Innovation, Inquiry, Math

A Geometry Inquiry with Attitude

One of my concerns is that math in my classroom is not as inquiry based as I would like it to be.  My students and I just began a unit on Geometry.  I gave the pretest and for the most part, students had a spattering of knowledge and the test was completed with much hair pulling and cries of “Man! I KNOW this….but….I forgot!”.  When we went over the paper, I could see a collective “aha” from the majority of the students as they started to dust off the vocabulary sitting at the back of their minds.  So, what to do?

I did some scouring of the internet and came up with a couple of really interesting reads: Angle Measurement – An Opportunity for Equity, and Inquiry Maths: A Parallel Lines Inquiry.

After reading these articles, the next day my students and I sat with the pretest and pulled the vocabulary from it.  They spent the lesson with math dictionaries, math tools, the Khan Academy, and various math text books from the classroom shelves in order to create an understanding of what these terms meant.  They found all these connections that I wanted them to know but didn’t want to just tell them: that perpendicular lines were also intersecting lines but not all intersecting lines are perpendicular.  Same with equilateral triangles and isosceles triangles (all equilateral triangles are isosceles but all isosceles are not equilateral). Some asked if they could work on their “Math Dictionaries” at home.  Others took screen shots of Khan Academy videos and added their own notes.  I told them they were preparing for an inquiry and they needed to be well equipped!

The next day, we discussed the idea of using math as a language. I drew a rectangle on the board in purple marker.  If this were to be described using the English language, I would call it “Purple Box”.  If it were to be described using Math language, I would call it “A rectangular quadrangle with interior angles of 90 degrees each (right angles) formed by a set of horizontal, parallel line segments and a set of vertical, parallel line segments. They got the idea.

I told the class that this was an open, collaborative inquiry. That meant they were free to consult any source they needed in order to extend their inquiry and that the work was collaborative in that I wanted them to build off each others ideas.  I have 18 students (I know, I am blessed!) and so I printed off 9 pictures (3 of each image) so that children had a choice of where to work in the small room.  I also wanted to be able to have them come together with other groups during the next lesson to share and compare their findings. Before I showed them the images, I shared the Success Criteria for the lesson:

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Here are the images I used, the first one from the Parallel Lines inquiry and the second and third from another Inquiry Maths Inquiry:





My students were shown the pictures, reminded of the Success Criteria for the lesson,  and were off!  It was fascinating.  They partnered up in a similar fashion to the day before when they were creating their vocabulary understandings and quickly started to use their knowledge to describe their image. I “casually” asked if anyone wanted a protractor (YES! YES! YES!).  As I wandered around I saw children reading, questioning, measuring and using their math language to describe the image in front of them.  “Can we draw on it?”  Yes!  For one group who had the star shaped image, this led to some pretty crazy coloring/marking which to my naive eye looked more like silly scribbling than serious math but I let them keep going.  One group started talking about symmetry and I found some mirrors and laid them on their table which started another investigation into where that line of symmetry actually was.

This was supposed to last 15 minutes but it was clear they had more than 15 minutes of math language in them!  As the end of the lesson neared, I asked them to briefly group with the other people who had the same image to get an idea of what others had done. Cries of, “I was going to do that next!”, “I hadn’t thought of that!”, “I forgot to put that, too! ” and  “Where did you get a mirror from?!” were heard around the room.

I have a really great class of kids but like all kids they need to be asked to think about why they do what they do and how they are behaving.  As a PYP school, we offer a values-laden curriculum so teaching about attitudes is part of what we do.  We are currently working on the culminating project of the PYP – the Exhibition.  It requires a lot of group work and one of the things I am noticing is that students need more than to be physically placed in a pod of four students, for group work to be successful.  We have been looking at the type of attitudes we expect to see at our school and I wanted them to see the connection to this in math class so I gave them the following exit slip for the lesson:

Success Criteria


Their comments were so insightful:

curiosity…because I wanted to see just how much I could write in math language

confidence…because I knew I knew a lot about this and I knew I could describe the picture in a lot of details

respect…because I listened to the ideas of the person I was working with and also added my ideas

and the student that I thought was goofing off:

creativity…because I was able to add really colorful and interesting designs to our star and it looked really good and then it also helped my group see patterns within the star and we were able to add a lot more information


I think the students were not the only ones learning something today!

To download a PDF of the lesson plan and materials used today, click here



Brain Research, Mindset

One Step Towards A Growth Mindset

A lot has been said of developing a growth mindset.  Carol Dweck is a great person to start with if you are unfamiliar with this term.  Essentially, it means that:

people believe that their most basic abilities can be developed through dedication and hard work—brains and talent are just the starting point. This view creates a love of learning and a resilience that is essential for great accomplishment. – Carol Dweck

The Khan Academy have jumped with this idea and are incorporating one simple step within their already fabulous program that has  led to a 5% increase in problems attempted, proficiencies earned, and return visits to the site.  What is this step?  One simple line of text added to a page with a math problem on it:

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Research indicates that our brains have a high degree of plasticity and as teachers, we should take every chance we can to tap into that.

What could this look like in your classroom?

I have followed the Khan Academy example and added a growth mindset quote to any printed work I hand out to kids.

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I know this is only one small step, but hey, if it is good enough for Sal Khan…

What else can we do? Everything from framing your questions, giving kids more information about HOW and WHY we are doing what we are doing, and framing your classroom with statements for the development of a growth mindset such as these examples from MindSetWorks

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Get Started: Khan Academy


Starting at a new school means re-evaluating the things I “used to do back at my old school”.  One of these “used to do’s” is the Khan Academy, which aside from sporadic usage by some teachers, is relatively new to our Junior School.

As I began looking over the (recently updated) Khan Academy website, I was once again blown away by what an awesome tool this is for teaching math.  I put together a packet of information for my team as we begin discussions on the role the Khan Academy will play in our classroom.  The majority of the packet comes from me pulling from the mountain of resources available on the Khan Academy website.  There is a ton of stuff there to help you get started on your Khan journey.  Try clicking here for a good place to begin. 

The packet I put together contains links to KA materials on their website as well as some materials that I downloaded from the website and (confession time) reformatted so the packet would look more organised.

One of the things I like the most about the Khan Academy is that it (in its own words) elevates the role of the teacher:

Teachers are critical in a Khan Academy classroom

Teachers provide the human element to inspire, motive and guide students through their learning paths. There are some people who believe that technology in education diminishes the importance of teachers. In our experience, nothing could be further from the truth. The fantastic teachers we have seen implement Khan Academy are bright, innovative, creative, and they take their classrooms to new heights. They realize that technology is a tool to help make classrooms more personalized, more mastery-based and more individualized.

Shifting from lecturer to domain expert, master coach, and mentor

In a personalized, mastery-based learning environment, the role of the teachers is elevated. Responding to the needs of many learners goes beyond classroom management and a lesson plan. Teachers must be responsive to the data and must problem solve for every learner individually.

Technology allows the role to evolve to allow teachers to minimize time grading homework or giving the same lecture, and instead use real-time data to ensure time is used most effectively. Khan Academy provides the tools so you can do what you went into teaching to do – to personally interact with and provide guidance to every student, and to engage students in collaborative activities and interactions. Technology will not replace teachers, but empower you to be great mentors to your students.


We are about to have a back-to-school night and later a parent information session regarding the use of the Khan Academy and this will definitely be information I want to share with my parents.  It has also reminded me that I need to be on the lookout for really good collaborative activities and interactions in math for my students. Which is where YOU come in! Please share your gems for thought-provoking, discussion based problem solving for math in the comments box below.  I need your help!

For the Khan Academy packet I put together for my team, click here.

21st Century, Innovation, Math

Reason #4565 Why I Love the Khan Academy

I couldn’t have said it better myself!  This is what I want to keep on an index card to pass out to those people who either have never heard of the Khan Academy (seriously?) or who scoff at it’s relevance or purpose in our education system.  In fact, I just may whip a few up when I finish this post!  Leave a comment below if you want me to send you your own!

“This has absolutely nothing to do with replacing teachers. When we talk about getting lectures out of the room, that’s because we think we can move teachers up the value chain. That they are better off forming the bonds and connections. That’s what you need a human being to do and for a really great teacher to do. Khan Academy takes some of the more traditional stuff off the plate and now, all of a sudden, the classroom becomes a richer and more stimulating experience.” ~ Sal Khan

For more of Sal’s comments, click here.

21st Century, Creativity, Innovation

More from the Khan Academy

Khan Academy Co-Deans for Art and History Beth Harris and Steven Zucker have created over 100 videos as part of the new Google Art Project, which launches today.

What is the Google Art Project?

The Google Art Project attempts to provide more access to Museums and works of Art. The Google Art Project is an initiative to provide thousands of high quality, high resolution images from museums across the globe in one place, making art’s history, meaning and beauty available in ways never possible before.

Harris and Zucker have added 90 Khan Academy videos, like the one below, to the project.  The videos offer a unique opportunity to view paintings and works of art that would otherwise be inaccessible unless you were to travel to their museum home.  The videos have close up frames – so close, that in the case of the Van Gogh Bedroom video, you can see the layers of paint on the canvas as they describe his thick, heavy style.


In addition to the videos, there is a section called Look Like An Expert which offers tutorials on topics such as The Birth of Avant-Garde, Signature Strokes and Hidden Meanings, in which you can learn all kinds of tips that you would normally pick up from the friendly docent at your museum.  Here is an example from the Signature Strokes tutorial:

Look closely. Can you read these signature styles? Which eye is from the seventeenth century, before artists experimented more aggressively with color and brushwork to construct the forms of the face?

The coolest part (in my opinion) is that once you have explored the extensive galleries, learned all these inside tips from the experts and really brushed up (pun intended) on your art history and appreciation skills, there is a whole list of DIY choices for you to actually USE this resource in hands on ways beyond simply looking.  The options are diverse and really cool, utilizing both online content, tech tools and good old fashioned art supplies – depending on which project you choose.  Here is the current line up of DIY projects:

  • YouGallery Try on the role of curator by creating an Exhibit in the Google Art Project, choosing a style or artist and deciding the order in which people will view your collection. 
  • Rebus Create links between one piece of art to the next.  Create your own collection but with a twist – have your audience try and find the key that connects one piece to the next. 
  • Remix If you could bring one piece of work home, how would you display it?  What would the room look like?  And what would someone wear whilst in your room, looking at your art, in order to complement the style of the art work?
  • Wildlife Photo Expedition You are on safari – in the art museum!  How many creatures can you find amongst the art works?  Take pictures for your scrapbook! 
  • The Lens of Now How would the works of art from history be transformed if they had been created today?  What would their narrative be?
  • Materials Matter The materials that art is made from can change the way we view it.  Pick a favorite from the gallery and then grab some materials and recreate it in a different media.
  • Inventing Color Create your own gallery based on the Color Wheel, choosing images that represent segments of the wheel.
  • Scavenger Hunt Send your friends on a Scavenger Hunt through the Google Art Gallery! 
  • A Funny Thing Happened on the Way to the Museum Every piece that makes it to a museum has lived somewhere else beforehand.  This information is included in the ‘details’ section of each art work.  Think beyond this and create your own story behind how a piece of art made it to the museum. 
Whilst looking through all of this amazing stuff, I came across something I hadn’t seen before which I thought was too cool not to pass on: Sesame Street’s brilliant, “Three Primary Colors”. Maria Popova of the fabulous blog,Brainpickings, wrote of this short stop-motion film, “it might just be the finest treat for budding designers.”  Enjoy!

21st Century

Parents Guide to 21st Century Learning

I just downloaded the Parents Guide to 21st Century Learning from Edutopia. It is a relatively short document that would be a good starting point for any parent (or teacher) wanting to wrap their head around this whole ’21st Century’ education business.  Primarily, it’s purpose is to highlight the importance of the four C’s:



critical thinking


both in and out of the classroom.  Page 7 of the document offers the following suggestions for parents:

As I read through this, I thought of some other options – at least for the first three on the list:






Idealist is on a mission and has a clear vision for where they are headed.  This is a great organization and a great way to help connect you to your community – wherever in the world that may be.  In their own words:

Our Vision

We would like to live in a world where:

  • All people can lead free and dignified lives.
  • Every person who wants to help another has the ability to do so.
  • No opportunities for action or collaboration are missed or wasted.

Our Mission

Idealist connects people, organizations, and resources to help build a world where all people can live free and dignified lives. Idealist is independent of any government, political ideology, or religious creed. Our work is guided by the common desire of our members and supporters to find practical solutions to social and environmental problems, in a spirit of generosity and mutual respect.










I am a huge fan of the Khan Academy and so excited to hear of their Summer Camp – the Khan Academy Discovery Lab (even the name is awesome!).  I believe that spaces are filled for this coming summer but it would definitely be something to look into if you have a curious 6th through 9th grader in your life. Back in June last year, Sal Khan blogged on Bringing Creativity to the Classroom.  Here are some projects he had kids doing in the name of math and science – projects that convinced him that this is how math and science in schools ‘should’ be:

  • Having the students estimate the value of coins in a a large jar given any tool at their disposal (including sample coins, scales, rulers, spreadsheets).  We then took pictures and posted them on Mechanical Turk to see if the “crowd” estimate was better (we payed 0.01 per guess) even though their information was not as good.  This led to a fascinating discussion of information and noise and when crowd estimates could be better than experts.
  • Six students played a variation of Risk called “Paranoia” Risk where every player had the secret mission to eliminate one other player from the board (you only knew who you had to eliminate; you had to try to figure out who was charged with eliminating you). Once a player is eliminated, the winner is declared and the game is over.  All of the other non-playing students were each given $500 in monopoly money and a colored strip of paper representing each of the players on the board.  At the end of the game, a colored strip is worth 100 if that player won and 0 if they lost.  The students then traded slips as the game progressed.  Several students independently developed spreadsheet models based on the probabilities involved in the game.  Led to a deep discussion around information in markets and when bubble behavior develops (for example, several securities irrationally traded above 100) .
  • We played a variation of freeze tag where we changed the size of the playing field and the number of “freezers”.  Students predicted and observed how the dynamics of the game changed as more freezers were added and the threshold needed to freeze everyone.  They were quick to draw analogies to other areas of science.

And here is what ‘camp’ looks like, Khan Style:








For a really good, first hand look at how we can help people in economically challenged environments, is a great place to start.  Kids (or adults or families) can loan out money and then be paid back once the person taking the loan has used the money to set up whatever entrepreneurial venture they have proposed. The repayment rate is 98.94% and once repaid, you can pull your money back out or you can reinvest it in someone else to make a difference in another persons life.  Recently featured an article on Kiva and similar programs, indicating that there was bias in the way donors chose their recipiants.  This would be an interesting article to read before donating.  More interesting, would be to first go to Kiva and pick out your potential recipiant and then read the article to see if you fit the pattern suggested by the research!   (Either way, go to Kiva and give someone a hand).


OK…that covers items 1-3 on the list above.  Stay tuned for further investigation and alternative ideas for the other items on the list!

Design, Innovation, Math

The Best Way To Teach Math

I read an article that expressed extreme dislike (understatement) for the Everyday Math curriculum.  This is the same curriculum we currently use in school and was the same I used in my last school, as well as another of the schools I have taught in.

Do I love it?  No. Do I loathe it?  No.  I actually find it to be quite adequate as a basis for developing math skills within a class of students IF you make a few adjustments and additions to support student learning.  Having taught in one school where your progress through the math book was monitored on a day by day basis with repercussions for not being in line with ‘state expectations’, I know that words like adjustments and additions are challenging if not impossible to entertain.  If you are fortunate enough to work in a school that allows you the freedom to teach and facilitate learning amongst your students AND uses the Everyday Math curriculum, here are some points I have picked up along the way…

  • If you teach in an IB school, you are going to be asked to look for ways to authentically integrate data handling, measurement and geometry into your units of inquiry.  This is difficult but not impossible to do if the basis of your math program is EDM.  I haven’t done it yet, but I would love to sit down and look for the connections between these areas of math and our units and pull them into their own ‘math unit’.
  • Again, for IB school teachers, you are not expected to integrate number, pattern and function into your units of inquiry.  Think of these components of math as the ‘alphabet’ – addition, subtraction, multiplication, division, fractions, decimals, percentages…these all should be taught in their own right and EDM provides multiple ways of sharing the ‘why’ behind these processes.
  • Use your discretion!  As a teacher, you can decide how you present the content to your students.  I find the EDM methods for some things quite confusing.  I also find some of them really help me explain the ‘why’ to kids who then really get it.  If I have kids who are already solid in a method that works for them, great!  Just like we make kids eat a balanced diet, I ask my kids to at least follow along but the freedom is theirs to use whichever method works for them -mostly it is the one they are comfortable with but sometimes they will change it up and go with something new that makes even more sense to them.
  • Play the games – or at least some of them.  They are great for engaging discussion and when disagreements arise, perfect grounds for ‘forcing’ kids to demonstrate and explain their understanding of a concept.
  • Don’t start at the very beginning of each lesson.  It is just a guide, a plan.  It doesn’t say that if you don’t follow it precisely, the world will end!  There is actually a lot of good problem solving and analytical thought required in much of the enrichment and study link pages.  Flip things around and have students work collaboratively on these in class when they are supported by yourself and other students and can engage in discussion.
  • Use the Khan Academy!  It is brilliant.  I can’t figure out why more people don’t use it.  I have had a few parents say they don’t like it.  That they don’t want their child sitting in front of a computer and “can’t they just have a worksheet for once?”.  Seriously?  Who doesn’t want quality online support by way of the videos, hints that show the ‘how’ of solving a problem, instant feedback on their work and some really fun badges!  If you assign Khan videos/exercises for homework, your students are prepared for the more analytical problem solving challenges at school that ask them to apply their understanding.    We are fortunate, here in Boise, to have Sal Khan coming to town to share his thoughts on the changing face of education.  Educators were asked to share their experiences with the Khan Academy – here is an excerpt from my letter:

I never liked math as a kid.  As a teacher, I love it.  I love the mystery of it, the methods and the madness!  I am really grateful to Sal and his Academy for helping me develop a greater love for math.  I love that my kids get the chance to challenge themselves and review their work on a regular basis.  I love that the videos really explain the concepts in a clear way.  I love that I have more time for doing and less need for telling.  I love that I can see my kids whiz off into the math matrix with a huge interest and passion for developing their math skills.  It is “cool” to love math in my classroom and I have had kids log on each week, on average, for about 2 and half to 3 hours between Monday and Thursday nights.  The “requirement” is 15 minutes a night.

I think my own classroom math program could do with some work.  I would like to be a little more dedicated to integrating with our units in a more meaningful way and relying less on the unit tests at the end of each unit of work in the EDM book.  I would love there to be a few more hours in the day so we could play more of the games together.  Perhaps these could also become homework tasks?  What I would also like to do is to focus more time and importance on remembering to talk math with kids.  This article reminded me that more important than which book or website your ideas come from, make sure you are engaging kids in math related discussions – even (or especially) when it is not ‘math’ time.  A great way to do this with kids is to tap into the booming market of infographics.  They are everywhere and are a perfect way to bring math into the discussion.  Take a look at these as examples:

I see two uses for these – discussion and inspiration for creation.  I can already hear a few cries of ‘but we don’t have the technology to do that!”.  Who says you need technology to make an infographic? Ultimately, it is a graphic that shares information (we used to call those ‘charts’ or ‘diagrams’ or even ‘graphs’ when I was at school).  There are people in the wings, ready to launch infographic capabilities to the masses: here are ten sites you could use, a great resource called, or five kinds of infographics and free tools to make them.  (I can see another post coming soon after I try some of these out!).

Until then, take a leaf out of the book of this Portugese design firm, who created the following ‘low tech’ infographics that are awesome and totally do-able in any classroom and would again, encourage and stimulate mathematical conversations both in their construction and sharing:

What works for you in your math classroom?  Or as a parent, what works for you at home?

I know people may have extreme feelings towards Everyday Math, and if I were given no leeway in how to implement in my classroom, if I were not allowed to supplement where necessary (in my professional opinion) and to skip parts that seem somewhat pointless, then I might have some of those feelings too. Words like accountability and consistency are just the words that got Everyday Math its stronghold in many schools.  In order to have a prescribed set of units of work, complete with tests that even a muppet could pick up and spew forth over the the classroom, schools needed to subscribe to ‘something’ – and Everyday Math has become that ‘something’ for many schools.   What is missing – and can’t even be found by substituting the Khan Academy (gasp!) – is good math teaching.  I am not sure which comes first but if teachers were more confident in teaching math, pulling from a variety of resources to create a balanced math diet, student performance in math would rise, parents anxiety about their child’s level of achievement would dissipate  and administrators would breathe, have confidence in their faculty and let them do their job.

Am I going to throw my hands up in the air and rage against the Everyday Math machine?  At this stage no.  Am I going to blindly plod through each page in the text (do I even have to answer that one?). Ultimately, I trust that educators and administrators and parents can work together for the greater good – the kids.