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How might you apply the TPACK model when planning a unit of work? 

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How might you apply the TPACK model when planning a unit of work? 

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Content: Have students compile what they know about a topic before a unit and again after the unit
Pedagogy: Use a concept map or mind map, which may also connect with skills development such as critical thinking, communication, collaborating, and creativity
Technology: Use a digital mind mapping tool, such as this, to connect ideas (before, after, and connecting what was the same)

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How might you use the SAMR model in your own practice?

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How might you use the SAMR model in your own practice?

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Substitution and Augmentation: Convert lessons/notes to PDFs and upload online for easy reference; allow students to do assignments/answer questions online
Modification and Redefinition: Use websites like Canvas to manage the class and Zoom to allow for students to ask questions when they normally would not feel comfortable doing so in-person in front of their peers; use websites like Twitter and Skype to collect data/conduct interviews (such as in statistics projects)

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Good technology integration isn't about using the fanciest tool, it's about being aware of the range of options and picking the right strategy-or strategies-for the lesson at hand.

The biggest obstacle to teaching online probably isn't the technology. Teachers seek out educational technology, in fact, because it "can have considerable positive impacts on student performance," according to a 2016 study -improving test scores and allowing teachers to assess student achievement more efficiently. The big problem is how to integrate it: Beyond the sheer number of tech tools available, the same researchers identified "inadequate professional development and training" as the primary obstacle to using technology productively in classrooms.

Understandably, the emergence of the coronavirus has dramatically accelerated the process of integrating edtech, as educators around the country race to get online as fast as they can. But as many of our teachers have noted, the current state of K-12 online learning is more like triage-a form of crisis management-and not at all like skillfully managed distance education.

What's missing from the conversation right now, which makes sense given the immediacy of the need, is a focus on the bigger question of what the highest quality technology integration actually looks like. That's an important conversation to have, because in a post-coronavirus world there's likely to be a greater emphasis on digital learning, even as we return to physical schools and take advantage of face-to-face opportunities.

A Hierarchy of Tech Uses

A powerful conceptual tool to think about technology integration-and edtech's best uses-is the SAMR model, developed in 2010 by education researcher Ruben Puentedura, who was the 1991 recipient of a Phi Beta Kappa teaching award.

The SAMR model lays out four tiers of online learning, presented roughly in order of their sophistication and transformative power: substitution, augmentation, modification, and r edefinition. When switching to an online format, teachers often focus on the first two levels, which involve replacing traditional materials with digital ones: converting lessons and worksheets into PDFs and posting them online, or recording lectures on video and making them available for asynchronous learning, for example.
Creative Commons
Those are important steps, especially when teaching online for the first time, but in classrooms where tech integration has moved to the mastery level, the last two levels of the SAMR model-modification and redefinition-should also be in the mix. Students in classes where this kind of mastery is embedded find more novel and immersive uses for technology. They are creators and publishers of their own work across multiple forms of media, for example, or they are inviting professionals to provide feedback on their work products, or participating in digital forums with other peers around the world.

It's tempting to think of SAMR as a mountain to be summited. But good technology integration isn't about living at the top of the SAMR model; it's about being aware of the range of options and picking the right strategy-or strategies-for the lesson at hand.

Here's a closer look at good classroom practices at each level in the model:

Substitution

"Substitution" means replacing traditional activities and materials-like in-class lectures or paper worksheets-with digital versions. There is no substantial change to the content, just the way that it is delivered.

The goal here is to keep things simple: there's no need to reinvent the wheel. Scan your lessons and worksheets, convert them into PDFs, and post them online using Microsoft OneDrive, Google Drive, or a similar file-sharing service. Think about the information you have on your walls, such as the classroom norms, the daily schedule, or vocabulary lists, and convert them into digital formats that students can easily reference.

It may also help to provide synchronous as well as asynchronous versions of your lectures. If you're holding class meetings over a videoconferencing service like Zoom or Skype, provide a recording for students who can't attend. You can also create your own instructional videos for students to view at their own pace.

Augmentation

This level involves incorporating interactive digital enhancements and elements like comments, hyperlinks, or multimedia. The content remains unchanged, but students can now take advantage of digital features to enhance the lesson.

For example, students can create digital portfolios to create multimedia presentations, giving them more options to demonstrate their understanding of a topic. And instead of handing out paper quizzes, you can gamify your quizzes with tools like Socrative and Kahoot.

Teachers can also create virtual bulletin boards-using an app like Padlet -where students can post questions, links, and pictures.

Modification

At this level, teachers can think about using a learning management system like Google Classroom, Moodle, Schoology, or Canvas to handle the logistical aspects of running a classroom, like tracking grades, messaging students, creating a calendar, and posting assignments.

Teaching online opens up new channels of communication, many of which can help students who have traditionally been marginalized. Research shows that girls may be less likely to speak up in class, for example, so they may benefit from backchannels -alternative conversations that can run alongside instruction-that encourage participation.

Zoom's text chat feature, meanwhile, gives students an opportunity to write their questions out, which can feel less intrusive if there are dozens of students participating in the call. Also, students who prefer to collect their thoughts may benefit from slower-paced, asynchronous discussions in an online forum or email threads.

Redefinition

Learning is fundamentally transformed at the "redefinition" level, enabling activities that were previously impossible in the classroom.

For example, virtual pen pals can connect students to other parts of the world, whether it's with other students or experts in a field. Virtual field trips enable students to visit locations like the Amazon rainforest, the Louvre, or the Egyptian pyramids. After reading a book in class, you can invite the author to chat about their work and answer questions.

Technology also provides an opportunity to bring authentic audiences into your virtual classroom, and can make publishers out of your students. Kids can write their own wikis or blogs for public consumption and feedback-and platforms like Quadblogging can connect distant classrooms together so students both write and respond. Students can tackle local problems-like investigating the water quality of a nearby river-and invite members of the community to assess their digital proposals.

Going Beyond SAMR

Finally, consider how technology can be used not just as a way to deliver content, but also to strengthen relationships with your students. For elementary school teacher John Thomas, Seesaw and Flipgrid allow him to share a daily greeting, which his students then respond to-or to each other-with a typed or voiced comment. It's a good way to provide some stability while they're isolated.

How can my lesson be improved using technology?

How can I engage and empower students through technology?

How can online learning more closely resemble authentic, real-world learning?

Remember, again, that SAMR is sometimes thought of as a mountain to summit, but it's really more like a toolbox. The goal isn't to use the most sophisticated tool, but to find the right one for the job. More importantly, however, it's a way to reflect on your technology integration by thinking about a few key questions:

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A Powerful Model for Understanding Good Tech Integration

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Good technology integration isn't about using the fanciest tool, it's about being aware of the range of options and picking the right strategy-or strategies-for the lesson at hand.

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How will you lead the transition toward concept-based math curriculum for teaching deeper understanding?

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How will you lead the transition toward concept-based math curriculum for teaching deeper understanding?

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Use the 8 Mathematical Teaching Practices:

Establish mathematics goals to focus learning.

Implement tasks that promote reasoning and problem solving.

Use and connect mathematical representations.

Facilitate meaningful mathematical discourse.

Pose purposeful questions.

Build procedural fluency from conceptual understanding.

Support productive struggle in learning mathematics.

Elicit and use evidence of student thinking.

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What is the role of a teacher? Is it to transmit information and knowledge? Or is it to inspire, guide, and facilitate the social process of learning? 

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What is the role of a teacher? Is it to transmit information and knowledge? Or is it to inspire, guide, and facilitate the social process of learning? 

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The role of a teacher is to inspire, guide, and facilitate the social process of learning. While teachers do transmit information and knowledge, these are the very basics in which the role of a teacher is built upon. A teacher does not simply teach. We make connections with our students that allow us to inspire, guide, and facilitate the social process of learning.

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Students aren't consumers of facts. They are active creators of knowledge. Schools aren't just brick-and-mortar structures -- they're centers of lifelong learning. And, most important, teaching is recognized as one of the most challenging and respected career choices, absolutely vital to the social, cultural, and economic health of our nation.

Today, the seeds of such a dramatic transformation in education are being planted. Prompted by massive revolutions in knowledge, information technology, and public demand for better learning, schools nationwide are slowly but surely restructuring themselves.

Leading the way are thousands of teachers who are rethinking every part of their jobs -- their relationship with students, colleagues, and the community; the tools and techniques they employ; their rights and responsibilities; the form and content of curriculum; what standards to set and how to assess whether they are being met; their preparation as teachers and their ongoing professional development; and the very structure of the schools in which they work. In short, teachers are reinventing themselves and their occupation to better serve schools and students.

New Relationships and Practices

Traditionally, teaching was a combination of information-dispensing, custodial child care and sorting out academically inclined students from others. The underlying model for schools was an education factory in which adults, paid hourly or daily wages, kept like-aged youngsters sitting still for standardized lessons and tests.

Teachers were told what, when, and how to teach. They were required to educate every student in exactly the same way and were not held responsible when many failed to learn. They were expected to teach using the same methods as past generations, and any deviation from traditional practices was discouraged by supervisors or prohibited by myriad education laws and regulations. Thus, many teachers simply stood in front of the class and delivered the same lessons year after year, growing gray and weary of not being allowed to change what they were doing.

Many teachers today, however, are encouraged to adapt and adopt new practices that acknowledge both the art and science of learning. They understand that the essence of education is a close relationship between a knowledgeable, caring adult and a secure, motivated child. They grasp that their most important role is to get to know each student as an individual in order to comprehend his or her unique needs, learning style, social and cultural background, interests, and abilities.

This attention to personal qualities is all the more important as America continues to become the most pluralistic nation on Earth. Teachers have to be committed to relating to youngsters of many cultures, including those young people who, with traditional teaching, might have dropped out -- or have been forced out -- of the education system.

Their job is to counsel students as they grow and mature -- helping them integrate their social, emotional, and intellectual growth -- so the union of these sometimes separate dimensions yields the abilities to seek, understand, and use knowledge; to make better decisions in their personal lives; and to value contributing to society.

They must be prepared and permitted to intervene at any time and in any way to make sure learning occurs. Rather than see themselves solely as masters of subject matter such as history, math, or science, teachers increasingly understand that they must also inspire a love of learning.

In practice, this new relationship between teachers and students takes the form of a different concept of instruction. Tuning in to how students really learn prompts many teachers to reject teaching that is primarily lecture based in favor of instruction that challenges students to take an active role in learning.

They no longer see their primary role as being the king or queen of the classroom, a benevolent dictator deciding what's best for the powerless underlings in their care. They've found they accomplish more if they adopt the role of educational guides, facilitators, and co-learners.

The most respected teachers have discovered how to make students passionate participants in the instructional process by providing project-based, participatory, educational adventures. They know that in order to get students to truly take responsibility for their own education, the curriculum must relate to their lives, learning activities must engage their natural curiosity, and assessments must measure real accomplishments and be an integral part of learning.

Students work harder when teachers give them a role in determining the form and content of their schooling -- helping them create their own learning plans and deciding the ways in which they will demonstrate that they have, in fact, learned what they agreed to learn.

The day-to-day job of a teacher, rather than broadcasting content, is becoming one of designing and guiding students through engaging learning opportunities. An educator's most important responsibility is to search out and construct meaningful educational experiences that allow students to solve real-world problems and show they have learned the big ideas, powerful skills, and habits of mind and heart that meet agreed-on educational standards. The result is that the abstract, inert knowledge that students used to memorize from dusty textbooks comes alive as they participate in the creation and extension of new knowledge.

New Tools and Environments

One of the most powerful forces changing teachers' and students' roles in education is new technology. The old model of instruction was predicated on information scarcity. Teachers and their books were information oracles, spreading knowledge to a population with few other ways to get it.

But today's world is awash in information from a multitude of print and electronic sources. The fundamental job of teaching is no longer to distribute facts but to help children learn how to use them by developing their abilities to think critically, solve problems, make informed judgments, and create knowledge that benefits both the students and society. Freed from the responsibility of being primary information providers, teachers have more time to spend working one-on-one or with small groups of students.

Recasting the relationship between students and teachers demands that the structure of school changes as well. Though it is still the norm in many places to isolate teachers in cinderblock rooms with age-graded pupils who rotate through classes every hour throughout a semester -- or every year, in the case of elementary school -- this paradigm is being abandoned in more and more schools that want to give teachers the time, space, and support to do their jobs.

Extended instructional periods and school days, as well as reorganized yearly schedules, are all being tried as ways to avoid chopping learning into often arbitrary chunks based on limited time. Also, rather than inflexibly group students in grades by age, many schools feature mixed-aged classes in which students spend two or more years with the same teachers.

In addition, ability groups, from which those judged less talented can rarely break free, are being challenged by a recognition that current standardized tests do not measure many abilities or take into account the different ways people learn best.

One of the most important innovations in instructional organization is team teaching, in which two or more educators share responsibility for a group of students. This means that an individual teacher no longer has to be all things to all students. This approach allows teachers to apply their strengths, interests, skills, and abilities to the greatest effect, knowing that children won't suffer from their weaknesses, because there's someone with a different set of abilities to back them up.

To truly professionalize teaching, in fact, we need to further differentiate the roles a teacher might fill. Just as a good law firm has a mix of associates, junior partners, and senior partners, schools should have a greater mix of teachers who have appropriate levels of responsibility based on their abilities and experience levels. Also, just as much of a lawyer's work occurs outside the courtroom, so, too, should we recognize that much of a teacher's work is done outside the classroom.

New Professional Responsibilities

Aside from rethinking their primary responsibility as directors of student learning, teachers are also taking on other roles in schools and in their profession. They are working with colleagues, family members, politicians, academics, community members, employers, and others to set clear and obtainable standards for the knowledge, skills, and values we should expect America's children to acquire. They are participating in day-to-day decision making in schools, working side-by-side to set priorities, and dealing with organizational problems that affect their students' learning.

Many teachers also spend time researching various questions of educational effectiveness that expand the understanding of the dynamics of learning. And more teachers are spending time mentoring new members of their profession, making sure that education school graduates are truly ready for the complex challenges of today's classrooms.

Reinventing the role of teachers inside and outside the classroom can result in significantly better schools and better-educated students. But though the roots of such improvement are taking hold in today's schools, they need continued nurturing to grow and truly transform America's learning landscape. The rest of us -- politicians and parents, superintendents and school board members, employers and education school faculty -- must also be willing to rethink our roles in education to give teachers the support, freedom, and trust they need to do the essential job of educating our children.

Judith Taack Lanier is a distinguished professor of education at Michigan State University.

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Redefining the Role of the Teacher: It's a Multifaceted Profession

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A closer look at what being an educator really means.

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What does ideal math learning look like in your classroom? 

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What does ideal math learning look like in your classroom? 

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Students are highly engaged in the learning and there is excitement in the room.

This looks like knowing my students (their interests, backgrounds, and lives) so I can plan my lessons according to this.

All students are presented with some challenge, regardless of ability or background.

This looks like providing challenges based on where they are at (meeting each student individually where they are at).

Students have a deep understanding of the concepts involved and are able to apply their understanding to different contexts.

This looks like the real-world problems I give them, the problems they present, and how they connect each situation.

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How will you use the rubric for developing as a concept-based teacher and developing concept-based instruction in your own practice? 

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How will you use the rubric for developing as a concept-based teacher and developing concept-based instruction in your own practice? 

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Strive to meet mastery in each category!

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How will you foster a growth mindset in your classroom? 

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How will you foster a growth mindset in your classroom? 

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Support productive struggle!

allow students to struggle (individually and collectively)

"Not yet" Statements

allows students to realize the ability to grow

Demonstrate mistakes and celebrate corrections

allows students to see that mistakes are made by everyone and a part of learning and growing

Set goals

allows students to individually have something to strive for

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Of the many challenges educators face, one of the most formidable-and least obvious-is student mindset. Teaching students to comprehend, absorb, and apply new material and concepts is challenging under any circumstances. The undertaking is much greater, however, when students doubt their ability to learn.

Can simply believing you have the ability to learn increase your ability to learn? According to growth mindset researchers, the answer is yes.

A more detailed look at mindset theory, as well as some of the ways that teachers can foster growth mindset in the classroom, illustrates mindset development's importance in education.

What Is Growth Mindset?

One theory of intelligence holds that people can be categorized according to their implicit beliefs about ability. People with a fixed mindset believe that abilities are innate, while people with a growth mindset believe that they can acquire abilities through effort and study.

Carol Dweck, a psychologist, developed the fixed versus growth mindset theory. The Stanford University professor's pioneering research, as well as her 2006 book, Mindset:The New Psychology of Success, popularized the concept of fixed and growth mindsets.

To gain a better understanding of what growth mindset is, compare the differences between fixed and growth mindset outlooks and behaviors.

Fixed Mindset vs. Growth Mindset

Fixed and growth mindsets profoundly affect the way that individuals respond to failure. People with a fixed mindset view failure as a result of their lack of ability, while people with a growth mindset see the opportunity to expand their abilities and to work harder and smarter.

The fixed mindset vs. growth mindset dichotomy can also be expressed in terms of goals. People with a fixed mindset wish to appear intelligent because they believe that failing to do so reveals a weakness. People with a growth mindset are much less apprehensive about how their intelligence is perceived because they believe that knowledge and ability can be improved through effort. These outlooks produce almost diametric goals related to learning: a person with a fixed mindset wants to avoid appearing unintelligent; a person with a growth mindset wants to overcome challenges.

While the basic concept of fixed mindset vs. growth mindset is simple, mindset designations are much more nuanced in reality. An important factor of the mindsets is that they can be different in different domains. For example, a person may have a growth mindset in athletic or artistic endeavors but have a fixed mindset in academic endeavors.

Furthermore, mindsets aren't absolute, immovable states. The fixed or growth mindset exists on a spectrum, and even those positions can vary. A person may have a predominant growth mindset in one area-math, for example-but an especially challenging problem may trigger a response that's more consistent with a fixed mindset.

Dweck has described the growth mindset theory as a response to the self-esteem movement, which held that lavishing students with praise builds confidence that leads to improved achievement.

Mindset researchers propose a different approach. Instead of giving students general praise or praising outcomes-an ineffective practice in Dweck's view-teachers should focus on praising students' efforts and the steps they take to overcome challenges and make progress. By doing so they reinforce the simple but powerful idea that ability can be improved.

Benefits of a Growth Mindset in Education

Dweck's early research demonstrated multiple potential benefits to encouraging a growth mindset in education. One of her studies showed that praising students' efforts rather than their intelligence made them more likely to pursue more difficult challenges. Another study demonstrated the influence of simple mindset interventions. Her research showed that students who were taught about developing intelligence went on to perform better in school and exhibit more motivation in the classroom.

Growth mindset has been widely embraced inside and outside of education. (Growth mindset has become a popular topic in the business world, for example.) However, the concept isn't without its critics. Skeptics point out that few researchers have been able to replicate Dweck's results, and much of the corroborating research that's been done has been done by Dweck and her colleagues. Still, mindset research continues to advance, and evidence of the benefits of a growth mindset in education continues to mount.

How to Foster a Growth Mindset in Students

Learning how to foster a growth mindset in students requires time and practice. The approach requires consistent instruction that reinforces and demonstrates the idea that students can improve their ability. At the most basic level, mindset techniques involve shifting emphasis away from outcomes and toward efforts and process. Rather than praising an accomplishment, the instructor praises the efforts and learning steps that led to the positive outcome.

Looking at examples of how instruction and language can be altered to encourage a growth mindset illustrates how teachers can change the way that students perceive the learning process:

Fixed mindset statement: "It's OK if you're having trouble. Maybe algebra isn't one of your strengths."

Growth mindset statement: "When you learn how to do a new kind of problem, it develops your math brain."

Another example stresses the importance of encouraging students to work through problems:

Fixed mindset statement: "Great effort. You tried as hard as you could."

Growth mindset statement: "The goal isn't to get it right immediately. The goal is to improve your understanding step by step. What can you try next?"

Teaching students to be encouraged when they experience difficulty can also reinforce growth mindset:

Fixed mindset statement: "Don't worry, you'll get it if you keep trying."

Growth mindset statement: "That feeling you're experiencing of algebra being hard is the feeling of your brain developing."

A growth mindset approach still demands optimal performance from a student-simply rewarding efforts isn't effective, and, in fact, it can be harmful if the student's efforts are ultimately fruitless.

Part of growth mindset instruction is teaching students to ask for help and to use any resources that can assist them. Just as students are still responsible for learning the material, teachers are still responsible for giving them the tools they need to learn.

10 Strategies for Fostering a Growth Mindset in the Classroom

Helping students develop a growth mindset requires deliberate effort from teachers, but many of the methods can be easily integrated in their existing practices. The following strategies and tips can help educators foster a growth mindset in the classroom:

1. Normalize struggle. Struggle is part of the learning process, and emphasizing and reinforcing that idea helps students react positively when they feel challenged.

2. Encourage engagement with challenges. Portray challenges as fun and exciting, and easy tasks as boring.

3. Embrace the word "yet". If someone makes the statement "I'm not a math person," adding a simple qualifier will signal that a process exists for gaining ability. "You're not a math person yet."

4. Tout the value of hard tasks to the brain. Promote the idea that brains are malleable "muscles" that can be developed. Research on brain plasticity supports the idea of neural growth, and mindset research has shown that believing the brain can grow has a demonstrative effect on behavior and achievement.

5. Demonstrate mistakes and celebrate corrections. Mistakes should be viewed as learning opportunities. Teachers can model this outlook in reactions to their own mistakes and steps they take to correct a mistake.

6. Set goals. Having students set incremental, achievable goals demonstrates the attainability of growth and progress.

7. Develop cooperative exercises. Working together to solve problems emphasizes process and reinforces the importance of getting help and finding solutions. It also deemphasizes individual outcomes.

8. Provide challenges. Part of developing a growth mindset is teaching students to overcome obstacles. A particularly hard math problem or complex writing assignment that stretches their abilities can provide opportunities for growth and further instruction that emphasizes problem-solving.

9. Avoid praising intelligence. This may seem counterintuitive, but praise for "being smart" reinforces the idea that intelligence is a fixed trait. This can be demotivating for the students being praised ("I'm smart; I don't have to try harder"), as well as for those who don't not receive the praise ("That student is smart; I'm not").

10. Don't oversimplify. "You can do anything!" may feel like harmless encouragement, but if students aren't put in a position to overcome challenges, they'll conclude that such statements are empty, and the educator will lose credibility.

How to Foster a Growth Mindset in Teachers and Administrators

Extending the concept of growth mindset theory to teachers and administrators is critical. Research suggests that growth mindset approaches are more effective in supportive environments. However, individuals interested in promoting growth mindset initiatives can encounter resistance. Cultural changes in any institution or system are difficult, but understanding those challenges is the first step to finding common ground and moving forward.

Overcoming Challenges

Changing the instructional practices and policies that teachers and administrators have established can be difficult for many reasons:

Teachers and administrators who have invested time and effort in their current methods of instruction and classroom culture may resist change, particularly if their methods have been successful.

Teachers and administrators who are routinely asked to make changes or adopt new teaching methods may feel skeptical about a new method's benefits.

Teachers and administrators commonly work under tight time constraints that make it difficult to learn new methods or implement new policies.

Teachers and administrators may have fixed mindsets themselves and discount the mindset theory's validity.

Efforts to overcome resistance to growth mindset initiatives should focus on the demonstrated benefits to students. A shared goal of helping students can ease the challenges of culture change.

Five Tips for Creating a Growth Mindset Teaching Culture

Working together to reframe their shared challenges and goals can help both teachers and administrators develop growth mindsets and culture:

1. Value the process over the result. Valuing the process over the result can be difficult in a results-driven environment, but teachers and administrators who value the growth mindset and its long-term benefits are more likely to stay the course.

2. View challenges as opportunities. Students aren't the only ones facing challenges; teachers and administrators can benefit just as much from an outlook that celebrates the process.

3. Experiment with different teaching methods and learning strategies. Helping students build the skills they need to improve is a key element of the growth mindset approach. Teachers and administrators who embrace a growth mindset shouldn't be afraid to experiment.

4. Celebrate growth. Finding teaching strategies that reach students and help them learn should be celebrated with the students, as well as with other teachers and administrators. This reinforces the emphasis on effort and process. It also provides opportunities to share valuable learnings.

5. Communicate about your growth mindset. Sharing successes and failures among teachers and administrators helps to establish a culture that values challenges and a growth mindset.

Providing Tools for a Lifetime of Learning

A growth mindset's defining characteristic-the belief that intelligence is malleable-provides a powerful formula for improving student outcomes. Students who believe that they can get smarter and that effort makes them smarter will put in the effort that leads to higher achievement.

The understanding that they can always learn new skills and overcome challenges through hard work is a lesson students can carry well beyond their school years. As the COVID-19 pandemic demonstrated, many of the learning challenges that students will face can't be foreseen. By helping them develop a mindset that doesn't shrink from challenges and teaching them learning methods that can be applied to any subject or obstacle, educators prepare students for increasingly dynamic work environments and an increasingly complex world.

Educators interested in how education and learning can be life changing should explore American University's School of Education degree programs, including its Online Master of Arts in Teaching (MAT) and Online Master of Education (MEd) in Education Policy and Leadership. Preparing teachers and policy makers to transform the education system to benefit all learners is a core principle of American University's School of Education.

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How to Foster a Growth Mindset in the Classroom | American University

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Embracing a growth mindset can help students learn more and overcome challenges. Explore strategies for fostering a growth mindset in the classroom.

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Chapter 8: What Do Ideal Concept-Based Mathematics Classrooms Look Like?

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Chapter 8: What Do Ideal Concept-Based Mathematics Classrooms Look Like?

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When would you use technology in your practice and when would you not?

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When would you use technology in your practice and when would you not?

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Technology is a necessary part of the classroom to enhance the content and make it more relatable. It can be used in every aspect in a lesson (from planning, to the lesson itself, to assessments). However, technology should not be used in the classroom if it does not enhance learning. This could be when:

it is merely used as a substitution tool

it does not support connecting and sharing

when it reduces the chance of failure

when it creates harmful shortcuts

when it undermines basic or deeper learning

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When Not to Use Technology: 15 Things That Should Stay Simple In Education | InformED

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Most of us know better than to use technology for technology’s sake. The Shiny New Tech Syndrome is taking the world by storm, and with the added pressure of finding new ways to improve educational outcomes, we try our best not to be tempted. But there are some things–certain methods, activities, an…

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How do you integrate technology in the math classroom in order to promote and enhance learning?  

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How do you integrate technology in the math classroom in order to promote and enhance learning?  

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Promoting and Enhancing Learning:

Create more interactions/collaboration/communication between students

Allow for student discovery

Keep students engaged

Allow for different learning styles

Prepare students with life skills

Examples:

Google Applications (Slides, Docs, Sheets)

Exit and Entrance Ticket Apps

Collaborative Thought and Concepts Maps (like this!)

Multimedia Projects

Instructional Videos

Mathematical Software (my favorite: Desmos)

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Drexel University School of Education
Technology provides instant accessibility to information, which is why its presence in the classroom is so vital. Smart phones, computers, and tablets are already an omnipresent element of everyday life for students and teachers alike. It's only natural that the use of technological devices in the classroom are explored to create meaningful learning experiences for students of all ages.

Utilizing different types of technology in the classroom, including a virtual classroom, creates learners who are actively engaged with learning objectives. The implementation of technology also creates pathways for differentiated instruction to meet the unique needs of students as individual learners within a broader classroom climate.

What Does Technology Integration Mean in Education?

Integration of technology in education simply refers to the use of technology to enhance the student learning experience. Utilizing different types of technology in the classroom, including a virtual classroom, creates learners who are actively engaged with learning objectives. The implementation of technology also creates pathways for differentiated instruction to meet the unique needs of students as individual learners within a broader classroom climate.

How to Integrate Technology in the Classroom

There is a common misconception that the integration of technology in the classroom can be a financial burden for school districts, but students do not necessarily need their own tablets or laptops to succeed with technology. The use of technology during whole-class instruction can foster student engagement for auditory and visual learners. Integrating simple technologies Power Points, games, internet homework assignments, or online grading systems can be difference makers in students' growth in the classroom.

Power Points and Games

Powerpoint presentations can be used to introduce a classroom concept while providing the opportunity for engagement. Along with the use of graphics and bulleted information, links to videos that accompany the ideas presented in the Powerpoint can be embedded within the slides.

Educational apps in the classroom like Kahoot can be used to review information after a lesson or unit. Teachers can create and share Kahoots with one another while students can create anonymous user names to participate in the game. This allows for whole-class participation from students who may usually be reluctant to participate in class. Kahoot is accessible to play on phones or computers and teachers can determine if they want students to work independently or be assigned to teams.

Internet Homework Assignments

Posting homework assignments online (via learning platforms like Blackboard, Brightspace, and Moodle) is one way many teachers can begin to integrate technology in the classroom. Assignments are easily accessible, which can increase student engagement and help students become more organized.

Online grading Systems

Communication is a key element in education that helps teachers, administrators, parents, and students recognize a student's strengths and areas for improvement. Online grading systems such as PowerSchool open and facilitate lines of communication where teachers can post grades, analyze student attendance patterns, and manage transcript data.

Classroom Tablets

For classrooms that are fortunate enough to have tablets for students, technology can allow teachers to implement differentiation throughout instruction. Students can work at their own pace during assignments and teachers have the opportunity for one-on-one instruction.

Listserv

Software such as Listserv allows parents to manage and organize their emails. Parents can receive updates from teachers about important announcements, newsletters, and discussions that keep frequent lines of communication open.

Strategies of Implementation for Age Groups

The benefits of technology can enhance any contemporary classroom. However, the way technology is implemented and used in classrooms of various grade levels and content areas will differ.

Ways to Incorporate Technology in the Elementary Classroom

For younger students, technology can be used to build fundamental skills to prepare them for future independent learning. Students can use interactive games to reinforce math, spelling, phonetic, and reading skills. Sites like Spelling Training permit students or teachers to upload their own word lists to practice word pronunciation and create interactive games. Parents can also use these sites to exercise fundamental skills beyond the walls of the classroom.

Using Technology in the Middle School Classroom

As students begin to take steps to transform into independent thinkers, they can use technology to develop basic life skills. Students at the middle school level will gain independence by having different teachers for each subject. Using technology to acquire skills such as conducting research can be applied to any content area. Websites like Easy Bib guide students to find credible sources through a variety of search engines and teach students to correctly cite those sources to avoid plagiarism.

Technology in High School Classrooms

Once students reach their secondary education, they can discover ways to use technology that can be beneficial for college and career development. Familiarization with Microsoft Office and Google Drive teach students to make spreadsheets, slide show presentations, and share documents where they can receive fluid feedback on their work. Many careers use these elements of Microsoft and Google to organize information and collaborate between colleagues or clients.

Why is it Important to Integrate Technology in the Classroom

Teachers often find success when they present the opportunity to use technology in the classroom. There are various benefits and effects when technology is used for educational instruction and some may argue that not all of the effects are positive. Having an infinite flow of information and entertainment available at any given time could be seen as a distraction, but if technology is integrated into the classroom with routines in place that are monitored or assessed, the pros of using technology in the classroom outweigh the cons.

Keeping students engaged

Active engagement is a key part of any lesson plan. Whether students are working independently or collaboratively, technology engages students because it is interactive.

Helps students with different learning styles

Not all students learn and retain information in the same way or at the same speed. Technology is an opportunity for teachers to differentiate instruction to modify information for the appropriate learning capabilities of their students. The use of technology can also allow students to work at their own paces.

Prepare students with life skills

Technology has become its own form of literacy because of how often it is used in everyday life. Many careers use at least one aspect of Microsoft Office or Google Drive on a daily basis: balancing budgets on spreadsheets, creating decks or slide shows to be presented, or attaching documents to emails to communicate important information. Allowing students to learn and refine these skills prepares them for life beyond the classroom.

For additional information about programs or certifications associated with technology in the classroom, visit Drexel University's School of Education Instructional Technology Specialist Certification program.

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How to Integrate Technology in the Classroom - School of Education

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Check out our list of tips for the best ways on how to use technology in the classroom. Learn the benefits & effects of technology with Drexel University.

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Three main knowledge areas for effective technology integration:

Content knowledge (CK) represents the subject knowledge we are teaching (the "what")

Pedagogical knowledge (PK) refers to the approaches to teaching, such as inquiry-based, problem-based, or collaborative learning, to name a few (the "how")

Technological knowledge (TK) refers to the technological tools that we use to make the content more accessible to students (the "partner")

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Substitution, Augmentation, Modification, and Redefinition

Hierarchical levels of technology use

Substitution and Augmentation fall under "enhancement" of material

Modification and Redefinition fall under "transformation" of material

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Chapter 7: How Do I Integrate Technology to Foster Conceptual Understanding?

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Chapter 7: How Do I Integrate Technology to Foster Conceptual Understanding?

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How do you develop core skills such as communication, self-management, and research skills in the math classroom? What might you do in the future to improve your practice in developing these skills? 

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How do you develop core skills such as communication, self-management, and research skills in the math classroom? What might you do in the future to improve your practice in developing these skills? 

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It is necessary to develop core skills such as communication, self-management, and research skills in the mathematics classroom. I have attempted to develop these skills by:

using group work so students can learn to communicate with each other using mathematical terms

using self-assessments so students can gauge themselves on a topic or behavior

using real-world examples to relate the content to students and allow them to practice research skills (such as discovery activities and hands-on data collection)

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What math topics lend themselves well to performance assessment tasks? 

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What math topics lend themselves well to performance assessment tasks? 

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With the right approach, all math topics lend themselves well to performance assessment tasks. A performance assessment task assess what the students understand (generalizations), what students know (factual knowledge), and what students can do (skills) in a particular unit. They engage student intellect and develop conceptual understanding in a real-world context.

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Performance Tasks

A performance task is any learning activity or assessment that asks students to perform to demonstrate their knowledge, understanding and proficiency. Performance tasks yield a tangible product and/or performance that serve as evidence of learning. Unlike a selected-response item (e.g., multiple-choice or matching) that asks students to select from given alternatives, a performance task presents a situation that calls for learners to apply their learning in context.

For more information of the design and use of performance tasks and performance task assessment, we've published a seven part series at https://blog.performancetask.com/ , authored by Jay McTighe.

Defined STEM Performance Task Project Outline

1. Explore the Background (GRASP)

1. Goal - Each performance task begins with a Goal. The goal provides the student with the outcome of the learning experience and the contextual purpose of the experience and product creation.

2. Role - The Role is meant to provide the student with the position or individual persona that they will become to accomplish the goal of the performance task. The majority of roles found within the tasks provide opportunities for students to complete real-world applications of standards-based content. The role may be for one student or in many instances can serve as a small group experience. Students may work together or assume a part of the role based upongroup dynamics. These roles will require student(s) to develop creative and innovative products demonstrating their understanding of the content through the application of the content and a variety of skills across disciplines.

3. Audience - The performance tasks contain an Audience component. The audience is the individual(s) who are interested in the findings and products that have been created. These people will make a decision based upon the products and presentations created by the individual(s) assuming the role within the task.
Click here for an article on the importance of audience within a performance task.

4. Situation - The Situation provides the participants with a contextual background for the task. Students will learn about the real-world application for the task. Additionally, this section builds connections with other sections of Defined STEM. This is the place that may invite students to consider various video resources, simulations, language tasks, and associated websites when appropriate. This section of the performance task will help the students connect the authentic experience with content and concepts critical to understanding.

5. Products - The Products within each task are designed using the multiple intelligences. The products provide various opportunities for students to demonstrate understanding. Based upon each individual learner and/or individual class, the educator can make appropriate instructional decisions for product development. These considerations may include:

having a student complete all products within a task;

having students complete a number of products based upon content application and/or student interest;

having a student complete certain products based upon the educator's decision to maximize content, concept, and skill application;

having student work as part of a cooperative group to complete a product or the products; and/or

having students complete products based upon the strength of their multiple intelligences.

Each task contains between four and six products for development consideration. The products are meant to provide students with experiences aligned with 21 st Century Learning and Innovation Skills. Additionally, each task contains at least one product that requires the utilization of a technology-based experience. This provides the student with experiential learning opportunities based upon software and web resources and the application of these tools to demonstrate their knowledge and understanding. The products provided within each task are closely aligned with the multiple intelligences and the set of products within the task will help educators meet a number of multiple intelligences.

2. Design Process & Product Creation

3. Final Reflections

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Performance Tasks - Defined Learning (formerly Defined STEM)

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A performance task is any learning activity or assessment that asks students to perform to demonstrate their knowledge, understanding and proficiency. Performance tasks yield a tangible product and/or performance that serve as evidence of learning.

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Are there other graphic organizers you use that highlight students’ conceptual understanding? 

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Are there other graphic organizers you use that highlight students’ conceptual understanding? 

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I have used Venn diagrams for students to show relationships between topics. In the Venn diagram, students are to connect what the concepts used in the new unit that were used in the old unit (middle), concepts/skills done in the previous unit that were not touched on in the current unit (left), and completely new concepts (right). This has been very beneficial to me by showing me what students remember and how well they understand the relationships.

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How would you use a visible thinking routine? 

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How would you use a visible thinking routine? 

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Think, Pair, Share

Allow students time to think, discuss with a partner, and then share with the rest of the group

Example: Think about a problem given in class, discuss possible processes and solutions with a partner, and discuss these different processes and solutions as a class

"I used to think _______ Now I think _______" statements

Allow students to identify their misconceptions and how their learning/understanding has changed over the unit

Example: Have students write their misconceptions on an idea and what is actually true

Headlines

Have students summarize the key ideas and concepts in a lesson

Example: Students write a headline, like in a newspaper, that summarizes the main idea for the lesson

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What are some advantages of visible thinking routines? 

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What are some advantages of visible thinking routines? 

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Useful for diagnostic and formative feedback

Allows students to express and communicate their personal thoughts and biases

Gives teachers an insight to student's thinking

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Visible Thinking is a flexible and systematic research-based conceptual framework, which aims to integrate the development of students' thinking with content learning across subject matters.

Visible Thinking began as an initiative to develop a research-based approach to teaching thinking dispositions. The approach emphasized three core practices: thinking routines, the documentation of student thinking, and reflective professional practice. It was originally developed at Lemshaga Akademi in Sweden as part of the Innovating with Intelligence project, and focused on developing students' thinking dispositions in such areas as truth-seeking, understanding, fairness, and imagination. It has since expanded its focus to include an emphasis on thinking through art and the role of cultural forces and has informed the development of other Project Zero Visible Thinking initiatives, including Artful Thinking, and Cultures of Thinking.

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Visible Thinking is a flexible and systematic research-based conceptual framework, which aims to integrate the development of students' thinking with content learning across subject matters. Visible Thinking began as an initiative to develop a research-based approach to teaching thinking dispositions. The approach emphasized three core practices: thinking routines, the documentation of student thinking, and reflective professional practice.

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Choose three of the assessment strategies listed in this chapter. When, why, and how would you use each in your own classroom practice? 

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Choose three of the assessment strategies listed in this chapter. When, why, and how would you use each in your own classroom practice? 

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Zero, One, Two, or Three

Students write their understanding level in their book/notes to convey how well they understand a concept (0 being did not understand, 4 being completely understand).

I can use this in my own classroom practice when introducing new concepts so that I can gauge how well my students are understanding a concept and if they need extra guidance or more challenging tasks. At the end of each lesson, I could create an exit ticket that would allow me to see the different levels of understanding so I can differentiate where needed.

Agree, Disagree, Depends

Students pick whether they agree, disagree, or depends on a factor when given a statement. Their choice reflects their level of understanding.

I can use this when reviewing concepts so I can see the levels of understanding and hear the justifications of my students as to why they made their choice. This may be done during a test review or even simply during a lesson.

Use of Multiple-Choice Questions

Students justify their choice of the multiple choices to reflect their thinking and understanding.

This is a tool I use already for assessments. During the practice of some lessons, students are given multiple-choice questions during a game. While walking the classroom, I ask them to explain their justification for their choice.

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Chapter 6: How Do I Know My Students Understand the Concepts? Assessment Strategies

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Chapter 6: How Do I Know My Students Understand the Concepts? Assessment Strategies

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What role do you think the structure of your classroom plays in student learning?

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What role do you think the structure of your classroom plays in student learning?

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Individual student chairs/tables paired in groups of 4-5

Encourages group work

Instructional grouping

Teacher's desk at the back of the classroom with the whiteboard and projector in front for presenting content

Focus is taken off of the teacher, which allows for higher student autonomy

Small group table in the back for quiet, individual work, students that were absent the previous day, or students that need extra guidance

Allows students an extra space to get the individualized instruction they need

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What differentiation strategies do you use to cater to the individual needs of your students? 

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What differentiation strategies do you use to cater to the individual needs of your students? 

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Give tiered tasks of varying levels of difficult

Example: Allowing students to focus on a topic in a lesson that they need more practice with (probability choices: converting into decimals, vocabulary, finding probability)

Math games with various topics

Allows teacher to see which areas need improvement for individual students

Tri-Mind activities

Example: Students choose how to represent a summary at the end of the lesson (takes into account analytical, practical, and creative thinkers)

Notes based on varying academic ability

Example: Students can write in their own notes, fill in a note template with words missing, or use a fully-filled note template (for students that struggle with writing or focusing, those that are ELLs, those that need to be fully engaged, those that need more time to focus on what the teacher is saying)

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Math classrooms are mosaics of strengths and experiences. When we have students with diverse backgrounds-with various languages, achievements, and interests-in the same space, everyone learns from each other and broadens their world.

On the flip side, though, teaching math to a broad array of students can be challenging. Do you struggle to reach all of your students? Are you a newer teacher who is looking to improve your practice? The strategies for differentiated instruction provided here might help you out.

What Is Differentiated Math Instruction?

Differentiated math instruction refers to the collection of techniques, strategies, and adaptations you can use to reach your diverse group of learners and make mathematics accessible to every single one. Dr. Timothy Kanold, former president of the National Council of Supervisors of Mathematics (NCSM)-and HMH author- clarifies that differentiation in a math lesson is "differentiation on the entry points into the task for support or the exit point to advance student thinking."

By applying various tools and strategies, such as incorporating technology, assigning hands-on projects, and teaching in math small-group formats, you can help every student meet expectations. We know that there are different schools of thought regarding what differentiation means. When we use the term, we are talking about providing student choice, voice, and agency. Differentiating instruction isn't meant to add more work to your day. Quite the opposite, in fact; it's meant as a teaching approach that will help you to reach more students in terms of accessibility and equity, making your job both easier and more effective in the long run.

Why Is Differentiating Math Instruction Important?

Some people think that math, more than any other subject, is the best fit for differentiation. Even though a 2018 survey by Texas Instruments found that 46% of kids said they really liked math, there are hundreds of books, websites, and memes discussing the difficulty of the subject. From the anxiety caused by there being only one correct answer to the cultural buy-in to the myth of being-or not being-a "math person" to the fear of solving a word problem, many students struggle with math. In addition, many students and educators alike find it hard to make the connection between math and the real world, which only increases disillusionment with the subject. That's why it's especially important to be open to new ways of providing instruction.

The National Council of Teachers of Mathematics (NCTM) promotes differentiating math instruction for differences in learning as well as differences in achievement, interest, and confidence. NCTM advises that the need is greater in middle and high school, as higher-level math relies on more complex reasoning. When you differentiate your math instruction, you support all learners by targeting and addressing specific needs of groups and individual students.

Examples of Differentiated Instruction in Math

Do you need ideas for how to differentiate your teaching to be sure your math students are progressing? Below are seven differentiation strategies for math instruction, along with ways that you can use them in your math classroom. They serve as examples of differentiated instruction in math and may work better for some classrooms and math topics than others. Customize these ideas however you need to serve you and your students.

Strategy 1: Math Centers

For this, you'll need to come up with a few activities your students can rotate through (be sure to browse our library of free activities and resources!), such as watching a video, reading an article, or solving a word problem. We spoke with Kristy McFarlane, an instructional supervisor at Sandshore Elementary School in New Jersey, about differentiation. She says math teachers at her school spend about 10 minutes on a mini-lesson for the whole class and then students spend about 15 minutes at various math centers. "They might meet with the teacher in a small group for extra help, use math software, do a game or project at the hands-on station, or do seat work based on the day's mini-lesson," she says.

Math centers are a powerful way to facilitate independent and small group learning within your classroom. Our program, for example, is known for embedding resources and instructional time to math centers. If a select group of your students are all struggling to, say, add fractions, they may benefit from an activity that has them practice finding least common denominators. Think about ways to customize the groupings and centers so they're perfect for your students' strengths, misconceptions, and interests, and make use of tools that strategically group students and recommend activities for you.

Strategy 2: Activity Cards

Choice is an important part of differentiation, and letting students decide how they want to spend their time is a great way to appeal to various learning preferences. You'll need to come up with math problems, tasks, or questions. As much as possible, use or create cards that span several lessons and offer options to work independently, with a partner, or in a small group. Ask for feedback so you can adjust future learning accordingly. Many of HMH's math programs, including , , and include inquiry-based task and project cards that help teachers differentiate.

Strategy 3: Choice Boards

As we just mentioned, giving students the ability to make decisions about their learning is an important part of differentiation. A choice board is a graphic organizer that gives students activities to choose from. There are different types of choice boards, but they need to focus on specific learning needs, interests, and skills. Choice boards increase student ownership; students pace themselves and get to decide how to engage with information, along with how to demonstrate their learning. Some teachers create different versions of the same choice board; others will color-code options to signify topic, activity type, or expected level of challenge. Check out the choice board we developed for remote learning. This board covers all subjects but also includes a free template to get you started on a math-only version.

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7 Strategies for Differentiated Math Instruction

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These strategies provide differentiated math instruction to K-12 teachers looking for ways to personalize their mathematics teaching.

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Do you think it is important to balance whole teacher talk time with student working time? Why or why not? 

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Do you think it is important to balance whole teacher talk time with student working time? Why or why not? 

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YES
Balance is necessary because most students cannot listen or follow along for an extended period of time. It is well-known that students learn best when engaged and interested, so most of the class time needs to revolve around this. Students are not fully engaged and interested if the teacher is presenting the majority of the time. It also does not allow for individual practice so that students can make mistakes.

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How do you promote a growth mindset in your classroom and encourage mistakes as part of the learning process? 

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How do you promote a growth mindset in your classroom and encourage mistakes as part of the learning process? 

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Create a safe, secure learning environment

No "dumb" questions

Help students change their language (no more "I just don't get it," rather "I don't understand this topic yet")

Explain mistakes and common issues

"You are on the right track"

Model mistakes

Talk about personal mistakes and how they have helped growth

Make room for mistakes - let students know it is a necessary part of learning

Open communication - creative feedback when asking a question

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Making mistakes used to be a bad thing. It was, at one time, equated with weakness or apathy, or worst of all, stupidity. This was often especially true if you were a student. But it was a long time ago, and thankfully things have changed.

Nowadays, if there is one place that the term " useful failure " is setting an honoured place for itself, it's in our classrooms. It's great news for learning, and for our learners.

The methods we can use for helping our kids with embracing mistakes in the classroom are many. The point, however, is that we want to dispel the age-old idea that failure is a negative thing with harmful connotations in learning. Instead, we want to allow students to realize failure is the best opportunity to build up their confidence and engage them in powerful learning and teachable moments.

We're going to give you some solid strategies for guiding students toward adopting this mindset for school and for life. First, though, let's do some light reading.

The Power of Failure

Perhaps one of the most compelling articles written on the subject of encouraging mistakes in the classroom was written in 2015 by Helen Snodgrass. The article is titled In My Class, Failure is Not an Option-It is a Requirement. Already, just from the title, we understand how passionate this educator is about the idea of turning the word "mistake" into the word "opportunity" or her kids. She explains it this way:

"When students first walked into my classroom this fall, many of them immediately noticed a large quote on the wall above the whiteboard:'In this class, failure is not an option. It's a requirement' ... this does not mean that I simply sit back and watch students grasp at straws as they tackle really difficult material. What it does mean is carefully selecting tasks for students to work on that might not have one clear answer or only one possible approach and then providing them the space and the skills to work through the challenge and reflect on their process and struggles as they go."

This passage brings up a few encouraging points about embracing failure and mistakes in the classroom. Firstly, it's an indirect way of personalizing a task. After all, most students who are encouraged to take on challenges and consciously provided margin for failure will quite willingly forge their own pathway to find the answer. This is especially true if failure is demonstrated by the instructor as integral to the learning journey.

The second is that the idea of useful failure ties directly into inquiry-based learning. For instance, when we fail, questions usually arise such as:

Why did we fail?

What did we not consider?

What questions didn't we ask that we can next time?

Where are other routes we can take to get more information?

Whose experiences can we refer to for guidance?

These are just a few examples, but they are all part of the journey of inquiry-based learning. In such compelling activities, these questions go hand-in-hand with ones learners ask about the subject matter itself. The idea of failure then sort of fades away, and all questions simply become part of driving inquiry forward to create meaningful learning experiences. Also, the ability to adopt failure as an opportunity as opposed to seeing it as a final outcome is a hallmark of great critical thinking.

We want to allow students to realize failure is the best opportunity to build up their confidence and engage them in powerful learning.

Lastly, embracing struggle leads to insightful reviews and debrief sessions, which we know from Solution Fluency as being a vital part of all learning. Debriefing gives students the opportunity to look at their own outcomes and determine what was done well and what could have been done better. In our own experience, once students have done this, they start to pre-debrief by themselves and make their own solutions and products better. When this happens, students truly are taking responsibility for their own learning.

Tearing Down Scaffolding

She goes further in a discussion of the idea of scaffolding tasks. While a necessary teaching objective, her argument is that sometimes it is done to a degree that often sucks the realism out of learning:

"While scaffolding can be a very good thing in the classroom, it is sometimes taken to mean that all material must be broken down into such small and simple steps or chunks of information that students are all able to be successful every step of the way ... Students, as a result, often get the message from very early on in their education that if they do not immediately grasp how to solve a problem or get the right answer, they must not be very smart or good at that particular subject. With years of training in this way of thinking, it comes as no surprise that students often respond to challenging work by either immediately asking the teacher for help or by giving up."

So how do we scaffold effectively and provide room for mistakes as opportunities in learning? Perhaps part of it is, like Helen says, consciously creating room for struggle and failure. We can be there to guide students to get back on their feet and try again when they fall, much like helping a child learn to walk or ride a bicycle.

1. Make room for it. Perhaps the single best way to begin adopting useful failure in your classroom is to make sure students know it's allowed. Helen placed her quote about failure so it was the first thing students saw. Explain to your learners that making mistakes will be a required part of their time both in school and in life. What they do with those mistakes is what will be important.

2. Model failure. Too often teachers are seen as the keepers of the keys and the guardians to the doors of all knowledge. As such, they're expected to approximate perfection to the best of their ability. After all, some say, who wants to learn from a teacher who makes mistakes? But this isn't realistic. Teachers are humans too, and they make mistakes all the time. That's why they're in a perfect position to demonstrate how beneficial such instances can be for real learning.

3. Provide immediate feedback. It can be an incredibly powerful learning motivator. The benefits of highly constructive feedback in the moment are psychological too. Included as a consistent factor in the learning journey, it allows students to get over mistakes easier as well. It has more than just benefits tied to formative assessment strategies.

4. Embrace messy learning. It's a non-linear and constantly transforming journey for all of us. Einstein said it best when he suggested anyone who's never made a mistake has never tried something new. And if we can't make mistakes when learning new things, then when can we make them?

5. Think of learners as your allies. They are with you every day and are beside you at every step. You keep each other company and support each other in many ways. Most of all, you are learning together. Such is not the time for separation or an "us and them" mentality. Show your readiness to struggle, fail, try again, and succeed as a team.

6. Talk to them. Sometimes when your learners make mistakes, they'll get stuck and feel lost. Finding the next step to take can be hard when you're so immobilized. This is the time to make sure the lines of communication are wide open. Ask them how you can help or what they need from you at that moment. The quality of their answers might surprise or even inspire you.

7. Encourage them. No matter what mistakes they make, they can find a way over them. Sometimes all it takes is just a little nudge in the right direction. Our learners are dealing with tremendous social and personal pressures. Sometimes making a mistake can seem like the end of the world for them. You're there to tell them it's not; actually, it's the beginning of a whole new journey.
Editor's note: This post was originally published in 2019 and has been updated for comprehensiveness. Originally published May 3, 2019, updated September 21, 2021

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7 Ways of Embracing Mistakes in the Classroom for Inspired Learning

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Making mistakes used to be a bad thing. It was, at one time, equated with weakness or apathy, or worst of all, stupidity. This was often especially true if you were a student. But it was a long time ago, and thankfully things have changed.

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How do you keep students on task in a social learning environment? 

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How do you keep students on task in a social learning environment? 

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Provide them with activities that they can all engage in

Promote discussion among students

Allow for group work (my students have shown that they are most on-task when working in groups)

Allow for creativity - drawing a picture to represent a problem, solving in different ways, allowing choice when creating a project, etcetera

Peer teaching/coaching

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There's been a lot of talk about social learning, yet most of what has been shared involves online or blended learning. Why not social learning in the classroom? Many of the recent instructor-led training (ILT) trends have elements of social learning. Below are at least four ways to incorporate social learning into the classroom.
Attention and the Flipped Classroom
Social learning is based on the behavior modeling theory where people learn new things by observing others. The first step of this learning through observation and the modeling process is that you have to pay attention, otherwise you aren't observing anything. Isn't that the whole reason that many university classes and now corporate training classes have moved to the flipped classroom method? There's no better way to get groups to pay attention than to put them in control of the discussion and allow the learners to determine the agenda. So assign pre-work, often online e-learning modules that include knowledge check questions that you can track or not, but that the facilitator uses to gauge the level of knowledge. Then, the facilitator asks, "What content was most challenging? What questions do you have?" Let the learner's questions guide the class. By gathering answers to questions from the class or even assigning questions to teams or pairs, you encourage peer collaboration and group problem solving. Learners today come to class for an experience - not a lecture or instruction.
Retention: Peers Teaching Peers and Peer Coaching
Another way to incorporate social learning is to have peers teaching peers and include peer coaching. In order to model behavior, learners must retain what they observed. Take away large group lectures and replace them with group discussions or knowledge huddles. If learners are in teams, they attend different discussions and return to their teams and 'teach' their topic to their team mates. This closely mirrors the real-world. According to Bersin by Deloitte, 80 percent of workforce learning occurs through on-the-job interactions with peers, teammates and managers. To ensure all key points are taught around the topics, online or printed materials can be created as reference material, or even follow-up online learning.
Imitation through Real Plays
Next in the social learning modeling behavior process is performing and practicing the positive behavior that is observed. An important adjustment to truly apply social learning is to transform the role play into a 'real play.' Bring in individuals who can realistically take on the roles, either managers, colleagues, or clients. And for the social aspect, have learners prepare together. Don't tell them which person is going to conduct the real-play until a few minutes before. Then, during the real play, anyone not real-playing is an observer filling out an observer scorecard. This allows the observers to become actively involved when it comes time to debrief. If there is a score being assigned at the end, allow the observers 'score' to be part of the final. Time permitting, the real play can be conducted and debriefed multiple times. Not only does this process incorporate social learning, it also appeals to the growing numbers of young leaders who when surveyed said that their teammates are the most important people at work.
Reinforcement and Motivation through Simulation and Gamification
The final way to incorporate social learning into the classroom is through simulation and gamification. Gamification has been a growing trend, and when combined with simulation, can really bring social learning theory to life. The ultimate goal of classroom training, and social learning, is learning. However, observation does not necessarily equate to behavior change or learning. Yet, there is a higher likelihood for learning and behavior change when participants see a person taking an action and being rewarded for that action (or conversely, punished for an inappropriate action or inaction). This is the basis of simulation, and more specifically what is often called "best practice" computer-based simulation. Place learners in teams and have them collectively take on a role in a computer simulation. As they make decisions, they follow a decision-tree where they experience the outcome of their decisions - good or bad - in terms of reactions of other characters in the simulation, future events, and on their scores.

With all teams starting from the same place, final scores can create competition and be used to incorporate gamification elements such as a leaderboard. According to Bersin by Doloitte, tor a young generation of future leaders eager to receive training and development, even more so than flexible hours and cash bonuses, using simulation and gamification can immediately engage learners and create the 'pull' necessary to ensure learning and behavior change.

With leadership development spending increasing, but instructor-led training decreasing, what are the most effective organizations doing? High performing organizations today are transforming their instructor-led training into a more engaging and effective experience by incorporating social learning. Millennials and adult learners alike want to learn together.

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How to Incorporate Social Learning into the Classroom - Training Industry

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There's been a lot of talk about social learning, yet most of what has been shared involves online or blended learning. Why not social learning in the classroom? Many of the recent instructor-led training (ILT) trends have elements of social learning. Below are at least four ways to incorporate social learning into the classroom.

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What strategies do you employ to engage and motivate the hearts and minds of your students? 

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What strategies do you employ to engage and motivate the hearts and minds of your students? 

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Comparing methodologies and seeing there are several pathways through a problem (such as how to solve a simple addition problem without a calculator)

Promotes collaborative and active learning, in which students learn best through engagement

Allow for student choice (such as allowing them to choose if they want a computer game, group work, or project with manipulatives)

Allows students to have a voice in their learning

Classification of mathematical objects and encouragement of discussion by comparing each other’s ideas (such as classifying quadrilaterals by their different properties)

Allows open discussion for students to present their ideas and thinking

Connect ideas and topics to the real world

Allows students to see relevance and make connections

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When student engagement is going downhill, we often find ourselves reaching for the prop box.

We pull out random videos, time-sucking resources, overly complex activities... anything to get and keep attention. But even these can't compete with our students' pen tricks, doodling, and discussions of what happened at recess. And there's a simple reason why:

Our students aren't engaged by things. They're engaged by us. That's why the best (and easiest) ways to increase student engagement come from you.

We compiled 20 essential strategies that generate lasting engagement, without the flashy props and hours of post-school prep.

Let the countdown begin!

20. Connect learning to the real world

We've all heard it before: "When am I ever going to use this?" Answer this question and you'll engage students with content that they know is relevant to life beyond school. Use anecdotes, case studies, and real-life examples from outside the classroom to root your teaching in "the real world".

19. Engage with your students' interests

Find out what already engages your students and build it into the learning process.

Using mathematics as an example, you could have students chart their performance in a video game over the week. You might even get your budding social media influencers to calculate a projected number of Instagram followers.

Learning what excites your students does more than just engage them. You'll build strong relationships and rapport, too.

18. Fill "dead time"

"Dead time" is any point in a lesson where students are left without something to do. You might be handing out a worksheet, getting a presentation set up, or waiting interminably for a YouTube video to load. These are brief windows that leave just enough time for students to tune out, after which time it can be very difficult to get them back.

Fill these blank spaces with low-order activities to hold students' attention. These should be quick, easy, and require minimal follow-up. For example:

Think Pair Share: students reflect on something, discuss with a partner, and then share with the rest of the class once everyone is ready

Quickwrite: write down three questions or points that have been raised by the lesson so far

What I know already: if you're just about to dive into new content, ask students to identify three things they already know about the subject and jot them down as bullet points.

17. Use group work and collaboration

Collaborating with small groups gives students a welcome break from solo bookwork. They'll benefit from each other's perspectives and the ability to express their ideas.

Use your judgment and knowledge of who works well together when organizing group work. Engineering the groups might avoid troublesome partnerships, while allowing students to work with friends might generate the buzz you need for more productive activity.

16. Encourage students to present and share work regularly

Giving students a regular opportunity to share their thoughts and demonstrate learning in front of their peers drives engagement in two ways:

it makes students accountable

it lets them hear from someone other than their teacher.

If your students quiver in fear at the thought of speaking in front of the class, combine presentations with group work. A few ideas:

Have students present in groups after a group task.

Let students share each other's work within smaller groups before asking them to choose one piece to share with the rest of the class.

Let students read or present their work while sitting down. It avoids the pressure of having to "stand and deliver".

Ask for one contribution from each group after discussion, with each group nominating a "spokesperson".

Above all else, make presenting and sharing a regular part of class activity. Your class will become an equitable and engaging space that echoes with the voice of every student, not just your own!

15. Give your students a say

If you don't know how to engage your students, let them tell you! Give your students a say in classroom activity by:

providing a choice of different activities (e.g. group work or solo)

seeking student input for assessment design (e.g. students can choose a final product, provided it meets the criteria)

periodic check-ins to monitor the pace of delivery (e.g. "do we need to go over this a bit more slowly or are we feeling pretty confident?").

Giving students a choice also fosters their sense of ownership over their learning. They'll move from passive consumers to active learners with a stake in classroom activity.

14. Use mixed media

Present learning content in a variety of mediums, including video, audio, and digital resources. Using such tech-rich resources is engaging for two reasons. It's a welcome change from the stacks of paper our students are usually saddled with, and it establishes a direct and relevant connection with the digital world they inhabit.

13. Get your students moving

If your students struggle to sit still for an entire lesson, get them moving. All that pent-up energy can be channelled into a learning activity that puts them on their feet. Try the following.

Have students come to the front and brainstorm together on the whiteboard.

Have students rotate through different stations around the room over the course of an activity.

Have students split into groups or arrange themselves in different areas of the room.

Take a stand: have students move to a particular area of the room to indicate their thoughts on an issue (e.g. "everyone who thinks x, move to the right side of the room; if you think y, stand on the left").

Movement works equally well to engage sluggish or weary students. A quick bit of physical activity will leave them more alert for the next phase of learning.

12. Read the room

If you're steadily losing students to doodling, off-topic chatter, and the pervasive "need to tear and ball up little pieces of paper", it's time to shake things up.

Cut the activity short if it's dragging, clarify instructions if there's confusion, or switch to a more student-centered activity for greater engagement.

Remember: it's impossible to have every student engaged 100% of the time. The next best thing we can do is to notice disengagement and respond to it quickly.

11. Scaffold tasks with checkpoints

If you dump all your instructions on students at the start of a lesson before turning them loose with an activity, confusion and disengagement will likely follow.

That's why it's important to scaffold larger tasks by breaking them into achievable steps. Each of these can be separated by brief "checkpoints" of instruction reorienting students and reminding them of what needs to be done next. They also serve as a periodic call to attention when students are liable to go off track.

10. Focus on discovery and inquiry

Sometimes the best thing you can do for engagement is to get out of your students' way.

Let them discover learning for themselves without being spoon-fed. They'll exercise critical and creative thinking, and pursue the lines of inquiry that interest them.

This doesn't mean you should retreat behind the teacher's desk. Observe your students, listen to them, and talk to them about what they're thinking. Be their guide as opposed to their instructor.

9. Ask good questions

Ask good questions of your students and you'll drive rich, engaging discussions that are open to everyone.

Good questions should be:

open-ended: to avoid "yes/no" answers

equitable: open to answers of varying depth and complexity

legitimate: asked because you want to hear students' thoughts and opinions, not because you're fishing for a correct answer.

When students answer a question, engage with their response. Even if it's incorrect or misinformed, recognize their effort and use it to refine the question further (e.g. "you're on the right track, but could we also think about...").

8. Allow for think time

It's gratifying to see hands shoot up as soon as you ask a question, but letting your students think it over has two benefits. It leads to more considered responses that drive engaging discussions, and it also makes the conversation accessible to those who don't have an instant answer.

After you ask a question, insist on a twenty-second pause and give students an opportunity to extend their standard responses further. For example, you might ask "See if you can explain how you came to your answer, too". You'll receive better answers and start to notice some new hands going up.

7. Shake things up

Predictability is safe, but it can get boring. Mix up your staple teaching strategies with new and novel activities from time to time. Talk to other teachers for ideas. In addition to engagement, you'll also be giving your students an example of what it means to take a risk and try something new.

Experimenting with some new parts of the teacher toolkit also makes it easier to differentiate your instruction. A new activity or delivery method might be the trick to engaging that student who has been a tough nut to crack all year.

6. Give brain breaks

Periodically give students a breather with brain breaks. These are short activities that allow students to stretch their legs before returning to work feeling focused. You can find a list of 20 brain breaks at Mind Bloom.

Meritopia in Mathletics is gamified learning at its best and keeps students engaged! When students earn enough points in Mathletics, they get the perfect brain break in Meritopia by discovering new lands and unlocking characters.

5. Be personable

Relationships and rapport are pillars of lasting engagement, and you can't have either without being personable. This means getting to know your students and letting them get to know you.

While enthusiasm for the learning content might ebb and flow, your smile, laughter, and conversation will engage students every time they walk through your classroom door.

4. Encourage friendly competition

Use in-class games, quizzes, or gamified learning programs to engage students with friendly competition. For example, Live Mathletics (a feature in Mathletics) allows students to test their mathematical skills by competing in 60-second quizzes against peers in their class or around the globe in real time.

"Friendly" is the keyword here. Make sure competitive activities are low stakes and put the emphasis on learning instead of winning.

3. Start lessons with introductory hooks

Engage students from the outset of your lesson with an introductory hook. This could be anything that piques interest, establishes relevance, or inspires curiosity in the subject of the lesson, for example:

a personal anecdote connected to the subject

a brain teaser or challenge question

a historical example

a multimedia source.

Keep your hooks short and segue them directly into an overview of the learning goal. It's an effective opening that engages while setting students up for the main instructional component.

2. Laugh together

Weaving humour throughout your lesson lightens the mood and makes for a more fun experience. Laugh with your students, and don't be afraid to let them laugh at you from time to time!

1. Gamify learning

Games are the most powerful source of engagement for students outside of class, and they're equally effective at driving engagement in learning. Transform activities into games by including levels of difficulty, rewards, and competitive elements. You can read more about gamifying learning here.

You don't have to invest hours in creating rules and drawing up game boards, either. Gamified learning programs can do the work for you. The learning programs in the 3P suite, for example, provide a host of student-friendly games for mathematics, literacy, and science. All you have to do is select the appropriate curriculum and grant your students access.

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20 Student Engagement Strategies for a Captivating Classroom

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Here's our list of the best student engagement strategies to help you create an engaging classroom and better connect with students.

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KUDs: What Students Know, Understand, and Do

Know: factual content

Understand: generalizations

Do: skills specific to the unit

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How will I encourage collaboration when unit planning?

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How will I encourage collaboration when unit planning?

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State the benefits:

saves time

ownership of the material produced

use the experience and expertise of the WHOLE team

everyone is given an opportunity to contribute

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Written by Megan Davenport and Elizabeth Clarkson, Ed.D, Faria Education Group
As professional development facilitators at Faria Education Group, we spend a tremendous amount of time with school leaders and teachers discussing curriculum and unit plans. We coach teachers to build strong content and skills that are aligned to standards while using the unit plan as the major organizing structure in their curriculum. We facilitate conversations around the components of a solid backwards design unit plan. We support school processes and structures to ensure unit plans are reviewed and revised on a consistent basis. It's a natural question to step back and ask ourselves, "But why plan at the unit level?"

Great question! To listen to an audio clip from Curriculum Conversations click here, or keep reading to understand the benefits of unit planning from an administrators' and teachers' perspective. We'll also offer three tips on building buy-in from teachers that will help them to invest their time and effort into the process of unit planning.

How is curriculum organized?
Let's review a few key terms and orientate ourselves to the conversation. Think of these components as building on each other.

Curriculum Mapping is the process of capturing the written curriculum.

A Unit of Instruction is a section of the curriculum map and involves dividing the course and its information into logical segments, usually about 2-4 weeks in length.

A Lesson Plan describes one learning session within that unit.

Depending on the perspective of one's roles and responsibilities in the school, unit planning offers different benefits that can help achieve the goals.

1. Efficiently Provide Evidence of Alignment:
School leaders, including instructional coaches, department heads, or building principals, are responsible for certain curricular tasks, such as vertical alignment. They are also responsible to agencies and organizations, such as accreditation and licensing agencies. When they have systems in place to capture their unit planning work, they can quickly and effectively provide evidence of various parts of their curriculum alignment to these agencies.

2. Ability to Review and Revise:
Curriculum work is constantly evolving over time. When schools have a strong practice of unit planning in place, their curriculum is easier to review and revise for alignment. WIthout that unit structure, information can be disorganized. It can be difficult to identify gaps, redundancies, opportunities for collaboration, and spiraling. With a unit planning structure in place, the process of curriculum review and revision becomes a manageable task and allows for multiple people or teams to contribute at the same time.

3. Time Management:
Teachers need strong lesson plans to help them organize their day, but an instructional leader might have limited time to review these lessons in meaningful ways. Unit plans offer a higher level look at Stage 1 and Stage 2 components of a backwards design unit. Reviewing a unit plan is more realistic and allows time for quality feedback and thoughtful reflection that will be more helpful to the teacher.

4. Structuring your PLC Time:
Educators all know how valuable PLC or grade level meeting minutes are. When a school has an agreed upon unit plan and format to guide them, teachers can make the most of their time together. This is a benefit for teachers and administrators!

5. Smooth Transition:
Teacher transitions in schools will inevitably happen - teachers move from grade to grade, maybe even changing subjects. International schools work with visa issues requirements and routinely transition teachers in and out of their school communities. When schools are using unit plans to organize their written curriculum, that scope and sequence of information becomes part of the educational fabric of the school and its curriculum. Students will ultimately benefit because incoming teachers have a solid place to pick up for that grade or subject and outgoing teachers leave behind the best of their own contributions. A smooth transition for teachers helps keep a strong curricular alignment and a viable and guaranteed curriculum.

1. Wise Investment of your Time:
Think of units as a teachers' professional portfolio, a collection of one's best professional work. An initial investment in time quickly pays dividends in the future. Although curriculum review and revision are a healthy part of a school's culture, with solid units in place from the year before, teachers can easily adapt, adjust, and improve their units based on new resources, feedback from observations or students, and reflection on the unit's overall effectiveness from the year before. If you are a new teacher to a grade or subject, the units are a great place to quickly get up to speed on the expectations for that grade level.

2. Pacing:
Units are organized in a pacing calendar and can be easily adjusted at the beginning of the year to ensure appropriate time is given to each unit. By pacing these units in advance, teachers can ensure the more challenging units are given appropriate time and/or spiraled throughout the year. Once this pacing calendar is complete, teachers can quickly see if their teaching is on track and can make adjustments for future weeks.

3. Less Dependent on Specific Resources:
Teachers might find that a textbook's material is not a match for students' academic level at a given time in the year. We also know that information, especially in subjects like science or social studies, change and printed material could be out of date. When units are written to address standards, content, and skills, a teacher has more flexibility to change and update resources as better ones become available. The units are the basic structure of a grade or course and they remain the backbone.

4. Collaboration:
Teachers, in addition to administrators, also see planning at the unit level as beneficial because it provides an established structure to PLC and grade level meetings. Conversations can stay focused and allows for more productive collaboration during these meetings.

1. Make the process of unit planning a useful and authentic part of your school culture:
We want to make the process of unit planning part of the long term operations of the school. Once this process is embedded and sustainable, schools can quickly get the information they need to inform conversations and decisions. They could use the curriculum maps, organized by units, to 1) review assessment data, 2) offer feedback during classroom observations, 3) talk to prospective students and families at the admissions level, and 4) showcase grade level content during open house sessions or share with the public on other platforms. Unit planning then becomes helpful to both teachers at the instructional level but also for administrators within the larger school context.

2. Get creative and find the time for teachers to work together:
Finding time during the busy year for unit planning conversations can be a challenge and will require some creativity and persistence. Unit planning should not be a stand-alone initiative, but something that is built into existing school processes. There may need to be additional time dedicated to documenting curriculum for the first time, but after that, teachers should focus on maintaining and revising units as one way to support larger goals. Get creative! Think about formal (early release and late start days, long term PD schedules, PLC meetings) and informal times (support from specialist teachers, early morning donuts and coffee, after school snacks) and focus on collaboration!

3. AVOID unit planning as a compliance task:
Schools want to avoid treating the task of unit planning like a one time only initiative or ONLY for compliance purposes. Strive for building it into the school culture (see Tip #1). Teachers get discouraged if they see unit planning as the "next new thing" added to their to-do list, and will wait for the next new initiative to land on their plate tomorrow. Unit plan with a purpose and tie that purpose to student success at your school!

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As professional development facilitators at Faria Education Group, we spend a tremendous amount of time with school leaders and teachers discussing curriculum and unit plans

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How do the three types of essential/guiding questions—factual, conceptual, and debatable—support student learning? 

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How do the three types of essential/guiding questions—factual, conceptual, and debatable—support student learning? 

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Purpose is to help students to connect the factual support with the conceptual understandings

Each question corresponds to a generalization with the purpose of drawing out conceptual understandings

Typically 3-5 factual and conceptual questions and 2-3 debatable questions

Examples

Factual: What are the different forms for the equation of a line?

Conceptual: How can you express a linear relationship in different ways?

Debatable: How well does a linear function fit all situations in real life?

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What is the purpose of inquiry questions? What is important to know about developing inquiry questions in an IDU?

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What are the main components for a math unit web? 

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What are the main components for a math unit web? 

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Unit Web: contains critical content topics and concepts and gives an overview of the depth and breadth of the unit of instruction

Knowledge strands

Concepts in mathematical processes (process generalizations specific to the unit)

Meso concepts

Micro concepts

Conceptual lenses: purpose is to create a synergy between the factual and conceptual levels of thinking (synergistic thinking)

Also known as key concepts (form, logic, and relationships, etc.)

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How does unit planning empower teachers?

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How does unit planning empower teachers?

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take control of the learning in the lesson

have an input into designing curriculum

allows for creativity

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Abstract

Time for a Change

A Philosophical Approach

The Foundation: Building Trust

Pillar 1: Consistency and Cohesion

Pillar 2: Adult Learning Theory

Pillar 3: Time for Collaboration

Pillar 4: Innovative Design Thinking

Pillar 5: Autonomous Learning

Putting Ideas into Action

1. Making Time for Teacher Learning

2. Guiding Groups and Areas of Focus

3. Establishing Parameters and Accountability

4. Sharing and Using Results

5. Administrative Participation

Reaping the Rewards

How one high school reframed professional learning around autonomy with structured support-and sparked innovation that proved critical during the pandemic.

Abstract

Time for a Change

A Philosophical Approach

The Foundation: Building Trust

Pillar 1: Consistency and Cohesion

Pillar 2: Adult Learning Theory

Pillar 3: Time for Collaboration

Pillar 4: Innovative Design Thinking

Pillar 5: Autonomous Learning

Putting Ideas into Action

1. Making Time for Teacher Learning

2. Guiding Groups and Areas of Focus

3. Establishing Parameters and Accountability

4. Sharing and Using Results

5. Administrative Participation

Reaping the Rewards

As we labored throughout the summer of 2019 to reconceptualize our school's professional development plan for the upcoming year, little did we know we were also preparing to face COVID-19 and, consequently, one of the most challenging years of our careers. Not that we were unaccustomed to challenges. Two of us, Joel and Michael, were emerging leaders (an acting and assistant principal) in a New York City high school serving a 100-percent immigrant student population, while the other, Kristina, provided coaching support to the school. We found every school day presented us-and our faculty-with a series of productive struggles.

As our summer planning progressed, we realized we needed to strike that same balance between challenge and support with our teachers. Unbeknownst to us, the plan we would develop would ignite teachers' passions and energize their efforts, ultimately sustaining their professional learning trajectories throughout the upheaval of the then-looming pandemic.

Time for a Change

We knew our school's professional development model needed improvement. Traditionally, administrators selected topics, books, or theories for study and planned one-size-fits-all workshops around those topics. Observations revealed teachers seldom integrated those ideas into their instruction. In addition, our school's 2018-2019 stakeholders' survey revealed that our faculty desired a clearer vision for professional development, a relationship of greater trust with the administration, and more opportunities to work with their colleagues.

We decided a revamp was in order. But rather than announcing a problem and telling teachers how to solve it, we wanted to engage teachers around self-selected problems of practice. We aimed to focus our attention on empowering and supporting teachers, which in turn, we hoped, would improve students' sense of well-being and academic achievement.

A Philosophical Approach

Before making functional plans for changing staff development, we fleshed out the philosophical and structural principles on which those plans would be based. To that end, we developed a conceptual framework reflecting the school's values and setting the compass for year-long decision making. As shown in Figure 1, this framework featured five pillars for professional learning built on a foundation of trust and reaching toward teacher agency and impact.

Figure 1. Conceptual Framework Guiding Planning for Professional Learning

The Foundation: Building Trust

Our year-end stakeholder survey had revealed that establishing trust with our teachers was an imperative first step. We took deliberate actions in this regard. First, we dedicated ourselves to communicating a message of trust by giving teachers the freedom to choose the problems of practice they wanted to explore. Second, we committed ourselves to supporting teachers by regularly visiting classrooms and planning meetings to provide continuous, nonevaluative feedback. We believed these two measures would result in staff feeling more comfortable taking risks and being innovative.

Pillar 1: Consistency and Cohesion

To guide teachers in framing their problems of practice, we selected three schoolwide instructional goals, derived from data analysis and included in our comprehensive school plan:

1. Design robust projects that challenge students to create high-quality work.

2. Develop structures for the teaching and learning of literacy across the curriculum.

3. Plan lessons and assessments that establish highly successful thinking routines.

These goals would drive our scheduling, community building, messaging to parents and students, teacher reflections, coaching models, and our professional development plan.

Pillar 2: Adult Learning Theory

One of our aims was to empower staff to push each other toward growth. Adult learning theory asserts that, in an ecosystem of trust, teachers are "willing to expose their challenges, mistakes, and questions-as well as their hopes, strengths, and successes-and be comfortable discussing all of these with their colleagues" (Drago-Severson & Blum-DeStefano, 2016). We believed that such vulnerability should be modeled, so we launched our plan by sharing an overview with teachers before the school year began and asking for their feedback. We then opened the year with a full-staff meeting in which we discussed the teachers' feedback, shared the prior year's data, and used both to shape a staff-wide discussion of instructional goals.

Pillar 3: Time for Collaboration

Our district dictates that schools designate time for professional learning each week. To make the most of these meetings and create opportunities for authentic collaboration, we decided on a number of protocols designed to overcome barriers to engagement. These included:

Encouraging teachers to select "problems of practice" (aligned to school goals) that intersected with challenges arising from everyday instruction.

Providing flexible time and space to pursue inquiry.

Structuring purposeful collaboration between colleagues around shared areas of challenge.

Ultimately, we wanted professional learning to be so intrinsically motivating that it expanded beyond the confines of a weekly meeting and into every area of professional interaction.

Pillar 4: Innovative Design Thinking

We were inspired by the premise, "When people are operating in conditions of high-trust, collaboration, and effective leadership, they are more willing to innovate and take risks" (Fullan & Quinn, 2016, p. 26). Giving our teachers permission to "fail" felt counterintuitive; still, we were committed to encouraging our teachers to take instructional risks. We were certain that if we implemented nine-week learning cycles and embedded opportunities for on-demand coaching and feedback, teachers would have the time and liberty to experiment, analyze results, regroup, and try again. Our goal was to balance autonomy with support, thus encouraging teachers to expand beyond their comfort zone.

Pillar 5: Autonomous Learning

Ultimately, we wanted to signal that professional development would not happen to teachers as an audience; rather, it would be driven by their interests and expertise as professionals. In addition, staff development would not be "accomplished" within a short time period; rather, it would become a sustained effort toward progress in becoming more versatile and well-equipped to handle instructional dilemmas. Professional learning would be transformed into a shared culture that unfolded daily throughout a year-long process, personalized according to interest, pursued in the company of peers, and supported by the administration.

Putting Ideas into Action

With our philosophy and goals firmly in place, we turned to the plan itself, taking care that the foundations of the plan mirrored our beliefs. Specifically, we outlined the following five priorities.

1. Making Time for Teacher Learning

No plan will succeed without time carved out for its success. Accordingly, we incorporated both long-term cycles of inquiry and weekly opportunities for collaboration. For our year-long plan we adopted continuous cycles of improvement (Bryk et al., 2015). During each nine-week learning cycle, staff members were asked to make plans based in their interests, targeted toward our school instructional goals, and supported by relevant literature. Teachers implemented one or more instructional changes each cycle, analyzed their results, and shared their findings.

We maintained our regular weekly meeting dedicated to PD but reconfigured it to provide time for teachers to collaborate in smaller groups on self-designed projects. In addition, we created more flexibility in teachers' daily schedules, gaining time by rearranging class offerings and restructuring teaching loads. Our external coaches' visits were scheduled, based on requests from teachers, to provide resources to support teams and/or individuals, both during PD time and throughout the school day. These proactive adjustments deliberately reinforced our premise that professional learning is most effective when driven by organic inquiry rather than by mandated topics du jour.

Inquiry would flow through four distinct nine-week cycles. The first three cycles of the year would focus on our schoolwide instructional areas. The final cycle was geared toward connecting the initial three cycles by consolidating all of the work from the year, sharing lessons learned, and assembling materials created for future use.

2. Guiding Groups and Areas of Focus

To determine collaborative groups for each cycle, each faculty member developed an inquiry question to pursue growth in self-designated areas of challenge. These "challenges" had to be (1) geared toward promoting student growth; (2) central to improving teaching effectiveness; and (3) instrumental in improving our instructional practices toward at least one of our three schoolwide instructional goals.

Once each teacher identified a unique challenge, we created faculty groups of four-to-six members around common themes that emerged from teacher choices. Each group chose a problem of practice (Bryk et al., 2015) that encompassed individual members' personal challenges. They then sought a book, article(s), or other relevant research to ground their inquiry (Fisher et al., 2020). Teams directed their own vision and work, supporting one another as they pursued their individual questions within their common area of inquiry. Teams were changed every cycle, so teachers had opportunities to work with a variety of colleagues throughout the year.

3. Establishing Parameters and Accountability

While we wanted our plan to be malleable enough to bolster autonomy and energize staff, we realized we had to provide reasonable parameters to frame productive work. Accordingly, we shared examples of inquiry projects in action, sample problems of practice, frameworks for pursuing inquiry, and avenues for sharing professional lessons learned (see options in bulleted list below). To foster motivation and a sense of agency, we ended our weekly PD time by gathering teacher teams into a full group where they could share their progress and problems. In addition, we asked each team to present a more formal summary of their findings at the end of each cycle. Figure 2 illustrates the stages of inquiry and reporting in Cycle One.

Figure 2. Professional Learning Schedule Example

Cycle 1: Design robust projects that challenge students to create high quality work

Empowering Teacher Growth - table

Calendar

Weeks 1-9

Outcomes

4. Sharing and Using Results

For their formal presentation of findings at each cycle's end, teams were asked to showcase how their work (1) illustrated practical, unique applications of research and (2) impacted student outcomes. To do so, they could use any two of the following work products:

Analysis of student work/data, with interpretations shown in written or visual form.

Analysis of actions taken in the classroom through annotated running records.

Annotation and analysis of video recorded teaching practices.

Annotated before-and-after snapshots of lesson/unit plans and teaching materials.

Screencasts, annotated whiteboard videos, or "panel interviews" illustrating student reactions to work.

Debrief of classroom peer-visits (conducted by members of the same team), including resulting action items.

Any proposed alternate product that aligns authentically with area of inquiry.

Groups shared their practices and findings with members of different teams in a full-staff expo, using a round-robin protocol so that each teacher shared the work of their entire group. Teachers were able to learn about the work of the other groups while presenting their own.

5. Administrative Participation

As an administrative team, we fully participated in the learning alongside our staff. We used our own inquiry projects to generate conversations with teachers about their practice and to solicit feedback about schoolwide areas for improvement. Most important, we modeled leading from a vulnerable standpoint. We followed the same foundational structure and participated in each cycle, sharing like the other teams but with a leadership focus on school structures and culture. We sought input from faculty and incorporated their feedback into our work.

Reaping the Rewards

Overall, the returns exceeded our expectations regarding teacher engagement, creativity, and collaboration. One of the most exciting parts of this journey was seeing the steadfast motivation of our staff through the fall and winter. Along with adopting innovative practices and making progress toward instructional goals, the most significant outcome was how our staff began to operate as a bonded team. Over the course of the year, we saw marked increases in staff initiative to support students in innovative ways, staff support for each other, nonstructured collaboration, and staff happiness.

These positive impacts were felt most fully in the spring when, overnight, schools closed their doors and were forced to reimagine classroom formats in the wake of the pandemic. We believe that our professional learning plan's momentum allowed our staff to adjust in stride, continuing the work we had pursued all year. Our plan was purposefully designed to build staff capacity; as a result, teachers felt empowered to meet challenges head on. This sense of self-efficacy propelled them to develop innovative models of remote instruction.

Cycle Four was originally designed for teachers to regroup into teams teaching the same cohorts of students, synthesize lessons learned from the first three cycles, and apply them to new innovations. Though not in the format we imagined, this is exactly what happened-even without the structure and accountability established in the first three cycles.

Throughout all four cycles, we conveyed a message of trust with our words and actions, emphasizing autonomy and engagement and encouraging teachers to share their reasoning along with their results. Professional development became efficient to plan, smooth to implement, and rewarding to undertake. Our hope was that "teachers who were motivated, connected, and collaborative would be happier about pursuing meaningful work and therefore be more innovative" (Fullan & Quinn, 2016, p. 26). That hope was realized.

Because of COVID-19 disruptions, students did not take their state exams, nor did we receive a 2019-2020 year-end review. Recognizing this lack of hard evidence, our faculty crafted and shared an open letter (signed by every teacher) testifying to improvements in professional learning and school culture. It ended with this statement:

Through their actions and genuine efforts, Dr. Heckethorn and Mr. Giovacchini have shown that teachers at [our school] are valued, trusted, and respected as professionals. It is through these efforts that we have witnessed tremendous student growth and engagement in student activities both inside and outside the classroom.... We would like to thank both of them for leading the charge to be better practitioners while respecting us as individuals with needs, preferences, and concerns. Upon returning in the fall, we are hopeful that our school environment continues to be one of invention, promise, and pride.

References

Bryk, A., Gomez, L., Grunow, A., & Lemahiue, P. (2015). Learning to improve: How America's schools can get better at getting better. Cambridge, MA: Harvard Education Press.

Drago-Severson, E., & Blum-DeStefano, J. (2016). Tell me so I can hear you: A developmental approach to feedback for educators. Cambridge, MA: Harvard Education Press.

Fisher, D., Frey, N., Almarode, J., Flories, K., & Nagel, D. (2020). PLC +: Better decisions and greater impact by design. Thousand Oaks, CA: Corwin.

Fullan, M., & Quinn, J. (2016). Coherence: The right drivers in action for schools, districts, and systems. Thousand Oaks, CA: Corwin.

ASCD is a community dedicated to educators' professional growth and well-being.

Let us help you put your vision into action.

ASCD empowers educators to achieve excellence in learning, teaching, and leading so that every child is healthy, safe, engaged, supported, and challenged.

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As we labored throughout the summer of 2019 to reconceptualize our school's professional development plan for the upcoming year, little did we know we were also preparing to face COVID-19 and, consequently, one of the most challenging years of our careers. Not that we were unaccustomed to challenges

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Chapter 4: How Do I Plan Units of Work for a Concept-Based Curriculum? 

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Chapter 4: How Do I Plan Units of Work for a Concept-Based Curriculum? 

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Topical: subject or topic specific; focus on a particular insight we want our students to acquire within a unit of study
Overarching: understandings beyond the specifics of a unit; larger, transferable insights we want students to acquire (often reflect year-long program understandings)

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Providing Opportunities for Students to Demonstrate and Communicate Conceptual Understanding

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Providing Opportunities for Students to Demonstrate and Communicate Conceptual Understanding

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Key to inductive teaching is providing students with specific examples from which to draw generalizations.

Guiding students, through inquiry, to discover generalizations and principles for themselves

Looking at specific examples and seeking patterns in order to devise and write generalizations

Writing to communicate how the students have processed information and skills and whether they possess the conceptual understandings of a unit

Using a graphic organizer to show a visual representation of how the concepts are related

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According to Adding It Up (H/T Dan Meyer):

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)

Essentially, conceptual understanding is knowing more than isolated facts, it is also knowing connections between those facts and having those facts well organized.

This image represents someone who understands some isolated facts.

And this image represents what conceptual understanding looks like:

I propose that those line segments between the points, representing the connections between isolated knowledge, are themselves a type of knowledge, consequently our second image is really more like the first image.

What is clearly missing in this analogy is that conceptual understanding is knowledge that is well-organized, but I claim that organization of knowledge is itself another type of knowledge. Also missing is knowing why the knowledge is important and the contexts in which an idea are useful, but again these also seem just like other types of knowledge.

Let's take a look at a problem I had to solve recently and see if we can use it to unpack what conceptual understanding is.

According to the solution key, the writers of this question are assessing students' ability to solve quadratic equations. However, that likely requires applying the quadratic equation to all four equations given, which would be time-consuming and potentially error-prone.

When I solved this problem, I saw four quadratic functions, each transformed from y = x ², and based on those transformations, I was able to eliminate (1) and (3) as options since they have real solutions. I also know that if 1 - i is a solution, then so is 1 + i, and that the sum of the roots of a quadratic function are related to the coefficient of x in the equation. I could not remember if the sum of the roots is equal to the coefficient of x or to -1 times the coefficient of x, so I multiplied out ( x - a)( x - b) to double-check and determined that if a and b are the roots of a quadratic equation, then -( a + b) is the coefficient of x in that equation (assuming a lead coefficient in the equation of 1). 1 - i + 1 + i = 2 so (4) must be the answer.

My actual solution strategy doesn't matter all that much, what matters is that in order to produce it, I needed to know some smaller things and that those smaller things could be seen as connected together to create a larger idea. Instead of seeing this as the difference between an expert and a novice, I prefer to think of knowing things as being a gradient between knowing a little bit and knowing more than a little bit.

Conceptual understanding is knowing ideas in an interconnected and organized way and knowing more ideas along the novice-expert spectrum.

The reason why conceptual understanding is an important goal is because otherwise we might be tempted to rely on teaching kids tricks instead of mathematics.

What mathematics does a child learn from learning this procedure for adding fractions?

Very little! It is extremely difficult for a child to connect this procedure to other mathematics that they know, consequently using a trick like the butterfly method is likely to lead to students knowing an isolated idea and not much else. Note that I am not opposed to memory aids, I just think they should be based on solid mathematical reasoning and they definitely should not be taught instead of the mathematics.

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According to Adding It Up (H/T Dan Meyer): Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful.

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So what? What is the significance of effect?

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How or why?
instructional focus

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Students will understand that...

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Scaffolding!
Level 1: Generalizations that are weak and do not help learners to fully understand what and why they are learning
Level 2: Generalizations that address how or why a student is learning something
Level 3: Generalizations that explain the purpose/significance of what the student is learning

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Why Should Students Know Principles and Generalizations?

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Why Should Students Know Principles and Generalizations?

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Generalizations are statements of conceptual understanding and represent our goals for learning. They are also known as essential understandings, enduring understandings, or the big ideas in a unit of work. Principles are cornerstone truths that the students should know in order to meet the learning goals.

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Difference Between Principle and Generalization

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Difference Between Principle and Generalization

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Generalizations: two or more concepts stated as a sentence of relationship; understandings that transfer through time, across cultures, and across situations in the context of mathematics; overriding ideas we would like our students to understand and transfer
Principles: laws or foundational truths that hold all the attributes of generalizations and commonly describe real-life situations

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Chapter 3: What Are Generalizations in Mathematics?

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Chapter 3: What Are Generalizations in Mathematics?

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Six Mathematical Processes Broken Down Into Skills and Strategies

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Six Mathematical Processes Broken Down Into Skills and Strategies

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Relationship between Structure of Process and Structure of Knowledge

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Relationship between Structure of Process and Structure of Knowledge

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Complementary, three-dimensional, concept-based models that have a symbiotic (interdependent) relationship

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How is mathematics a language of conceptual relationships made of macro, meso, and micro concepts?

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How is mathematics a language of conceptual relationships made of macro, meso, and micro concepts?

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Macro concepts give a breadth of understanding and possess the greatest transferability across most disciplines. These macro concepts can be broken down into meso and micro concepts.
Macro examples: calculus, geometry, algebra
Meso examples: differentiation, trigonometry, functions
Micro examples: limits, unit vectors, quadratics

From a sociological standpoint, this would be individuals, encompassed in medium systems, encompassed in large systems

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Sociologists study different levels of society: the macro level, the meso level, and the micro level.

Sociologists define three major levels of society. They use data to understand types of behavior that occur at the different levels and the interconnections of these levels. Micro-level analysis, detailed examination of one-to-one interactions between individuals, includes studying people's behavior during negotiations, confrontations, and everyday conversations. Micro-level analysis allows for nuanced analysis of particular dynamics of social phenomena but may fail to consider broader social forces that impact such processes. Meso-level analysis, detailed examination of a specific group, community, or organization, studies certain parts of a society. Also referred to as network analysis, this approach examines the patterns of social ties among people in a group and how those patterns affect the overall group. Macro-level analysis, examination of society as a whole, looks at the broad systems, institutions, hierarchies, and patterns that shape a society. Macro-level analysis takes into account the social, political, economic, and other forces that impact societies and individuals but might not capture important facets of social interactions that occur on the micro level.

Macro-level analysis is used to look at societies at large. Meso-level analysis involves the study of groups, communities, and institutions. Micro-level analysis focuses on the social interactions of individuals or very small groups.

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Macro-Level, Meso-Level, and Micro-Level Analysis - Course Hero

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Sociologists study different levels of society: the macro level, the meso level, and the micro level. Sociologists define three major levels of society. They use data to understand types of behavior that occur at the different levels and the interconnections of these levels.

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What are the key categories of processes in mathematics? Provide examples for each process. 

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What are the key categories of processes in mathematics? Provide examples for each process. 

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Examples
Problem-Solving: Guess and check and look for a pattern
Reasoning and Proof Example: Make generalizations and ask "why"
Communicating: Question and discuss as part of a group
Making Connections: Make connections between facts and connect new problems to old
Creating Representations: Use drawings and symbols and include geometric and algebraic forms of representation
Investigating: Create different lines of inquiry and investigate unfamiliar situations

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What is the distinction between processes, algorithms, and skills in mathematics? 

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What is the distinction between processes, algorithms, and skills in mathematics? 

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Strategies are a number of skills that learners use in a methodical and systematic way to support learning.
Skills are small operations or actions that are embedded in strategies and, when appropriately applied, "allow" the strategies to work; e.g. plotting a graph
Mathematical processes are complex, sophisticated performances, which are composed of strategies, algorithms, and skills; e.g. problem-solving
Algorithms are sets of computational rules to produce a specified outcome; e.g. PEMDAS

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This article shares examples of how classroom routines could be one of the effective tools to foster learners' agency and raise their voice and choice while learning mathematics.
During the last months, the concept of "agency" has dominated the discussion between educators since it is one of new terminologies that appeared in PYP: From principles into practice. Although it is not "new" in the programme because PYP educators were aware of student-centered learning, supporting learners too have voice, choice, and ownership is a challenging journey.

Being responsible for assisting and supporting teachers to implement the Mathematics curriculum effectively and for sustaining the growth of the programme, we started to experiment with practices in mathematics that build student agency. We chose one tool for teaching Math: classroom routines.
Mathematical proficiency and routines
The National Council Teachers of Mathematics (NCTM, 2001) defines five strands that are interwoven and interdependent in the development of proficiency in mathematics: conceptual understanding, strategic competence, adaptive reasoning, productive disposition, and procedural fluency. One of the known tools or activities that help learners develop procedural fluency and conceptual understanding, as well as reasoning and problem-solving skills, is mathematics routines, where learners work together on a strategy or skill to develop specialized knowledge and language needed for conceptual understanding and applying mathematical concepts.
Dynamic Routines
Supporting learners to guide their learning required making a shift in the way routines are delivered and instructional practices that foster sense-making in Mathematics. Accordingly, I transferred the traditional routines into dynamic ones and put the needed criteria or the key elements that guide us toward planning meaningful experiences that support the learner's voice and choice.

To apply dynamic routines, teachers create a safe learning environment where learners are provided with opportunities for metacognition, are confident and motivated, contribute to a positive attitude toward Mathematics, are encouraged to use mathematical language and test others' strategies to assess their validity.
Examples of Dynamic Routines
We have started implementing the dynamic routine this year with fourth grade students. We chose the learning outcome: describe mental and written strategies for adding and subtracting two-digit numbers- Strand Number (Phase 2), and we extended it to teach adding and subtracting three-digit numbers.

Learners were asked to find one or more strategies and justify their answers. They shared their ideas with peers and then in groups using appropriate mathematical language. We used posters in classroom as samples to encourage them communicate and explain their thinking.

Figure 1: Sample of Class Poster

The following are samples of learners' thinking:

Figure 2: Vertical and horizontal Calculations

Figure 3: Decomposing and Skip Counting

Figure 4: Use of Soroban

Figure 5: Same Change Method

Research suggests that once students have memorized and practiced procedures that they do not understand, they have less motivation to understand their meaning or the reasoning behind them (Hiebert, 1999). Therefore, we aim to develop our understanding of how to create opportunities for students to develop agency in the mathematics classrooms by encouraging them to participate as creative agents, who think and reason for themselves in mathematics, rather than being passive recipients of knowledge. Our routines reflect our beliefs, so all what we need to do is to choose the best way to apply these routines, reflect on their effectiveness in the sense of raising agent learners, and then act to go beyond and deeper.

References:

National Research Council. 2001. Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press. https://doi.org/10.17226/9822.

Hiebert, J. (1999). Relationships between research and the NCTM standards. Journal for Research in Mathematics Education, 30(1), 3-19.

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Samia Henaine is a PYP Math coordinator who assists and trains teachers to implement the curriculum effectively, deepens their understanding of how teaching and learning are viewed from the perspective of new educational approaches, and support them to adapt teaching methods and materials to meet students' varying needs and interests. You can follow her on Twitter @HenaineSamia.
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The Mathematics routines that cultivate learner agency

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This article shares examples of how classroom routines could be one of the effective tools to foster learners' agency and raise their voice and choice while learning mathematics. During the last months, the concept of "agency" has dominated the discussion between educators since it is one of new terminologies that appeared in PYP: From principles into practice.

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What is the difference between formulae and theorems in mathematics? 

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What is the difference between formulae and theorems in mathematics? 

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Formula: an equation that uses mathematical symbols or variables to show a relationship; represented by the facts in the Structure of Knowledge
Theorems: statements that have been proven and connect explanations of conceptual understandings; represented by the principles in the Structure of Knowledge

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In context|mathematics|lang=en terms the difference between formula and theorem

is that formula is (mathematics) any mathematical rule expressed symbolically while theorem is (mathematics) a mathematical statement of some importance that has been proven to be true minor theorems are often called propositions'' theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called ''lemmas .

In context|logic|lang=en terms the difference between formula and theorem

is that formula is (logic) a syntactic expression of a proposition, built up from quantifiers, logical connectives, variables, relation and operation symbols, and, depending on the type of logic, possibly other operators such as modal, temporal, deontic or epistemic ones while theorem is (logic) a syntactically correct expression that is deducible from the given axioms of a deductive system.

As nouns the difference between formula and theorem

is that formula is (mathematics) any mathematical rule expressed symbolically while theorem is (mathematics) a mathematical statement of some importance that has been proven to be true minor theorems are often called propositions'' theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called ''lemmas .

As a verb theorem is

to formulate into a theorem.

Other Comparisons: What's the difference?

Noun

(wikipedia formula)

(mathematics) Any mathematical rule expressed symbolically.

x = \frac {-b \pm \sqrt{b^2 - 4ac}}{2a} is a formula for finding the roots of the quadratic equation ax² + bx + c = 0 .

(chemistry) A symbolic expression of the structure of a compound.

H₂O is the formula for water (Dihydrogen monoxide)

a plan or method for dealing with a problem or for achieving a result

The company's winning formula includes excellent service and quality products.

A formulation; a prescription; a mixture or solution made in a prescribed manner; the identity and quantities of ingredients of such a mixture.

The formula of the rocket fuel has not been revealed.

Drink given to babies to substitute for mother's milk.

(logic) A syntactic expression of a proposition, built up from quantifiers, logical connectives, variables, relation and operation symbols, and, depending on the type of logic, possibly other operators such as modal, temporal, deontic or epistemic ones.

Synonyms

* ( in mathematics) mathematical formula * ( in chemistry) chemical formula

Derived terms

* ( l) * formulate * formulation * Formula One * formula racing * multiformula

Noun

()

(mathematics) A mathematical statement of some importance that has been proven to be true. Minor theorems are often called propositions''. Theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called ''lemmas

(mathematics, colloquial, nonstandard) A mathematical statement that is expected to be true; as, (as which it was known long before it was proved in the 1990s.)

(logic) a syntactically correct expression that is deducible from the given axioms of a deductive system

Synonyms

* ( proven statement): lemma, proposition, statement * ( unproven statement): conjecture * See also

Derived terms

* central limit theorem * Pythagorean theorem * binomial theorem * * intercept theorem

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Formula vs Theorem - What's the difference?

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is that formula is (mathematics) any mathematical rule expressed symbolically while theorem is (mathematics) a mathematical statement of some importance that has been proven to be true minor theorems are often called propositions'' theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called ''lemmas .

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Composed of mathematical processes, algorithms, strategies, and skills. Like the Structure of Knowledge, the goal is to have students progress to the point of conceptual understanding.

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We've written specifically about the discipline of mathematics several times - the two specific to concept-based are here and here. To me, it's the discipline that requires the greatest shift in how it is traditionally taught. Many experts talk about how we need students to be better problem-solvers, allowing them space to figure things out without giving them the formula or only the specific information needed to solve the problem. I agree. At the same time, we need to be sure we are teaching the conceptual relationships behind the "how" of math. Students need to discover the "what" of math.

Like English class with drills about the rules of grammar, procedures are often taught without much thinking behind why we are doing them. But very much unlike English class, there is lots and lots of content behind the procedures. In 2013, Dr. Lois Lanning published her book with the helpful illustration of the Structure of Process which is the complement to Dr. Erickson's Structure of Knowledge. She describes strategies and skills associated with complex procedures as similar to specific facts associated with complex ideas. Many people see the two images below and think mathematics falls on the Structure of Process side. This is the wrong way to think about it.

Here are some tips that might help:

If we find ourselves writing out goals for student understanding that start with "how to...", we know we are missing the mark for conceptual understanding. Those are skills.

If we find ourselves writing goals that sound like things we want kids to do, again missing the mark.

We should ask ourselves, what is the content behind this mathematical procedure? How can I write that in a clear, powerful statement of conceptual relationships.

An example:

What is the content behind pi?

The circumference of a circle is roughly 3 times its diameter.

How can I write that in a statement clear, powerful statement of conceptual relationships that doesn't use the weak verb "is"?
The ratio of circumference to diameter in all circles represents a fixed constant, π.
Once we have this powerful statement of conceptual relationships, we then set to work on writing lesson plans that allow students to discover it.

One thing I've noticed from my old school and from the MYP IB maths documents is that when we aim to write inquiry lessons, it's very easy to veer into "how to" statements rather than statements of conceptual relationships with content or knowledge as the driving force.

Stay tuned for more details on writing those lesson plans. Also, be sure to check out my friend Jennie Wathhall's book, due out in January!

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Grappling with transforming mathematics? Us, too!

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We've written specifically about the discipline of mathematics several times -- the two specific to concept-based are here and here. To me, it's the discipline that requires the greatest shift in how it is traditionally taught. Many experts talk about how we need students to be better problem-solvers, allowing them space to figure things out without...

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The levels are facts, topics, concepts, and generalizations/principles. As students progress, they should move up each level until they have a deep understanding of the particular knowledge (e.g. understanding the fundamental theorem of calculus).

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When examining the elements of knowledge, facts, topic, concept, generalization/principle, theory, H. Lynn Erickson determined that knowledge has a hierarchical structure. Understanding there is a hierarchical structure in the construction of knowledge will assist us in designing our classroom instruction.

The " Structure of Knowledge " diagram provides an instructional framework to apply to curriculum. This framework promotes students ' higher-level thinking and is the basis of concept-based instruction. It illustrates the relationships between the levels of knowledge.

Many teachers begin planning units based on facts and knowledge and are topic-based. However, by "standing back" and reflecting on bigger ideas and concepts, knowledge-based information can be extended beyond recall and used to support generalizations and theories that connect to other, and broader, topics.

Explore this framework on the following page as it relates to the dinosaur unit.

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concept-based curriculum: structure of knowledge

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Concept-based Curriculum: Structure of Knowledge

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When examining the elements of knowledge, facts, topic, concept, generalization/principle, theory, H. Lynn Erickson determined that knowledge has a hierarchical structure. Understanding there is a hierarchical structure in the construction of knowledge will assist us in designing our classroom instruction. The " Structure of Knowledge " diagram provides an instructional framework to apply to curriculum.