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Cybrary Man's Educational Web Sites The internet catalogue for students, teachers, administrators & parents. Over 20,000 relevant links personally selected by an educator/author with over 30 years of experience.

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Rigor doesn't simply mean giving students more or harder work. Instead, it's the result of work that challenges students' thinking in new and interesting ways.

You would think that it would be more prevalent than it is. But it appears only four times in the Common Core State Standards. Why has a word that is mentioned so little caused such dread, anxiety, and confusion among teachers?

I'm talking about rigor.

When We Say Rigor, What Do We Mean?

Comb through all 66 pages of the ELA standards, and you will find it hiding amid larger conversations about analyzing author's choice, evaluating sources, and writing arguments. Look in the math standards, and you will not find it at all.

"Our lesson must be more rigorous."

"We must increase the rigor of our assessments."

"Does this book possess the necessary rigor for that grade level?"

Yet rigor is all the buzz:

These are all things that I have heard at conferences, in faculty meetings, and through conversations with colleagues. It is a term used often, but I am still not sure if it has been clearly defined.

Some mistakenly assume that rigor means making things more difficult. Others believe it means piling on the work. A few say that they can't define rigor, but they know it when they see it.

If teachers are to achieve rigor, we must aspire to something more specific. Too bad the dictionary is of little help:

Rigor
1. (a) Harsh inflexibility in opinion, temper, or judgment: severity. (b) The quality of being unyielding or inflexible. (c) An act or instance of strictness, severity, or cruelty.
2. A tremor caused by a chill.
3. A condition that makes life difficult, challenging, or uncomfortable.
4. Strict precision or exactness.

It is this understanding that has led to the push-down and pile-on syndrome. College-level books are now being taught in high schools. Middle school students are tackling works and ideas once assigned to high school students. Now, 20 minutes of homework for elementary kids has become two hours of cruelty.

Rigor is not defined by the text -- it comes from what students do. It is not standard across a curriculum -- it is individual to each student's needs. It is not quantified by how much gets crammed into a school day -- it is measured in depth of understanding.

Rigor is a result, not a cause.

Rigor and David Foster Wallace

For proof, we need look no further than the great 20th-century novelist, David Foster Wallace. In 1994, he taught English 102 (Literary Analysis: Prose Fiction) at Illinois State University. His syllabus does not feature the heavyweights of literature that are recommended by the Common Core. No Hamlet. No Crime and Punishment. No Canterbury Tales.

Instead, his required texts were Mary Higgins Clark's Where Are the Children?, Thomas Harris' The Silence of the Lambs, Stephen King's Carrie, and Larry McMurtry's Lonesome Dove, among others. Interesting that a writer known for being a little pretentious might have known a thing or two about rigor. In the syllabus, Foster-Wallace writes:

Don't let any lightweightish-looking qualities of the texts delude you into thinking this will be a blow-off-type class. These "popular" texts will end up being harder than more conventionally "literary" works to unpack and read critically.

There it is. Rigor is the result of work that challenges students' thinking in new and interesting ways. It occurs when they are encouraged toward a sophisticated understanding of fundamental ideas and are driven by curiosity to discover what they don't know.

Foster-Wallace makes a point in his syllabus to say that his course will not be what many would expect: "heavy-duty lit-crit or Literary Theory." Instead, he has the broader and more practical aim to develop students that can. . .

. . . read fiction more deeply, to come up with more interesting insights on how pieces of fiction work, to have informed intelligent reasons for liking or disliking a piece of fiction, and to write -- clearly, persuasively, and above all interestingly -- about stuff you've read.

Let us aspire to something greater than making difficult work for our students. Let's take them to that intersection of encouragement and engagement, where they confront ideas and problems that are meaningful. Let's stretch their thinking. Let's unleash their sophistication. And let's foster a love of deep knowledge.

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Rigor doesn't simply mean giving students more or harder work. Instead, it's the result of work that challenges students' thinking in new and interesting ways. You would think that it would be more prevalent than it is. But it appears only four times in the Common Core State Standards.

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The Rigor/Relevance Framework is a tool developed by the International Center to examine curriculum, instruction, and assessment along the two dimensions of higher standards and student achievement. It can be used in the development of both instruction and assessment.

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EduHacker Mark Barnes, creator of the Hack Learning Series, brings you cutting edge, progressive-minded education strategies. This is the stuff you can do tomorrow, because teachers, school leaders, and parents, interested in problem solving, need practical right-now answers--not 5-year plans. Check out http://hacklearning.org. It's what you can do tomorrow.

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rigor - definition, examples, related words and more at Wordnik

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rigor: Strictness or severity, as in temperament, action, or judgment.

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Let's start by clearing up a misconception: Rigor isn't unfriendly. Adding it to your class doesn't mean you become boring, a techie, or an overseer of a fun-free zone.

In fact, done right, rigor fills your class with Wow!, those epiphanies that bring a smile to student faces and a sense of well-being to their school day.

Rigor provides positive experiences, is an emotional high, and engenders a pervasive sense of accomplishment students will carry for years--and use as a template for future events.

Rigor is NOT lots of homework, projects, resources, or rules. When those four nouns are used to define rigor, the teacher is flailing, thinking quantity is quality. Rigor is not about adding a column of data or remembering the main characters in a Shakespeare play. It's seeing how knowledge connects to life, to circumstances, and to daily problems.

Simply put, adding rigor creates an environment where students are:

Expected to learn at high levels.

Supported so they can learn at high levels.

Cheered on as they demonstrate learning at high levels.

It also helps students understand how to live life using brainpower as the engine. Sure, it will ask them to collect evidence and draw conclusions with which their peers may find disagreement. But it will insist they defend a position or adjust it to reflect new information, and it will often move them outside their comfort zone. It will also prepare them to solve the problems they will face in the future.

How do you as a teacher add rigor? Here are some ideas from my PLN colleagues:

1. Make it easy to decode words by adding an online dictionary or a backchannel that students can access with unknown words.

2. Build grit in students. Let them know they are capable, competent, that thinking hard does not mean they didn't understand. It means they want to understand.

3. Expect inquiry. Be prepared. Make time for it.

4. When you ask a question, pause. Let students think before answering. This wait time isn't delaying your lesson. In a rigorous classroom, it is the lesson.

5. Expect rigor -- don't allow students to be satisfied with a superficial answer. Build habits that look for understanding.

6. Face unknown questions with a smile. Make the classroom a safe, non-judgmental learning environment where students can practice their critical thinking skills.

7. Require students do projects with a high degree of precision and skill.

8. Make it normal to invent solutions that haven't been thought of before, but satisfy requirements.

9. Expect students to doublecheck both the approach and result (according to Grant Wiggins -- great article).

10. Create an environment conducive to personal growth. Know what that means for each student.

11. Focus on exceptional expectations.

12. As students participate in class conversations, expect them to use academic and domain-specific vocabulary. If they use words like "something," "you know," "that," or "like," prod them to come up with specifics. Like what? No I don't know. This immerses them in learning, discovering, and thinking critically.

13. As students answer questions, expect evidence to support their answer. This can come from personal experience, but more often will find relevance from learning resources.

14. Listen to student answers. Pay attention. Challenge them to be thorough.

15. Don't draw conclusions for students. Present them with evidence. See where it takes them.

16. Expect students to build on classmate comments and understand their perspectives.

17. Support students so they can reach high expectations. If they are thinking outside the box, don't pull them back in. Ask questions: Is this approach going to meet expectations, solve problems, achieve desired results? If it is, let them do it!

18. Provide a way for each student to demonstrate learning even if it isn't a way you've thought of. Applaud them if they devise an approach that works for them.

19. Differentiate for student needs. If you have a project that seems to fit everyone, it doesn't. How do I know that? Because no project does.

20. Expect students to analyze data during reading. Ask why -- determine if the author provided proof or simply made a statement -- and know the difference.

21. Expect problem solving. Don't jump in to solve problems. Provide students with strategies they can use and let them try them.

22. Remind students that "easy" occurs through hard work. Discuss this -- how they excelled at soccer, piano, or the violin through tenacity and drive. Have students share their experiences.

For more on higher-order thinking, check out NWEA's education blog. As I researched this post, I kept running across Barbara Blackburn's name. You might want to check her out for more in-depth materials.

has been teaching K-8 technology for 15 years. She is the editor of a K-8 technology curriculum, K-8 keyboard curriculum, K-8 Digital Citizenship curriculum, and creator of dozens of technology training books that integrate technology into education. She is webmaster for six blogs, CSG Master Teacher, an book reviewer, Editorial Review Board member for Journal for Computing Teachers, CAEP reviewer, CSTA presentation reviewer, freelance journalist on tech ed topics, a tech ed columnist for , and a weekly contributor to .

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Let's start by clearing up a misconception: Rigor isn't unfriendly. Adding it to your class doesn't mean you become boring, a techie, or an overseer of a fun-free zone. In fact, done right, rigor fills your class with Wow!, those epiphanies that bring a smile to student faces and a sense of well-being to their school day.

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I thought I had a pretty good handle on universal design for learning (UDL), but after chatting with Katie Novak, Ed.D., I realize I didn't understand the framework at all. Novak, author of UDL Now! A Teacher's Guide to Applying Universal Design for Learning in Today's Classrooms, Second Edition and assistant superintendent of schools at the Groton-Dunstable (MA) Regional School District, helped me truly understand what UDL is and, perhaps more important, what it isn't.

Q: What exactly is UDL, and why does it matter?
A: Our classrooms today are incredibly diverse. As we embrace equity and inclusion, we have to meet the needs of all students. To do this, we have to change the way we "do" school. When I was young, we were tracked, starting in first grade, into "high" or "low" reading groups, gifted, etc. Now we know that's not good for anyone. Classes have a wide mix of strengths and weaknesses, and a one-size-fits-all curriculum does not meet most children's needs.

In 1990, the Individuals with Disabilities Education Act (IDEA) and the Americans with Disabilities Act (ADA) renewed our national focus on the least-restrictive environment. As more and more students were educated with their peers, we started to realize that having all students read the same book and take the same test doesn't work. We began providing accommodations through differentiated instruction and teachers figured out what to change or modify to accommodate "disabled" learners. Although this allowed students to access knowledge, they weren't empowered to become learners and make choices for themselves. Instead, the curriculum was compartmentalized and decisions were consistently made about students without their voice.

The UDL framework starts with the belief that every student is different and that's the norm. We call these differences "variability" and we embrace it. When students come to us differently, and they face barriers to learning, it's our curriculum that's disabled, not our students. All students have assets and strengths and goals and interests. UDL lets us offer them options and choices to create personalized pathways to meet very rigorous goals. Our job is to teach them as they come.

Q: Can you give me an example?
A: Sure. In English language arts, students need to analyze how complex characters develop over the course of a text, interact with other characters, and advance the plot. Ten to fifteen years ago, the whole class would read The Old Man and the Sea, analyze how Santiago develops over the course of the text, and write an essay. But that doesn't tell the teacher if the students understand characterization; it only proves they can answer questions about that novel.

A UDL practitioner helps demystifies the framework

With UDL, you look at a traditional lesson and start by identifying the barriers. For this example, the barriers of handing out the same novel to every student could be that the reading level is too rigorous, or not challenging enough, the students lack necessary background knowledge, or visual processing is challenging. To eliminate those barriers, I would provide options like listening to the novel, providing visuals or supporting background knowledge through a video, reading with a partner, or choosing a different book. To teach about characterization, I'd provide students with the option to work individually or in small groups and offer resources and videos so they can personalize their learning and connect it to a text that's relevant, authentic, and meaningful to them.

After that, I'd ask students to express what they learned about character development by either writing an essay, creating an infographic, doing a presentation, creating a vlog, working alone or with partners, and so on. In UDL, I start with the goal and consider which options and choices students need to achieve that goal. There are numerous pathways to reach the same destination.

Q: That sounds like differentiated instruction.
A: UDL empowers students to recognize their own interests and needs and personalize learning to their standard. Differentiated instruction is about what the teacher will do, based on her perceptions of students' needs. With UDL, teachers proactively design the curriculum to eliminate barriers. Differentiated instruction emphasizes the teacher's role to address students' needs; the teacher constructs activities based on different groupings of students. UDL empowers teachers to design lessons for the broadest possible range of students.

Q: Wow; the two are really quite different. Tell me more about UDL.
A: With UDL, we let students make choices and then participate in a self-assessment. They learn how their choices allowed them to work toward their goals so they can make adjustments and learn about what works and what doesn't.

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I thought I had a pretty good handle on universal design for learning (UDL), but after chatting with Katie Novak, Ed.D., I realize I didn't understand the framework at all. Novak, author of UDL Now!

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Practice makes perfect, so the cliché says, although like many clichés, it has also spawned corrective variants. "Practice makes permanent," a common one of them goes, and what it lacks in catchiness it may well make up for in neuroscientific truth. We've all recognized that, when we do things a certain way, we tend to keep doing them in that certain way; in fact, the more we've done them that way before, the more likely we'll do them that way next time. What holds true for simple habits, formed over long periods of time and often inadvertently, also holds true for deliberately perfected - or anyway, permanent-ified - tasks. But what happens in our brains to cause it?

"Practice is the repetition of an action with the goal of improvement, and it helps us perform with more ease, speed, and confidence," says the narrator of "How to Practice Effectively... for Just About Anything," educators Annie Bosler and Don Greene's TED Ed video above. It then goes on to explain our two kinds of neural tissue, grey matter and white matter. The former "processes information in the brain, directing signals and sensory stimuli to nerve cells," and the latter "is mostly made up of fatty tissue and nerve fibers." When we move, "information needs to travel from the brain's grey matter, down the spinal cord, through a chain of nerve fibers called axons to our muscles," and those axons in the white matter "are wrapped with a fatty substance called myelin."

Myelin, and the sheath it forms, is key: "similar to insulation on electrical cables," it "prevents energy loss from electrical signals that the brain uses, moving them more efficiently along neural pathways." (You've probably read about the weakening of myelin sheaths as a factor in ALS and other movement-related neurological disorders.) Recent studies performed on mice suggest that repeating a motion builds up the layers of those axon-insulating myelin sheaths, "and the more layers, the greater the insulation around the axon chains; forming a sort of superhighway for information connecting your brain to your muscles." This, though it has no direct effect on our muscles, may be what we're building when we say we're building "muscle memory."

All interesting facts, to be sure, but how can they help us in or own practice sessions, whatever those sessions may find us practicing? Bosler and Greene provide a series of tips, each quite simple but all in alignment with current neuroscientific knowledge. They include:

Focus on the task at hand. "Minimize potential distractions by turning off the computer or TV and putting your cell phone on airplane mode."

Go slow. "Coordination is built with repetitions, whether correct or incorrect. If you gradually increase the speed of the quality repetitions, you have a better chance of doing them correctly."

Frequent repetitions with allotted breaks. "Studies have shown that many top athletes, musicians, and dancers spend 50-60 hours per week on activities related to their craft. Many divide their time used for effective practice into multiple daily practice sessions of limited duration."

Practice in your imagination. "In one study, 144 basketball players were divided into two groups. Group A physically practiced one-handed free throws while Group B only mentally practiced them. When they were tested at the end of the two week experiment, the intermediate and experienced players in both groups had improved by nearly the same amount."

If you'd like more suggestions on how to practice effectively, have a look at the list of twelve tips from Wynton Marsalis we featured here on Open Culture last year. He takes a more expansive approach, encouraging those who practice - not just music but sports, art, or anything else besides - to adopt strategies like writing out a schedule, avoiding showing off, and staying optimistic. We must also stay realistic: optimism, even optimism backed by science, can't make our skills perfect. None of our skills are perfect - not even Wynton Marsalis' - but with the right techniques, we can at least give them some degree of permanence.
Related Content: Playing an Instrument Is a Great Workout For Your Brain: New Animation Explains Why Wynton Marsalis Gives 12 Tips on How to Practice: For Musicians, Athletes, or Anyone Who Wants to Learn Something New What Are the Most Effective Strategies for Learning a Foreign Language?: Six TED Talks Provide the Answers How Buddhism & Neuroscience Can Help You Change How Your Mind Works: A New Course by Bestselling Author Robert Wright
Based in Seoul, Colin Marshall writes and broadcasts on cities, language, and culture. His projects include the book The Stateless City: a Walk through 21st-Century Los Angeles and the video series The City in Cinema. Follow him on Twitter at @colinmarshall or on Faceboo k.

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How to Practice Effectively: Lessons from Neuroscience Can Help Us Master Skills in Music, Sports & Beyond

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Practice makes perfect, so the cliché says, although like many clichés, it has also spawned corrective variants. 'Practice makes permanent,' a common one of them goes, and what it lacks in catchiness it may well make up for in neuroscientific truth.

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by Terry Heick

Rigor is a fundamental piece of any learning experience.

It is also among the most troublesome due to its subjectivity. What does it mean? What are its characteristics? Rigorous for whom? And more importantly, how can you use to promote understanding?

Barbara Blackburn, author of "Rigor is not a 4-Letter Word," shared 5 "myths" concerning rigor, and they are indicative of the common misconceptions: that difficult, dry, academic, sink-or-swim learning is inherently rigorous.
Myth #1: Lots of Homework is a Sign of Rigor
Myth #2: Rigor Means Doing More
Myth #3: Rigor is Not For Everyone
Myth #4: Providing Support Means Lessening Rigor
Myth #5: Resources Do Not Equal Rigor Why Rigor Matters
Rigor matters because it imposes cognitive load on students, forcing them to confront misconceptions, reconsider positions, separate the implicit from the explicit, and other critical thinking practices that distinguish shaky familiarity from true understanding.

As such, it's different for every student. If students can't consistently negotiate rigorous tasks, either understanding or thinking habits should be more closely examined. But if work is beyond their Zone of Proximal Development, students are only being setup for disengagement, frustration, and ultimately failure.
10 Strategies To Add Rigor To Any Lesson, Unit, or Assessment 1. Necessitate a transfer of understanding
Several common classroom tasks are inherently rigorous, including reading idea-dense literature, taking notes, and using the writing process itself, but these are rarely engaging, and don't always fit with a given academic standard or task. But the following 10 strategies can be used to add rigor to almost anything.
2. Require students to synthesize multiple sources
By definition, transfer requires a student to apply knowledge in new and unfamiliar situations, an inherently rigorous process. If you can encourage self-initiated transfer (unprompted or coached), all the better.
3. Design tasks with multiple steps that build cognitively
In rigorous tasks, learners will often need to synthesize data, positions, or theories from multiple sources or perspectives. Whether these are literary perspectives, scientific viewpoints, religious ideas, mathematical theories, or any other fundamentally subjective content, when learners have to analyze, internalize, and reconcile multiple perspectives to create a new position or perspective, rigor is a safe bet.

Not all tasks that require multiple steps are inherently rigorous ( fill out the worksheet, turn it in, read the book, answer the questions, talk to your partner about your answers and turn them in is neither a unique or rigorous approach to learning).
4. Use divergent perspectives
If the tasks build (somewhat parallel to Bloom's Taxonomy), rigor is more likely. In this approach, a student might define "conflict," analyze cause-effect of a specific conflict, research the sources of said conflict, then design some kind of short-term solution to one critical cause of said conflict. This approach starts simple, becomes more complex, and is likely to challenge any student no matter how "proficient" their understanding.
5. Use divergent media forms
Use authors, philosophers, artists, content experts, or other thinkers who make authentic cases of their own that offer contrasting perspectives. Not only does this encourage argument analysis, credibility, etc., but also models how elusive and illusory "truth is," a rigorous idea of its own.
6. Break away from content-area convention
Rather than two (or more) texts, require students to analyze a conversation, a poem, and a tweet; a YouTube video, an encyclopedia resource, and an interview with an expert. The more (seemingly) awkward and divergent, the more learners are challenged to develop new strategies to find solutions.
7. Require design thinking (often in project-based learning)
Use literature to frame math. Use science to promote political discussions. Extract the philosophy from cartoons. Find poetry in the stars. Use Google Earth to make sociological observations. These approaches force students to revise schema for new situations, a key characteristics of rigor.
8. Require long-term observation or analysis
Build design thinking into rubrics or scoring criteria, supply exemplars, or model their use, but whatever you do, be sure that elements of design thinking, creativity, and the "tinker culture" aren't just "encouraged" but required for the student to find success.
9. Study nuance
Another potential use of project-based learning or learning simulations, when students are required to observe long-term, cognitive actions such as identifying patterns, cause-effect analysis, and problem-solution thinking are natural by-products.
10. Require students to take and defend positions
Nuance is often overlooked, and offers a world of rigor due to the unique thinking habits it requires.
Rigor is Always Accessible
This can be done first in small groups, then socialized to larger groups (hopefully outside the classroom). A "position" requires a kind of cognitive ownership that is not only indirectly engaging, but also intellectually stimulating and even emotionally demanding, requiring students to think "Why?" as much as "What?," "When?," and "Where?"

As Strong, Silver, and Perini explain in "Teaching What Matters," rigor is a "quality of content, not a measure of the quantity of the content we cover." Certain content areas may be more inherently rigorous than others (Astrophysics comes to mind), but rigor can be added to anything.

Watch an episode of Spongebob, have students reconcile Patrick's presumptions about friendship with Whitman's ideas on the rugged individual, ask students to study the nuance of Spongebob's body language and speech patterns over the course of several episode to reveal patterns, then require them to socialize their thinking in small groups to present a new theory on interpersonal relationships from America's inception to today. You've used #s 2, 4, 5, and 10.
Image attribution North Carolina Department of Public Instruction and flickr user woodleywonderworks
The above is a purposefully absurd idea, but the premise is clear: Rigor is always accessible.

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by Terry Heick Rigor is a fundamental piece of any learning experience. It is also among the most troublesome due to its subjectivity. What does it mean? What are its characteristics? Rigorous for whom? And more importantly, how can you use to promote understanding?

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harsh inflexibility in opinion, temper, or judgment : severity; the quality of being unyielding or inflexible : strictness; severity of life : austerity... See the full definition

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To illustrate the value of metacognition and how it actually plays a role in learning, we can consider an example from mathematics, where it has been shown that metacognition plays a central role in learning and achievement. Specifically, when novice students were compared to seasoned mathematicians, the students selected a seemingly useful strategy and continued to apply it without checking to see if the strategy of choice was actually working well. Thus, a significant amount of time was wasted in fruitless pursuits. The more experienced mathematicians on the other hand, exercised metacognition, monitoring their approach all along the way to see if it was actually leading to a solution or merely to a dead end.Being aware of how one is engaging with the process of learning influences how the student interprets the task at hand, and what strategies are selected and employed in service of achieving learning goals. It can help optimize the problem-solving experience at a very high level, and is thus applicable across a large range of contexts. These metacognitive strategies are powerful tools for any discipline, inter-discipline or for learning in general.

Of course, with such an abstract learning goal, it is important for educators to be precise with how they teach it. Traditional methods for improving students' learning strategies often focus on prescribed procedures (note-taking, self-testing, scheduling, etc.) and typically result in initial motivation and some short-term improvement, but ultimately a reversion to old habits happens. While these tactics may work in the short term (e.g., to cram for an exam), once the context changes, successful transfer of these methods is less likely to occur. More strategic methods that focus on metacognition for deeper learning- such as developing a growth mindset (discussed later), setting and monitoring one's learning goals, and growing one's capacity to persist despite difficulties-have been shown to result in more permanent learning gains.

It is important to note that since metacognition involves higher-level thinking overseeing lower-level thoughts, there is actually a range of mental processes that fall under its definition. Effects of metacognitive training vary based on what kind of lower-level thoughts are being overseen, and how they are being overseen. Research has identified three levels of reporting on metacognitive processes:

1. Verbalization of knowledge that is already in a verbal state (such as recalling what happened in a story).

2. Verbalization of nonverbal knowledge (such as recalling how one solved a Rubik's Cube).

3. Verbalization of explanations of verbal or nonverbal knowledge (such as explaining how one makes use of the rhetorical structures of a story as one reads).

Only this third level of metacognitive process has been linked to improved results in problem solving.

Metacognition can be developed in students in the context of their current goals and can enhance their learning of competenciesas well as transfer of learning, no matter their starting achievement level. In fact, it may be most useful for lower-achieving students, as the higher-achieving students are already employing strategies that have proven successful for them. For learning disabled and low - achieving students, metacognitive training has been shown to improve behavior more effectively than traditional attention-control training.

Students who have higher levels of self-efficacy (more confidence in their ability to achieve their goals) are more likely to engage in metacognition and, in turn, are more likely to perform at higher levels. This strongly indicates a positive feedback loop for high-achieving students-they are more successful by using metacognitive strategies, which increases their confidence and in turn leads them to continue to increase their performance. Metacognition is an integral part of this virtuous learning cycle, and one that is amenable to further improvement through instruction.
Charles Fadel is founder of the Center for Curriculum Redesign, Bernie Trilling is founder of 21st Century Learning Advisors and Maya Bialik is researcher at CCR.

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The Role of Metacognition in Learning and Achievement

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To illustrate the value of metacognition and how it actually plays a role in learning, we can consider an example from mathematics, where it has been shown that metacognition plays a central role in learning and achievement.