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Puzzles, Activities and Lesson Plans - Mathigon

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These activities and lesson plans are designed to engage higher order thinking using the interactive and visual tools Mathigon Provides in their poly pad (see my draw on high-end digital Math applications and tools

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Puzzles, Activities and Lesson Plans - Mathigon

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Explore our free library of tasks, lesson ideas and puzzles using Polypad and virtual manipulatives.

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Mathigon courses offer a free, visual, interactive and engaging way of learning content.  Teacher accounts are free and there are safe ways to set up student accounts.  Courses can be selected to suit content and shared via class link with students, allowing them to digitally engage in content at their own pace.

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Mathigon's innovative courses cover everything from fractions and trigonometry to graph theory, cryptography, prime numbers and fractals.

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This higher order thinking task from nrich could be modified digitally using 3D revolutions in Geogebra or enhanced with 3D modelling in Tinkercad for deeper understanding and spatial reasoning

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What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

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Teacher Desmos is a free platform where teachers can share their custom math activities, adapt those of others or use Desmos created activities to engage students in deeper visual and interactive learning in mathematics.  Activities are allocated to custom classes and shared via code or link.  This is highly recommended for problem-solving and modelling needs - particularly where graphing and mathematical functions are involved.  Students work shows on the teacher dashboard in real-time, and student work can be shared with each other with anonymity, adding a safe element for number talks to accomodate students with math anxiety.  Automatic marking and reporting on teacher designed activities is also an option.  The sky is the limit.

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Visual Math Improves Math Performance - YouCubed

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This article discusses how new brain research shows that integrating visuals in mathematics can even help develop numeracy skills, recommending the use of visuals at all levels and not just for understanding abstract concepts.

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Mathematics educators have long known that engaging students in visual representations of mathematics is extremely helpful for their learning. When youcubed offered " How Close to 100 " as an activity for learning math facts with visual representations, teachers across the world were thrilled and responded with thousands of tweets showing students learning by playing the game.

Research shows the importance of visual thinking for mathematicians.

When researchers have considered the brains of mathematicians, and compared them with similarly high achieving academics who worked in non-mathematical fields, they find something fascinating (Amalric & Dehaene, 2016). We might assume that when people are thinking numerically, they are thinking "with language" and that high-level mathematical reasoning would draw from language processing areas of the brain. What the researchers found was that the brain activity that separated mathematicians from other academics came from visual areas of the brain - and this was true whatever the mathematical content. It was not only subjects like geometry or topology that caused activity in the visual brain areas, but algebra and calculations, with minimal use of any language areas. This led the researchers to offer the possible explanation that the mathematicians' achievement in math came from early childhood experiences with "number and shape."

Visual mathematics is an important part of mathematics for its own sake and new brain research tells us that visual mathematics even helps students learn numerical mathematics.

In a ground breaking new study Joonkoo Park & Elizabeth Brannon (2013), found that the most powerful learning occurs when we use different areas of the brain. When students work with symbols, such as numbers, they are using a different area of the brain than when they work with visual and spatial information, such as an array of dots. The researchers found that mathematics learning and performance was optimized when the two areas of the brain were communicating (Park & Brannon, 2013). (for math questions that encourage this use of visual and symbolic representations see our Tasks). Additionally, they found that training students through visual representations improved students' math performance significantly, even on numerical math, and that the visual training helped students more than numerical training.

What is Visual Mathematics?

At youcubed we provide many different mathematics tasks that engage students in visual mathematics. Through decades of work with students, teachers, high-tech companies, politicians and others we have learned that people are excited and inspired when they see mathematics as pictures, not just symbols. For example, consider how you might solve 18 x 5, and ask others how they would solve 18 x 5. Here are some different visual solutions of this problem.

Each of these visuals highlights the mathematics inside the problem and helps students develop understanding of multiplication. Pictures help students see mathematical ideas, which aids understanding. Visual mathematics also facilitates higher-level thinking, enables communication and helps people see the creativity in mathematics.

Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, openness, visualization, and flexibility, the mathematics comes alive.

Teachers can create such mathematical excitement in classrooms with any mathematics question by asking students for the different ways they see and can solve the problems and by encouraging discussion of different ways of seeing problems.

For an example of visualizing algebra see here.

When we don't ask students to think visually, we miss an incredible opportunity to increase students' understanding and to enable important brain crossing.

This extract contains excerpts from Jo Boaler's book Mathematical Mindsets, and her new, forthcoming book: Mathematical Diversity.

References:

Amalric, M., & Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematicians. Proceedings of the National Academy of Sciences, 113(18), 4909-4917

Park, J., & Brannon, E. (2013). Training the approximate number system improves math proficiency. Association for Psychological Science, 1-7.

Mathematics educators have long known that engaging students in visual representations of mathematics is extremely helpful for their learning. When youcubed offered " How Close to 100 " as an activity for learning math facts with visual representations, teachers across the world were thrilled and responded with thousands of tweets showing students learning by playing the game.

Research shows the importance of visual thinking for mathematicians.

When researchers have considered the brains of mathematicians, and compared them with similarly high achieving academics who worked in non-mathematical fields, they find something fascinating (Amalric & Dehaene, 2016). We might assume that when people are thinking numerically, they are thinking "with language" and that high-level mathematical reasoning would draw from language processing areas of the brain. What the researchers found was that the brain activity that separated mathematicians from other academics came from visual areas of the brain - and this was true whatever the mathematical content. It was not only subjects like geometry or topology that caused activity in the visual brain areas, but algebra and calculations, with minimal use of any language areas. This led the researchers to offer the possible explanation that the mathematicians' achievement in math came from early childhood experiences with "number and shape."

Visual mathematics is an important part of mathematics for its own sake and new brain research tells us that visual mathematics even helps students learn numerical mathematics.

In a ground breaking new study Joonkoo Park & Elizabeth Brannon (2013), found that the most powerful learning occurs when we use different areas of the brain. When students work with symbols, such as numbers, they are using a different area of the brain than when they work with visual and spatial information, such as an array of dots. The researchers found that mathematics learning and performance was optimized when the two areas of the brain were communicating (Park & Brannon, 2013). (for math questions that encourage this use of visual and symbolic representations see our Tasks). Additionally, they found that training students through visual representations improved students' math performance significantly, even on numerical math, and that the visual training helped students more than numerical training.

What is Visual Mathematics?

At youcubed we provide many different mathematics tasks that engage students in visual mathematics. Through decades of work with students, teachers, high-tech companies, politicians and others we have learned that people are excited and inspired when they see mathematics as pictures, not just symbols. For example, consider how you might solve 18 x 5, and ask others how they would solve 18 x 5. Here are some different visual solutions of this problem.

Each of these visuals highlights the mathematics inside the problem and helps students develop understanding of multiplication. Pictures help students see mathematical ideas, which aids understanding. Visual mathematics also facilitates higher-level thinking, enables communication and helps people see the creativity in mathematics.

Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, openness, visualization, and flexibility, the mathematics comes alive.

Teachers can create such mathematical excitement in classrooms with any mathematics question by asking students for the different ways they see and can solve the problems and by encouraging discussion of different ways of seeing problems.

For an example of visualizing algebra see here.

When we don't ask students to think visually, we miss an incredible opportunity to increase students' understanding and to enable important brain crossing.

This extract contains excerpts from Jo Boaler's book Mathematical Mindsets, and her new, forthcoming book: Mathematical Diversity.

References:

Amalric, M., & Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematicians. Proceedings of the National Academy of Sciences, 113(18), 4909-4917

Park, J., & Brannon, E. (2013). Training the approximate number system improves math proficiency. Association for Psychological Science, 1-7.

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Visual Math Improves Math Performance - YouCubed

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Mathematics educators have long known that engaging students in visual representations of mathematics is extremely helpful for their learning. When youcubed offered "How Close to 100" as an activity for learning math facts with visual representations, teachers across the world were thrilled and responded with thousands of tweets showing students learning by playing the game.

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NRICH - Mathematics Resources for Teachers, Parents and Students to Enrich Learning

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This site by Cambridge University is full of ideas to transfer content knowledge to real-life context and challenge those who need it. It ha rich math problems for all ages categorised by age, content and level of difficulty.

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search

Enriching mathematics for all learners

A free-to-use collection of mathematics activities, lessons and problems designed to nurture curious, resourceful and confident learners of mathematics. Produced by The University of Cambridge.

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The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.

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Login required | ICT in Everyday Learning

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I found this site on Scootle.  It has some great ideas for modifying mathematics learning in the Australian Curriculum with digital technology.

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This site is designed as part of the ICT in Everyday Learning: A Toolkit for Teachers project, and is available to all Australian education jurisdictions.

Users are required to login to access content. To view the site please login to Scootle or your jurisdiction's content portal. For further information on accessing the ICT in Everyday Learning please view Access information.

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Login required | ICT in Everyday Learning

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This site is designed as part of the ICT in Everyday Learning: A Toolkit for Teachers project, and is available to all Australian education jurisdictions. Users are required to login to access content. To view the site please login to Scootle or your jurisdiction's content portal.

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This site is always worth investigating for adaptable lessons, and their methods used to formatively assess, providing feedback as reflective questions rather than a score, is designe to further encourage deeper thinking.  Well worth a look!

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Classroom Challenges are lessons that support teachers in formative assessment. There are 100 lessons in total, 20 at each grade from 6 to 8 and 40 for 'Career and College Readiness' at High School Grades 9 and above. Some lessons are focused on developing math concepts, others on solving non-routine problems.

The Brief Guide for teachers and administrators (PDF) is recommended reading before using these lessons for the first time.

Each lesson is downloadable as an all-in-one PDF to print out, and a supporting PowerPoint presentation.

Finding a Lesson

There are several ways of navigating:

1. Use the menu on the left to browse by grade and topic area, or search for a key phrase.

2. Go to the Standards tab to find lessons linked to a particular content standard or practice.

3. The complete set of lessons is listed below in alphabetic order.

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Classroom Challenges are lessons that support teachers in formative assessment. There are 100 lessons in total, 20 at each grade from 6 to 8 and 40 for 'Career and College Readiness' at High School Grades 9 and above.

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The Eucator s Guide to design Thinking by the D School Stanford

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I scooped this article on tops draw resources here https://padlet.com/kbirt4/resources. It has some great tips on scaffolding students through the design process, complete with graphic organisers explaining each step.  Done by teachers for teachers - check it out!

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Mathematics Archives - Annenberg Learner

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Although designed for the American curriculum, this website is one of my go-tos for including different higher order thinking perspectives.  Free lesson plans and resources can be adapted for the Australian curriculum.  There are often teach-the-teacher video perspectives that give great insight into the value of activities. Not teaching maths? They have other learning areas also, so check it out!

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Mathematics Archives - Annenberg Learner

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This video workshop for middle and high school teachers presents strategies for improving how typical Algebra 1 topics are taught.

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Five Principles of Extraordinary Math Teaching | Dan Finkel | TEDxRainier

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In this inspirational Ted Talk, Dan Finkel shares his 5 principles for inviting thinking into the classroom, including not rushing to give students the answer - letting them engage in creative struggle to deepen curiosity and ability to take a risk.  Importantly for math teachers nervous to venture into student-led higher order thinking, the teacher is not the answer key - learn together!

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Five Principles of Extraordinary Math Teaching | Dan Finkel | TEDxRainier

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Dan Finkel is an inspirational math teacher with a goal to sparking curiosity and higher order thinking, with a focus on student-led seeking of knowledge and the teacher as guide - not the answer key.  His site has some great ideas and rich tasks to build from and make your own.

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Math came naturally to me from a young age, and I excelled without having to work that hard. I was a relentless game player, puzzle solver, and maze drawer as a kid. In elementary school I was accelerated until there was nowhere to put me. I was done with calculus as a freshman in high school, and didn't have any math to do for half of high school.

Even though I was good at math, what I learned in school often felt a little pointless to me. The teachers explained how to do some process ("Here's how to use the quadratic formula.") and then we copied them. Who cared? If that was all I had seen of math, I probably would have left the field to do something more interesting. But I was lucky: in the summer of ninth grade I was accepted into a math camp at Hampshire college. The math I did there, and the opportunity of getting to work with kids like me under the tutelage of real mathematicians changed my life.

That experience inoculated me against all future problems with math. Because I knew what the real thing looked like, because I had done math like a real mathematician and had seen the beauty, I was protected when someone told me that boring busywork was all there was to math. The deeper I went into the subject, the better it got. I graduated from Swarthmore College with a major in mathematics, and taught math for two years at Saint Ann's School in Brooklyn Heights. In 2004 I came to graduate school at the University of Washington, and in 2010 I graduated with my PhD in algebraic geometry. The whole time I've been wondering why kids never get to see the good stuff in math until they're graduate students. Why is a typical math class at school more likely to be a place where students learn to despise math than to love it? Math for Love is my response to this state of affairs. I want to to give kids the opportunity to be mathematicians right now. I want to teach math to groups, to individuals, to parents, adults, and children. I want to give everyone the chance to fall in love with mathematics.

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Team Member - Dan Finkel - Math For Love

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Math came naturally to me from a young age, and I excelled without having to work that hard. I was a relentless game player, puzzle solver, and maze drawer as a kid. In elementary school I was accelerated until there was nowhere to put me.

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Brilliant is an online text with a difference - catered more at the augmentation level, it incorporates interactive visuals for making sense of content concepts.  It is free for teachers, and lessons or activities can be shared to students via class code, allowing teachers to view progress in real-time. It can be used for student-paced lessons, or teachers could use the interactive snippets to demonstrate. Math is not the only learning area focus, so it may be of interest to other areas. I recommend it to supplement teaching and learning.

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Brilliant - Build quantitative skills in math, science, and computer science with hands-on, interactive lessons.

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Polypad - Virtual Manipulatives - Mathigon

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The Mathigon Polypad is a mathematical treasure box of tools with everything you need to engage students in learning.  It of course requires creativity on the teachers part for higher-end digital activities, however the sky is the limit, and the accompanying mathigon lessons are a good start. (See My shelf on Favourite Websites with Rich Math Lesson Plan & Activity Ideas)

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Desmos Graphing Calculator provides an interactive visual tool to explore cause-effect for changing the properties of functions such as equations of a straight line, quadratic, cubic, quartic, exponent and logarithmic equations.  If it can be graphed, this tool is for you.  Teachers can create interactive graphs and share using a class code or link for students to explore in think-pair-shares, or students can problem solve using functions to map and model surveyed profiles such as bmx tracks or shape or inland rivers I problem solving tasks.

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Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.

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Tinkercad | From mind to design in minutes

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This program is free by Autodesk - the makers of AutoCAD and AutoCAD Inventor, is reasonably user friendly and provides an introduction to the world of engineering design.

It can be used in mathematics to develop visual/spatial understanding, exploring 3D shapes, spatial reasoning ability and design skills, where students can come up with a 3D-printable prototype design for a math problem-solving project.

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Tinkercad | From mind to design in minutes

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Tinkercad is a free, easy-to-use app for 3D design, electronics, and coding.

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Geogebra Classic is a higher-end tool for creating and interacting with visual geometric shapes, including 2D and 3D.
Like desmos, teacher accounts allow teachers to create, modify and allocate activities to the class.  It is fiddly in the early stages of learning but worth persevering with.  Our digital natives may need guidance in these early stages to get them to the point where they can engage without frustration, however it is invaluable for visualising trigonometric concepts.

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Welcome to GeoGebra Classic! Learn how to use this integrated bundle of math apps for graphing, geometry, 3D, algebra, CAS, data (spreadsheet) and probability. Choose one of the quick and easy tours in the chapters below and get started with the apps.

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Welcome to GeoGebra Classic! Learn how to use this integrated bundle of math apps for graphing, geometry, 3D, algebra, CAS, data (spreadsheet) and probability. Choose one of the quick and easy tours in the chapters below and get started with the apps.

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This multiplayer site is for developing fluency in arithmetic and can be used from a smart phone.  Teachers can assign students tasks via a class code and students can compete for scores, with statistics results uploaded to the teachers dashboard.

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The easiest most fun way to practice math facts in a classroom! Instantly engaging and perfect for all-class activities. No students' accounts required.

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Math Whiteboard | Collaborative Whiteboard for Math Teaching & Learning

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Introduce math students to math whiteboard for providing instant math solutions and visual interactives where math comes to life in front of your eyes.  It is addictive to use and incorporates error detection and some nifty animation tools. A must to check out!

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Math Whiteboard | Collaborative Whiteboard for Math Teaching & Learning

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Math Whiteboard is a FREE collaborative whiteboard designed specifically for Math teaching and learning. Give it a try today!

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Castle Photogrammetry using Polycam, iPhone 12 Pro and iMovie

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Polycam is an easy to use app for IPhone 12 Pro incorporating LiDAR 3D surveying that students can measure directly from scans, comparing to own manual measurements, import into Tinkercad and other engineering drawing applications for building 3D printable models, or explore modern day surveying application of mathematical content as they learn about surface area, volume and composite area. The video above demonstrates my own experimentation.  More details can be found here: https://poly.cam/

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Castle Photogrammetry using Polycam, iPhone 12 Pro and iMovie

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Create Venn Diagrams with this Online Generator from ClassTools.net

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With this online tool, students can collaborate to create Venn Diagrams.  Using higher order thinking of compare and contrast to classify.

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Create Venn Diagrams with this Online Generator from ClassTools.net

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Create your own Venn Diagrams at ClassTools.net

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Introduce your students to this app to explore quadratic functions in the real-world, using it in conjunction with Desmos to track, model and optimise their own sporting action mathematically

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Video Physics brings automated object tracking and video analysis to iPhone, iPod touch, and iPad. Capture video of an object in motion, then tap to track the object automatically. Video Physics instantly creates trajectory, position, and velocity graphs for the object.

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Creating Collaborative interactive presentations using PowerPoint https://www.microsoft.com/en-us/microsoft-365/powerpoint and Google Slides https://www.google.com/slides/about/

The above demonstrates a collaboration between Power Point and Google Slides, showing how it could be used for whole-class presentations, such as Stem-fair, to a wider audience.  They are similar programs but both offer different functions.  Together, they create beautiful, versatile interactive presentations that students can create together and share to a wider audience.

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Digital Stem fair - a PowerPoint and Google Slides Collaboration

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Kahoot! Is a program for teachers to host and share online competitive tests where students receive immediate feedback. The statistics of results are recorded on the teacher dashboard for instant assessment. Students can play on any device individually and in teams.  They can choose from a library of shared test resources on any content or construct their own.

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Join a game of kahoot here. Kahoot! is a free game-based learning platform that makes it fun to learn – any subject, in any language, on any device, for all ages!

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Quizizz - The world's most engaging learning platform

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Like Kahoot!, teachers can host fun, engaging and competitive online Quizizz for instant feedback and assessment data.  They can choose from a library of shared test resources on any content or construct their own.

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Quizizz | Free Online Quizzes, Lessons, Activities and Homework

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Motivate every student to mastery with easy-to-customize content plus tools for inclusive assessment, instruction, and practice. Engage students with Quizizz today!

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What is a Google Site? - Google Sites Tutorial

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Google sites is a webspace that provides templates to make it simple for anyone to build a quality Website. In class, it could be used for students to express their ideas individually or collaboratively at low risk.

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What is a Google Site? - Google Sites Tutorial

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Google site tutorial Tutorial for building a Google site. Create and build a Google site. Includes template tips, video tutorials, additional Google resources and template examples.

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This site contains visual interactive activity ideas for exploring changes of volume with linear dimensions using proportional reasoning and compare/contrast

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Interactivate: Surface Area and Volume

Surface Area and Volume: Manipulate dimensions of polyhedra, and watch how the surface area and volume change. Parameters: Type of polyhedron, length, width and height. Surface Area and Volume one of the Interactivate assessment explorers.

On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science. Additionally, the National Computational Science Institute (NCSI) provides nation-wide workshops portraying resources and instructional ideas to middle school, high school, and undergraduate instructors for use in the classroom. Resources and materials offered to these instructors are available free of charge from Shodor's website and are largely developed by Shodor student interns. Shodor's academic program efficiently guides participants from excitement to experience to expertise through computational explorations, research opportunities, and service.

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Interactivate: Surface Area and Volume On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science.

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8-Ways Pedagogy is about the process of teaching rather than what is being taught - incorporating Story Sharing,
Learning Maps, Non-verbal, Symbols and Images, Land Links
Non-linear, Deconstruct/Reconstruct, Community Links into teaching to enhance learning and development of cultural understanding for all students while helping Aboriginal and Torres Strait Islander students feel at home in the classroom.

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Aboriginal Pedagogy

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Murri Maths - NEW EDITION COMING SOON - Sally Lawrence

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This wonderful online article provides teachers with ideas for implementing cultural perspectives in mathematics teaching, categorised by age.

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$0.00

Embed Aboriginal Perspectives within the Australian Curriculum - Maths.
Murri Maths: Connecting to Country provides teachers with a rich collection of stories, images, and experiences designed and collated to support teachers as they embed the Cross-Curriculum Priority of Aboriginal and Torres Strait Islander Histories and Cultures within Teaching and Learning.

This is a resource developed North Coast Region's Indigenous Education Unit, project lead Nicole Simone. Developed in partnership and consultation with local Kabi Kabi/Gubbi Gubbi community for the Department of Education Queensland. Putting the culture into Maths.

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Murri Maths - NEW EDITION COMING SOON - Sally Lawrence

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Murri Maths: Connecting to Country provides teachers with a rich collection of stories, images and experiences designed and collated to support teachers as they embed the Cross-Curriculum Priority of Aboriginal and Torres Strait Islander Histories and Cultures within Teaching and Learning.

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Mathematical Mindset Teaching Guide, Teaching Video and Additional Resources - YouCubed

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Mathematical mindsets focuses on a growth mindset in math, having high expectations to make math challenging, and  tackling issues of math anxiety and fear to take risks.  This site has resources to help implement this mathematical mindset mentality in class.  It is recommended elements of these concepts are implemented to tackle math anxiety and low self concept in math.

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Mathematical Mindset Teaching Guide, Teaching Video and Additional Resources - YouCubed

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We have designed a Mathematical Mindset Guide to help teachers create or strengthen a growth mindset culture. This guide contains five Mathematical Mindset Practices along with links to teaching videos. […]

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Two-way learning has students investigating and appreciating Indigenous perspectives along side contemporary practice, building understanding and respect.

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Two-way learning connects people from different cultures and backgrounds and enables them to learn with and from one another.

Learning with and from one another

At Yiramalay, learning goes both ways. The Studio School's Aboriginal students come to Yiramalay from communities across Australia and learn about each other's cultures, traditions and languages.

Two-way learning also takes place between the Studio School's Aboriginal students and Wesley College's Melbourne-based students. This sharing of knowledge, backgrounds and experiences is mutually beneficial. It helps all students become wiser, better informed global citizens, and also fosters respect, understanding and empathy.

Teaching the teachers

Two-way learning between students is supported by the Studio School's staff of teachers, mentors and local Elders.

Regardless of their background or position, all staff at Yiramalay understand their role as both teachers and learners. Our staff constantly participate in two-way learning - learning from Yiramalay students, the land, the local community and each other.

An approach that brings people together

The two-way learning approach builds confidence and knowledge and supports the Studio School's overarching vision - to offer a hands-on educational experience that champions:

intercultural understanding that encourages the union of ideas and solutions to strengthen communities

respect for cultural heritage and protection of the environment

feelings of solidarity and equity for all.

Related links

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Two-way learning connects people from different cultures and backgrounds and enables them to learn with and from one another. Learning with and from one another At Yiramalay, learning goes both ways. The Studio School's Aboriginal students come to Yiramalay from communities across Australia and learn about each other's cultures, traditions and languages.

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ClassKick is an amazing collaborative learning site.  Teachers can upload PDFs and insert automatic answer keys, start collaborative online number-talks, and students can communicate with anonymity, reducing anxiety and creating a safe space where everyone feels safe to contribute.  What I especially love is the option for the teacher to allow peer tutoring when a student puts their virtual hand up.  It’s free and all you need is a class code. I hope you love it as much as I do!

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Helping Teachers Be Awesome See all your students working and give high-quality feedback–from anywhere. Sign Up (Free!)

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Exploring the Ratio of Surface Area to Volume in Cubes - Mathigon

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Mathigon has a great interactive surface area tool for digitally investigating 3D shapes. This activity by mathigon.org includes higher order thinking concepts for a deeper understanding.  This also complements it: https://mathigon.org/task/exploring-the-ratio-of-surface-area-to-volume-in-cubes

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Overview and Objective

When comparing sizes of objects, it is common to make a statement such as one object is twice as large as another object. In this activity, students will discuss the importance of clarifying if the size comparison is based on length, area, or volume by exploring that the length, area, and volume of similar objects do not grow or shrink using the same scale factor.

Students also learn that this property is called Square-Cube Law by discussing its effects on cells, organisms, and animals.

Warm-Up

Enlargement is a transformation that changes the size of an object by a scale factor. Share this polypad with students and have them use the custom polygon to enlarge the square. Click here to learn how to share polypads with students and how to view their work and click here to learn more about using the custom polygon tile.

Take some time to discuss that when the length of a square is doubled, the area becomes four times bigger, when it is tripled, the area becomes 9 times bigger and so on.

Main Activity

Share this canvas with students and let them create the nets of cubes using given squares. After creating each net, they will use the "Fold Net" button to create the cubes.

Encourage students to use a table to record the side length, surface area, and volume of each cube. Discuss as a class at what side length of square the area became equal to the volume, and what happened after that point? Proportionately, which grows faster: surface area or volume?

Let them express the area and volume algebraically using the equation tool. Then, sketch the graphs of area and volume functions by dragging the equations on to the coordinate plane.

After discussing the graphs, invite students to add a column to their table to record the ratio of surface area to volume. Discuss what they notice and wonder. Which cube has the most surface area in proportion to its volume?

Closure

Share with students that this property is called the square-cube law. It is a mathematical principle, applied in a variety of scientific fields, which describes the relationship between the volume and the surface area as a shape's size increases or decreases.

It was first described in 1638 by Galileo Galilei in his book of Two New Sciences. It helps explain phenomena including why large mammals like elephants have a harder time cooling themselves than small ones like mice, and why building taller and taller skyscrapers is increasingly difficult.

More on the Biology Connection

Cubes can be used to serve as a model cell. In order for cells to survive, they must exchange nutrients and waste with their environment using the cell membrane that covers the surface of the cell. When the volume of the cell increases, more reactions will be needed. As a cell grows, its SA:V decreases. As SA:V decreases, the reactions on the surface of the cell fail to meet the needs of its volume. At this point the cell cannot get any larger.

For example, if a human becomes 10 times taller, wider, and longer, the skin would cover 100 times more surface area, but the volume (therefore the weight) would be 1,000 times bigger. It means only 10 times bigger bones needs to carry 1000 times bigger weight. Although science fiction writers have fun playing with the idea, any giant animal or human would break its legs with the first step it took .

To close the lesson, ask students to think about how flatteningan object while keeping the volume constant impacts the surface to volume ratio?

Why leaves are very thin and flat? Why do desert plants generally have smaller leaves?

For animals, heat loss occurs only at the exposed surfaces. If animals need to maintain a constant body temperature, how can you explain the large, almost flat ears of elephants or fur covered skins of mice?

Support and Extension

For students ready for additional extension in this lesson, consider asking the following question:

If you cut a cube in half, how does the volume, surface area, and SA: V change comparing the original?

Polypads for This Lesson

To assign these to your classes in Mathigon, save a copy to your Mathigon account. Click here to learn how to share Polypads with students and how to view their work.

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Exploring the Ratio of Surface Area to Volume in Cubes - Mathigon

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Explore our free library of tasks, lesson ideas and puzzles using Polypad and virtual manipulatives.

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This site by the University of Melbourne incorporates higher order Aboriginal and Torres Strait Islander perspective in math

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This online article uses 3D digital media images to understand development of surface area and volume equations.  It could be modified a step further by having students create a Yangma in Tinkercad and building a square from it themselves. It would make a good simple introduction to Tinkercad without the technology taking over from the math content.

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Article by Emma McCaughan

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

(The ideas used in this article are also used here to find the volume of a sphere and the, impressively named, mouhefanggai shape.)

What you need to know.

You need to be familiar with the formulae for the area of a square, the area of a circle, and the volume of a cuboid. We will also use ideas of enlargement and ratio.

The yangma.

When we think of a pyramid, we usually think of one where the vertex (the top point) is above the centre of the base, a right pyramid . However in this article, we will begin with a yangma. Yangma is an ancient Chinese name for a rectangular-based pyramid whose vertex is vertically above one of the vertices of the base. We will take a yangma with a square base of side length $a$, and height also length $a$. Three of these can be fitted together to form a cube.

Since we know that the volume of the cube is $a \times a \times a = a^3$, it follows that the volume of each of these yangmas is $\frac{a^3}{3}$.

Enlargement

Consider, for a moment, a cube with edges 1 unit long. You know that the volume is $1 \times 1 \times 1$. Now stretch that cube in a horizontal direction so that it measures a by 1 by 1. Its volume is now $a \times 1 \times 1$. If we imagine the cube sliced horizontally, there will be the same number of slices, but each will be a times as long.

If we stretch the cube in a perpendicular direction, so that it measures a by b by 1, the volume will be $a \times b \times 1$, or $b$ times larger. If we stretch the cube in the third perpendicular direction, by a scale factor of $c$, the volume will be $a \times b \times c$, which we know as the formula for the volume of a cuboid.

There are three independent directions in which we could enlarge a 3D shape. If we enlarge it by a scale factor $k$ in one of these directions, the volume will be $k$ times as large.

Back to our yangma:

Suppose we want the volume of a yangma whose height is different to the base lengths, perhaps height $h$ instead of a. Well, this is just an enlargement in the vertical direction.

The scale factor? If we are turning $a$ into $h$, we have multiplied by $\frac{h}{a}$.
So the volume of our new yangma is $\frac{h}{a} \times \frac{a^3}{3}$, or $\frac{a^2h}{3}$.

Sliding the slices

We now have a formula for the volume of any square-based pyramid whose vertex is above one of the vertices of the base. What about if the vertex is somewhere else - the middle, for instance? What we are going to do is to imagine the pyramid cut into lots of slices horizontally. We are going to slide these slices across, so that the top of the pyramid is above the middle of the base.

If we had an infinite number of slices, our pyramid would have nice straight edges. You can probably imagine that with more slices, it would look smoother than in this illustration. Has the area of any of the slices changed? So has the volume of the shape changed?

We can now see that the volume of any square-based pyramid is $\frac{a^2h}{3}$.

Comparing a cone with a pyramid

We will now look at a cone. We'll start with a right cone , whose vertex is above the centre of the base. In fact, by slicing it as in the previous section, we can show that the same formula applies for any cone.

Imagine a cone whose base is a circle radius $r$, and height is $h$. This cone will fit exactly inside a square-based pyramid with base length $2r$ and height $h$.

Suppose we take a slice of the pyramid with the cone inside, from some way up the pyramid. This will look like a square with a circle fitting inside. We don't know the radius of the cone at this point, so we'll call it $x$.

The area of the circle is $\pi x^2$.

The area of the square is $2x \times 2x=4x^2$.

The ratio of the circle to the square is $\pi : 4$.

The same is true for every slice we take: the area of the circle is $\frac{\pi}{4}$ of the area of the square.

So, if each slice is $\frac{\pi}{4}$ the size, the volume of the cone will be $\frac{\pi}{4}$ the volume of the pyramid.

The pyramid's volume is $\frac{(2r)^2h}{3} = \frac{4r^2h}{3}$

So the cone's volume is $\frac{\pi}{4} \times \frac{4r^2h}{3} = \frac{\pi r^2 h}{3}$.

Non-square-based pyramids

We can use the same principles to find the volume of any pyramid.

Rectangular-based pyramid

If we have a pyramid with rectangular base measuring $a$ by $b$, and height $h$, then this can be obtained by stretching our square-based pyramid by scale factor $\frac{b}{a}$. The new volume will be $\frac{b}{a} \times \frac{a^2 h}{3} = \frac{a b h}{3}$.

Triangular-based pyramid

If the base of the pyramid is a triangle with base $a$ and perpendicular height $b$, it will fit exactly in the rectangular pyramid above.

Any slice will look like this:

Although we don't know the measurements of the rectangle in this slice, the sides will still be in the ratio $a:b$ (this may take some thinking about). Let's call them $k{a}$ and $k{b}$ (where $k$ is less than 1).

Area of rectangle = $k{a} \times k{b} = k^2 a{b}$

Area of triangle = $\frac{1}{2}{k}{a} \times {k}{b} = \frac{1}{2}k^2{a}{b} = \frac{1}{2} \times \mbox{area of rectangle}$.

If each triangle is half the size of the rectangle, the volume of the triangular-based pyramid will be half the volume of the rectangular-based pyramid, or $a{b}h/6$.

Generalisation

The formula for the volume of any pyramid is $\frac{1}{3}\mbox{base area} \times \mbox{height}$.
Verify that this works for the pyramids above (and indeed for the cone). Can you convince yourself that this is always true?

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Article by Emma McCaughan These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts. (The ideas used in this article are also used here to find the volume of a sphere and the, impressively named, mouhefanggai shape.)

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Pear-deck is an add-on platform to google slides or PowerPoint, designed to help students interact with Presentations.  Teachers build presentations with added interactive checkpoints, higher order questions and surveys. Students can respond to questioning anonymously, allowing them to feel safe to contribute.  They share their ideas by drawing on whiteboard, answering for whole class responses to display as an overlayed survey and many other options, leading to opportunities for immediate, low risk feedback, whole-class discussion, anonymous ‘my favourite no’ number-talks etc.  Data is automatically recorded for the teacher to look back on. I trialled it, and I love it.

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Pear Deck is on a mission to help teachers create powerful learning moments for every student, every day - using the tools you already know and love.

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This article discusses 3D scanning to input models into Tinkercad for 3D printing.  Qlone is one option, and Polycam will also produce the same file type.  This technology could support learning of scaling in measurement and trigonometry, investigation of changes in volume and surface are to volume SA/V raitios.  Students could import models of themselves or real objects into Tinkercad, use composite standard prisms to build within their model and estimate their surface area to volume ratios using mathematics.  These concepts have real-life application in biology, optimising packaging in industry (reduced cost, materials and environmental concerns)

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One of the easiest ways to get started with 3D design is to start with a 3D scan and import the model into Tinkercad. We have been working with the team over at Qlone and wanted to share this easy simple project. Check it out and let us know what you think.

Never heard of Qlone? Qlone is an all-in-one 3D tool for everyone. They have made it easy and fast to scan real objects, using your phone's camera, modify them in app and seamlessly export the result to many platforms, 3D file formats and 3D printers... all on your iPhone or iPad.

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One of the easiest ways to get started with 3D design is to start with a 3D scan and import the model into Tinkercad. We have been working with the team over at Qlone and wanted to share this easy simple project. Check it out and let us know what you think.

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This Higher order thinking concept from nrich.com can be modified and redefined to have students explore the optimisation of beverage packaging within functional constraints, having students collect and measure existing bottles, contact soft drink suppliers and packaging designers and investigate to minimise surface area to volume ratio (SA/V), modelling and presenting final designs to industry experts.

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An Australian Stonehenge? - Australian National Maritime Museum

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Aboriginal and Torres Strait Islander eel-traps are an Ingenious design, utilising simple 3D shapes for a complex concept,  meeting fit-for-purpose functions. With permission from Traditional Owners, I’d like to work with owners to implement an investigative task for students to consider geometric properties of different historical eel traps and their composite 3D shapes, model, analyse and compare surface area to volume rations within necessary design constraints, while learning of tradition aqua-farming practices and challenging previous views of First Nations people and math.  This would likely require conferring with traditional owners and the museum to obtain measurements, researching and collaborating with Gunditj Mirring Traditional Owners Aboriginal Corporation using special protocols recommended by Oxfam Australia, (n.d.)

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Traditional owners will find out next month if their push for a 6,000-year-old network of eel traps in south-west Victoria is to be supported for inclusion on the UNESCO World Heritage List. The eel-farms were built by the Gunditjmara people in south west Victoria to manage eels in Lake Condah and nearby Darlot Creek. They are among the earliest surviving examples of aquaculture in the world.

The eel farms cover more than 75 square kilometres and include artificial channels and ponds for separating eels, as well as smoking trees for preserving the eels for export to other parts of Australia. Just to be clear, this industry and the complex of stone arrangements including houses began around 6,000 years ago - before Stonehenge and the Pyramids.

Known as Budj Bim, the site received national heritage listing in 2004 and has been put forward for inclusion in the Australian government's nomination to the UNESCO World Heritage Council by the Victorian government, at the instigation of the Gunditjmara people.

The Budj Bim landscape was the area where Robbery Under Arms author Rolf Boldrewood described the Eumeralla Wars. From the late 1830s as squatters rushed to take the fine grazing lands in southwest Victoria that Major Thomas Mitchell had written glowlingly about after his 1836 journey through the area, the Gunditjmara people fought against the squatters for their lands.

Rolf Boldrewood - himself a squatter in the area in the 1840s (his real name was Thomas Alexander Browne) recalled the warfare conducted amongst the rocky and intricate landscape of the Budj Bim - a landscape that was important during guerrilla warfare conducted for over twenty years. After a bitter and bloody struggle including several massacres of Aboriginal people according to Boldrewoode-Browne, some remaining Aboriginal people were relocated to reserves. But many refused to move from their traditional lands.

In 1868 the Victorian government established a mission for the people who had survived the settlers onslaught at Lake Condah, close to some of the eel traps and within sight of Budj Bim. The mission station was officially closed in 1919 and the mission church destroyed by the 1950s. The Gunditjmara continued to live in the area and protect their heritage. According to elder Ken Saunders the Gunditjmara people had continued to trap eels as well.

The mission lands were returned to the Gunditjmara in 1987. They were awarded native title over the area in 2007 and proceeded to restore the lake that non-Indigenous farmers had drained in the late 19th century. The eel traps began to fill again.

The traps are a series of canals and graded ponds, running for some 35km around the lake. Gunditjmara people manipulated water levels to encourage eels to swim into holding ponds and placed funnel-shaped baskets at the spillway between ponds to ensure that smaller eels could slip through and larger eels be harvested.

Eels were transferred from one pond to another, just like a fish hatchery today, and they would be grown until of an edible size. They may also have been stocked from outside the area as well. When the eels were harvested they could be eaten there and then, or smoked in hollows of the Manna Gum for later preservation and trade with other communities.

The traps and other abundant wildlife provided by the lake allowed the Gunditjmara people to remain in one central place, rather than following the food resources over the seasons. Accordingly, they built stone houses around the lake. Archaeologists have suggested there were even villages of stone buildings among the potentially hundreds of stone structures in the area. These are also included in the world heritage status application.

Since the 1970s, archaeologists had suspected that the stone remains in the Lake Condah region were evidence that the local Aboriginal people had lived in villages. But it was not until an eight-year research project was carried out by a Flinders University archaeologist, Heather Builth, that the real importance of the remains became clear.

The area was naturally a wetland, but Builth discovered that the Gunditjmara had modified it with weirs, channels and dams to make the landscape eel-friendly. She estimated the output from these eel farms would have been enormous - perhaps feeding up to 10,000 people, an ancient fishing industry more than a subsistence farm.

Monash University professor of Indigenous archaeology, Ian McNiven, said that carbon dating of charcoal found during an excavation of one of the fish traps found it was 6,600 years old. McNiven said that particular site, called the Muldoons trap complex, appeared to have been remade a number of times over the intervening six millennia, indicating it was continually used.

McNiven and his team of archaeologists working with the Gunditj Mirring Traditional Owners Aboriginal Corporation say their findings show that Muldoons Trap Complex underwent phased redevelopment and major elaboration over the past 800 years. "Muldoon trap complex is currently the oldest known stone-walled fish trap in the world and amongst the world's oldest known fish traps," he told Guardian Australia. "It is also the oldest continuously used fish trap in the world ... Indeed, the trap was still being used by Gunditjmara people at the Lake Condah Mission in the late 19th century."

McNiven noted that the extensive network of traps are important evidence in destroying the myth that Aboriginal and Torres Strait Islander people did not farm or improve the land. The deliberate manipulation of the environment to ensure better production of marine resources was a sophisticated form of agriculture, which McNiven said was of world heritage significance.
UPDATE - 20 January 2017 - Budj Bim Cultural Landscape a step closer to UNESCO World Heritage status
The Federal Government has submitted a nomination for the Budj Bim Cultural Landscape at Lake Condah to be included on Australia's World Heritage Tentative List.

If the nomination is successful, Budj Bim would be the Australia's 20th World Heritage site and the only place listed solely for its Indigenous cultural value.

The Budj Bim Cultural Landscape must be on Australia's World Heritage Tentative list for at least 12 months before the United Nations' World Heritage Committee make a determination on its nomination.

Read more here from ABC News and an excellent overview by Ian McNiven here.

For further information see the 2011 award winning book The People of Budj Bim - Engineers of aquaculture, builders of stone house settlements and warriors defending country written by the Gunditjmara people with Gib Wett enhall

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An Australian Stonehenge? - Australian National Maritime Museum

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Traditional owners will find out next month if their push for a 6,000-year-old network of eel traps in south-west Victoria is to be supported for inclusion on the UNESCO World Heritage List. The eel-farms were built by the Gunditjmara people in south west Victoria to manage eels in Lake Condah and nearby Darlot Creek.

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Here is an example of a design activity relating to discussed concepts above.

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An interesting take on the development of critical and creative thinking

I have scooped this article from https://padlet.com/kimberleygraham/2iwhf845813505og
I really like the use of analogy and colourful visuals to scaffolding the thought process.  I can see this valuable for supporting students with ASD.

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This slideshow requires JavaScript.

Quite often we'll ask children to "Put your thinking cap on". Empowering learners with the skills to think creatively and critically is a gift which we can facilitate as teachers. The Developing Thinking and Reading Between the Lines PowerPoints in the Literacy Toolkit provide professional learning resources for staff to engage with materials that promote thinking.

Many of our posts have referred to the work of Bloom; however, when I was at an Active Literacy CPD session in Falkirk recently, the Six Hat materials were mentioned during a round table discussion.

Edward de Bono's Six Thinking Hats is a tool which can be used by practitioners to promote thinking. Each hat colour represents a different aspect of thinking.

Some of the resources, tailored to middle/upper Primary and Secondary, suggest physically wearing coloured representations of the hats when beginning to engage with the materials. This way, learners are physically moving between the hats; therefore they are moving between the different modes of thinking.

There are banks of free resources that are available to support the teaching of thinking skills using the De Bono's Six Hats materials. The Six Hats materials can be used in conjunction with other Higher Order Thinking materials. Check out the resources below:

CLICK HERE - Using Six Hats as a reflection tool
CLICK HERE - Using the Six Hats to Respond to Literature
CLICK HERE - Six Hats Display
CLICK HERE - Thinking Display
CLICK HERE - FREE De Bono Thinking Resources

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Quite often we'll ask children to "Put your thinking cap on". Empowering learners with the skills to think creatively and critically is a gift which we can facilitate as teachers. The Developing Thinking and Reading Between the Lines PowerPoints in the Literacy Toolkit provide professional learning resources for staff to engage with materials that promote thinking.

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Indigenous engineering and technology at Budj Bim

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This YouTube video shows how digital technologies are being used to investigate one of the worlds oldest technologies.   University of Melbourne work in conjunction with Gunditj Mirring Traditional Owners Investigating Indigenous aqua-cultural systems at Budj Bim.

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Indigenous engineering and technology at Budj Bim

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Problem-Based Learning is a strategy where students learn through problem solving, taking a constructivist student-led approach to learning, with teacher as a guide in the background.

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Problem Based Learning

What is it?

Problem based learning (PBL) is a teaching strategy during which students are trying solve a problem or a set problems unfamiliar to them. PBL is underpinned by a constructivist approach, as such it promotes active learning. Activities are carried out with groups of students, typically in a tutorial or seminar setting. PBL fosters self-directed learning, effective problem solving, communication and collaboration skills. Students tend to work in groups to problem solve with the lecturer or tutor taking a backstage role during the activity, their involvement is usually limited to guiding and monitoring the process.

Resources

Links to relevant PD:

Case Studies:

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Problem Based Learning Problem based learning (PBL) is a teaching strategy during which students are trying solve a problem or a set problems unfamiliar to them. PBL is underpinned by a constructivist approach, as such it promotes active learning. Activities are carried out with groups of students, typically in a tutorial or seminar setting.

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This site has invaluable research information on packaging design optimisation in industry, with a focus on reduced costs, reduced materials and environmentally friendly design - it can be incorporated in problem based learning for students to extend deeper learning of measurement and geometry concepts in math.

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This lively podcast series from Packaging Digest reveals what's new or what's next for packaging executives and engineers, designers and developers.

Listen now!

Brand makes sustainable changes, retortable PCR food pouches, BK's reusables, rice brand redesigned, olive pits into bioplastic, AromaPak...

Hybrid polypropylene cup in test would help move the restaurant chain beyond virgin petrochemical-based ...

These Continental tires recycle about 10 plastic bottles per tire.

Sign up for the Food & Beverage Packaging newsletter.

Hybrid polypropylene cup in test would help move the restaurant chain beyond virgin petrochemical-based ...

These Continental tires recycle about 10 plastic bottles per tire.

Sign up for the Food & Beverage Packaging newsletter.

Anheuser-Busch's sustainable initiative allows MLB, NFL fans to recycle aluminum cups, cans, and bottles without leaving their seats. What can we learn?

Who's embracing recyclable aluminum cans for sustainability and new graphic designs that pop? Stalwarts like PepsiCo (twice) and Coors and newcomers like Machu Picchu.

I wanted to talk about shelf-life performance, and we did. But Absolut's Director of Future Packaging veered into marketing and manufacturing - because they're all connected.

Constellation Brands' beautiful, intricately textured glass bottle required some technical wizardry from Ardagh Glass Packaging.

JBT-Avure's FlexiBulk system unlocks exciting new options in high-pressure pasteurization (HPP) that's attracted food and...

Brands' program with rePurpose Global has removed 2,000 tons of plastic waste from the environment, the equivalent of more than...

Packaging sustainability isn't about single-use plastics, it's about single-use packaging...and that changes everything.

Themed packaging for beer, whiskey, specialty coffee, and more - all designed with varying amounts 'o Irish green - celebrates the March 17 holiday.

1,000,000 plastic PET bottles have been recycled through ZeLoop, a blockchain- and cryptocurrency-based smartphone app.

What happens to a can of Pepsi Max, glass bottle of Coke, or empty PET bottle under extreme pressure? Let's go to the video...

These Continental tires recycle about 10 plastic bottles per tire.

Sign up for the Food & Beverage Packaging newsletter.

Anheuser-Busch's sustainable initiative allows MLB, NFL fans to recycle aluminum cups, cans, and bottles without leaving their seats. What can we learn?

Who's embracing recyclable aluminum cans for sustainability and new graphic designs that pop? Stalwarts like PepsiCo (twice) and Coors and newcomers like Machu Picchu.

Who's embracing recyclable aluminum cans for sustainability and new graphic designs that pop? Stalwarts like PepsiCo (twice) and Coors and newcomers like Machu Picchu.

I wanted to talk about shelf-life performance, and we did. But Absolut's Director of Future Packaging veered into marketing and manufacturing - because they're all connected.

Constellation Brands' beautiful, intricately textured glass bottle required some technical wizardry from Ardagh Glass Packaging.

JBT-Avure's FlexiBulk system unlocks exciting new options in high-pressure pasteurization (HPP) that's attracted food and...

Brands' program with rePurpose Global has removed 2,000 tons of plastic waste from the environment, the equivalent of more than...

Packaging sustainability isn't about single-use plastics, it's about single-use packaging...and that changes everything.

Themed packaging for beer, whiskey, specialty coffee, and more - all designed with varying amounts 'o Irish green - celebrates the March 17 holiday.

1,000,000 plastic PET bottles have been recycled through ZeLoop, a blockchain- and cryptocurrency-based smartphone app.

What happens to a can of Pepsi Max, glass bottle of Coke, or empty PET bottle under extreme pressure? Let's go to the video...

Constellation Brands' beautiful, intricately textured glass bottle required some technical wizardry from Ardagh Glass Packaging.

JBT-Avure's FlexiBulk system unlocks exciting new options in high-pressure pasteurization (HPP) that's attracted food and...

Brands' program with rePurpose Global has removed 2,000 tons of plastic waste from the environment, the equivalent of more than...

Packaging sustainability isn't about single-use plastics, it's about single-use packaging...and that changes everything.

Themed packaging for beer, whiskey, specialty coffee, and more - all designed with varying amounts 'o Irish green - celebrates the March 17 holiday.

Packaging sustainability isn't about single-use plastics, it's about single-use packaging...and that changes everything.

What happens to a can of Pepsi Max, glass bottle of Coke, or empty PET bottle under extreme pressure? Let's go to the video...

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This lively podcast series from Packaging Digest reveals what's new or what's next for packaging executives and engineers, designers and developers. Listen now! Brand makes sustainable changes, retortable PCR food pouches, BK's reusables, rice brand redesigned, olive pits into bioplastic, AromaPak... Hybrid polypropylene cup in test would help move the restaurant chain beyond virgin petrochemical-based ...

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Using TinkerCad, Google Blogger and a Google Slides Presentation to inform higher context problem solving and Modelling - Year 10
To accompany a Unit in measurement (solving problems involving surface area and volume of composite solids), and trigonometry (applying knowledge of sine, cosine and area rules to authentic problems), year 10 students will be given a Problem-based-learning (PBL) Problem Solving and Modelling Task  (ECU, 2022; Queensland Curriculum and Assessment Authority (QCAA), n.d.) to apply their mathematical knowledge of 3D composite shapes, volume and surface area, to design and create either optimized beverage packaging, or a prototype of a First Nations Eel trap (Gunditj Mirring Traditional Owners Aboriginal Corporation RNTBC, 2022).

By the makers of AutoCad and Inventor, TinkerCad(Autodesk, 2022)is an introductory 3D design freeware for preparing students for developing 21st century critical and creative thinking, ICT and numeracy skills, measurement, apply ratios and scaling, and work with transformations in a 3D space. They can import and modify common shapes to construct 3D composite prototypes, developing spatial reasoning to visualize and analyze and make decisions analogous to real world of engineering design. This will complement manual calculations and provide perspectives to compare and contrast and make inferences to solve complex problems relating to surface area and volume of composite solids.

In the sequence, Google Blogger(Blogger.com, 2022) and an adapted version of a teacher-designed interactive virtual design-fair Google Slides presentation (Google.com, 2022) provide complementary digital technologies to support real-world interactions through systematic, chronological communication of ideas to community experts and other students. Google Blogger allows students to project manage their work, communicating ideas from multiple perspectives using written, and visual literary methods, receiving two-way feedback and collaboration with teacher, other students and community experts as they progress, promoting high expectations and holding themselves accountable for project management, quality, legal, safe and ethical expectations in their work, while collaboration in a virtual Google Slides design-fair allows students to display their creations to a wider expert audience, promoting ownership and further building world-wide community connections (Queensland Curriculum and Assessment Authority (QCAA), 2018).

Used together, these digital technologies provide a transformative learning experience where students can plan, think critically and creatively on a shared purpose, creating an authentic design using ICT, communicating ideas with community experts, applying literacy, numeracy and content knowledge in a real-life, constructivist, student-led PBL approach to promote active citizenship and a deeper understanding (Hobdell, 2022), meeting the General Capability requirements in the mathematics context (Queensland Curriculum and Assessment Authority (QCAA), 2018) (Australian Curriculum Assessment and Reporting Authority (ACARA), 2021).

Australian Curriculum Assessment and Reporting Authority (ACARA). (2021). General capabilities in the Australian Curriculum: Mathematics. https://docs.acara.edu.au/resources/Mathematics_-_GC_learning_area.pdf

Autodesk. (2022). Tinkercad. Autodesk Inc. https://www.tinkercad.com/

Blogger.com. (2022).  https://blogger.com

ECU. (2022). Problem based learning. https://intranet.ecu.edu.au/learning/curriculum-design/teaching-strategies/problem-based-learning

Google.com. (2022).  https://sites.google.com/

Gunditj Mirring Traditional Owners Aboriginal Corporation RNTBC. (2022).  https://www.gunditjmirring.com/

Hobdell, G. (2022). Topic 2: The nature of digital pedagogy. CQU Moodle. https://moodle.cqu.edu.au/mod/book/view.php?id=2413826&chapterid=179797

Queensland Curriculum and Assessment Authority (QCAA). (2018). Capabilities and skills frameworks across senior curriculum phases. https://www.qcaa.qld.edu.au/downloads/aust_curric/ac_cap_skills_frameworks_snr_phases.pdf

Queensland Curriculum and Assessment Authority (QCAA). (n.d.). Problem solving mathematical modelling. https://www.qcaa.qld.edu.au/downloads/senior-qce/mathematics/problem_solving_mathematical_modelling.pdf

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Blogger.com - Create a unique and beautiful blog. It’s easy and free.

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Blogger is a collaborative tool where students can blog chronologically on their projects and ideas, sharing with peers for constructive feedback.  This could be extended to group or whole-class blogs, taking turns reflecting in own words on what they learned today. Here is an article on how that might look in a math class:  https://ditchthattextbook.com/kids-blogging-in-math-class-why-how/

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Blogger.com - Create a unique and beautiful blog easily.

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Publish your passions your way. Whether you'd like to share your knowledge, experiences or the latest news, create a unique and beautiful blog.

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The SAMR Model Explained (With 15 Practical Examples) - 3P Learning

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If like me, you often have to refer back to the levels of the SAMR model and how each look in the classroom, this article is a handy go to.  It also provides tips on finding your purpose, with questions to ask to avoid ‘technology for technologies sake’

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What is the SAMR model?

The SAMR model is a framework that orders strategies for classroom technology implementation into four stages:

In each stage of the SAMR model, the use of educational technology (EdTech) becomes increasingly sophisticated.

Substitution and augmentation strategies use EdTech to enhance learning and add value, whereas modification and redefinition stages harness EdTech to transform learning and make new things possible.
The effectiveness of any of these four strategies depends on when and how it is used.
Here's how you can harness each stage for more powerful classroom learning:

Substitution

This is the simplest stage, where EdTech is used as a direct substitute for traditional practices.

Substitution strategies can save you time and space by cutting back on laborious pen and paper tasks. Instead of printing out twenty-plus paper resources that clog the cupboard, you can use technology to manage resources with just a few clicks.

Substitution also provides a much more accessible introduction to technological soft skills than modification and redefinition. It's the perfect opportunity for students to get comfortable with new technology before you start to transform their learning.

Examples of substitution

having students type their work instead of handwriting it

using online quizzes and programs instead of pen and paper

uploading a worksheet in PDF for student access, as opposed to photocopying

using a digital interactive whiteboard as opposed to a traditional whiteboard and saving the results as a document.

Remember: Substitution strategies do not change the learning process or outcome. Ask yourself what you and your students will gain from the technology before implementing it. If it adds nothing but hassle, stick to pen and paper.

Augmentation

At the augmentation stage, technology adds something to the learning process beyond just convenience. It might give your students a clearer understanding of a complex topic or making it engaging in a way that traditional methods can't.

It also allows for the introduction of more independent and student-centric learning. By using technology as a source of information, students can start actively learning without requiring constant teacher-led instruction.

Examples of augmentation

Students give more informative and engaging oral presentations accompanied by a PowerPoint or Prezi containing multimedia elements.

Students use the internet to independently research a topic, as opposed to relying on teacher input.

Students use an EdTech program that gamifies curriculum content for student engagement and allows students to track progress in an accessible way.

Teacher instruction is supplemented with a video that clarifies a particularly hard to explain concept.

Modification

At the modification stage, technology is used to design interactive and dynamic tasks that go beyond the limitations of a traditional classroom.

For example, students can collaborate on shared documents, or work in large groups, allowing for seamless collaboration and knowledge sharing. This peer-to-peer work nurtures a more cooperative and dynamic class culture.

Technologically modified tasks also allow students to generate inspired and innovative work that isn't confined to paper. Some students will jump at the opportunity to get in front of a camera, while others will do anything to make it onto the global Mathletics leaderboard.

Examples of modification

Students produce podcasts summarising a topic, which can then be accessed by other students as a revision resource.

Students create an informative video presentation in place of a standard oral presentation. They can use their voice alongside a broader variety of creative multimodal components.

Students use a technological tool that makes an abstract concept visible in a hands-on, responsive way (e.g. voyaging on Google Earth to better understand measurement and geography).

Redefinition

The most sophisticated stage of SAMR, redefinition sees you using technology to make entirely new learning opportunities possible.

Redefining learning has the potential to connect learning with the real world and produce authentic outcomes. It also gives students strong technological soft skills such as digital collaboration, communication, technological literacy, and the ability to adapt to new systems and processes.

Most importantly, using technology to redefine learning in your classroom translates to vibrant and engaged students who embrace a growth mindset.

Examples of redefinition

connecting your students with other people around the world as part of the learning journey

having students publish their work online where it can be viewed by peers and the broader community

recording students as they deliver a presentation or practice a physical skill, then using this recording to prompt student reflection

experimenting with tasks that use extensive multimodal elements (e.g. producing documentaries or short films, webpages, print documents with creative layouts).

Remember, redefinition isn't the same as 'high tech'. A learning experience is redefined when it integrates technology seamlessly and meaningfully to open new doors for student learning - regardless of how sophisticated that technology might be.

In an English classroom, this might involve learning how to critically assess online media during web-based research. In mathematics, students might look at data represented through amazing visuals and get inspired to create their own.

Find your purpose

Starting with a purpose avoids the 'technology for technology's sake' that sometimes infiltrates the classroom.

Ask yourself:

What am I hoping to achieve by using this technology?

How will it make a difference to my students' learning?

Why is it preferable to not using technology?

How equipped are my students and I to use this technology?

How much time do I have to invest in making it work?

These simple questions can make it simpler to determine which stage of the SAMR model suits your needs. Remember that you don't always have to redefine your students' learning. A few simple technological additions to an already effective teaching strategy might be all you need to make a difference.

Using technology in the classroom sends a clear message to students. It shows them that you're connecting learning to the rapidly changing world outside the classroom walls, and you aren't afraid to adapt to a new challenge. Get creative with technology and you'll inspire as well as teach.

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The SAMR Model Explained (With 15 Practical Examples)

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What is the SAMR model and how does it apply to learning? Here's how you can harness each stage for more powerful classroom learning.

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This is an interesting article I found in my own search for applying Puentedura‘s SAMR model in math.  It highlights the critical importance of ensuring digital technologies invoke higher order thinking, enhancing curiosity, with substitution levels alone risking removing the element of an enthusiastic teacher which can’t be underestimated when it comes to engaging curiosity, it also cautions that digital technology can take focus away from math learning if not carefully considered.
There are some great examples of modifying numbertalk - style activities using SAMR, and redefining mathematics learning using Padlet as a collaborative tool. I recommend this article for those looking to implement digital technology in mathematics!

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Introduction to PBL and benefits.

Scooped from https://padlet.com/kimberleygraham/2iwhf845813505og
A great resource that provides a lot of insight into the importance of PBL and authenticity for engaging deeper learning that has purpose.  I hope to capture these ideas in my sequence.

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Authentic learning is real life learning. It is a style of learning that encourages students to create a tangible, useful, quality product/outcome to be shared with their world. Once an educator provides a motivational challenge, or student selects their initiative, it is imperative to nurture and support the necessary criteria, planning, timelines, resources and support to accommodate student success. The teacher becomes a guide on the side or an event manager, a facilitator that co-creates with their students. Processes become the predominant forces while skills, knowledge and behaviours are activated in real-world, relevant contexts.

Authentic learning is actualized learning. It is only when student initiatives are directed towards a genuine audience, beyond the classroom, that learning becomes truly activated. When learning initiatives are designed for initiatives that permeate the classroom walls, then lasting, transferrable learning begins to take hold. It can be achieved by going out to the community, by inviting a community audience into the school setting, or directing it towards an audience in the global, digital or electronic domain.

When relevant and meaningful learning initiatives are skilfully designed and implemented for a genuine audience, in a realistic context, then innate neurological receptors to learn are heightened. The closer learning is to real-life scenarios, the more optimal the learning experience will be.

Authentic learning engages all the senses allowing students to create a meaningful, useful, shared outcome. They are real life tasks, or simulated tasks, that provide the learner with opportunities to connect directly with a relevant world.

Instead of vicariously discussing topics and regurgitating information in a traditional industrial age modality, authentic learning provides a learner with support to achieve a tangible, useful, quality product worth sharing with an audience in their world. Instead of prioritizing the curriculum expectations and its contents over experience rich processes, authentic learning chooses fertile, student driven, community connected audiences to create experience rich products for and then strategically align relevant curricula connections to them. with our environment that creates experience

As Albert Einstein stated, "The only source of knowledge is the experience." Yes we are complex, intricately wired beings with a vast network of neuro-receptors that serve us on our unique journeys through life. Regardless of our analytical talents, physical prowess, perceptual abilities or health status, we all come with plethora of innate skills and deficits. It's only when we're exposed to opportunities to explore, engage, create, reflect, interact socially, emotionally and cognitively

exposure to Authentic learning emphasizes 'experience nurturing' that provides longer lasting, transferable skills to an individual. It is more relevant, motivational and personal.

Our greatest short coming in education these past years has been to ignore the brain research that is richly available to us that affirms that implementing multi-sensory activities, pursuing meaningful tasks, exploring a variety of skills with real world applications is optimal learning and that it needs to be practiced regularly.

A student sitting at a desk, taking notes and regurgitating curriculum content uses a minute amount of their potential. In general, students learn to sit quietly, respond in turn, follow instructions and complete tests for a control teacher. Without relevance, personal connection, multi-sensory engagement and purpose, content driven curricula begins to evaporate once the test has been taken. You remember studying for exams, cramming, finding memory cues and rote to consolidate information? Do you also remember how fast you started to forget those brain bytes once you left the exam room? If it's not relevant to you, it not lasting, transferrable learning. If learning is not based in the rich processes that only experiences can provide, it's static learning.

This classic, industrial age approach has been used since the mid 1870's to produce a work force to facilitate mass production lines that were to become prevalent in the next hundred years. That era ended in the 1980's if not before. If all we do is sit at a square table, with a square piece of paper, in a square room with departmentalized lessons and timetables, then what are our education systems really producing? Brain-based research shows that by using all senses it maximizes the learning experience. The key being - the experience!

Interacting, manipulating, exploring, collaborating, discussing openly and sharing for meaningful reasons while having ample time to nurture a greater depth of reasoning and creativity is optimal learning. It's learning that sticks. It's learning with roots!

In an authentic learning model the emphasis is to prioritize the processes over the content. It's about allowing students to pursue individual learning opportunities that are unique to their interests, through relevant, real life journeys. The emphasis isn't just about regurgitating content for a unit test, it's about developing a set of culminating skills sets, within a realistic timeline, using self-motivated, natural inquiry methods to create a useful product to be shared with a specific, meaningful audience. It's about engaging students into full fledged, multi-sensory contexts that nurture their learning experiences in significant, lasting ways.

More importantly, the learning journey that ensues is life altering as students are aligned to meaningful processes that nurture non-perishable skills rather than the short term memory exercises that a content driven curriculum demands. Don't get me wrong, content is important but when using an authentic learning approach content is recognized as it is properly intended to be - a portfolio of records, research, plans, lists, notes, reading, contacts, drafts and support information that will become the artifacts of the authentic journey.

With the advent of the internet, content is readily available to everyone. We can reference, cross reference, research any topic at rapid speeds and do it at any time when it's relevant for us. We can access it through mobiles, tablets or computers as we access information as it is needed and when it is needed. Is it still relevant for a teacher to predominately dispense content to students, then evaluate the regurgitation of that content given as the main assessment of a person's learning ability?

Authentic learning allows for students to demonstrate their skills through a series of unfolding learning processes with a definitive, quality product that they can actually demonstrate. At no time in history has information (content) been so readily available to the masses, but it's the synergy and processes of engagement that defines our human experience. That's where the real value in education lies. Students need purpose. They need to feel useful and be a valued part of a community. They need to feel connected and real-world contexts allow for these elements to be activated and groomed. By nurturing 'purpose' we are providing the channels for students to recognize their unique talents and see their inherent value in society. Thus, individual and community self-esteem becomes a priceless byproduct of a successful authentic learning initiative.

I believe that education systems must reevaluate the weighting of its' curricula as it must reflect a working balance between learning processes and properly aligned content? Remember, processes take time to learn but once they're consolidated they're impact lasts for a long time. Processes require skill sets that are developmentally prioritized and require skilled instructors to transfer these skills. If you want students to engage in deeper thinking opportunities, then time is one of the most valued commodities. Are education systems willing to forgo the number of content expectations in their curricula in favour of embracing processes that require more time commitment? Good teachers are quick to point out that their curricula covers so many topics and expectations that there isn't enough time to cover everything appropriately. The more topics, the more content and the more time it takes to cover everything in the curriculum, and mathematically it just doesn't work. The rules dictate how the game is played, so to cover every expectation educational systems rely on the "here's the information" and "here's the test" mode of education. It's an efficient way to deliver content quickly so that all expectations are lawfully covered. It's an over used exercise in short term memory skills and cognitive cueing tricks, but its not a fulfilling, meaningful way to learn. Unless students are immersed in relevant, multi-sensory activities with specific, audience directed outcomes and culminating skills sets, then what students are learning is not sticking.

With the recognition of STEM, STEAM, Maker education platforms in the past decade, it demonstrates a deeper appreciation and commitment to relevant processes. This is very promising!

The quality of research, subject knowledge, skill development, expert consultation, dedication and resourcefulness that goes into making a product is directly related to the quality outcome of that product.

Is It Authentic Learning or Not?

Remember, in true authentic learning an outcome designed to interact successfully with a community is the goal. When this happens a whole new layer of emotional, academic and skill set developments take place. There's a big difference between preparing a skit for some classmates verses a dramatic production for the public. There's a much larger personal investment required when a student becomes an ancient Egyptian tradesperson producing tools and wares, while working at a market stall to an interacting audience than holding up a Bristol board display and explaining ancient Egyptian information on it. Best of all, the student retains information from this type of multi-sensory, authentic learning experience longer because it's real, it's connected.

Authentic learning is not project-based learning nor is it constructivism. These models of education were designed within the classroom context. Although at times they stepped successfully into the world of authentic, and are extremely useful tools in moving closer to an authentic learning approach they are not authentic learning models. Authentic learning is intended to successfully interact with a community; by going out to interact with a community or the community coming into the student community to interact. That's why it's authentic! It's a quality outcome or product of significance that is intended for community consumption or betterment.

Kudos to those teachers who provide project-based, simulation, game-based learning opportunities, Maker, STEM and STEAM. These are significant, engaging initiatives that provide the best kind of classroom learning and education systems need more of you. They are fantastic opportunities in providing working instruction and to model inquiry method, team work and sound skill development. What I'm encouraging is to take the next step. Consider a project to be developed for real world applications beyond the school walls, for a viable audience. Not every time, but perhaps once in a school year attempt an authentic event. For those who have stepped into the realm of authentic you'll understand how this impacts everything. It will impact the initial discussions, the consultation, the refined skill sets, networking and the professional development a student and teacher requires.

Think about it. Every important, relevant thing a person requires to achieve a goal in life has a quality outcome or product attached to it with an intended audience. Pitching a business proposal requires an in depth presentation that reflects relevance, meaning and purpose. If it's backed by tangible proof, even better. A prototype of a new invention requires a quality, tested, working device or apparatus to gain support in the marketing world. An environmental assessment requires extensive research and impact studies of connected, like-situations from around the ecological world.

A grade five teacher once shared a wonderful collection of student written and recorded persuasive "Saving Energy" stories with me. They even designed CD covers in art lessons for their project. The work, learning and planning by teacher and student was tremendous. Upon being asked however, "What did you do with all that great work" she replied she'd listened to the CD in class and put the CD covers up on the bulletin board. This is a great example of integrated learning but it's not authentic learning.

By planning beyond the classroom, a whole new level of understanding and relevant learning takes place. What if the CDs were intended for sale at the nearest mall? Now, an in depth degree of consultation needs to be pursued. Mall managers, local radio stations, graphic designers, bar codes for selling, copyrights local recording studios and marketing experts will need to be considered. That's the major leap an educator needs to take to enter the realm of authentic learning. What if the recordings were uploaded onto a website and promoted to leaders of energy conservation intended to be used to educate the surrounding community on "Energy Saving"? What about an "Energy Saving in Our Homes" campaign within your community? These are only suggestions but with a little brainstorming and with the right people, it wouldn't be long until a list of exciting possibilities could be created. Why are they exciting possibilities? Because they're real, relevant and worthy!

One of the reasons I started to use the term "authentic learning" was that I recognized that the term "outcomes-based" education, William Spady's model which shared a number of characteristics, was being recognized mainly for it's outcome statements. His strategy of reforming an education system, built on students performing a hierarchy of skills was diluted down to the content-based, outcome statements. Again, it became more about the content than pursuing the meaningful journey of authentic education. It isn't hard to understand why this happened because that's how it would be perceived by policy makers who's successes were affirmed in a content driven system.

Authentic learning is not another edu-trend or buzz word. It is real life - real world education and when appropriately implemented it can provide optimal learning opportunities to all students. I have noticed that some educators are misrepresenting the term "authentic" to qualify their student's inferred learning. If learning is engaged solely in classroom settings, referencing real world examples only, it's vicarious learning, - it's not authentic learning. Authentic learning is relevant learning, activated in a real world context to a real world audience.

​If you've ever initiated a business or an organization, you'll have a far better understanding of the enormous work and planning an authentic learning event requires. As in real life - these initiatives take an enormous amount of time, dedication, consulting, flexibility, networking, research and communication to launch. Yes, to launch! Once the launch has been initiated, then the real work starts. As guide on the side, or project manager you're now having to coordinate, educate, coach, consult and communicate with your students and parents in shaping their successful roles while pursuing the larger goal of your signature event. As well, you're going to require some additional professional development directly connected to the skills required in your authentic initiative.

Creating a Signature Event

A signature event is when tangible product is shared with a genuine, relevant audience. For example; if your high school senior class planned to design, construct and test their solar cars, organizing a Solar Car Derby with a spectator audience at a nearby racetrack would be a worthwhile signature event.

To fully immerse the students in an authentic learning experience they would be expected to prepare for the event with community experts, consultants, parents, teacher colleagues and community leaders. The more the students participate in the organization of the event, the more ownership and pride the students will have towards the event.

By providing students with the opportunity to have input at all levels allows for the distribution of learning at all levels. If there are rules involved, then student representatives need to be part of the derby rules development, refinement and approval. Students will possess a better grasp of all aspects of the authentic event when they are personally connected to the complete journey.

While the signature event is being planned, the students are honing their knowledge and skills to create their own quality product to share. It all depends on the type of product being created that will dictate how much class time can be used to design and create their products.

This style of learning is labour intensive and time consuming but it is the most worthwhile education you can provide students. Once you've implemented an authentic event, or simulation, the impact on student development is remarkable. The great thing about authentic learning is, it's a context that allows for student's uniqueness to be highlighted. Whether it's a specific skill, academic or a social/behaviour issue, an authentic setting, appropriately implemented, provides a very rich personal growth setting.

This new, authentic learning approach, however, requires another breed of educator. An academic as well as a life skilled, co-creator. A real, "life" long learner with skill sets and experiences that can facilitate a creative, diverse, well structured, team-based, open-ended, criteria based and globally directed education program. Ask yourself if you're on the way down the enlightened authentic learning road? Are you working towards sharing a tangible product beyond the class or bringing the world into your schools either electronically or for real? The authentic experiences you provide your students is also the beginning of an unforgettable learning journey in your life.

The Tale of Two Lyres

Pictures of Quality Products (below)

I had just completed an authentic event, a Medieval Market Living Museum one April evening. It was the culmination of one and half months of preparation as the students demonstrated their personas (roles) as medieval trades people in medieval Coventry, UK. The students worked diligently away at their trades during this open house, all academic levels to their potential. Their goal was to create an authentic medieval trade persona, an authentic costume that reflected their trade role, an authentic working trade tool or product and a market stall to share their wares. The room was decorated with beautiful tapestries, murals, personalized heraldry banners and a large book display of the student's collective work/activity sheets carefully spiral bound.

In authentic learning paper collections are referred to as portfolios. These portfolios were carefully done and included an extensive variety of curriculum content and authentic learning preparation sheets. Spelling, creative writing, math, art, drama, technology, problem solving and reading comprehension work sheets were all included. Nearly five hundred visitors came through our themed room and only about ten people even looked through the spiraled bound unit booklets. They were clearly displayed, beautifully presented and featured eye pleasing, colorful covers.

Visitors were far more interested in interacting with the trades people, hearing their stories, watching them work, than reading the notes that helped them attain their goal. Really, it is this kind of instance that affirms the reality that content and paper curriculum isn't as important as we believe them to be. It's important, and it certainly has its place, but the applied knowledge is far more important. The doing consolidates all the learning that went into the event.

​Somewhere along the continuum of learning development the jot notes became more important than the tangible, meaningful applications of the education purpose itself. In the end, it's the applied demonstration that counts. "It's all in the doing".

Student Behaviours When Immersed in Authentic Learning

Is it realistic to think that authentic learning of this nature can be done, ongoing, throughout the school year? Of course it can and there are school's that engage in authentic approaches already. Maybe one day authentic learning may be the motis operandum of the majority of schools, but realistically not yet. With the current school classroom structures, resources, timetabling, standardized test demands, physical classroom limitations, administration directives and the plethora of curriculum expectations as they are, it's difficult to do.

If, however, an educator can attempt one or two authentic learning events in a school year then that's a fantastic start. At year's end you and your students won't regret it and it won't be long until you're looking forward to starting your next. There are different degrees of authentic learning events from simulations shared with your community to complete, real world integrations. Start small, then develop your authentic learning undertakings over the years to come. Let other educators, parents and community members observe your authentic learning events. Remember, a well implemented authentic unit/event is the best kind of promotion for this rich style of learning.

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Authentic learning is real life learning. It is a style of learning that encourages students to create a tangible, useful, quality product/outcome to be shared with their world. Once an educator provides a motivational challenge, or student selects their initiative, it is imperative to nurture and support the necessary criteria, planning, timelines, resources and support to accommodate student success.

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This has some great digital simulations to visual and model science concepts.

Scooped from https://padlet.com/janelleplant/t470muq4dno4phjr
this resource has some great ideas for virtual experiments in math and science

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Over 1.1 billion simulations delivered

160interactive simulations

97language translations

3001teacher-submitted lessons

Founded in 2002 by Nobel Laureate Carl Wieman, the PhET Interactive Simulations project at the University of Colorado Boulder creates free interactive math and science simulations. PhET sims are based on extensive education research and engage students through an intuitive, game-like environment where students learn through exploration and discovery.

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Founded in 2002 by Nobel Laureate Carl Wieman, the PhET Interactive Simulations project at the University of Colorado Boulder creates free interactive math and science simulations. PhET sims are based on extensive education research and engage students through an intuitive, game-like environment where students learn through exploration and discovery.

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A collation of 10 instructional evidence based teaching strategies heavily based on Hattie and Marzano research. A great reference to consider when unit and lesson planning.

Scooped from: https://padlet.com/janelleplant/t470muq4dno4phjr
Its a great resource to check on and compare with strategies to ensure we get the most out of our pedagogy.

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This article is about wearable technology and the integration into classrooms. These  devices have the potential to bring different perspectives to understanding physical activity while incorporating ICT. These devices also create variation and excitement which could increase student engagement and participation.

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HPE-Research in Health and Physical Education

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In this modern era filled with technological advancements, the way we interact and learn is evolving around technology. Today, it is hard to go anywhere and not see some sort of wearable technology in use. These devices are not limited to just social influence but can have a significant impact on learning outcomes in the education setting as well. As we know, building healthy habits from a young age is critical to developing healthy behaviors across a lifespan to increase performance, longevity, and well-being. Healthy behaviors include actions that have a positive influence on markers of health and wellness. Physical educators strive to positively influence their student's health by promoting physical activity in various forms to build lasting habits and skills to thrive.

Pedometers are commonly used for tracking step count and accelerometers can add the ability to evaluate the intensity of movement as well. Physical educators can incorporate these new devices to bring a new perspective to comprehending physical activity in an innovative and quantifiable manner. There are many benefits to integrating tech use into the school curriculum but there are some things to consider as well. Data is data but what we do with it will ultimately depict what message is sent to the user. Although there is a negative association with obesity and markers of health, young people can begin to make false connections with having to look a certain way to be healthy. Also, some people view this method of data collection as invasive and are reluctant to participate in the new trend. We now know that there are some potential benefits and drawbacks to incorporating wearable technology, but now the question is, how is this actually being incorporated into the physical education curriculum and what are the perspectives of the teachers who are using it?

This study in the Greater NYC area collected data with 6 different public schools and 12 certified physical educators using interviews, observation, and classroom artifacts to determine teachers' perceptions of incorporating digital technologies in PE and how they influenced pedagogical practices. These next sections represent the primary themes of digital technology in the physical education curriculum.
Seamless Integration
With the rise in technology and young students already having an attachment to their mobile devices, integrating technology into the curriculum was a seamless transition. The integration of digital technology was not a method of replacing the current lesson plans but rather an augmentation of their current objectives. Notably, however, the technology became a novelty and although the students embraced the addition of the devices, the enthusiasm wore off as time went on and the devices became part of their routine. Variation and excitement are key to sparking the interest of young students and keeping them motivated to participate.
Ideologies and Pedagogy
Teacher's practices are typically shaped based on ideologies and experience instead of full on philosophies. This means that pedagogical decisions are influenced by context and come from various backgrounds. Many, however, are influenced by the central mission of SHAPE America to impact the health of young children through physical education. Finding innovative ways to educate and promote physical activity is at the heart of what many physical educators stand for. Using technology gave a unique way to comprehend the benefits of physical activity and allowed students to recognize the connection with their everyday lives. For example, this study noted a particular participant connecting the number of steps that they accumulated simply by going with their family to go pumpkin picking.
Barriers to Implementation
With every good idea comes barriers to implementation. In this study, the devices were accompanied by an online portal that would be visually inspected as part of instruction. Some teachers faced the challenge of having little experience with computer-based instruction and utilizing computers in the curriculum. Although the lack of experience was a potential barrier, most teachers embraced the changes and were willing to integrate technology into the classroom. Some issues reported may be noteworthy throughout general public-school environments. Firstly, the wealth of the school district along with available resources is a limiting factor to successfully integrating technology into education. Devices can be costly and training to be able to properly implement new strategies should be provided to allow for the highest likelihood of successful implementation. Lastly, parental control of device use and caution on privacy concerns should be understood as a challenge to bringing technology into the physical education curriculum.

Citation: Marttinen, R., Landi, D., Fredrick, R. N., & Silverman, S. (2019). Wearable digital technology in PE: Advantages, barriers, and teachers' ideologies. Journal of Teaching in Physical Education, 39(2), 227-235. https://doi.org/10.1123/jtpe.2018-0240

You can also read the full article here for free: https://www.researchgate.net/publication/336079937_Wearable_Digital_Technology_in_PE_Advantages_Barriers_and_Teachers'_Ideologies

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Wearable digital technology - HPE-Research in Health and Physical Education

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Physical educators are challenged to integrate technology into their curriculum during the current growing tide of technological advancements. This entry brings teachers' perceptions of incorporating digital technologies in physical education and the influence this had on their pedagogical practices. The author will discuss teachers' barriers and students' learning and engagement using pedometers in physical education classes.

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Range of ICT related Classroom Management Tools

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Range of ICT related Classroom Management Tools

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Various types of classroom management tools that can assist in behaviour management, timing, and name selectors.

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Behaviour Management Tools

Tools such as ClassDojo are great for making students alert and keeping them on task.

ClassDojo

ClassDojo helps teachers improve behavior in their classrooms quickly and easily. It also captures and generates data on behavior that teachers can share with parents and administrators.

Teachers can use this app to give students realtime feedback while in class and Parents can now view their child's progress via the ClassDojo app via a special invitation code.

The latest version of ClassDojo now includes a Timer and can also be used as a "Random Name Picker" (See below).

Visit ClassDojo.com to find out more.

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Tools such as ClassDojo are great for making students alert and keeping them on task. ClassDojo helps teachers improve behavior in their classrooms quickly and easily. It also captures and generates data on behavior that teachers can share with parents and administrators.

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https://www.cultofpedagogy.com/contrasting-cases/

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When I ask my students what color this is, most of them say "green." Then I contrast that green with similar colors, and ask them, "How would you describe the color now?"

Then kids generally shift to more specific descriptors like "dark green," "forest green," or "emerald green," and they add names for the other kinds of green they see.

This shift occurs because it's easier to understand the characteristics of a thing when you contrast it with other things that are just slightly different (Gentner; 1999; Schwartz, 2016). Cognitively speaking, contrasting similar things activates and helps add to your mind's map (also called a schema) of those things, which helps you recall more details, make more connections, and develop more general rules about what something is or how it works.

This is why, when someone asks you a broad question like "How would you define learning?" or "What is learning?" you might draw a blank or go to vague generalizations. But let's say someone asks, "How did you define learning 10 years ago, and how do you define it now?" or "How would you compare learning to ride a bike and learning to write a paper?" or even "What's a bad definition of learning compared to a better one?" Depending on the contrasting case, you will think more specifically about different parts of learning. You will generate more ideas and maybe make insights you would not otherwise have made.

Contrasting cases is a valuable tool for any cognitive work-in any subject area-that involves going beyond surface traits and considering deeper connecting principles, or reflecting on specific features of a thing. You can read a really helpful introduction to this approach in Schwartz et al.'s book, The ABCs of How We Learn. ( Amazon | Bookshop.org)

Here are a few examples of what this approach looks like in different disciplines:

writing instruction: Having students compare pieces of writing at different levels of proficiency improves the writing they produce (Lin-Siegler et al., 2015).

math: Having students compare two or more ways of solving a problem improves their procedural knowledge and problem-solving flexibility (Rittle-Johnson & Star, 2007).

science: Comparing similar structures in organic chemistry and phenomena in physics helps students strengthen their conceptual knowledge (Chin et al., 2016; Graulich & Schween, 2018).

In my field of study, literary interpretation, using contrasting cases is especially useful to help students enrich their reading experiences and build interpretations of many kinds of texts, including fiction, poetry, or political speech. But you can use this approach in any subject where you're studying rhetorically powerful texts, like advertisements, headlines, art, primary historical documents, and film.

In this post we'll drill down specifically to using contrasting cases for language study. I'll show you three variations of this approach: (1) Synonym Scale, (2) There and Not There, and (3) Bare Bones vs. Dressed Up.

1. SYNONYM SCALE

Let's say you're teaching about climate change in a science class, or propaganda in a social studies or English language arts class. You are interested in having students consider the rhetoric around climate change. Maybe you start by asking them to focus on the common phrase global warming, and you ask them how that phrase might affect public understanding of climate change.

As a teacher, you probably already have a mental framework, or schema, for language about climate change, so you already know other phrases people use to describe climate change (like extreme weather or climate crisis). Because of this knowledge, you are probably already using contrasting cases when you consider the rhetorical effects of the term global warming. So you might already have in mind that, compared to other terms, the phrase global warming could make climate change seem less urgent than it is, or might make people think positively about it, since people like to be warm.

But in many cases, our kids don't yet have those schemas to draw from. And it's really hard to think about the effects of something in isolation. So when you ask how global warming might affect public understanding of climate change, your kids might say kind of obvious things like, "The phrase global warming will remind people that the earth is getting warmer," Or "it will make them think about how the globe is changing." Here is where you can use contrasting cases to help students build their schema and better understand how language can work to mislead, manipulate, rally people to action, and more.

First, look for synonyms that help highlight the ideas you are interested in exploring. If you want to help your kids think about rhetorical effects, you'll look for synonyms that highlight or downplay the urgency of climate change. You can draw on your own and your students' knowledge, or go the interwebs for possibilities. For example, after a quick synonym search, I chose mild and extreme synonyms like climate change, climate emergency, climate fluctuation, climate arson, extreme weather, increase in average temperature, and global heating.

When you share these synonyms with students (and ask them to contribute more), they can better conceive of and articulate the effects of the original phrase, as well as all the others, and further understand tone, connotation, symbolic meaning, and other aspects of textual and rhetorical interpretation that contribute to people's understanding of science. You can offer them sentence stems that help them articulate these effects, if you like.

Next, you can teach kids to put synonyms on a scale anchored by positive effects on one side and negative effects on the other (a similar approach is used in Up-Down-Both-Why). In pairs or groups, kids can debate the order of those terms (like, is warming more alarming than heating?) based on their interpretations of the terms' connotations. They can also consider how context matters (positive for whom? negative effects on what?), how familiarity with a term might diminish its impact, or how different words suggest different levels of agency or blame.

The strategy is especially useful for helping students understand the relationship between language and power.

For example, try contrasting carefully chosen synonyms for "woman" as more positive or more negative can help students think about gender, power, and language: female, muñeca, gal, doll, woman, broad, sister, lady, chick, señora, chica.

Contrasting recent news headlines about political or social issues can help students reason about bias more generally, as with this comparison of the left and the right's framing of Florida's "Don't Say Gay" bill.

Contrasting cases works beyond individual words and phrases. You can compare whole essays, poems, short stories, or news articles to help students develop general criteria for writing, or see nuances in authors' choices. For example:

If you want kids to identify the conventional features of a college essay, show them a bunch that all have the same fundamental elements. Ask them to see what the essays have in common. By identifying those commonalities themselves, they are more likely to remember and be able to implement them, as compared to you listing the elements or showing them just one example of a good essay.

Let's say you want to help kids explore the effects of some aspect of personal narrative writing, such as descriptive imagery. Write 3 to 5 short beginnings or endings that are all about the same topic, but are different because they have too little or too much detail, strong verbs and weak verbs, and so on. Ask students to debate which narrative works best or is most powerful, and why.

Contrasting several translations of a text can help students explore the nuances of each, and the effects of particular authorial choices. Many language teachers do this kind of work already. In my ELA classroom, many students speak Spanish and English, while some others speak only English. In a unit on versions of love, all my students looked at two translations of Pablo Neruda's "Tonight I Can Write (The Saddest Lines)." The multilingual students discussed which translation they felt best captured the meaning and tone of the original. The English-only students learned from their classmates and were still able to compare the two translations and explain which they felt worked better for them. I usually look online for texts with more than one translation, like this one with a bunch of translations of Neruda, or these contrasting translations of a poem by Wang Wei. Also, this book contains a number of poems in different languages with several English translations each.

2. There and Not There

If you'd like kids to think about the effects of specific authorial moves, you can use contrasting cases to present the original version alongside one where you've gotten rid of those moves. Then your students can compare the "there" and "not there" versions. Doing this helps students interpret and maybe even more deeply appreciate or critique the effects of the original.

For example, consider an excerpt from Martin Luther King's classic speech, "I Have a Dream." Let's say you want students to reflect on the effects of King's use of repetition in the text:

But one hundred years later, the Negro still is not free. One hundred years later, the life of the Negro is still sadly crippled by the manacles of segregation and the chains of discrimination.

But if you ask kids, "What are the effects of this repetition?" they may say something like "It really helps you see how many years later it is," or "It makes you see how things are still the same after all this time." Both of those observations are valuable. But again, it's really hard to comment on the effects of something without being able to compare a rich stock of examples that are already in your head.

Try getting rid of the repetition and asking your kids which speech seems more effective, moving, persuasive, and so on.

The comparison gives kids more to work with. When I have done this with my students, one or two kids usually say they prefer the shorter speech-it's more efficient. Or I myself make that argument. Then other kids can counter with their interpretations of the effects of the structure of the speech, which means they are building an interpretation for a reason-to help someone else see what language can do.

3. Bare Bones vs. Dressed Up

In another version of contrasting cases, you can help your kids learn to boil a text down to its essence, and then compare it to the original to better understand the effects of the original. This approach works well with longer pieces of literature, essays, or speeches. For example, I asked my students to try to articulate the essence of Toni Morrison's Beloved. One wrote, "Slavery is really, really awful, and it can ruin you, and you have to be super strong to find yourself even after you've escaped."

This student captured the essence of the book really well. But if you can capture the essence of the book so well, what's the rest of the book doing? Why not just read that theme statement and skip the novel?

You can use contrasting cases to have students explore how authorial moves and specific details of a story contribute to the overall effects and themes. With Beloved, for example, you can compare the addition of each event, each setting, each character, and each turn of phrase to the bare bones version of the text. What does the ghost of Beloved actually add to the story? How does it help a reader feel and think? Why include the setting of 124? What does that add to the bare bones of the story? And so on.

Or back to the college essay. No matter the particulars, every essay is essentially saying, "Take me!" If you ask your students to compare the bare bones version of an essay with other dressed up versions, you can help them evaluate the effects of different features, like sentence structure, tone, or specific narrative choices.

Here are some final sentences of college essays (no room here for entire essays). I made them up with a particular purpose: to have kids think about the effects of tone and specificity in different endings. Along with the bare bones version, I created a generic version that is also kind of humble (#1), a detailed one that seems to connect with a story told in the body of the essay but is not as humble (#2), and one using an arrogant and inadvisable tone (#3). Kids will probably think all of them are pretty cheesy. Some will point out that the bare bones version might actually do the job. Either way, students can contrast them based on how well they paint a picture, how humble they are, or other criteria you choose. They will develop more sophisticated understandings of writing for effect, and they will build up the schemas that good readers and writers need.

When students are working on their own college essays, contrasting cases can work as a writing tool. Have them write terrible versions and then revise.

There are many ways to tweak contrasting cases for your kids. The big take-aways are: 1)It's easier to think about the effects of a thing if you can compare it to other, similar things; 2) Contrasting cases allow you to see underlying rules and big ideas; and 3) You create contrasting cases by finding or creating examples that differ in one important way: different solutions to the same math problem, different photographs of the same subject; different headlines about the same event. When we want our kids to do deep and critical thinking, let them contrast different examples. One is often not enough.

References

Chin, D. B., Chi, M., & Schwartz, D. L. (2016). A comparison of two methods of active learning in physics: inventing a general solution versus compare and contrast. Instructional Science, 44(2), 177-195. https://doi.org/10.1007/s11251-016-9374-0 [ PDF]

Gentner, D., & Namy, L. L. (1999). Comparison in the development of categories. Cognitive Development, 14(4), 487-513. https://doi.org/10.1016/S0885-2014(99)00016-7

Graulich, N., & Schween, M. (2018). Concept-Oriented Task Design: Making Purposeful Case Comparisons in Organic Chemistry. Journal of Chemical Education, 95(3), 376-383. https://doi.org/10.1021/acs.jchemed.7b00672

Lin-Siegler, X., Shaenfield, D., & Elder, A. D. (2015). Contrasting case instruction can improve self-assessment of writing. Educational Technology Research and Development, 63(4), 517-537. https://doi.org/10.1007/s11423-015-9390-9 [ PDF]

Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561. [ PDF]

Schwartz, D. L., Tsang, J. M., & Blair, K. P. (2016). The ABCs of how we learn: 26 scientifically proven approaches, how they work, and when to use them. W. W. Norton & Company.

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Contrasting Cases: A Simple Strategy for Deep Understanding

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Listen to the interview with Sarah Levine: Sponsored by Listenwise and Read&Write by Texthelp This page contains Amazon Affiliate and Bookshop.org links. When you make a purchase through these links, Cult of Pedagogy gets a small percentage of the sale at no extra cost to you. What's the difference between Amazon and Bookshop.org?