Project-based learning is a terrific way to link your curriculum with real world events and applications of concepts that your students are learning. There are two pages in this section to help you and your learners:
Project Based Learning (Page 1 of 2): An essay
Projects on the Web (Page 2 of 2): Our list of project sites
Projects help students personalize their learning and are ideal for gaining key knowledge and understanding of content and answering the question: Where am I ever going to use this?" Among the greatest benefits of project-based learning (PBL) are gains in students' critical-thinking skills and development of their interpersonal and intrapersonal skills. PBL is also an ideal way to help learners gain speaking and presentation skills indentified in the Common Core Standards. PBL in mathematics, particularly when completed in teams, helps learners "model with mathematics" as they "apply the mathematics they know to solve problems arising in everyday life, society, and the workplace," "use tools strategically," and "construct viable arguments and critique the reasoning of others," as noted in the Common Core Standards (2010) for Mathematical Practice.
However, as Bryan Goodwin (2010) found in reviewing the literature, a major shortcoming in many student projects is that educators tend to assign projects just for the sake of doing them. "Educators can avoid this phenomenon and realize the potential of projects to promote students' critical-thinking by framing projects around a driving question" (p. 81). This challenging question or problem is just one of the essential elements of meaningful projects, according to John Larmer and John R. Mergendoller (2010) of the Buck Institute for Education (BIE), who in 2014 updated their PBL model to what they called a "gold standard" (See their 2015 blog post on why). Every good project needs significant content, meaning tied to standards so that students gain key knowledge and understanding. Students also need to perceive the work as meaningful to them. A clear connection to an entry event adding this meaning might be via almost anything: "a video, a lively discussion, a guest speaker, a field trip, or a piece of mock correspondence that sets up a scenario" (2010, p. 35). Students need a voice and choice in fulfilling project requirements, keeping in mind that limited choices be considered and that "teachers should design projects with the extent of student choice that fits their own style and students (p. 36). Projects should give students opportunities to build 21st century skills (or success skills) and to use technology that will be useful to them in life and the workplace. Projects should enable learners to conduct real inquiry. This has to do with authenticity or how real-world a project is. With "real inquiry comes innovation--a new answer to a driving question, a new product, or an individually generated solution to a problem" (2010, p. 37). Learners should receive feedback to use in revision, as learning that real-world work often involves revision. Outside experts or mentors can also provide input (Larmer & Mergendoller, 2010). Teachers should not be the only ones to provide this feedback. Peer-editing sessions with the aid of appropriate rubrics or checklists can be useful for students to present their rough drafts to each other (Pahomov, 2014).
Students should publicly present their work, as they will be more motivated to produce a quality product when knowing a real audience will view it (Larmer & Mergendoller, 2015, 2010). As Larissa Pahomov (2014) pointed out, "Why should students put so much effort into a product that is only going to be viewed by one person?" -- the teacher (p. 92). Although there might be live presentations to share projects, "they should also be designed to stand on their own, after the formal presentation has ended" (p. 87). A venue for presenting completed projects might be "as simple a setting up a gallery in the hallway or a landing page for links to projects" (p. 92). A blog or wiki is ideal for posting online presentations, which elevates projects beyond the school walls. "What makes this process meaningful, however, is when students have been creating their project with a particular audience in mind and then they target that audience once the content has been posted" (p. 98). Projects might be entered into contests and competitions, or presented to real-world professionals for feedback.
If projects involve teamwork, educators will need to emphasize commitment to the team as an essential component for success of group work. Larmer (2014) noted that this may not automatically emerge, but a "sense of responsibility to their peers can be one of the most powerful motivating factors for students working on a project in teams" (p. 45). To help support teamwork, teachers might consider "constructing list of norms or a rubric with students; having students write contracts for how they will work together; providing them with tools, such as task planners and online collaboration platforms; and teaching them how to resolve conflicts and make decisions. During a project, have team members frequently check in with one another—and the teacher—to be sure things are going smoothly" (p. 45).
Finally, projects should include the element of reflection. Per Larmer and Mergendoller (2015), "students should reflect on what they’re learning, how they’re learning, and what they have accomplished in a project" (Two new elements section). Self-Reflection on Project Work from BIE is one such template.
With the above being said, Volker Ulm (2011) offered teachers some sound advice regarding math project-based learning:
Enriching classroom teaching with projects is certainly the most challenging, but at the same time the most beneficial form of independent learning. It is challenging because it requires high-level skills on the part of the students, e.g. skills in applying methods, self-management, and social competence. So project-based learning should never degenerate into a teacher-centered training course where ultimately the teacher still does all the planning, structuring and organizing, prepares and procures all the materials, or even produces and presents the results. (p. 44)
What do we mean by building 21st century skills?
Numerous documents have referred to the need for this or that activity to build 21st century skills needed for career and college readiness. However, what does that mean stated in terms that everyone can easily remember?
The National Research Council (2012, p. 2) suggested three broad domains as a way in which to organize competencies--such skills and abilities: cognitive, intrapersonal, and interpersonal. They are intertwined based on human development and learning, however.
While one might elaborate on specific skills or competencies within each, those domains are easy to remember. Briefly:
What might not surprise readers, however, is that "Precise definitions of the many terms used for “21st century skills” are not possible at this time, in part because there is little research to support such definitions" (p. 2).
Thus, projects should result from students' attempts to answer essential questions. They can take many forms: products, presentations, performances. They might fit any of three structures: interpersonal, information sharing, or problem-solving. When selecting an existing project, or creating one of your own, consider the following. In terms of math projects:
Is the project devoted only to mathematics (or a single subject area), or is there a link to other curricular areas?
Is the project tied to standards for the curricular areas addressed, such as those from the National Council of Teachers of Mathematics, the Common Core Standards, or the National Education Technology Standards?
Does the project come with classroom instructional materials (e.g., teacher resources, student activities, rubrics and assessment tools)?
Can all students in your class participate? Projects should not be reserved for your talented and gifted students, as all students should be able to benefit.
What is the total time for project completion?
Is the project collaborative in nature? A collaborative project, particularly involving students outside your own school setting, will take more time and monitoring to help students learn how to be a part of a team and communicate appropriately with others.
How will students benefit both academically and personally from their involvement in the project? Consider that when students interact with other students and experts across the country or internationally, they get a broader feel for diversity. Their participation in an actual real world activity might encourage them to do their best work, and see the relevance of mathematics in their daily lives. If students have input into project selection, and like the topic, they will tend to become more involved and excited about their learning.
Is there a cost involved to participate?
Don't forget the home-school connection!
If you decide to engage in a project of significance to your learners, parents would appreciate knowing about it. Consider sending parents a letter to explain PBL, the nature of the project, how learners will be assessed, and what they can do to help. BIE has a Template for a Letter to Parents.
Help learners reach project milestones by recording oral feedback along the way using tools such as Screencast-O-Matic.
Adding to the above, educators need to decide on what they mean by project-based learning (PBL) and which model is needed. In his book World Class Learners: Educating Creative and Entrepreneurial Students, Yong Zhao noted three forms of PBL, according to J. Robinson (2013). In an academic model, all elements are controlled by the teacher. The goal is to teach prescribed content and skills. Products are not meant for authentic consumption. In a mixed model of PBL, the artifacts students create (sometimes for consumers outside of school) and prescribed content are valued. Still under the control of the teacher, students have a degree of freedom within the project. The focus is more on learning real world skills, rather than transmitting knowledge. Both of these latter models are about teaching the curriculum. The third PBL method is a product-oriented Entrepreneurial Model, which is the most valued for developing 21st century skills and an entrepreneurial mindset. Students who work individually or collaboratively are more in control of products, which must meet an authentic consumer need. Students develop a business plan and the teacher takes on the role of consultant.
These resources are for those who need to know more before engaging in projects and inquiry-based learning.
Buck Institute for Education has an overview of project based learning (PBL), plus numerous resources for conducting PBL, which are "organized into three broad categories: things to read, to watch, or to interact with." A PBL handbook and other PBL books are available to purchase. For example, John Larmer, John Mergendoller, and Suzie Boss of the BIE authored Setting the Standard for Project Based Learning (2015) in which they provide the step-by-step method to create, implement, and assess PBL in classrooms, systemwide, and in informal settings. Sample projects in various grade levels and subject areas are included. ASCD published this book.
Designing Learning is a section within the Galileo Educational Network Association, which includes a series on the nature of inquiry-based learning. Learn about what inquiry is all about, choosing a topic, essential questions, inquiry and assessment, and then go to the section for Classroom Examples of projects for elementary, middle, and secondary students and a rubric for assessing inquiry projects.
Intel Education has a three-hour hands-on free workshop on project-based learning, which has a guided self-study module as an accompaniment. Learn about this method as you examine It's a Wild Ride, an extended interdisciplinary project that studies roller coaster design in science, mathematics, and language arts classrooms. It's a Wild Ride provides a case study experience. Intel then offers a series of K-12 projects categorized in mathematics, science, language arts, social studies, and interdisciplinary.
PBL-Online has all the resources you need to design and manage high quality projects for middle and high school students. You can learn how to design your own project, what project based learning is all about, search for projects developed by others or contribute your own, review research on PBL, and access other web resources on the topic. PBL-Online was created under the leadership of the Buck Institute of Education, with major contributions from the George Lucas Foundation and others.
HOT: Project-Based Learning: A Resource for Instructors and Program Coordinators from the National Academy Foundation and Pearson Foundation provides comprehensive coverage on this topic. You'll find a definition of PBL, questions to consider for when to use PBL, conditions and research supporting PBL, PBL examples and links to resources and training. Commentary on PBL includes: "The best projects skillfully weave together opportunities for students to engage in classroom activities (Level 1) that address content standards (Level 2) while encouraging them to develop habits of mind (Level 3) and the ability to take responsibility of their own learning (Level 4)" (p. 12). The 6 A's of PBL are addressed: Authenticity, Academic Rigor, Adult Connections, Active Exploration, Applied Learning, and Assessment Practices. Finally, within Project Delivery you'll learn about the scaffolding that teachers must consider when implementing sophisticated projects in their classrooms.
Project-Based Learning Professional Development Guide at Edutopia.org from the George Lucas Educational Foundation can be used as a two- to three- hour learning module, or expanded to day-long workshops. Find out what project based learning is, why it is important, how it works, and get some supporting resources. Also see Edutopia's Project-Based Learning section.
Project Based Learning explained by Common Craft is a short video with an easy explanation of PBL, courtesy of BIE.org.
Project Approach to Teaching and Learning in school addresses the foundation theory for using projects, strategic planning, and project development structure. This is an award winning site by Sylvia Chard of the University of Alberta, Canada. You might also be interested in the interview of Dr. Chard addressing project-based learning, which is available from Edutopia of the George Lucas Educational Foundation.
In Using the Internet to Promote Inquiry-Based Learning, authors D. Jakes, M. Pennington, and H. Knodle describe a structured approach to inquiry-based learning that uses the World Wide Web. They address an intuitive 8-step process that begins with an essential question and ends with a knowledge product produced by students, typically completed in a cooperative setting. They discuss the skills that students and teachers require to make inquiry-based learning and the Internet a successful endeavor; and the components of a Project Page, which include the scenario, task, resources, product students will build, and assessment.
What is Project-Based Learning? is a video tutorial from New Tech Network, which provides an in-depth overview of this method. Related videos are also available.
Muschla, J., Muschla, G., & Muschla, E. (2012). Teaching the Common Core Math Standards with Hands-On Activities, Grades 6-8 has over 100 activities correlated to the CCSS math standards.
Yetkiner, Z. E., Anderoglu, H., & Capraro, R. M. (2008). Research summary: Project-based learning in middle grades mathematics. Available from http://bie.org/object/document/pbl_in_middle_grades_mathematics
Volker Ulm (2011) had several ideas for potential math projects. Selection, of course, depends on the skills of your learners for meaningful project-based learning:
Use the following to get other ideas:
Applied Math Science Fair Projects posted at Education.com included illustrative images of projects and is searchable by grade level. You can then download the project.
Buck Institute of Education (BIE) Project Search, a database of projects in multiple content areas. Per BIE: "The projects you will find here have been curated by BIE and were gathered from online project libraries. These projects are meant to inspire your own ideas or may be adapted to fit the needs of your classroom" (Project Search webpage). There are over 160 math projects, which can also be identified by Common Core Math standards.
Hands-On Math Projects With Real-Life Applications: Grades 6-12 by Judith Muschla contains 60 math projects.
Math Project Ideas: Examples of Project-Based Learning, posted by EdTecher Heidi Reina at HubPages, includes an extensive collection of engaging projects for all grade levels from general math to money and finance, algebra, geometry, pre-calculus, calculus. Many have cross-curricular connections.
There are multiple tools in a variety of categories on the Web for project-based learning activities. Certainly, to help them get started, learners will benefit from a list of resources that you anticipate they will need. Deal (2009) included specific examples within categories such as collaboration suites, course management systems, dedicated project management tools, wikis, web-conferencing tools for real-time communications, collaborative concept mapping tools, presentation and slide sharing tools, online collaborative writing tools, and task management tools. The choice would depend on the objectives of the assignment and whether or not the project is to be completed by individual learners or collaboratively within groups.
If you are planning your own project, CT4ME provides an essay on project development phases and tips at our page Technology Integration: Multimedia in Projects: About Multimedia and Project Development. In addition consider:
Bubble.us is a free web application that allows groups to brainstorm online and create mind maps. You can embed the map in a blog or website, or save the mind map as an image. This is a great way to start projects.
ClockingIT provides free space for project management, collaboration, and time and task management.
Glogster EDU provides a secure space online for educators and their learners to have an outlet for their creative posters or glogs. A glog is a virtual online poster that incorporates multimedia. In addition to text, learners can embed images, audio, video and hyperlinks. Posters are great projects for students to illustrate what they have learned. You'll find many examples of student glogs for math at this site to give you ideas. The site is free.
Wikispaces for Education enables free, private, and secure collaborative group work in classrooms.
Tools for Creativity and Innovation
Little Bits--build STEM and STEAM skills with electronic building blocks.
Makey Makey--an invention kit. "Turn everyday objects into touchpads and combine them with the internet."
Scratch--a programming language for kids (ages 8 to 16) to create stories, games, and animations, which was developed by MIT. It is free. Note: Members of the ScratchEd research team at the Harvard Graduate School of Education developed a free guide, Creative Computing: An Introductory Computing Curriculum for Scratch. The version for educators includes plans, activities, and strategies for using Scratch and the workbook version for learners has activities and reflection question templates.
Tickle--free app in iTunes for this drag-and-drop programming language, that can also teach kids to code. "Learn to program Arduino, drones, robots, connected toys, and smart phone devices, all wirelessly." Create games and interactive stories with a library of animated characters and sounds.
TinyTap is free and allows you to play and create interactive games, lessons, and quizzes. Download it for iPad and Android devices, or for play on your desktop.
Tynker--Kids 6-12+ learn to code with this programming language. Build an app (e.g., games), control a drone, program a robot. Self-paced beginner, intermediate and advanced courses are also available (not free), which include core concepts: "coding fundamentals, animation, conditional logic, patterns, coordinates, angles and direction, variables, functions, physics, gravity." School pricing is also available.
Tools for Group Projects
Group Project Tools from Carnegie Mellon include team contracts, team roles, self assessments, peer assessments, and group assessments.
Consider a WebQuest, an inquiry-oriented activity in which most or all of the information used by learners is drawn from the Web.
Math WebQuests help students to develop reasoning and critical thinking skills, as advocated in the process standards of the NCTM Goals 2000.
According to Tom March (2003) in The Learning Power of WebQuests, there are six elements of a real WebQuest:
Some activities may be designed to use Internet resources to produce a product, but can't be classified as WebQuests. These non-Webquest activities are those that enable learners to gather information that can go from a browser directly to a product without altering or involving students' understanding, and reflection on their own metacognitive processes. To assess the real value of a WebQuest, ask "Is this WebQuest real, rich, and relevant?"
Creating a WebQuest: It's Easier Than You Think! from Education World
Tom March/ozline.com Tom March, one of the original developers of WebQuests, has his own site with his blog, numerous articles and examples of WebQuests.
WebQuest.org training materials. This site is maintained by Bernie Dodge, who with Tom March developed this model at San Diego State University in 1995.
WebQuest 101 from TeachersFirst.com will help you get started.
WebQuest Template from Cape Breton-Victoria, Regional School Board, Education Centre. Includes directions within the sections: Introduction, Task, Process, Evaluation (Rubric design), and Conclusion.
HOT: Zunal WebQuest Maker is free web-based software for creating WebQuests in a short time without writing any HTML codes." The site includes 100 templates. You can include unlimited files and pictures, embed video, add a pre-test and post-test, create a quiz, add a table/rubric, and more. A bonus is shared webquests in multiple content areas to use or give ideas for designing your own webquest.
Assessment plays a key role in PBL. If the nature of your project is collaborative, Ashley Deal (2009) noted that three areas can be assessed. "Instructors can evaluate the process students use in approaching a given problem and finding solutions; they can assess the final product or end result of the project; or they can evaluate the individual student’s learning outcomes" (p. 4). However, a single approach to assessment of group work poses a problem, as "a satisfactory final product does not necessarily indicate that students approached the problem according to the preferred process. Similarly, even using the correct process to arrive at a satisfactory final product does not indicate that individual students grasped relevant concepts" (p. 4). So, more than one level of assessment is recommended.
Susan Brookhart (2013a) provided valuable advice in Grading and Group Work: How do I assess individual learning when students work together? noting the struggle that many teachers have between the need for learners to engage in cooperative learning and the need to provide individual grades. "Many teachers use group work in the general sense, assigning students to collaborate in groups that result in one undifferentiated product. But true cooperative learning requires individual accountability. No matter what kind of cooperative learning or group work you employ, it is important not to give group grades" (Does group work have to mean group grades? section). She further explains what's wrong with group grades and grading cooperative work versus group work. She answers: How can I assess learning and process skills? How can I assess and grade individual students' achievement? How can I adapt group projects to enable individual grading?
Mary Allen (2003) noted that rubrics can be used to evaluate the quality of “virtually any product or behavior, such as essays, research reports, portfolios, works of art, recitals, oral presentations, performances, and group activities. Judgments can be self-assessments by students; or judgments can be made by others, such as faculty, other students, or field-work supervisors. Rubrics can be used to provide formative feedback to students, to grade students, and/or to assess programs” (para. 1). Essential features include "evaluative criteria, quality definitions, and a scoring strategy" (Popham, 1997, Rudiments section).
There are three common types of rubrics: analytic, developmental, and holistic. An analytic rubric is often designed in column format with multiple performance criteria listed in the first column and the quality indicator levels, which vary on a continuum, listed in the first row across remaining columns. Each quality level might contain one point value or a range of points that can be assigned. Each criteria might also be accompanied by a comment space for additional feedback on the element. Cells within the rubric might be blank or contain elaborations of each performance criteria for the specific quality level. The developmental rubric is a subset of the analytic rubric and is based on a theory of development. It is useful when the goal is to determine a level of development (e.g., of a skill, value, ability, etc.) rather than the quality of a final product. When using a holistic rubric, a single score (e.g., from 1-4 points on a scale, or 1-6 points on a scale) is assigned based on the overall quality of work in consideration of all criteria. The evaluator matches the student's work to a single description of it. For example, ratings might be above average, sufficient, developing, needs improvement with elaborations on criteria for selecting each (DePaul University, Teaching Commons, Types of Rubrics, n.d.).
Students benefit from access to a rubric while they are working on a project. Knowing the criteria on which they will be assessed is a plus to help them develop, evaluate and tweak their own work prior to submitting it.
Rubrics are valued because they enable objectivity in assessment. However, when educators design their own rubric or select an existing one, they need to consider the potential increase in cognitive load it might present when actually using it. The greater the number of criteria and granularity within quality levels, the longer it takes to apply the rubric for one project. Ranges of point values within a quality level also add a degree of subjectivity to using the rubric. Then consider the number of projects to be assessed. Per Popham (1997), if a rubric is to be instructionally useful, its length makes a difference. Teachers will avoid using overly detailed rubrics, particularly for routine tasks.
Rubrics that include quality indicators requiring evaluators to count errors before assigning a point-value also increase cognitive load and require a level of granularity in grading that not all educators are willing or even able to do. Of course, this would depend on the objective of the assignment or project. Scores derived from a rubric might also be affected by terms within quality levels that might be understood differently among evaluators who use the rubric. Quality elements that differ by only a word or two are particularly challenging to use. For example, consider quality levels such as: … is not stated, … is stated, …. is clearly stated, … is clearly and concisely stated.
Rubrics might also pose a challenge if the numeric score obtained from it will be translated into letter-grades.
Learn how to create rubrics to help measure quality and student performance on projects with these additional resources. Or, use an existing rubric.
How to Create and Use Rubrics for Formative Assessment and Grading by Susan Brookhart (2013b) includes two sections. In Section 1, Brookhart lays out the basics of rubrics, explains their importance, discusses common misconceptions, and how to write or select effective rubrics. She identifies various kinds of rubrics and their essential components. In Section 2, she explains how to use rubrics for formative assessment and grading.
The following existing rubrics from various locations are of value in projects:
Teach-nology's Rubric Makers allow you to make grading rubrics by filling out a simple form. The materials are made instantly and can be printed directly from your computer. You can also customize your own rubric. Rubrics of interest include homework, class participation, math projects, oral presentations, WebQuests, team work, writing, research reports, and reading.
The Scoring Guide for Student Projects is also an excellent resource developed at the North Central Regional Educational Laboratory (MCREL). This Web tool helps teachers evaluate student products that are created with technology. It focuses on the student’s content knowledge and effective technology use.
Student Checklist and Judges' Rubric from Multimedia Mania, which is an annual award program sponsored by ISTE HyperSig.
Often students have difficulty seeing the math that is around them everyday or they lack motivation for studying mathematics because they don't see the connection of it to everyday life. One way to help them is to show them the beauty of math found in nature. These can also provide great topics for projects.
View The Great Math Mystery (2015, about 50 minutes) from PBS.org. Per the program description:
"Join NOVA on a mathematical mystery tour—a provocative exploration of math's astonishing power across the centuries. We discover math's signature in the swirl of a nautilus shell, the whirlpool of a galaxy, and the spiral in the center of a sunflower. Math was essential to everything from the first wireless radio transmissions to the prediction and discovery of the Higgs boson and the successful landing of rovers on Mars. Astrophysicist and writer Mario Livio, along with a colorful cast of mathematicians, physicists, and engineers, follow math from Pythagoras to Einstein and beyond. It all leads to the ultimate riddle: Is math a human invention or the discovery of the language of the universe?"
The image of the Fibonacci series shown is from Flawlesslayout.com within a post called Flawless Layout Logo, the Golden section proportion and the Fibonacci series formula where you'll find more images of the golden section in nature.
View the short video, Nature by Numbers, created by Cristóbal Vila, which was inspired by numbers (e.g., Fibonacci and Golden Ratio), geometry, and nature. This is truly beautiful and eye-opening. Then visit:
Dr. Ron Knott's web site contains more on Fibonacci numbers and the Golden Section in nature. You will also find activities to do with your learners. Note: The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ... (add the last two to get the next).
Who was Fibonacci? Fibonacci or Leonardo of Pisa lived from 1170-1250. Read a "brief biographical sketch of Fibonacci, his life, times and mathematical achievements."
Golden Ratio is about 1.618 and is given the Greek symbol "phi." Math is Fun provides learners with an easy to understand explanation of the Golden Ratio, the Golden Section, and Fibonacci numbers.
Learn more about the golden ratio at GoldenNumber.net and see where it is used in everyday applications.
Omnicalculator.com has multiple calculators. Use the Golden Ratio Calculator to find two lengths that would compute to give the golden ratio. Use the Golden Rectangle Calculator to input one side of the rectangle and automatically see the length of the adjacent side and area of the rectangle that forms a golden rectangle, one whose sides are in the golden ratio.
The Exploratorium in San Francisco features the Geometry Playground, which will change how you view geometry in nature. It contains three sections: The Geometry of Seeing with a photo exhibition of the invisible geometry of light; The Geometry of Moving on the arcs, angles, and shapes created when people and things are in motion; and the Geometry of Fitting Things Together. Each includes hands-on activities for grades K-8. The Geometry Garden features curiosities found in nature from crystals to seashells to sculptures. The site was made possible by the National Science Foundation and the Gordon and Betty Moore Foundation.
Imagery: A Key to Understanding Math by Holly Korbey at Mindshift (2013, October 31) is a quick eye-opener to helping students find the beauty in math and overcome math anxiety. It leads one to a valuable book by Michael Schneider (1994) on the topic: A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science. Schneider focuses on the numbers 1-10 and provides a fascinating historical discussion of their relationship to geometry found in nature, art, science, architecture, and so on over time. Using a compass, straightedge, and pencil he demonstrates how you can introduce this beauty to your learners. For example, "ancient philosophers conceived that the Monad [The First] ... creates all subsequent numbers (11111111 x 11111111 = 12345678987654321)" (p. 3). And the circle "represents a wonderful first glymph into nature's alphabet" (p. 4).
Learn about fractals.
The PBS television series , NOVA, has a one-hour program about fractals called Hunting the Hidden Dimension. There are five chapters, which can be viewed separately. Learn about fractals in nature, including those in the human body. There are links and books, a teacher's guide, and an email newsletter for learning more. You can also design your own fractal using NOVA's interactive generator.
Fractal Keys: A Pattern Paradise from the MathScience Innovation Center is a must see site for an adventure into learning all about fractals and where you can find them--everywhere, including music! Learners in grades 5-12 will benefit. Begin by clicking on the Welcome Center and viewing What are fractals?
Explore fractals with this unit by Cynthia Lanius, which is appropriate for elementary and middle schools learners and even adults. You will learn about the importance of fractals, properties of fractals, create a few, and get a series of links to other sites on the Web that address this topic.
NCTM Illuminations Fractal Tool applet, designed mainly for middle school learners, is a virtual manipulative to see how various shapes are fractals. Users can play with shapes that grow, shrink, and change over several stages and explore self-similarity and patterns in fractal measurements.
Amazing Seattle Fractals! will benefit high school learners and above. The developer provides tutorials to learn more about fractals and how to create fractal art. Users can download free fractal software programs and view some fractal art galleries.
If you wish to generate 2D or 3D fractals and those with animation, consider ChaosPro, which is freeware for MS Windows. It includes tutorials and a gallery of examples.
Pomegranate Software offers the program called Fractals, which is designed for use on iPad, iPhone, and iPod Touch. It “renders as you move and pinch to explore Mandelbrot and Julia set fractals in real-time.” See the exciting displays and learn more about fractals at this site.
Allen, M. (2003). Student learning outcomes and assessment. Retrieved from http://www.calstate.edu/itl/resources/assessment/rubrics.shtml
Brookhart, S. (2013a). Grading and group work: How do I assess individual learning when students work together? Alexandria, VA: ASCD.
Brookhart, S. (2013b). How to create and use rubrics for formative assessment and grading. Alexandria, VA: ASCD.
Common Core State Standards. (2010). Standards for Mathematical Practice. Retrieved from http://www.corestandards.org/Math/Practice
Deal, A. (2009). Collaboration tools. Carnegie Mellon University. Retrieved from http://www.cmu.edu/teaching/technology/whitepapers/CollaborationTools_Jan09.pdf
DePaul University Teaching Commons (n.d.) Types of rubrics. Retrieved from http://teachingcommons.depaul.edu/Feedback_Grading/rubrics/types-of-rubrics.html
Goodwin, B. (2010). Choice is a matter of degree. Educational Leadership, 68(1), 80-81. Retrieved from http://www.ascd.org/publications/educational-leadership/sept10/vol68/num01/Choice-Is-a-Matter-of-Degree.aspx
Larmer, J. (2014). Boosting the power of projects. Educational Leadership, 72(1), 43-46.
Larmer, J., & Mergendoller, J. R. (2015, May 11). Why we changed our model of the "8 essential elements of PBL" [Web log post]. Retrieved from http://bie.org/blog/why_we_changed_our_model_of_the_8_essential_elements_of_pbl
Larmer, J., & Mergendoller, J. R. (2010). Seven essentials for project-based learning. Educational Leadership, 68(1), 34-37. Retrieved from http://www.ascd.org/publications/educational_leadership/sept10/vol68/num01/Seven_Essentials_for_Project-Based_Learning.aspx See the 2012 update: 8 Essentials for Project-Based Learning at https://bie.org/tag/8+essential+elements
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Learn about the technical side of creating multimedia projects, including working with images and video.
See CT4ME Technology Integration