Monday, October 17, 2016

Does Class Size Matter?

The simple answer is yes.

Let's look at some data. An analysis released a couple of years ago by the National Education Policy Center uses an econometric take on the issue, where they try to disentangle data to find causal links. In their work, share evidence that class size increases harms students. This is from data that is "econometric" friendly, where the savings today in increasing class sizes (K-12) is offset in the future by far greater costs. (Penny wise, pound foolish?)

The point I want to get to is a very simple one. Another perspective of this issue is as a classroom teacher.  Let's look at the sequence of class sizes to see how teaching decisions are affected as class size increases. (The assumptions here include assuming we are talking about the college math setting.) Also I'm not going to list all possibilities. The goal here is to see a pattern as class size increases.

Class size of 10: Anything can be done at this class size. Projects, full IBL (where students present proofs or solutions), team/group work, seminar or whatever comes to mind.

Class size of 20: At 20, things are still manageable, and an instructor can get to know all of the students, customize materials and learning experiences, projects are still doable. IBL is doable in with small groups/pairs and students presenting individually.

Class size of 40: Things start to get to a point where there are too many students to let them present their work to the whole class.  Each student may only go to the board a couple of times a term. Somewhere between 30 and 40, instructors tend to switch away from some student-centered methods (such as students presenting their ideas at the board). Small groups can still be used, and the instructor may not be able to get to all groups on a particular task.

Class size of 80: Individual student presentation of mathematics is almost surely off the table. Small groups and peer instruction remain, so instructors can develop this area to support an active learning environment. Projects are highly difficult to implement at this class size, especially at institutions without teaching assistants. The instructor can still visit with some groups and make it around the room (depending on the space) say one trip per class period (approximately).

Class size of 160: At this point, you are well into large lecture territory.  Peer instruction (Think-Pair-Share) and similar methods (with or without clickers) remain in play, although the size or difficulty of the task may be on the less challenging side of the spectrum. Presentations by students, class discussions, projects, instructor visits to each group regularly are almost surely off the table.

As you go up in class size, you lose the implementability of teaching strategies that engage students and also anything that does get implemented needs to be done at a higher skill level and attendant time in preparing for class. Managing a discussion with 160 students takes skill, and most all instructors may be inclined to avoid it.

Consequently, what we are seeing is that there exists research evidence suggesting that a negative, material impact is a consequence of increasing class size AND from a practitioner standpoint instructors have fewer, high-impact strategies at their disposal as classes get larger. We have said nothing yet about how class dynamics can change as you get to larger class sizes. For example, it's easier to hide in a large class, and it's easier for students to checkout, be distracted, and show up late.

Another facet of class size issues is that class size is often a policy decision, not in direct control by an instructor. Instructors can have input on class size, but it's not up to instructors to set class size limits. This is done by policy makers or administrators, yet these decisions have significant impact on teaching decisions and hence learning outcomes. Faculty and administrators should take into account the big picture when it comes to class size. Focus on things like nominal efficiency, should be viewed while also heavily weighing learning outcomes, DFW rates and long-term impacts on student learning.

Thursday, September 1, 2016

Effort and Circumstances in Educational Achievement

The educational achievement by a student is not only a result of personal effort, but is also dependent on circumstances. Student accomplishments are not acts by a single person, but are also deeply influenced by the circumstances (or environment) in which they live and learn. A factor that often doesn't get the attention it deservers are the circumstances of students as a critical component in student success. This is a multilayered topic, and the goal of this post is to shed some light on the issues.

Before we dive into the details, an obvious sign of the importance of circumstances is the stress parents feel when figuring out what schools to send their children. It's a clear signal that where kids go to school and the people at the school matter. Yet strangely and in near complete contradiction, the notion that education is a solely individual accomplishment exists.

Math Analogy:  There's a difference between functions of one variable and functions of two or more variables.  Symbolically let $x$ be student effort, and let $y$ represent a student's circumstances. Then what is being asserted is that $f$, a student's achievement (whatever that means), is dependent on $x$ and $y$. As math teachers, we may be prone to looking at teaching as $f(x)$ and not $f(x,y)$ perhaps tacitly or perhaps because we don't know what more can be done.

What does $y$ represent? There are of course the usual things. These are factors like location, family income, ethnicity, poverty, school quality, parents' level of education, and so on.  Additionally we can include schools within the broad category of circumstances. Class environment, curricula, daily schedules, the architecture of the buildings, the number of students in the classes, the teachers, ... all these in the aggregate make up $y$.

Claim: $f(x,y)$ is highly sensitive on $y$.

Rationale:  There's evidence that suggests the sensitivity of $f$ on $y$ is rather significant. In a recent article by Ellis, Fosdick, Rasmussen, evidence is presented suggesting calculus apprehensions can steer women out of the STEM pipeline at 1.5 time the rate compared to men. Simultaneously we also know that the use of IBL reduces sizable gender gaps between men and women compared to non-IBL, traditionally taught courses. (See Laursen, Hassi, Kogan and Weston.)  That is, even changing $y$ by only factors limited to classroom pedagogy can change $f$ in ways we can measure statistically.

Researchers in Germany dig partially into circumstances under the label "Error Climate" (Link to a description 1, Link to description 2).  Steuer and Dresel identify factors that support a positive learning environment, such as empowering students to be willing to experiment and try.  In their work they identify factors that are related to how teachers teach and pedagogy.

1.  Error tolerance by the teacher
2.  Irrelevance of errors for assessment
3.  Teacher support following errors
4.  Absence of negative teacher reactions
5.  Absence of negative classmate reactions
6.  Taking the error risk
7.  Analysis of errors
8.  Functionality of errors for learning

In active, student-centered classes these items can be integrated. Mistakes can be de-stigmatized, and students can learn growth mindsets. We can't do much of anything about factors like poverty, at least not directly via classroom instruction. We can, however, do something about our classroom environments that can minimize gender gaps and other inequities.  Small group work, student presentations, portfolios, projects, productive failure are just a handful of IBL strategies that can be used to create a significantly different set of circumstances for your students.

Additive improvement:  Improving teaching often is focused on $x$, or student effort, via things like books, ordering of topics, clearer exposition, better problem sets, getting students to do homework. These are aimed at the experiences of the student and their effort on the subject. Those are of course good places to expend energy, and what I am suggesting is to add.  Add consideration and teacher effort on $y$, without diminished hard won successes in $x$. That is, improving $f$ optimally includes working on $x$ and $y$, and this is not a zero sum game. Instructors do not have to give up proportionally one to gain in the other.

Francis Su eloquently makes the case in his talk, Freedom Through Inquiry. He shares the story of Gloria Watkins, who experienced two starkly different realities in her education during the change from segregated schools to bussing and integration. Su, an MAA President and accomplished mathematician, shares his personal story about perseverance. The environment in which he grows up in and his educational experiences have made a material impact on his career trajectory and life.
"And just like Watkins, I had professors who didn’t believe I was capable of making it through, especially when I failed my qualifying exams the first time... 
It’s because I had that inquiry-based Moore-method class with Starbird that I knew that I could do research. I already had the experience of discovering things for myself. I knew that I knew how to ask good questions, because we had the freedom to ask any question in Starbird’s class and figure out which ones were fruitful. And I knew how to use those questions as a springboard to independent investigation. 
And because of that, I knew, no matter what anyone said or believed about me, that I could push through. Today’s literature suggests that inquiry-based teaching methods confer significant benefits on underprepared students, and of course I believe it. Because I’ve lived it."
Teachers have opportunities to make transformative changes. Expanding our view to see more variables related to learning helps us see more opportunities to help students succeed. While there are limits, constraints, and societal-level issues that form daunting challenges to improving the circumstances surrounding our students, nevertheless there still exists real and significant opportunities for change, right here in front of us in our classes!

Laursen, Hassi, Kogan and Weston (Link #2) (Link #3)
Ellis, Fosdick, Rasmussen
Error Tolerance
Freedom Through Inquiry by Francis Su

Thursday, August 25, 2016

Students Voices: Taylor

Taylor is a Liberal Studies major (Elementary Education) at Cal Poly, and shares her thoughts about her IBL experiences in Professor Grundmeier's IBL Math for Elementary Teaching classes.

Transformative experiences come in different forms.  In this case, Taylor learned about herself. She learned that she is a math teacher and her experiences in IBL math classes showed her a pathway towards a career in secondary math teaching!

"There's not just one way to solve a math problem..."

Wednesday, August 24, 2016

Beginning of Fall: IBL Blog Playlist

I want to wish all teachers starting their terms now or in a few weeks the very best. The start of a school year is a busy time, and much thought and effort goes into getting up to speed with classes, advising, mentoring, committee work, and on and on. Upward and onward!

We recently compiled an IBL Blog Playlist. This playlist has some of the main ideas we have shared over the years, compiled on a single page. Blog posts were reactions to needs discovered in our work in the IBL community, and over time it has become hard to find the older posts that are still relevant.  We'll keep updating the playlist periodically to keep up with content.

Quick point: If you can do only one thing IBLish, try Think-Pair-Share!

Wednesday, August 17, 2016

CBMS Statement on Active Learning

Okay, this is a really big deal.  The Conference Board of the Mathematical Sciences has weighed in. CBMS supports active learning (CBMS Active Learning Statement)!

Just to be clear, this isn't one or two professors clicking a like button on social media. Let's take a look at the CBMS member societies:

  • AMATYC, American Mathematical Association of Two-Year Colleges
  • AMS, American Mathematical Society
  • AMTE, Association of Mathematics Teacher Educators
  • ASA, American Statistical Association
  • ASL, Association for Symbolic Logic
  • AWM, Association for Women in Mathematics
  • ASSM, Association of State Supervisors of Mathematics
  • BBA, Benjamin Banneker Association
  • IMS, Institute of Mathematical Statistics
  • INFORMS, Institute for Operations Research and the Management Sciences
  • MAA, Mathematical Association of America
  • NAM, National Association of Mathematicians
  • NCSM, National Council of Supervisors of Mathematics
  • NCTM, National Council of Teachers of Mathematics
  • SIAM, Society for Industrial and Applied Mathematics
  • SOA, Society of Actuaries
  • TODOS, TODOS: Mathematics for ALL

These are the main players in college-level mathematics (and PreK-12 mathematics).  They have all signed on to supporting active learning, because "A wealth of research has provided clear evidence that active
learning results in better student performance and retention than more traditional, passive forms of instruction alone. "

The statement goes on to say in bold, "...we call on institutions of higher education, mathematics departments and the mathematics faculty, public policy-makers, and funding agencies to invest time and resources to ensure that effective active learning is incorporated into post-secondary mathematics classrooms."

It needs to be stressed, that active learning and IBL are not fads or fashion statements. These are methods that have been developed over long time periods. Certainly it takes much more work and energy to successfully teach via active learning (e.g. IBL), and for people like me it's not worth it, if it doesn't work.  I have better things to do with my time than just do things for stylistic reasons in my classes.  But we have a lot more evidence now that students learn better, retain more, and inequities like gender bias can be mitigated via active learning strategies.

If you have not done so yet, I encourage you to take a step towards actively engaging your students!

Thursday, June 2, 2016

10+ Videos on Productive Failure (Playlist)

Productive failure is increasingly becoming an important aspect in teaching, in light of the growth mindset research that have been published recently.  Below is a short list of videos I find useful to share with students.

1. Michael Jordan "Failure" Commercial

2. Sal Khan interviews Carol Dweck on Growth Mindset

3. John Legend, Musician: "Success through Effort"

4. IBL Instructors discuss the importance of failure

5. Growth Mindset Animation

6. Mike Starbird: Study Skills and Making Mistakes

7. Study Skills: Learning From Mistakes (Jo Boaler)

8. Diana Laufenberg: How to Learn from Mistakes

9. Uri Alon: Why Truly Innovative Science Demands a Leap into the Unknown

10. Astro Teller: The Unexpected Benefit of Celebrating Failure


11. One more suggested by Bret Benesh: Ira Glass

12. Thanks to Jane Cushman for sending me this:  Karen Schultz, On Being Wrong

Thursday, May 26, 2016

For Parents of K-12 Students (U.S.)

Dear Parents,
Every weekday morning I drop off my son at school.  Every weekday afternoon I pick him up. I have a vested interest in the success of schools both personally and professionally. When I talk to parents about their children’s education, I have noticed, however, that most parents have major gaps in their understanding of how our education system works, Common Core, active learning, and the point of education. In this post, I hope to nudge you to learn more about the issue for the sake of our children.

Basics about Education
Before we can talk about the main issues, we need to be clear on the basics. Some parents I talk to do not understand the “ingredients" of education. Just like cooking, you need to get the right ingredients, and on an even more basic level be able to recognize what those ingredients are. 

Basic ingredients of Education (in the US):
  • The state standards
  •  Curriculum (books and textbooks)
  • Teaching (Instruction) and learning environment
  • Student Learning
  • Assessment
  • Other (e.g. counseling, sports, clubs, facilities, etc.)

Looking at a list of ingredients doesn’t convey what the final product is. If you get recipe and only look at the list of ingredients, it doesn't tell you what the final dish will be when served.  Education (as a system) is much more complex than cooking a meal.  Hence, we need a way to organize the information to help us make sense of what we perceive at our schools.

Models let us see better how the pieces fit together, by organizing them in a structure.  The simple model below shows more or less how the basic ingredients fit together. It's not meant to be a definitive model covering all aspects of education. We're going to use the model to illustrate key points.
Your state (not the federal government) sets the standards. School standards then eventually result in textbooks that cover the standards. From there, teachers must take these pieces (and other resources) and incorporate them into their teaching system, and design classroom activities. Teachers must also customize the learning experiences to the actual students they have in the classroom. Student experiences and mindsets vary significantly, and day-to-day instruction adapts according to student learning needs.

An Example of How Some Parents Can Blame the Wrong Thing
I hear a ton of Common Core bashing, and what I hear is a lack of understanding of what education is, what standards are, and what standards are not. I noticed a particular example of this recently. Let's first set this up.

In grades K-6, research on homework strongly suggests that there's no learning gains for homework. It does perhaps cause students to dislike subjects or learning in some cases, so in sum it's a bad idea. High performing nations like Finland essentially do away with homework.

So homework in K-6 doesn't do anything for some students, and for others it's a net negative. The policy that should be adopted is to eliminate homework or reduce it significantly in elementary grades. Despite this research, parent often ask for homework, because that is what they grew up with.

Where things go wrong is when a teacher (in the US) is implementing math, such as in Common Core, where in addition to learning skills, students are also asked to think, explain, experiment, problem solve. This type of curriculum needs a carefully designed class experience and parents who understand that doing math well means doing math like mathematicians. That is, making mistakes, experimenting, and taking time to think deeply about the concepts. This is math that is far beyond what most parents experienced. In contrast, in order to placate parents who want the usual homework assignments, teachers send students home with math homework that often has good problems, but these are problems that take longer and require thought. Getting stuck is likely.  So here we have (a) teaching methodology that isn't going to work, based on research evidence, and (b) homework that is challenging that parents, who may have math anxiety will get into a homework-frustration struggle with their kids (at the end of the day, with dinner to cook...) This is a classic "conflating implementation struggles in the early years" with "the standards are a bad idea."

An added layer is that most people think doing math faster means smarter, and doing math slower means dumber. Misconceptions about Math and the nature of learning feed anxieties further, and then the blame game starts. Something has to be wrong, if Johnny can't add!

The final act of this tragedy is that people then assign blame to the wrong thing. They blame the standards, not the environment of parental pressure and the asking for homework in elementary grades. The standards aren't the problem here. It's instruction and how our society doesn't fully let teachers do their job and apply evidence-based practices. If anything, we (parents) make it harder for teachers to switch to better teaching methods.
If the goal is to improve education, then it's important to focus attention and effort on the correct thing. Helping and supporting teachers implement their curriculum as they were intended is what should be the focus. Instead, people want to tear down the standards (which will not improve education), and likely see us return to methods and curricula that have been shown to be significantly flawed and problematic.

Conflating Assessment and Standards
Here's another example. One of the other large pressures putting teachers and staff in a tough place is assessment. In the U.S. we assess too much, and those assessments use huge resources, and are then tied to job security. This creates unintended, "perverse" incentives that make schools teach toward narrow tests, at the expense of a fuller, holistic education. The arts, music, dance, etc. get nixed, only making it harder for children to find their element. Their passions.

Assessment should be done in scientifically sound ways to see how our schools are doing. The reality is that testing is far too large a force, and actually dominates education choices. Parents sometimes see this. They see too much assessment as a problem, where test results are used to punish or threaten teachers or administrators.  Instead of pushing back at the testing regime (and the massive testing industry behind it), some parents blame the state standards.
The issue here is quite clear again. The wrong ingredient has been identified as the problem. When you go in for knee surgery, you want to make sure the surgeon works on the correct one. Well, this is the same sort of thing. The torn ACL is on the left, but we've done surgery on the right. Most people would agree that it taking a step back, and not solving the real problem.

Understand the Problem First
What can parents do? One of the ideas we try to teach students at all levels it to understand what the problem is saying first, before making conclusions. We want students to ask pertinent questions, and understand the components of the question and the context surrounding it.  I think this lesson applies broadly. Parents in particular can learn more about education and what the actual issues are, BEFORE forming an opinion. Otherwise, you're committing a basic intellectual mistake or sin, called intellectual indulgence. (Intellectual indulgence is when you believe something to be true, because you like how it sounds, and not because you have any good evidence to support it.)

Ignorance is not a virtue, especially when it comes to decisions and policies about education. In fact, ignorance is damaging. Our society's collective, group ignorance prevents us from achieving far more than what we have. People have strong opinions about education and can prescribe or criticize, even when they do not have even a very basic understanding of what education is, how it's constructed, and what it's for. I hope you can see the deep and tragic irony with ignorance about education.

Parents can start with the books and videos listed below to learn more about growth mindset, what math education is like currently, how it could improve in the US, and how international comparisons shed light on what we can do better. Parents obviously matter a great deal, and an informed group of parents with a positive, constructive attitude can be a powerful force in supporting and shaping our education system.

I have seen it time and again. When teachers are supported by their community and get it right, I see students transform from passive, disengaged people to eager, vibrant learners. Parents come up to me and say things like, "My daughter likes math this year!" There is good reason to be optimistic today, as we have learned much about how to effectively teach. The problem and challenge of improving education is a tractable problem. Parents can choose today to be a positive contributor to this process. You can say to your kids, "Hey, I'm going to show you how I learn about something by doing my own research..." And what a wonderful thing that would be to teach your children!

References for Parents
  1. Mindset, by Carol Dweck
  2. What's Math Got to Do With It, Boaler
  3. Mathematical Mindsets, Boaler
  4. The Teaching Gap, Hiebert and Stigler

1. Carol Dweck on Mindset

2. Jo Boaler on Common Core

3. Khan Academy interview Dweck

4. Dan Meyer on Teaching Mathematics

5. Ken Robinson on Education