching and learning theory, then they can read the TODOS/NCSM Joint Position Statement and the research that supports it. They can read the works of Eric Gutstein and other related authors. Then they can start making the suggested changes to their instructional strategies.

It would be helpful if mathematics instructors could work in collaboration with one another to change their practices by doing things like planning lessons together, observing each other’s teaching and providing feedback, and looking at student work together.

Mathematics instructors need to join organizations like the National Council of Teachers of Mathematics, The Association of Mathematics Teacher Educators, the American Mathematical Association, and the Mathematical Association of America, and attend conferences and workshops that focus on successful pedagogical approaches.

Has there been any pushback or backlash in this area of work?
There are people who disagree with this point of view.  people work in social justice in Mathematics. Pushback from people who want to keep the status quo. Think about school buy-in. Student buy-in, teacher buy-in, parent buy-in, administrator buy-in.

There is pushback from people who are not in agreement with the movement for access, equity, and social justice in mathematics education. There are traditional teachers of mathematics who do not see anything wrong with the current system of mathematics education, and feel no need to change their practices. There are mathematics teachers who are opposed to introducing social justice topics into the mathematics curriculum. Many teachers feel that their role is to serve as a gatekeeper, keeping students they don’t deem capable of taking advanced mathematics tracked into lower level classes.

There are parents who like a system that privileges their children over other children. In this case, they understand that there are sometimes only one or two advanced placement mathematics courses offered in a school. Traditionally these classes are kept to an enrollment of thirty students or less. Many parents feel that if more students are given access to advanced placement mathematics courses, there will be fewer opportunities for their children. Schools and districts have a responsibility to find ways to increase the pipeline of students who are prepared for advanced placement mathematics while finding ways to offer more spaces to accommodate these students.

1. Read the Joint Position Statement by TODOS and NCSM. https://www.todos-math.org/assets/docs2016/2016Enews/3.pospaper16_wtodos_8pp.pdf  Learn about equitable teaching practices that you can use in your classroom. Here’s a link to 21 practices you can use in your classes.  HERE Tanner, K. (2013). Structure matters: Twenty-one teaching strategies to promote student engagement and cultivate classroom equity. CBE Life Sciences Education, 12(3), 322-331.
2. California Math Project: https://cmpso.org/
3. TODOS: Excellence and Equity in Mathematics https://www.todos-math.org/
4. Mathematics for Equity (San Lorenzo, HS) https://www.nctm.org/store/Products/Mathematics-for-Equity--A-Framework-for-Successful-Practice/
5. Rehumanizing Mathematics for Black, Indigenous, and Latinx Students, National Council of Teachers of Mathematics https://www.nctm.org/Store/Products/Annual-Perspectives-in-Mathematics-Education-2018/
6. SERP 5x8 to assist teachers and administrators in improving math classes via observation.  https://math.serpmedia.org/5x8card/

Thursday, September 5, 2019

An Arstechnica article is making the rounds essentially about student buy-in. There are several layers to unpack here.

First, here's a link to the article: College students think they learn less with an effective teaching method They don't even realize they've learned more.

The key point of the article is that students liked lectures more, but did statistically better when taught via active learning. The article ends with suggesting that instructors give a short lecture on the benefits of active learning to deal with the issue.  Giving a pep talk is a good starting point, but not enough.

Why do we even need to work on student buy-in? The overarching reason is because teaching is a cultural activity. When people walk into a classroom, they have default, often unacknowledged assumptions about what is "supposed to happen." Deviations from the norms can create tension.

We need to unpack this further. In math, we have Math anxiety that messes up a lot of students. In general, we could describe this as having fixed mindsets about intelligence. "I'm not a math person" or "I learn best when shown all the steps that I can memorize..." come up as signals of this. This matters when we get to a point where students get stuck. Getting stuck is exactly the point where we have to confront our images of ourselves.  Getting stuck has been implicitly learned as equivalent to being stupid. The "smart" ones get it fast, and if you're not fast you're not smart. This is actually something that comes in education research (under the heading "Nonavailing beliefs", which are beliefs that inhibit or do not support learning).

When an instructor uses active learning that sets students up to have to make sense of something actively, then it's natural for students to get stuck sometimes. And when students get stuck, all those issues mentioned above get activated.

Another layer is the "answer getting" culture we've created. Much of school success has been about getting the answer. The most common questions that are asked in class are "What's the answer?" and "Is this right?" Rarely is it about, "Why is this true?" or "How else could we approach this idea?" So when we ask students to process ideas at a deep level, rather than crank out answers, then it creates yet another tension -- "Is this going to be on the test?"

Smart has been co-opted. Let's re-co-opt smart. Smart is working on ideas and problems, getting stuck, trying new ideas, collaborating, and so on... Smart is thinking of education as a journey. We need to educate students (and parents) what being smart is. With growth mindset research, we have a framework to have productive discussions about this.

Teaching is also a system. So if you teach X, but test Y, there's a problem obviously. But life is more subtle. If we focus on process in active learning, but test the easy-to-test things for whatever reason, then our assessments are saying we value Y, but our activities are saying we value X. Actions speak louder than words, and assessment is where you put your money where your mouth is as a teacher. The point here is the conditioning students go through is not just about what happens in class, but about the whole experience. Assessment is one of the big pieces of a class, and affects how students view learning. It's not just what activities we use in class. We also need to align our assessments (both summative and formative).

Perhaps one of the more troubling ideas from the article is that students can't identify they learned more. This made me pause.  This isn't new news, but it's a reminder.  Let's think about this.  In almost any other context this is truly odd. If you're learning to play the trumpet and are learning to hit high notes, you know when you've learned it. Of course, there's nuance in learning music, so I'm not trying to make it a binary learning outcome. But if students don't recognize they have learned more, it's a sign that there's more than just the specific teaching that's not right. One thing that jumps out is feedback and coaching. Students need regular feedback that they are learning and making good progress. Pointing out successes regularly and equitably is essential and goes a long way. "We learned this... Way to go, and we learned this because we worked on it, got stuck, and figured it out. That's smart!"

Now all this sounds like I might be blaming students to some extent. I'm not. Not in the slightest. This is about unpacking the layers of our system. Circling back and putting the layers together, we get the outcomes our teaching culture is designed to achieve, whether we realize it or not. We still have holdovers from the roots of the industrial revolution, where our model for education was created using a factory model. You know, bolt on the knowledge and you're good to go. But our goals are different today, and we are shifting towards humanistic education. That is education for developing people as human intellectuals.

Student buy-in is generally about this broader cultural shift. When students walk in the door it's our job as teachers to help them make this transition in mindset and purpose. If we just change the way we teach, and don't inform students, it's on us if they walk away with a bitter aftertaste.

How to get started? Let's get practical. After all that blabbing above, we need things we can do in class that work.  Linked below is a post from earlier this summer with a collection of links from what to do on Day 1 to ongoing strategies to digging deeper into Math Anxiety.

And just yesterday I wrote a letter to students that can be used as a starting point to get students on board.

Letter: Dear Student

Wednesday, September 4, 2019

Letter: Dear Student

This letter is to students in college math classes, but might apply in other settings such as secondary math or other subjects.

Dear Student,
I am writing this letter to you, because your instructor, other instructors, all of us care. We care deeply about your success. We care about your future, and the future of others. That's why we went into teaching in the first place, a profession notorious for long hours, high commitment, and not the highest wages. Teaching is a calling, and our calling specifically is to help young people today to prepare them to solve tomorrow's problems. Teaching is a social responsibility to young people, to prepare young people with the knowledge, creative thinking, and values needed to live healthy, successful, impactful and meaningful lives.

Consider this next idea for a few minutes about time... Children who are in kindergarten today will retire in about  60 years. I write this in 2019, and that means a student in kindergarten will retire around the year 2079.  2079!  Think of the difference between what your life is like today, compared to 1959. The world has changed drastically in ways that people in 1959 could not predict. No way they could foresee smart phones, google, climate change, automation, globalization, etc.   The main point I want to get to is that there are questions and problems that young people (you) will have to solve that have not even been thought of yet by anyone on the planet. Your education today must prepare you to solve these unstated, problems far out into future.

You can get help from various resources. Resource number 1 is your instructor. That's the person who is responsible to your learning. Next there are your classmates, your textbook, the tutoring center, and perhaps the internet. Try and talk to as many humans beings as you can first. Math is learned better with more human interaction.

Office Hours: Office hours are for you, and if you are stuck on something, even a small thing, go to office hours. It's not an imposition when you show up, and your instructors want to help you. Even if you instructor seems completely different than you, you can and should ask for help. While it might seem a bit scary, it's ok. I know a ton of math instructors, and so far all of them are human beings, and are really nice in office hours. Some even have a sense of humor! I know that might be shocking, but it's true. And when they go home, they go home to things like cats and children, and watch TV or text their friends about Friday plans, just like you.

Yes, there exist legends of math geniuses, who work in their attics for years to invent math. That works for them in certain contexts, but none of them worked alone or got to that  point all by themselves. They have collaborators, consultants, books,... they actually went to school with other human beings at some point in their lives (and interacted with them). They read journal articles, they attend seminars, they go to conferences. Some people gave them a job, so opportunities were given to them. No one is 100% self made. Therefore, work with other people regularly, even if you don't view yourself as "social" in the everyday sense of the word or are introverted. In this letter, I mean social in a school or workplace sense. We all have to communicate with others to give and receive feedback, as well as brainstorm new ideas.

And really I hope you get stuck a bunch of times in your learning process (in a safe learning environment, not on tests).  Get stuck??? Yes, get stuck. Because you need to push your personal abilities. Each time you get stuck and unstuck, you learn what works and what doesn't work and you get smarter. Through working on problem solving in math (or any field), you are doing something like going to the gym for your brain. Your brain will get stronger, and you'll learn new ways to think, see, and feel.

Math anxiety is a real thing. I've written about it many times on this blog. I've talked to hundreds of students about it. I'm sorry about this. Math anxiety should not exist. Not everything in the world is right, and math anxiety is one of those wrongs that we are trying to fix.  In the meantime, if you had experiences that led you to math anxiety, what you need to know is that it's not your fault! You're not dumb, you're capable, and there is a way out.

The way out is doable. It's shifting from a fixed mindset (where one views their math abilities as fixed at birth), to a growth mindset, where one views effort and practice as the ingredients for getting smarter.  Think about one of your hobbies or interests. How did you get better at it? You practiced. You might have had a teacher or coach or watched videos, but at the end of the day you put in the focused, dedicated hours, and did the work. That's being smart, and from now on, we are co-opting the word smart.  Getting smarter at math is exactly same. A good teacher will provide you with a positive class environment and support you through your specific learning challenges, and when you practice, think, ask questions, collaborate, and do all those things people do in every profession and hobby, you'll make real progress.

Only watching videos of other people doing the math isn't going to cut it. Look I get it. Khan Academy is one click away. It's a useful resource, and I even watch KA sometimes to see which ones might help my students. Whenever I need to fix something in my house, I find a video on how to fix it. That's a good way to get information that you lack. However, learning math is like learning to be a musician or athlete. It's not just information and facts, but also about developing thinking and problem-solving skills. Doing better at math requires thinking mathematically, which is analogous to learning to ride a bicycle. You can't be taught to ride a bicycle beyond the basics by watching a video, because there are things your brain and body have to construct by actually doing it in order to build that skill.  Mathematical thinking is the same in that you can't just get info uploaded into your brain like a firmware update. You also need to construct understanding and meaning for yourself, just like your body and brain have to construct things in order to ride a bicycle safely.

Another example is learning how to hit a baseball/softball. We could watch tutorials all day and understand what we need to do. Basically it's swing a bat and hit a ball. But only watching videos is obviously not enough.  We need to actually swing a real bat and hit a real ball and get ongoing feedback from coaches. And then practice, play games, strike out, reflect, rest, repeat.  It takes time to get good at it. That's the perspective you should have about Math and watching videos. Sure watch videos sometimes to get some info, but don't stop there. Start there, and do the work. Do your own reps on real problems. Otherwise, you'd just watching, and that is just sitting in the stands. You need to be on the field, because this is your life and your future. Get in the game!

Hopefully, your instructor will ask you to work with your classmates sometimes on a question or task. In education this is called active learning and is part of inquiry-based learning (IBL). These methods are designed specifically for you to engage and think for yourself. Listening to someone isn't enough. Sometimes we need to hear ideas from the instructor that we can't easily build ourselves, but like sports or music or any hobby, you ultimately need to be the one engaged in the process, asking questions, and taking ownership of your development.

Learning to work with others is critically important. Working in groups is not only about helping one another, although that's a good aspect of group work. One of the main benefits of group work is learning through discourse. Sometimes we need to talk things out in order to make sense of what is going on, and hearing other people's ideas can also benefit all of us, and helps us engage in the process of trying and refining new ideas. Another benefit of group work is learning to communicate. In an era when more and more repetitive tasks are being automated, the ability to do humanistic work, such as communication and problem solving, is much more important and valuable.

Try to contribute to group discussions and regularly invite your group mates to share... "So what do you think? What did you get? [smile]"  It's not about one person getting the answer for the group, and everyone else copies. It's about giving everyone a chance to think, try, share, refine, and see ideas from multiple perspectives. That's good for you!

In summary, focus on problem solving as a process, embrace and be patient with being stuck and not having answers right away, think about the long game of your personal intellectual development, develop a growth mindset, and work on learning with your classmates. These are things you need to prepare for your future. All people, especially young people, have immense capacity to learn, grow and get a lot smarter! Believe in yourself by actively investing in how you learn.

Best wishes on a successful new school year!

Sincerely,
Professor Stan Yoshinobu

Monday, August 26, 2019

Pronouns, Early Term Simple Survey

Many thanks to the NSF PRODUCT team for bringing this up in an email discussion! This short post is on pronouns, and a result of that discussion. These are not my ideas, and I'm sharing the main ideas from that discussion.

Pronouns matter, and it makes a difference if we ask students to tell us their pronouns without assuming what they are. Why does this matter? Here's a quote from www.mypronouns.org

"Often, people make assumptions about the gender of another person based on the person’s appearance or name. These assumptions aren’t always correct, and the act of making an assumption (even if correct) sends a potentially harmful message -- that people have to look a certain way to demonstrate the gender that they are or are not."

One way to get this information is to ask. And while you're asking, you might as well find out what they preferred to be called and what might help them succeed.
1. Your name____ (and email optional)
4. For me to be successful in this class, I need you to know that ____
Working towards an equitable classroom is an important consideration today (as the 4th pillar of IBL), and we should be more conscious of the issues and be action oriented. The last question on what would help students succeed can help you understand specific needs right at the start of the term, which can make a significant difference for student success.

One more thing. I put "he/him/his" in my email signature. An easy and impactful thing to do.

In short, I learned not to assume. I learned that we should simply ask and share with our students why we are asking, since they may need to learn as well.

Thursday, August 1, 2019

Be Like Branch Rickey and Create Opportunities: Equity and Inclusion

My favorite baseball player of all time is Jackie Robinson. I grew up in LA, and have been a lifelong Dodgers fan, and Jackie Robinson was for me a wonderful role model. He was from Pasadena, not far from where I grew up in Glendale, CA, and became a celebrated, historical figure. But this post isn't about #42. It's about something else. It's about what we can do.

This post is about making a case for ordinary people, specifically math instructors or educators, to act like Branch Rickey.  Who's Branch Rickey?  There are links at the bottom, if you don't know who he is. Let me say this first. Jackie Robinson had the goods to play hall-of-fame baseball, while taking in daily verbal and physical abuse. What he accomplished is truly amazing, but he did not have the power to create his own opportunity to play in major league baseball. That power was held by the Dodgers, specifically the Dodgers' general manager at the time, Branch Rickey. Branch Rickey is one who gave Jackie Robinson the opportunity to play major league baseball.

If baseball was a meritocracy before April 15, 1947, then players of Robinson's ability would've been on MLB teams by that time. While "equal opportunity for all" is one of America's high ideals, it isn't yet a level playing field. Hence, someone had to go out of his way and create opportunities for those not included.

A clip from the movie, 42, "Why did you do it?" (Staring Chadwick Boseman, Harrison Ford).

What does this have to do with math teaching and learning?  Mathematics classrooms and schools are not yet gender, race, disability, LGBTQ+, and socioeconomic status neutral as of 2019. Further, Mathematics is a gateway or barrier into STEM fields, where many of the economy's current and future high paying jobs are.  I've highlighted some other issues in equity and inclusion HERE in the 20 year challenges I posed to the profession two years ago. In short, women and people of color are underrepresented in STEM fields, and underrepresentation is not only wrong (which is bad enough by itself) it's also just plain dumb for Math and Science. Diversity of thinking is needed to bring new ideas and fresh perspectives to complex problems. We should be harnessing all the talent and interest in our society possible, because it's simply the morally correct and smart thing to do.

Some people have claimed that Math is a logical, abstract subject, and has no connection to culture, race, gender, and so on. I understand where this argument is coming from. In the pursuit of knowledge we try our best to remove personality and bias, and consider logic, reasoning, and data.  Theoretically our identifies should not matter in mathematical studies. This viewpoint though conflates Math as an academic subject and the human constructed systems that educate young people (i.e. schools). While Math itself is an abstract logical subject, separated far from what many would recognize as everyday culture, teaching is a cultural activity. Where people learn Math is almost always in formal education settings, and these systems are part of our culture. Math teaching and learning environments are not immune from the influences of the culture within which they exist. People teach the courses. People are the administrators and staff at the schools. People setup standards and assessments. People are at the decision points in the system for who gets into advanced math courses and who does not. People determine resource allocation that funds schools. There exists inequity in society... Cultural assumptions, biases, and inequities are baked into the education system, because our schools are a reflection of our society.

A lot of what we do in Math is good. We should and do value Geometry, Algebra, Trignometry, Calculus, problem solving, proof, and so on. And in some instances we are teaching these subjects really well! But just because there exists good, that does not mean we are done.  It's not all bad or all good. It's a mix. Strong communities are willing to look at the hard issues, and think of ways to fix them. The key is to keep the good and improve on areas where there are issues. Inclusion and equity is one of those areas that needs attention and improvement.

What I believe we need are multiple analogies for a "hero."  I don't believe in hero worship, thus what I mean to convey are examples of how we can act that improves equity and inclusion.  Branch Rickey is one (and not only) example. The reason why I like Branch Rickey is that he created an opportunity, and was an active ally.  He didn't just think about or complain about the problems. He did something. He created an opportunity for Robinson and supported him through the tough days. And that created more opportunities for more people of color in the coming years. We need more people like that in education (and the rest of society), who intentionally create opportunities for underrepresented groups. These heroes are pulling others up, and helping us achieve one of the highest ideals of our American democracy, a level playing field.

Let's say a person asks, "I'm just an instructor/educator. What can I do?"  You may not realize it, but you are powerful! Yes, instructors/educators are powerful.  Within your communities, especially if you are male and/or white, you have the power to influence. Specifically every instructor has the capacity to be like Branch Rickey in their own way, and create opportunities for others. As educators, I believe we should view ourselves as forces for good. If we are passionate about our subjects, then it's also our responsibility to help create learning environments, where all students feel included and supported.

Opportunities come in different forms and sizes. You might say, "I haven't been trained to do this." That's true. Most of us haven't, but there are things you can do, without having do anything outside of your job description or sacrifice all your free time.  If we teach better, we can do impactful things that make a difference.  Below is a starter list of 10 things you can do, beginning with easier steps.
1. Use inclusive teaching practices and frameworks (IBL), that encourage more students to be engaged more often. This is increasing daily learning opportunities for your students. 21 practices by Tanner are HERE
2. Add an equity statement in your syllabus to signify the importance of inclusion and equity. This helps create a positive learning environment in your class.  Imagine a student of color, sitting in a room full of people not like her.  Examples HERE
3. Post a inclusivity flyer or image in your office door to help establish a safe learning environment.
4. Point out non-inclusive behavior and shut it down (in a nice way of course. "Hi John, how are you? Let's chat after class. For now, stop X and do Y."). Hopefully this never happens in your classes, but if it does, it helps immensely if you speak up. The cost of inaction is high, because  silence can be viewed as tacit agreement with the behavior or not caring.
5. Show up to some campus events and learn about cultures different than your own. Showing up is valued. If you're not sure, just ask the organizers and be a humble, respectful participant.
6. Amplify voices of those who are marginalized. Some students or colleagues are quiet, not because they were born shy, but because they have not been invited in and have been conditioned over a lifetime that their voices won't be heard. "I like Maria's comments and agree with them..."
7. Mentor or be the advisor to students from underrepresented groups, including them in research projects, jobs, etc.
8. Support organizations that support women and people of color, and become a member.
9. Volunteer or do projects with local schools, such as organize a Math Circle or Math Teacher Circle.
10. Work on hiring committees and include implicit bias training for the committee members.
The list above is not complete. It's a starter list. I'd say start with doing what you can, and add on as you go. Perhaps you'll find other things you can do that match your personality that may not be on this list. Some of the items above do not require much effort, and yet they can make a lasting difference.

Let's address some of the common questions and concerns that come up...

"It's not my job. I teach math." I hear this sometimes. We teach math, so why do we have to think about things like inclusion or equity? Branch Rickey's job wasn't to integrate baseball. His job was to be general manager of the team. Sure he saw benefits from a business standpoint to integrate, but there were also risks and costs. Most importantly he saw that something was wrong and he could do something to right this wrong. The point of view of "It's not my job..." is understandable but misplaced in world that is not a level playing field. Our profession has flaws, and we can try and fix them. You have the power to make a difference!

"Is it my responsibility?" Let me put it this way. If you were eating lunch at restaurant, and someone next to you falls and breaks their arm and asks for help, you'd help the person. You don't go on eating your sandwich like nothing happened. It's normal human nature to help those in need. In the case of equity and inclusion, the tricky part is that you aren't being asked to help in clear, direct terms like the restaurant example. Some issues become known by looking at data, reading history, and learning about what is going on. Some of these issues are things you can't see in your everyday interactions, likely due to your identity and privilege. If you didn't know there are issues, then now is a good time to inform yourself. Check out the AMS page on Equity, Diversity and Inclusion.   See also this post on Math Ed Matters, MAA by Beth Burroughs, Montana State. (Side note, my ancestry is Japanese. None of my ancestors owned slaves or created the inequities we have in the U.S. Yet, I view this issue as all our responsibility. It's specifically my responsibility to help and do the right thing, where I can make a difference. I don't feel guilt or shame for having privilege, as an asian male in STEM, and neither should anyone else. Instead, I view my role as using my privilege to help others who didn't start life a second or third base.)

"What about white men, are we excluding them?" The answer is definitely no. Society isn't pie. Society is not a zero sum game, and in fact society is a team sport. We're actually competing with our international peers, not the guy down the street. All our successes are tied together. Your success is my success, and my success is your success.  If a woman or person of color learns more Math in a class, it's obviously not taking away learning from someone else. Learning isn't a fixed commodity across society, where only N people can learn topic Y. More people succeeding in school is better for team USA, which benefits all of us. We aren't ignoring or taking away opportunities from one group, by emphasizing the needs of other groups, who have historically had far fewer opportunities. Highlighting inequities in order to fix longstanding issues is something we do, because that's adulting. That's a grown up version of sharing. An analogy is that what we are doing is making room at the dinner table for everyone.  There's plenty to eat, so we can scoot over, make room, and break bread together in fellowship.

Speaking of fellowship, Pee Wee Reese is another person we can learn from. Pee Wee Reese was a great player and teammate of Robison, and more importantly a good human being. He famously put his arm around Robinson before a game, when the fans were being hostile to Robinson, in a show of visible inclusion. It wasn't enough that he agreed with Branch Rickey and supported Robinson in concept. He showed who he is, when it was his turn to come into the light.
But his [Pee Wee Reese's] most important action on a baseball field may have been prior to a game. In 1947, the Dodgers were visiting Cincinnati, and the fans and opposing players were getting on rookie Jackie Robinson. Reports of the game state that Reese calmly walked over to Robinson, put his arm around his teammate’s shoulder, and chatted. The gesture is remembered as an important moment in both Robinson’s career and the acceptance of African Americans in baseball—and American society.
Scene from 42, "Maybe Tomorrow We´ll All Wear 42." (Lucas Black, Chadwick Boseman)

"I'm a nice person. Isn't that enough?" Kindness alone isn't courageous. Kindness is part of basic manners. It's assumed. Being nice doesn't create opportunities.  This is why including people and ensuring qualified people are included is fundamentally important for progress. Robinson appreciated the kindness of Rickey and Reese, but really what changed the game and society was being included on a major league roster.  Kindness without action is sitting on the sidelines. The thing that frustrates me the most is seeing good people sitting on the sidelines in silence. Hence,

Kindness + Inclusion = An Act of Courage

Within each of us is the capability to positively change our communities. Within each of us is the power to understand why we must act, and the power to build up the necessary courage to take action and create opportunities for others. The good news is that instructors don't have to go outside of their job descriptions to do some of this work. It can be accomplished via good teaching and mentoring -- things we do as part of our jobs. Teaching is something we already care about, and that means that we all can make a difference today. That's an encouraging thought!

“It's a thrill to fulfill your own childhood dreams, but as you get older, you may find that enabling the dreams of others is even more fun.” -  Randy Pausch, The Last Lecture.

1. https://baseballhall.org/hall-of-famers/rickey-branch
2. https://www.biography.com/athlete/branch-rickey
3. https://baseballhall.org/hall-of-famers/reese-pee-wee
4. AMS page on Equity
5. 21 practices by Tanner are HERE
6. Examples of equity statements for course syllabi HERE

Wednesday, July 10, 2019

Student buy-in is one of the issues that comes up frequently at workshops and in hallway conversations. Student buy-in isn't a simple thing. I've written about it previously on this blog, and I think this topic needs to be visited regularly.

Getting stuck is hard. Fruitful struggle and productive failure aren't usually taught and learned. Making mistakes has often been equated with failure (in the negative sense of the term). Students aren't usually encouraged to explore, experiment, and tinker.  Thus, the conundrum is that in order for learners to grow, they need to be challenged appropriately, which means being stuck on some ideas, yet being stuck is equated to being dumb.

Luckily today we have the advantages that can help change learning experiences into authentically positive ones.  Growth mindset work has zeroed in on beliefs that lead to becoming smarter. We know more about how to use active learning to open up learning spaces, and we have a growing collection of videos on productive failure that can direct students toward successful mathematical practices. Instructors can assign videos as homework with reflective writing prompts every week or so for the first part of the term.

Day 1 of a course is important. Linked below is one way to open a course, by starting with students' hobbies and how they got better at the hobby.
https://theiblblog.blogspot.com/2019/01/opening-course-and-launching-winter.html

http://danaernst.com/setting-the-stage/

Ongoing strategies for student buy-in are posted here. It's not enough to only do something on day 1, because it's a journey.

Nudging students to engage more is one way to address student buy-in. We all need a break sometimes, and we can all use a bit of support. One way to keep students going is to nudge them.
http://theiblblog.blogspot.com/2018/11/nudges-as-teaching-technique.html

Attend to Math Anxiety, because knowing where students are coming from can help us be better teachers. Math anxiety is a thing, and most students have some level of anxiety. Ignoring it only limits student learning, so we might as well deal with it. Math anxiety is linked to (lack of) productive failure, and fixed mindsets. Here's a post on the iceberg diagram and math anxiety and how instructors can detect math anxiety and fixed mindsets from statements like, "I don't learn this way..."
http://theiblblog.blogspot.com/2018/01/iceberg-diagram-fixed-mindset-math.html

Digging deeper, math anxiety is something you can read about from students directly. Here's a collection of math anxiety quotes to give you a sense what lies underneath. If you've never asked, try adding a math autobiography assignment at the start of the term. Let students share their experiences.
https://theiblblog.blogspot.com/2015/03/math-anxiety-realities-student-voices.html

Sharpening your IBL skills is also important, because a well-taught class is part of the equation. Problems that are too hard or leaving students struggling for too long works against student buy-in.  Also making things too easy is also. The IBL Blog Playlist is collection of posts organize by topic. If you are new to IBL, we also have a video series to get you going.

Monday, June 24, 2019

Standards-Based Grading Example in an IBL Course

I'm sharing an outline of standards-based grading I've recently used in a course for future elementary school teachers, although much of this is generally applicable to other courses. I'll list the main features and then get into some of the details below.  Also this is just one example, so do not assume that what I am sharing is representative. It's really a form that works for the specific course.

Here are the main features:
1. Gateway exams
3. Homework assignments
4. Productive failure
5. Class contributions and participation
6. Final project
How these fit together is that if a student earns a passing grade on all items, they earn a B in the course. Students can raise their grade to an A/A-/B+ with an excellent grade on the final project. Students earning one or more non-passing scores in any of the categories will earn a grade lower than a B, with specific grades reductions based on the nature and quantity of the unsatisfactory grades.

1. Gateway exams: These exams are based on the IBL units we work on regularly in class, and are based on the math being learned in the course. Students are required to pass all of the problems on the gateway exams (i.e. get the correct). For any problem that was not successfully passed, a student must retake that problem on the retake exam.  The retake exam is given about two weeks after the initial exam. Problem done correctly do not have to be retaken.  The first retake is done in class. Subsequent retakes are completed in office hours or alternatively completed in writing and submitted for review.  This past term, I gave 2 exams, and was limited by the quarter system (10 week terms) in how many retakes can be given in class.

Retakes can become a logistical challenge for large classes or in courses where there is a significant amount of material to cover. One has to weigh the costs and benefits of this and plan accordingly. The strategy I've taken is to start with a course that I thought would be relatively easier to manage, and then work my way to other courses where I feel I would be better off with more experience.

2 and 3. Reading assignments and homework assignments are graded for process and completeness. Accuracy feedback is given, however, the goal of these assignments are for students to think and reflect on math and math knowledge for teaching. Points are not taken off for mistakes or incorrect answers, and instead feedback is given when necessary and points are awarded for good process. For example, if a student gets a problem wrong, but writes questions or explains what they did and what they still need to work on, then they earn full credit for the problem.

4. Each student is required to present one productive failure (i.e. #PF) per term (in a 10-week quarter) about a mistake or something the student was stuck on. The format is to discuss (1) the mistake or issue, and (2) to share what they learned from the process.  (In some courses the number of #PF presentations is 2.)

5. Student contributions to the class discourse is another component. Students work in groups and are expected to show up to every class, contribute to discussions, be effective group mates (i.e. be good at listening, supporting, and sharing), and present math ideas sometimes. More or less this is participation grade, but with stipulations about expected behavior.

6. In lieu of a final exam, students must submit a final report. The report is based on 4 tracks related to mathematics teaching in the elementary school and the course content (in this case fractions for teachers).  Each track has a lead source (article or book). Students are required to do library research, branching out from the lead source, to find learning challenges (for children) established in the Math Ed literature. Lastly, students are required to create rich mathematical tasks that address the identified challenges that build from starter problems to middle problems to goal problems.

I get asked if creating math tasks is pedagogy.  The answer is no. Creating math tasks to address specific math learning goals is a teaching specific math activity. Identifying the main math ideas, ordering and sequencing math problems, and building up from first principals is doing a math (applied to teaching children).

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Gateway exams require students to learn all the standards of the course. There's no partial credit for problems, and students are required to demonstrate they know the math they need as teachers. Students have multiple chances to make sure they get problem completely correct. The retakes can be logistically challenging, if you are not organized. Overall, the workload is about the same, because retakes eliminates the time needed to determine partial credit, and there's a tradeoff that more or less washes out (for me).

Further, the overall assessment structure aligns the class to the mathematical work of teachers and the philosophy of IBL. The focus is on learning, and guiding students to what they know well and what they need to work on further. What students need to work on is clear, and this I find one of the main benefits of standards-based grading.

The reading assignments, homework, productive failure, and class contributions, as an ensemble focuses on process and prospective teacher beliefs. The shift is away from "answer-getting" without deep understanding.

Final projects or final exams can be implemented in ways that work with standards-based grading. In this specific case, I decided on final projects, since it gives future teachers the opportunity to connect they math they are learning, the research literature, and connect that to the classroom. In other courses, I have used standards-based final exams.

If you're thinking about trying standards-based grading, I highly recommend giving it a go.  If you have been using standards-based grading, please share what you do!

Edit: Dr. Kate Owens published A Beginner's Guide to Standards-Based Grading the AMS Blog.

Thursday, April 25, 2019

IBL Blog Q&A: The TIMES Project, Karen Keene, Justin Dunmyre

This blog post is an Q&A session conducted via email with Dr. Karen Keene and Dr. Justin Dunmyre. They are sharing information about the TIMES project. Thank you Karen and Justin!

Karen Keene has her Ph.D. in Mathematics Education from Purdue University.  Karen was introduced to active learning in undergraduate mathematics education while she was a graduate student involved in the creation of the Inquiry-Oriented Differential Equations materials. She has been serving as a project leader on the TIMES project where inquiry-oriented instruction, one form of active learning since 2013. She is currently an Associate Professor of Math Education at North Carolina State University and is currently serving as a rotating Program Officer for the National Science Foundation.

Justin Dunmyre has his Ph.D. in Mathematics from the University of Pittsburgh, and is a Brown ’13 Project NExT fellow.  He is currently an Associate Professor and Chair of Mathematics at Frostburg State University. Through Project NExT, Justin got interested in active learning, and subsequently participated in the IBL Workshop.  This transformative experience led him to wonder what IBL would look like in his discipline (differential equations) and almost as soon as he had that thought he got an email through the IBL mailing list about this exciting TIMES project!

1. We'd like to learn about the TIMES project. What is the main idea behind this effort?

The TIMES project began as a collaboration of second-generation authors of varied inquiry-oriented (IO) classroom materials.  By second-generation we mean that Michelle Zandieh, Sean Larsen, and Chris Rasmussen wrote the original IO materials for Linear Algebra, Abstract Algebra, and Ordinary Differential Equations, respectively.  The TIMES Principal Investigators, Christy Andrews-Larsons, Estrella Johnson, and Karen Keene were each graduate students of these original authors and launched the TIMES project to study how they might support other faculty in using these materials.  That’s why TIMES actually stands for Teaching Inquiry Mathematics: Establishing Supports. These supports were three fold: providing the curricular material, a 3-day summer workshop on using the materials, and weekly online working groups. The three supported curricula were IOLA (Inquiry-Oriented Linear Algebra) and IOAA (Inquiry-Oriented Abstract Algebra formerly known as TAAFU - Teaching Abstract Algebra For Understanding) and IODE (Inquiry-Oriented Differential Equations).

Justin was in one of the first cohorts of TIMES fellows for IODE, and became involved in running the online working groups and became a coauthor on the materials, along with Nick Fortune whose dissertation research was supported by TIMES project.

2. What classes do you have materials for?

A first course in Linear Algebra, an abstract-algebra course focused on groups, and a first course in differential equations.

3. What is a typical day like in an IO class?

A typical day is centered around the guided reinvention of particular mathematical concept(s).  The tasks are based on the principles of the instructional design theory of Realistic Mathematics Education (RME), of which one of the tenets is that the material must be experientially real for the students.  The Inquiry Oriented materials are grounded in contexts that the students can initially understand and reason about, maybe from a less sophisticated viewpoint, even if they’ve never had that specific experience before.  For example, in IODE, one tasks early in the materials is focused on population growth of owls in a forest, which students may not know a lot about, but can understand. As in any IBL classroom, students are constructing the mathematics for themselves, and taking ownership of that mathematics. The students work through a series of tasks, often encountering the the tasks for the first time in class. Therefore, we rely on small group work for an initial translation from the context to concepts relating to the learning outcomes of the class.

The instructor facilitates this guided reinvention by primarily using four instructional components (Kuster, Johnson, Keene, Andrews-Larson, 2018) , not all of which may happen on the same day.  These four instructional components are general instructional goals: eliciting student thinking, building on student thinking, building a shared understanding of the mathematics reinvented in the classroom, and connecting the students’ mathematics to formal mathematics.

In any given day in our classes, one would see the students sitting in small groups, working on tasks. One would also hear whole class discussions where the instructor is soliciting student input, re-voicing, and, reshaping it, innocuously,  to guide the conversation with an eye on the instructor's mathematical agenda. Occasionally, the students would make presentations of their ideas, but this would not always happen. The whole class discussion and small group discussions do happen every day.  Finally, when the class has finished an idea, or perhaps developed a need for notation to express their ideas, the instructor connects their work with formal language and notation.

4. What are some of your best moments as a teacher in an IO class?

Justin:  There are so many!  One of my favorites came from the first time I taught our optional unit on bifurcation theory.  This unit starts with a task where students are challenged to model the introduction of a parameter that represents harvesting of fish from an otherwise logistic model.  After settling on a simple shift of the form dP/dt = 0.2P(1-P/25) - k, where k is the harvesting parameter, students are asked to come up with a one page report to explain to the owners of the fish hatchery the ramifications of varied choices in k.  The one page report is the trick! By requiring students to use space efficiently, they can actually invent the bifurcation diagram for themselves. What really surprised me was how many different forms this bifurcation diagram can take. I’ve seen students use spreadsheets that show if dP/dt is positive in green or negative in red (the bifurcation diagram then emerges as the change from green to red), carefully stacked phase lines, analytically drawn bifurcation curves using the quadratic formulas, and more.  When the students present these ideas to one another, they realize they’re all saying the same thing, and absorbing insights from other groups result in very deep understanding of this sophisticated concept. The first time I taught this unit I was giddy, I couldn’t believe that these bifurcation diagrams were emerging before my eyes, completely invented by my students!  We wrote about this task sequence in a PRIMUS paper here (Rasmussen, Dunmyre, Fortune & Keene, 2019).

The “salty tank” problem is practically a rite of passage for students in my differential equations classes.  It’s developed its own legend here on campus, because the discussions are so robust, and it is the first time that students are really asked to develop their own equation.  The students really marvel at how they can have an 50 minute long debate over a prompt as simple as “A very large tank initially contains 15 gallons of saltwater containing 6 pounds of salt. Saltwater containing 1 pound of salt per gallon is pumped into the top of the tank at a rate of 2 gallons per minute, while a well-mixed solution leaves the bottom of the tank at a rate of 1 gallon per minute.” This problem is the seat of another favorite moment of mine. Ideas were flying all around the room, what does it mean to be well-mixed, what should the input term look like, what should the output term look like?  A student said something to her group, but she didn’t want to cut into the whole class discussion. Her group thought it was important though, so one of her more outgoing group members interrupted the discussion and he said “I think we should all hear what Sarah has to say.”  Of course, it was a critical insight that helped reframe the conversation in a productive direction! But the act of one student elevating the status of another student, that was a powerful moment that will always stick with me.

Karen:  When I taught this course to math and physics majors together, my favorite times were when we talked about the difference between instantaneous rate of change and rate of change over a "very very small time interval".  The conversations were always spirited and deep, with the Physics Majors declaring it doesn't matter if there is a difference and the Math majors wanted it to matter and try to understand what a "limit" really is. Of course, I was always rooting for the Math Majors, but it didn't really matter, as it was a situation where the students were engaged in thinking deeply about the math and taking the authority of learning on themselves.  Of course, ultimately, we had to agree to disagree and the physics majors usually could go along with the idea of instantaneous rate of change as that being the foundation of a differential equation.

I can think of other times that after we had small group discussions, two groups would present their ideas about a particular task- and they were not the same.  When that happened, I would encourage each side to state their case. Then I would send the students into small groups to continue the discussion and decide what they thought.  This might go on for much of a class. I know it took up a lot of time, but it was worth it--- they were not waiting on me to tell them- but making their own mathematical judgments.  Most of the time, it all seemed to move the agenda forward. I do remember one time that the whole class agreed on something that I knew was mathematically wrong. I made the decision (there was no test or assignment the next day) to let it stand.  On the next day, I brought of the decision with a question that led them to believe they were wrong-- and all was forgiven!

5. If you could give some advice to math instructors thinking about using active learning, who have not tried yet, what would you say to them?

Justin: There is evidence in the research that supports your decision to try active learning, so you should proceed with confidence.  For the IO curriculum, it is extremely exciting to be a sort of curator of the conversation. You don’t know what students are going to say, and you get the exhilaration of thinking on your feet to fit their ideas into your agenda.  So, my advice is: this is hard work! Be kind to yourself when you just can’t reshape their ideas the first time. We have found that, although active learning is our main mode of instruction, there is a “time to tell.”

Karen: Take it a step at a time— some instructors might go all out first time around, but trying one or two days or tasks and seeing how it works in your classroom is just fine. Ultimately, students will be more engaged and take ownership of the mathematics they learn in your new active learning classroom.

The TIMES NSF grant has essentially run its course, so we are no longer running workshops.  You can find the course materials, including instructor’s notes, at these websites:

IODE:  iode.wordpress.ncsu.edu
IOAA:  www.web.pdx.edu/~slarsen/TAAFU/home.php
IOLA:  iola.math.vt.edu

When you begin to investigate these materials, please don’t hesitate to contact us; we are more than happy to help!  With sufficient interest, we may even run informal online working groups.

References:
George Kuster, Estrella Johnson, Karen Keene & Christine Andrews-Larson
(2018) Inquiry-Oriented Instruction: A Conceptualization of the Instructional Principles, PRIMUS,
28:1, 13-30, DOI: 10.1080/10511970.2017.1338807

Chris Rasmussen, Justin Dunmyre, Nicholas Fortune & Karen Keene (2019) Modeling as a Means to Develop New Ideas: The Case of Reinventing a Bifurcation Diagram, PRIMUS, DOI: 10.1080/10511970.2018.1472160