The IBL Blog focuses on promoting the use of inquiry-based learning methods in the classroom at the college, secondary and elementary school levels. Learn more about IBL at The Academy of Inquiry Based Learning

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..."

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.

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!

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

Edit:

11. One more suggested by Bret Benesh: Ira Glass

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

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 sayingfirst, 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!

“In art intentions are not sufficient and, as we say in Spanish, love must be proved by deeds and not by reasons. What one does is what counts and not what one had the intention of doing.” -- Pablo Picasso

It's my belief that teachers at all levels have good intentions and genuinely want their students to learn. What this post is about is the notion that instructor intentions are not sufficient, and getting students to do what is needed for authentic learning is what counts.

Despite multiple passes through the proof and explanations, students in the study have a difficult time pulling out the instructor's intended messages in the proof and the instructor's spoken comments. At first glance this makes sense, as the "information transfer" model doesn't work so well when the goal is "developing critical thinking."

As mentioned by Bressoud, Annie and John Selden and others have documented the difficulties students have with analyzing, proving, generalizing, packing/unpacking statements, etc. Learning higher mathematics is challenging, and most math majors struggle with learning proof.

Bressoud suggests options like flipped classrooms and clickers can work, but he notes that such methods have high initial investment in time and requisite knowledge and skill. While these methods are learnable, many instructors may be in situations, where implementing them is not practicably feasible. Other options are needed.

Another Option Exists!
Now to the main point of this post. There exists an "easy entry, high upside" method to help students come away with the intended messages. Put simply, instructors can take their list of intended messages and turn them into math tasks. These tasks can be deployed via small group work, homework, etc.

Let's take a closer look. The intended messages of the instructor in the study are

Cauchy sequences can be thought of as sequences that “bunch up”

One can prove a sequence with an unknown limit converges by showing it is Cauchy

This proof shows how one sets up a proof that a sequence is Cauchy

The triangle inequality is useful in proving series in absolute value formulae are small

The geometric series formula is part of the mathematical toolbox that can be used to keep some desired quantities small

Each of these five points can be turned into "after the proof" problems. They can be reformulated as

Explain using sentences and diagrams why Cauchy sequences "bunch up."

True or False and Explain: One can prove a sequence with an unknown limit converges by showing it is Cauchy.

In the proof, find the part that proves the sequence is Cauchy.

In the proof, find the part where the triangle inequality is used, and then identify a general strategy based on this specific instance that you can put in your mathematical toolbox.

Explain how the geometric series is part of a mathematical toolbox to keep some desired quantities small.

One way to deploy these in the classroom is to present the proof on the board or pass out a handout with the proof, and then have students work through some of the tasks in pairs and then share (i.e. Think-Pair-Share). If time is an issue, doing one or two and assign the rest for homework. Alternatively instructors could ask students to read the proof before class, and the instructor could highlight the proof and spend more time on student-centered tasks in class.

Several advantages exist with this option compared to flipped classes or using clickers. The first advantage is that it requires the least experience and least amount of pre-class planning or classroom equipment. One could be in classrooms like I sometimes teach in with no technology, old boards, and desks built 50 years ago.

Another advantage is that it does not require deep knowledge of common misconceptions. Instead, the instructor asks students to explain their thinking (in one way or another) and that's how the instructor gets insights into student thinking. That is, use an activity and gather formative assessment.

A third advantage of this method is that it does not deviate from the conventional class prep process used by most instructors. Preparing presentations of a proof with explanations is something that instructors have done many times. The method presented here tweaks the process by transforming the intended messages that instructors would normally say to students into concrete mathematical tasks for students to work on. This change is practically feasible and doesn't require a significant alteration of an instructor's workflow. It's also a step towards active, student-centered teaching and can be built upon over time into forms of teaching that more deeply engages students, such as IBL.

Instructors have good intentions and intended messages. I claim that it is how the intended messages are deployed in class that can be addressed in practical and effective ways. We can turn those amazing insights into amazing learning experiences, and "let problems do the talking!"