## DragonBox Math Apps: Teaching students to think like mathematicians

What it is: I adore the DragonBox math apps! I was first introduced to DragonBox through their Algebra app. The app takes away all math anxiety by teaching algebraic concepts without numbers or algorithms. It is genius! You can read my review of that app here. DragonBox Numbers is for kids who are new to numbers, counting, addition, and subtraction. DragonBox BIG Numbers is the next level app of Dragon Box Numbers. As they play, children learn about how big numbers work, and how to perform long addition and subtraction. In DragonBox Elements, students will secretly learn geometry. By playing 100+ puzzles, kids will gain a deep understanding of the logic of geometry. As they play, students will actually be recreating the mathematical proofs that define geometry. What I love about the DragonBox suite of math games is that they are unlike any other math apps. You won’t find mindless repetitions and quizzes of math facts, instead, these games help teach students how to think like mathematicians. This is learning math through exploration.

How to integrate Dragon Box math apps into your classroom: The DragonBox math apps take advantage of a student’s innate curiosity through play and exploration of math concepts. Each game engages students through exploration, reflection, and application.  If you have a 1:1 iPad classroom, use these apps daily! Students can play a game, and then come together as a class to reflect and write down the rules they have learned in that “chapter.” Then, take an equation from the game and solve it on paper. See if students can connect the cards and rules of the game to the equations on paper. Solve the equations using strategies and rules from the game.

In a one or two device classroom, play DragonBox as a class by connecting your device to a projector. Explore the rules together and let students take turns being in charge of a chapter. Review rules and make connections as a class. Set up DragonBox as a math center that students can visit in a rotation where one center is interacting with the game, the next is an opportunity to record rules learned and reflect, and the last is application of the rules to an equation.

Be sure to check out the DragonBox for educators where you can download teachers guides and printable resources for each game in the DragonBox math suite. You’ll also find a great “rules” guide for the algebra and geometry games.

These are seriously my favorite math apps of all time! They encourage kids who can think and apply like mathematicians rather than kids who simply memorize a formula. DragonBox creates a deep understanding of mathematical concepts and number relationships.

Tips: While the DragonBox suite of math apps aren’t free, you can purchase them at a deep discount for educational institutions and bulk purchases. Teachers, you can get access for free to make sure that the apps are right before committing to a purchase for your students. You can also purchase apps in bundles.

## Mathigon: engage, play, and explore math

What it is: Today I was working on our inquiry block framework for the 2017/2018 school year and, as often happens with inquiry, fell down a wonderful rabbit hole that led me to this site. Mathigon is a fantastic *newish* math site (it’s still being built and added to) that brings textbooks to life. I know you’ve probably seen this claim before, but this is unlike the other online interactive textbooks I’ve seen. It’s more…alive. It’s like a personalized tutor, combined with story, and exploration. Really, textbook is the wrong word, because this is something totally new. A chat bot tutor makes Mathigon like having an additional team of teachers in the room, ready to answer questions and support your learners in real time. Real life application and narrative is part of the Mathigon DNA. This means that beyond learning the “rules” of math, learners are actually invited to engage the concepts, play with them, explore them in context, and find out what other concepts they are linked to. Rather than a linear approach, Mathigon lets students explore math in a more organic way through interest, linked ideas/concepts, and in a ‘down the rabbit hole’ approach. There are very few math sites that I’ve come across that truly support an inquiry approach to learning math, Mathigon is one such site.

How to integrate Mathigon into your classroom:  There are several ways to use Mathigon. Students can get a personalized math curriculum that adapts to them and offers recommendations based on what they are interested in and their understanding of different concepts. They can begin from several places: exploring the applications of math in every day life, the link between math and origami, Eureka Magazine (published by Cambridge University), through problems and puzzles, through fractal fiction, or through courses for grades 6-college.

The Treasure Hunt is a complete PDF Kit that can be downloaded and printed out. Split your students into teams and send them on an epic math treasure hunt through your school (available in primary and secondary levels) where each of the clues leads them to another.

Fractal fiction is particularly cool because it lets students explore mathematical concepts through interactive narrative of popular films including Alice in Wonderland, Oceans 11, and Harry Potter (the latter two are coming soon). You really have to go experience these to really understand the brilliance of how Mathigon has combined story with math exploration.  From the site: “The key to successful teaching is captivating storytelling – through real life applications, curious examples, historic background, or even fictional characters. These interactive slideshows combine an engaging narrative with beautiful graphics – explaining mathematical ideas in the context of popular stories and movies. They can be watched individually or be presented in classrooms.”

I cannot say enough about how impressed I am with the vastness of what this site brings to the classroom. Even if you don’t have the capacity for each of your students to have an account with Mathigon, the site can be easily adapted for the one computer classroom (as a center activity). Much of the content could also be explored as a whole class with a projector-connected computer.

Tips: I’ve found that really well done content for grades 6-12 (and beyond) in math to be severely lacking. This is a welcome addition to the math teachers tool box of resources!

## Photomath: Scan math problems for immediate step-by-step instructions

What it is: Photomath is an app available on the App Store and Google Play. With Photomath, students can scan a math problem and learn how to solve it with step by step instructions and an answer. The app includes a photo calculator (take a photo of a math problem, it gets solved in an instant), handwriting recognition, step-by-step instructions, and a smart calculator. For added features, Photomath+ includes complete step-by-step instructions, colorful explanations of the math, and extra math knowledge.

How to integrate Photomath into the classroom: On the surface, the Photomath app might look like the ultimate way for students to “cheat” their learning. After all, they can snap a picture of any math problem, get step-by-step “show your work” instructions, and the answer in an instant. When I look at Photomath, I see each device with the app as another teacher in the classroom. When students get stuck, or need to check their work/understanding, not only do they have access to the answer, but also to the process. They can see exactly where a mistake has been made, and even get an explanation about why the process is what it is. It also changes the math class from being procedures-based, and empowers teachers and students to engage math from a problem-based, practical aspect. Since students have help with the procedure of how to solve a problem, they can engage math as a mathematician, identifying the problem that needs to be solved, using number sense to understand the problem, and with Photomath ensuring that the procedure they apply has been solved correctly. It definitely could change the goal of your math class, rather than just finding the answer, assignments may become more practical application in nature.

Consider using Photomath as a check-in station where students can go through their own work and identify where they may need support. The app offers immediate step-by-step guidance, when students don’t understand the guidance, you know instantly that more teaching is needed.

Tips: If your school has homework, this would be a great app to recommend to parents! When they get stumped, the app can be a sanity saver.

## Which One Doesn’t Belong? K-12 Critical Thinking and Problem Solving

What it is: Which One Doesn’t Belong? is a site I learned about from @TeamBaldwin today. This math site is for students of all ages and challenges them to look at a set of four images and determine which image doesn’t belong and justify their answer. The best part of this site is that each problem has multiple correct responses that can be justified. Students have to think through the differences that they see and then make logical decisions and be able to explain it to others. There are three different categories for Which One Doesn’t Belong including: Shapes, Numbers, Graphs and Equations.

Yesterday, @TeamBaldwin used the site this way:

This is a class of kindergarten and first grade students! @michellek107 will be blogging more about the class experience on the class blog, Architects of Wonder if you’d like to read more.

Tips: The graphs and equations appear quite challenging, but even young students can begin making observations about the types of graphs that could lead to some higher-level math discussions.

## We choose the moon: assessment without labels

“With willing hearts and skillful hands, the difficult we do at once; the impossible takes a bit longer.”- unknown

Yesterday, @Kevreadenn shared the quote above with me on Twitter. It had me thinking about assessment and John F. Kennedy’s, “we choose the moon” (embedded above – you should watch it again). The declaration that we WOULD send a man to the moon, that we would do the impossible, seemed like an unworkable task given the technology of 1969. It is astounding to me that the technology I hold in my pocket is significantly more capable than the technology that sent a man to the moon. Kennedy inspired a nation to dream together, to achieve the impossible using the resources that were available. How can we inspire kids in this way as they quest for learning? How can we help those students who look at the goal of reaching the moon and think, “this is impossible, I only have 1969 technology,” to be inspired, to look at their resources in new ways and believe that they can do the impossible?

Everyone wants to know that they are “winning” and contributing to something meaningful. The declaration that a man would go to the moon was a lofty goal. It seemed incredibly  important, as a nation, that we achieve this goal together.

It strikes me as strange that the majority of assessment that we focus on in education only shows lag data. The lag data reveals to us where a student landed in a given moment, but doesn’t offer any opportunity for course correction. It also doesn’t take into account outside influences (not enough sleep, home struggles, lack of nutrition, friendship stresses, etc.). If our goals are lofty, and we want learning to be better, we will stop focusing on the lag data as the most important, and instead focus on the lead data. Lead data shows us what leads up to the learning, it gives us insight in the journey and process so that we can adjust as we go.

Grading systems need to be upgraded so that they engage students in the process of reaching lofty goals. When students are able to help set their own goals and the assessment happens as a part of learning, students are continually pulled forward in their learning. Each small win adds learning momentum that begins to snowball into something bigger. For those who struggle, the small victories reveal that they can win and they are making forward progress. This gives the confidence that all learning is possible and worth engaging (even when the end goal feels impossible). The students who find that learning comes easily aren’t halted by artificial ceilings. They aren’t left believing that the learning has ended and are encouraged to keep moving forward.

This is the “report card” that I created. It is our attempt to help kids better understand what contributes to learning. It is a helpful way to show students that they aren’t “stupid in math,” but instead help them realize that they aren’t risk takers in math. The aversion to risk taking in math is what really holds them back.

The Latin root of Assessment is assidere, which means “to sit beside.” At Anastasis, we believe that assessment is more than just a measurement, it is an opportunity for apprenticeship, a time for us “to sit beside” and guide. We’ve used various tools for assessment in our short history, our search was for the assessment tool that would offer a more holistic picture of learning. We’ve used Mastery Connect and Jump Rope, but they fell short in giving us that holistic picture because they were tied to Common Core Standards alone (and limited by grade levels). What our students do at Anastasis every day is SO much bigger and deeper than these standards, and yet we didn’t have a good way to demonstrate that. Beginning in January, we rolled out our own grading system. We call it UpGrade because that is what it felt like, an upgrade!

Our goal for feedback on the UpGrade report is twofold:

1. To give students feedback that causes them to think, engage, and reflect on their own learning process as they learn.

2. To give families a detailed account of the student’s learning journey and forward progress.

The UpGrade reports are designed to explain what students know and are able to do, rather than determining grades based on point averages. This kind of grading allows teachers to more effectively tailor instruction to students based on what they actually know and can do, no floors and no ceilings.

The UpGrade report contains the proficiency marks 1-5 (explained below). These numbers are intended to demonstrate the process of learning, students move through the levels of proficiency as learning progresses. A “1” indicates that a child is new to the learning or requires a lot of guidance. A “5” indicates that the student is able to apply the learning to new situations and make connections to other learning.

 1- Novice Concept and/or skill is brand new, student is just getting started in the learning. Student requires much teacher guidance and prompting. 2- Apprentice Some prompting or guidance is needed for the new concept and/or skill. 3- Practitioner Concept and/or skill can be done consistently and independently. Student may require occasional prompting or guidance. 4- Scholar Student can apply concept and/or skill to new and/or different situations with little guidance. Student is ready to build on the learning (next level of standard). 5- Change Maker Concept and/or skill come second nature and can be used to make connections with other learning. Students understand concept/skill and can apply, evaluate, analyze, and create using the skill/concept. Student can use skill/concept for in-depth inferences and applications.

Of course when you create a report card that looks like this, you also have to craft a grading system to populate the report card. Dang it.

This is where I had to get creative and use my limited resources to make something (that seems impossible) work. I used Apple’s Numbers as the method for creating the grading system. It is tedious work, the user interface is messy, but it allows lofty goals and the impossible.

On the back end, teachers can fill in rubrics, or learning evidence pages. These compile and end up as the final 1-5 on the image that you see above. Why did I use Numbers? It is the only spreadsheet program robust enough to incorporate the graphic above. The evidences of learning (assignments) and scores are the easy part. Getting the rubrics to work the way I wanted them to, not so much! I sent out a call for help the other night on Twitter and had some AMAZING spreadsheet ninjas step in to help me find solutions. I promised to share what I figured out…for those not interested in the technical bits, feel free to skip ahead. 🙂

This is the rubric that I was working on. I wanted it to “self grade” and then for the score to transfer as a learning evidence. It seemed like it should be simple, but this honestly stumped me for days. Thankfully my PLN stepped in to point me in the right direction!

I started out by creating a highlighting rule. This is simple in Numbers, just click on the cell(s) that you want the rule on and in the “cell formatting” pane, choose “Conditional Highlighting.” In each cell where teachers could leave a score, I wanted the cell to highlight when the score was added. I chose to add the rule “Text contains” and then the number that corresponded to the row of the rubric. Then I chose a color to highlight.

Next came the tricky part. In the cells full of text, I wanted the last cell to recognize when a teacher had typed in a number, and to add up all the numbers of the row so that we could get a total that would populate on the Learning Evidence sheet. I could find all kinds of ways to accomplish this in a Google Spreadsheet or in Excel, but none of the solutions seemed to work in Numbers. I kept getting a syntax error. I finally solved it by using the following format: =COUNTIF(B2:H2,”=*1″) In each row, I changed the number to reflect the score that it would count. This got me most of the way to what I needed, but it was only giving me a count of how many of the number were in the row, not giving me the sum (=SUMIF didn’t work). SO, I added another column to find the product of the “Countif” with the row number. Success! I’ll hide these two columns for the final grade system and just have the total at the bottom show up. This total is easy to then transfer to the Learning Evidence sheet.

THANK YOU @mathlioness @katieregan88 @mrmatera @alicekeeler @royanlee @jasonschmidt123 @benlouey @malynmawby @thomascmurray, you all are truly wonderful for spending time to help me solve this. I am seriously elated that there was a solution! Anastasis teachers will be thrilled as well! 🙂 YOU ARE NINJAS!!

When people ask about Anastasis, they are usually curious to know how we’ve broken past the barrier of labels. We have students who are dyslexic, twice exceptional, have struggled in school, are gifted, know how to play the game of school, etc. Everyone of these kids chooses the moon. They choose to do the impossible and keep moving forward. They aren’t stifled by the learning labels. They know they are more than an “A” or “F.” They start to understand that learning is not the same as a grade. They begin to understand where their hangups actually are and can work on adjusting those instead of believing they are failures.  It is amazing what happens when you take away the labels and help kids understand that no matter where they start from, there is something to be learned, forward progress to be made. They choose lofty goals. They do the impossible.

## Numberphile: a series of numberly videos

What it is: Numberphile is a series of “numberly” videos by Brady Haran.  This is the same guy behind some other great projects including periodicvideos and sixtysymbols.  These videos reveal some of the mystery behind numbers and math in fun, short snippets!  I could give a long, drawn out explanation about the site…but really, you should go have a look and play a few videos. Or, try out the video below:

How to integrate Numberphile into the classroom: Numberphile can make a math geek out of anyone, myself included!  I don’t tend to geek out very much about numbers or math, but show me pattern and reveal some of the mystery that numbers hold and I am in.  This is what Numberphile does beautifully!  Numberphile would make a fantastic opening for math class.  Start each math class with these short videos to get your brain’s math muscles working.  I’ve watched 3 videos in a row and I am seriously geeking.

Ask students to each choose a different video to watch.  Students can learn a new math “trick” or pattern in math to teach their classmates.  The goal: creatively teach the concept!  They could create their own video, stop motion animation, infographic, story, illustration, etc.  Hold a math day  (3/14 would be fun…pi day) where students get to spend the day teaching one another.

You may assume that these videos are best for older students, not so!  At Anastasis, our 2nd and 3rd graders had a ball learning about Fibonacci and will happily explain it to any who enter their classroom.  Find an area of interest and share the passion!

Tips: All of the videos on Numberphile are YouTube videos.  If you don’t have access to YouTube in your building, try one of these methods for accessing the videos:

• SafeShare TV- This site lets students watch YouTube videos without ads, links, comments and related videos.  You also have the option to crop videos and share videos with a unique URL.
• YouTubeXL- This is a service that YouTube provides that lets you watch videos on large screens without the ads and comments. Neat tip: if you time “quiet” before the YouTube url, it takes you to a safe page where you can watch a YouTube video.  WAY cool and easy to do on the fly!
• Clean VideoSearch- This site lets students search through YouTube videos without the comments, ads and busy sidebar.  It has additional features like the ability to choose how many videos you want to see on each page in your search.
• Clea.nr– This service (a browser plugin) deletes all of the obnoxious extras that hang around videos (ads, comments, related videos). You can also search YouTube without all of the extras showing up.
• ViewPure– This site cleans out all the clutter and gives you just a video.  Bonus: There is a quick button that you can add to your browser so that you can go to a video, click on “Pure” in your bookmark bar and instantly have a clean video.
• Dragontape– This service lets you drag videos into a timeline and share them easily with students.  This is great for mashing up several videos, or cropping multiple videos into one.
• Movavi– This is a video conversion service. Wonderful for teachers who can’t or don’t want to access a video directly from YouTube.  Copy/paste the url you want to convert, choose a file type, done!
• Zamzar– This is another great video conversion service.  Works quickly and easily!

## Dance, Factors Dance: Animated Factorization Diagrams #mathchat

What it is:  Dance, Factors Dance is a fantastic site to visualize factorization diagrams.  The first “tango” is inspired by the digital clock, with a separate diagram for each of the hours, minutes and seconds.  The Factor Conga is a “promenade of primes, composites, and their constituents, arranged with an aesthetically-tuned variation of Yorgey’s rules, one per second.”  I love the way these math factorization dances help students visualize numbers and Prime numbers.  Brilliant!

How to integrate Dance, Factors Dance into the classroom: Dance, Factor Dance is a stupendous way for students to visualize and think about numbers.  I Love the way that the prime numbers are depicted…so easy to see why it is a prime number!  Ask your students to explore this site and identify the patterns they notice in the dance of numbers.  What happens when a number is prime?  When a number is odd?  Even?

Dance, Factors Dance is a fun way to learn more about numbers, it is also a wonderful inspiration for finding the art in math.  How can students use the site as inspiration to create their own math dance?  Could they use stop motion animation and manipulatives to do something similar?  What patterns in math do they notice?  How can they use color and design to help them better understand math?

This site is a great one to explore as a class, as a center on classroom computers, or individually on student devices.  Students can pause the dance, rewind, and fast forward as they explore.

Tips: Be sure to watch (or fast forward) to the three digit numbers…this is where things get really impressive!  As a side note, I learned something today from a fellow teacher.  When looking at numbers in grid form, you know if a number is prime if it can only make one rectangle.  This understanding would have been helpful in math class! Better late than never 🙂

## Brown Sharpie: Mathematical Cartoons Inspired by Sharpie Fumes

What it is: Math geeks, eat your heart out…Brown Sharpie is for you!  I found Brown Sharpie by accident today as I was perusing the app store in iTunes.  Brown Sharpie is a collection of “mathematical cartoons inspired by sharpie fumes” drawn by Courtney Gibbons.  Gibbons is an aspiring mathematician and the cartoons were created as she completed her undergraduate and graduate degree.  Many of the cartoons reference higher math and may be too complex to use in the k-8 classroom.  There are a few here and there that could be used with younger students with a little explanation.  High school, college math students and math geeks are the main demographic for these cartoons. The cartoons are shared blog-style so you can search through them using the “next” and “previous” buttons or you can view by tags using the “view by…” word cloud in the right side bar.

How to integrate Brown Sharpie into the classroom: The Brown Sharpie cartoons would be a fun start to math class.  Put a cartoon up on a projector-connected computer each day for a little math humor to kick off class.  The cartoons will give you the opportunity to discuss current math topics as well as give an introduction to math concepts not yet touched on.

Why not hold your own Brown Sharpie day?  Give each student a brown fine-tipped sharpie to create their own math cartoons?  These can be shared on a class blog, website or wiki.  This will help your visual learners and artists think about math in a whole new way!  Students of any age can create a Brown Sharpie cartoon of their own!

In addition to the blog, Brown Sharpie is also a free app in the iTunes app store.

Tips: Some of the cartoons are PG-13 with alcohol or relationship references.  Best to preview the cartoon before displaying before your class. 🙂

## Study Jams!

What it is: Scholastic Study Jams is a fantastic collection of over 200 learning resource collections. Study Jams are videos, slide shows, and step by step explanations for science and math that will have your students discovering everything from invertebrates to the water cycle and the rule of divisibility.  Each Jam includes a teaching video/step-by-step/slide show, key vocabulary, and a test yourself section where they can practice what they have just learned.  Each Jam also suggests related jams where students can expand their learning and dig deeper on a subject.  To be honest, this is more like the textbook of the future that I envisioned.  I love that each concept is introduced in the context of a story.  Students learn the concept from fun Study Jam characters and can pause and rewind the learning as needed.  In the test yourself section, students can check for understanding and receive immediate feedback on their learning.

How to integrate Study Jams into your curriculum: Study Jams is a truly incredible collection of learning opportunities for students.  Use Jams to introduce your students to a new concept, or reinforce learning.  In Math students can learn about numbers, multiplication and division, addition and subtraction, fractions, decimals and percents, algebra, geometry, measurement, data analysis, probability, and problem solving.  Each topic has several sub-topics for students to explore.  In science topics include: plants, animals, the human body, ecosystems, landforms, rocks and minerals, weather and climate, solar system, matter, force and motion, energy, light, and sound, and scientific inquiry.  Again, each science topic has several sub-topics.

Study Jams can be used with your whole class as an anticipatory set for learning using an interactive whiteboard or projector connected computer.  After viewing the step-by-step, video, or slide-show check for understanding by having your students complete the “test yourself” as a class.  This can be done with personal whiteboards where students write down their answer and hold it up, a raise of hands, or student response systems (clickers).  Use this as formative assessment to guide your lesson.  Study Jams can also be used as a center activity in the math or science classroom.  Students can visit the Study Jam as part of a larger group of related activities.  In a center, students can visit individually or in small groups and self direct their learning.  For those students who have already mastered the concept, they can view related Study Jams to extend their learning.

Study Jams is ideal for students in a 1 to 1 or lab setting.  Here students can explore at their own pace, pausing and rewinding as necessary.  They can also extend their learning based on their personal interests by choosing a related Study Jam.

Can’t find a Study Jam that fits what your students are learning? Ask students to create their own Study Jam video, slide show or step by step.  Students can use tools like Animoto, Voice Thread, or Domo Animate to create their own.  Students can create their own “test yourself” using a Google Form or survey tool.

Tips: I learned about Study Jams from someone in my blogging alliance (sorry I didn’t make note of who!) If you aren’t already following these amazing blogs, I highly recommend them (alliance #1, alliance #2).  I learn SO much every day from each one of them.  If I learned about Study Jams from your blog, leave me a comment so I can thank you here!