Insights on Mathematical Writing at EARCOS Teacher’s Conference 2017


Blackboard Classroom Symbols Lesson Pie Chart Math
Blackboard Classroom Symbols Lesson Pie Chart Math From Maxpixel CC0 1.0

Writing and Common Core: a Math Made in Heaven?

There are some people who might still argue that mathematics has little to do with writing and that both manners of thinking should be taught as separate disciplines.

These people are most likely not mathematics teachers teaching in schools that follow the Common Core Standards for math. (Based on this article by a physicist-father of two young Common Core math learners/writers, they may not be scientists in the fields that require the most amount of math understanding either.)

A person has only to see the types of word problems that grade 2 students are learning to understand and solve to realise a distinct connection between math and language learning. There is therefore a need to teach math IN CONJUNCTION with language, keeping in mind the functions language must serve in mathematical discourse. For example, in order to effectively EXPLAIN their thinking when discussing math, students need to use cognitive academic language to infer, describe cause and effect, evaluate, and sequence.

As language teachers, a job that especially teachers in international schools must embrace, we try to integrate speaking, listening, reading, writing and presenting as much as possible. We believe that language develops best when these strands are taught in tandem. As such, it is logical that if we want students to skilfully use language to explain their mathematical thinking, they need to have opportunities to write in the course of studying mathematics as well.

Writing in Mathematics: It Can Be Done! at EARCOS Teacher’s Conference 2017

I had the good fortune to attend a session at the EARCOS Teacher’s Conference this year entitled “Writing in Mathematics: It Can Be Done!” led by two proponents of teaching students to learn and write for math, Jessica Balli and Dr. Patrick Callahan of Callahan Consulting, an organisation dedicated to helping schools make solid bridges between math and communication.


The session leaders began with some surprising information about student perceptions of why we as teachers ask them to show and explain their math work; most students in interviews confessed that they believed teachers wanted them to prove they had not simply copied someone else’s answers.

Next they threw some shocking Common Core statistics out:

  • Only 37% of students produce written explanations of math work that show grade level understandings (based on exemplars provided with various CC based scope and sequences like New York Engage);
  • By 8th Grade, in one study, fewer than 4% of students could write a grade level mathematical explanation of an algebraic problem.

Writing for Mathematics as a Genre of Writing

We then began to discuss mathematical writing as a genre unto itself, which immediately perked my attention as I have been thinking along these lines for a few months now since I have begun teaching grade 2 EAL students strategies for comprehending math word problems. Colleagues and I have been developing lessons in which students learn to visualise both the information offered by word problems AND the question(s) they ask us to answer. Questions we have raised in this development are:

  • Why do we present especially younger students with levelled texts during reading instruction, but then expect students to decode and comprehend math word problems that are usually written at a higher level? It seems in order for students to understand math word problems, we have to intensify or modify our reading instruction.
  • Why do we teach students to draw a picture or diagram as a problem solving strategy without making explicit links to the reading comprehension strategy of visualisation?
  • How are pictures that students draw to help them solve a problem different from pictures they might draw to show they understand the information, numerical and otherwise, presented in the problem and the essential question the problem begs us to answer?

Ms. Balli laid out that math writing as a genre should include features and characteristics such as:

  1. precision of language and definitions;
  2. clarity and logic;
  3. the statement of assumptions;
  4. vivid description of quantities;
  5. accurate and logical comparisons.

In hearing this, I thought of how these features combine many language purposes: identifying, sequencing, comparing, explaining, hypothesizing, inquiry/seeking information (when it comes to understanding the question part of the problem), analyzing, and evaluating. One school district in the USA has even described an academic language function for problem solving (p.12) that incorporates many of these language functions.

In order to effectively describe math thinking in written form, students need to be aware of these language functions and they need have practiced listening, speaking, and writing for these purposes. As I listened to Ms. Balli and Dr. Callahan, it seemed to me that I had to keep this goal in mind as I planned instruction for my grade 1 and 2 EAL students. I felt that students also needed to be made aware of how to tie all these language purposes together when writing for mathematics.

Big job…but it is good to see the big picture.

A realisation hit me…at least some part of math class needs to look more like writing class (or vice versa)!

Practical Teaching and Learning Strategies for Mathematical Writing

Ms. Balli then led us through a series of activities she uses in her classes to encourage the development of mathematical writing through analysis, writing about, and discussing of Bongard problems.

(A Bongard problem (examples here) is a type of puzzle, named after the inventor of such puzzles – M.M. Bongard, in which the inquirer tries to identify rules that three pairs of figures on the left side of the puzzle follow by comparing them with three pairs of figures on the right side of the puzzle that do not follow the rule.)

Balli’s students analyze Bongard problems using a template she has devised and write out solutions to them in which thinking is described. She stresses that:

  • these Bongard puzzles do not need to explicitly connect to content being studied in math (remember the objective is for students to gain proficiency with writing about mathematical thinking);
  • the puzzle element of these problems are engaging for students;
  • writing about such problems, which may have more than one solution, is a low stakes activity for her students, a fact that she highlights throughout her lessons. Students have opportunities to go back and revise initial hypotheses about solutions using the template, much as they would revise any piece of writing to increase clarity and logic. This encourages students to be risk-takers in their writing and explanations. She also allows for variations in thinking and teaches students to tolerate and appreciate ideas that are different from their own.

(For more on using Bongard problems to teach math reasoning and writing, check out Jessica Balli’s blog!)

Beyond Bongard…

While Bongard problems may provide these classes with a low stakes introduction and practice of mathematical writing, Balli also has students solve and write about content based problems as well using another template she created that is meant to be used as part of a peer feedback exercise. You can see an example of problems on this template here. Again, in this situation, a problem is posed. Students write about their reasoning behind their solution. Students then share these written solutions with peers, who offer feedback based on a set of discussed criteria and using key sentence frames.

Guidelines for good feedback on math writing include:

  • Be specific;
  • Avoid opinions;
  • Think about what feedback you would find helpful;
  • Giving feedback does not equal being mean;
  • Good feedback pushes our peers to clarify their logic, language, and meaning;
  • Giving good feedback takes practice.

Sentence frames that Balli encourages students to use while writing peer feedback include:

  • I don’t understand…
  • How did you know _____ ?
  • I agree with ______ but…
  • What if you tried…?

Then, as a final step, students use this feedback to revise their initial written reflections.

Final Thoughts

I loved in this session how Callahan and Balli really helped clarify the connection between writing and mathematics. In particular, I think it is useful for teachers to understand that just as good writing instruction involves drafting, peer editing/reviewing, feedback and revision, so should writing instruction in terms of learning to express mathematical understanding. If the purpose of expository writing is to express understanding about a topic clearly, so is the purpose of mathematical writing to express understanding of math concepts and information found in problems and solutions in a way that allows a reader to understand the writer’s train of thought.

The main challenge, like usual, is where to fit everything in? It seems that sometimes the best way, as is often the case, is through integrating and blurring the lines between subjects. Again, we can devote some of our writing time to math writing. We can devote some of our math time to writing.

Lastly, having already discussed Bongard problems with lower Elementary teachers, I think that these problems may be slightly advanced to begin working on with younger students. Perhaps simple math analogies such as can be found in this series might be more appropriate to have students express their understanding through writing.


Google Virtual Reality Academy at EARCOS 2017 (Part 2 – Google Expeditions)

Sketchnotes of Google Virtual Reality Academy with Jay Atwood of EdTechTeam.

This is part 2 of a two part blog entry about my foray into the Google Virtual Reality Academy with Jay Atwood of the EdTechTeam at the EARCOS 2017 Teacher’s Conference here in Kota Kinabalu, Borneo, East Malaysia.

In part 1, I discussed how to assemble a Google Cardboard VR viewer and my initial exploration of the app necessary to use it. In this part, I will discuss how Google Expeditions works with the Google Cardboard apparatus. I’m also going to give my basic conclusions about what I loved about this app, potential places for development, possible ways to use the technology in classrooms now, and questions I have!

Google Expeditions – The Set Up

Using the Google Expeditions app, teachers and students can go on a virtual journey of visual discovery. In our VR session, in which Jay was the leader, we saw both aspects of various “Expeditions” currently available on the app: a guide (teacher) view and an explorer (student) view (although there is no reason why, for example, a student couldn’t be a guide as part of perhaps a summative assessment).

An expedition is, at this time, a Google-recognised and sanctioned “trip” through a slideshow of 360 degree photos, some of which might contain embedded sound recordings as well, seen in a seemingly 3D manner with a Google VR viewer of some sort, Cardboard or otherwise. (I say “Google-recognised” because at this time, normal users cannot simply create and post an Expedition. *We should all keep pestering Google, however, as Google, according to Jay Atwood, listens to user feedback.)

When users open the downloaded and installed Expedition app, they are first asked whether they would like to be guides or leaders; again, in most early cases, a teacher will be the guide while students will be explorers. Once a “role” has been chosen, Google offers some recent expedition suggestions, or you can search for a topic of your choosing. At this point, Jay pointed out a technical issue: it is best to download expeditions you want your students to explore onto ALL devices that will be involved – this includes the teacher’s phone, and all the phones (or iPads, more on this in a bit) that students will use in the course of this immersive, interactive experience.

(Note: Explorers and Guides need to connect to the same wifi router. While most schools will probably not allow these sorts of connections to be made on a school wide network, I got around this by making my iPhone an available wifi hotspot, linked my iPad and an Oppo phone to it and led an expedition using that technology set up. Otherwise, you can buy a small portable wifi router for connecting.)

Quick Procedure with Images for How to Set Up and Run an Expedition

Materials Needed:

  • Mobile phones and Google Cardboards for each student OR iPads/tablets to be shared between two students. Each device needs to have the Expeditions app downloaded.
  • iPad or tablet for teacher with Expeditions app downloaded (possibly with Personal Hotspot enabled).
  • If Personal Hotspotting is not a possibility, then you need a portable wifi router
Mifi 4g
Portable Wifi Router


  1. Teacher needs to log into the same wifi router that students will be using with the guide device, and/or enable a Personal Hotspot on the iPad.
  2. Teacher needs to open Expeditions app on the guide device, choose to Guide/lead, search for an Expedition topic of interest for students, and preferably download that collection of 360° degree photos.
  3. Students need to open Expeditions on their devices, choose to be Explorers/followers on an available Expedition (which by default will be the one available on the guide’s device if all parties are logged into the same router/hotspot). *If school does not have phones and Cardboard, students can go on an Expedition with a teacher/guide by hitting the “expansion” square towards the bottom right of the screen. If students have phones, they can insert phones into Cardboard now.
  4. Teacher waits to see the smiling faces of all his/her followers pop up.

    In this image, from the Guide view, you can see there are no followers yet by looking at the people icon in the top right corner.
  5. When all faces appear, teacher can rotate image and click on certain items. Students will see arrows pointing them where they need to go.

    In this image, in the Guide view again, you can see a follower in the form of a happy face. This happy face indicates where that follower is currently looking. The target is where the Guide has clicked on his/her screen as an area of focus for the followers.
  6. When all smiling faces of explorers are around an area of the image for discussion, teacher begins discussion.

What I appreciate about Google Expeditions (especially when combined with Google Cardboard):

Google Expeditions is a wonderful new way to engage in visual discovery with students. With carefully chosen Expeditions that connect to units of study, teachers can truly provide a much more immersive experience of many topics related to science, research, geography, history, and art. (A complete list in spreadsheet form can be found here. At this point, due to my relative inexperience with virtual reality, I find it truly amazing to put on the Cardboard apparatus. I’m sure this will be the same with students young and younger.

Google Expeditions could also be a fabulous tool, when in the research or “Finding Out” stage of inquiry, for students to describe the form of objects, or places they encounter while exploring. For example, my grade 1 students are currently inquiring into how the oceans are a resource that we have a responsibility to conserve. On the whole, through preassessments, I realised that most students have never experienced a coral reef through snorkelling. As coral reefs are such vital aspects of our oceans, I want students to have some experience in describing their characteristics and residents as a prelude to further learning. After a quick search of Google Expeditions, I found what seemed to be a perfect experience for my students to go on entitled “Preserved Oceans”. If we have the chance, I’m sure students would love to examine different corals and fauna in famous reefs from around the world.

What might be developed further:

As users, we cannot currently CREATE Google Expeditions that are fine tuned for our students. Likewise, students cannot create Expeditions to show their learning. I can think of possible reasons for this. Perhaps Google wants to keep the quality of Expeditions fairly high at this point in order to keep interest rising. To make a high quality near 360+°, you have to have a somewhat specialised camera rig up (which Google is allowing users to make 3D movies with that can be viewed online with Cardboard and YouTube – to be discussed in a further blog entry).

Still, I can see amazing possibilities in the future for students to demonstrate understanding of any variety of concepts through creation of VR videos and Expedition like experiences.

Also, the amount of technology necessary for students to fully engage in VR experiences through Google Expeditions can be prohibitive for many schools, despite the fact that you can get the viewers for a variety of prices (evidently for as low as $1.00 from some Chinese based manufacturers). For the best experience, a teacher would ideally have a class set of mobile phones and a class set of Google Cardboards. This can be gotten around somewhat if there is a half a class set of iPads or tablets with Expeditions, as I detailed above. However, the VR experience is much reduced in this case.

It would, in addition, be great if the quality of images presented to the eyes through Google Expeditions and Cardboard were slightly less pixelated. I realise that images must have begun as massive files that need to be compressed to go up onto the cloud in useful form, however it would be great to see a bit more clarity in the future and the potential for an explorer to zoom in while wearing Cardboard.

Another teacher at this conference mentioned that even for grade 7 students and after multiple uses of VR in class, it has been difficult to push students past the “wow!” factor of using apps like Google Expeditions into more calm usage that allows students to study some of the images presented through the app.

This might best show the need to use this type of experience even more, to get students and ourselves used to seeing the learning potential of Expeditions.

Stay tuned for further parts of this post in which I will discuss other apps related to Google Cardboard and Virtual Reality such as Cardboard, Cardboard Camera, and how to use Cardboard with other apps such as Within, Panoform, and YouTube.

Of course, I must explore further first.

A happy Monday morning reality, virtual or not, to all of you!

The author learning how to take hot air balloons into space.
The author learning how to take hot air balloons into space.

Blown Away by EARCOS 2017 Teacher’s Conference Keynote Speakers

Ms. Kim Phuc and her refashioned image to express what her life has become, a beacon of love and peace.
Ms. Kim Phuc and her refashioned image that expresses what her life has become, a beacon of love and peace.

EARCOS Executive Director Dick Krajczar and his team pulled together an incredible group of keynote speakers this year at the Teachers Conference in Kota Kinabalu.

We were left speechless at the Thursday morning opening of the conference with Kim Phuc’s presentation on her heart wrenching and ultimately transformative life experience.

The world learned of Kim Phuc after she became the victim of a horrendous napalm strafing of a South Vietnamese village in 1972. A Vietnamese reporter, Nick Ut, captured the image and helped get Kim medical attention as she fled the attack. Terribly burned but miraculously surviving this bombing, she recuperated for years going in and out of hospitals for reconstructive surgery. After Vietnamese unification, she led a strictly monitored life as a national icon and living memorial of the war before she defected to Canada in the mid 1980s.

Today, Kim Phuc has established a nongovernmental organisation, the Kim Foundation, dedicated to improving the lives of children from war zones around the world through projects that create schools, hospitals, and rehabilitation programs.
Révolutions Françaises - Christophe Galfard (22083202364)
On Friday, Dr. Christophe Galfard left us all gobstruck again with his frank and down-to-Earth discussion of astrophysics. Galfard gave educators a quick tour of his book, The Universe in Your Hand, simultaneously inspiring all of us to cultivate a love of mathematics and science in our students young and younger!

Not to let us down gradually, the EARCOS team had saved the dynamic duo, Aaron and Kaitlin Tate for Saturday’s last morning meeting in the Pacific Sutera Grand Ballroom. These two educators, entrepreneurs, entrepreneur enablers, superhuman humanists (can I say – SUPERHEROES?) wove an enthralling, horrifying, redeeming, and simply awesome tale with a message of how to save the world through finding and empowering local leaders.

(PS: Note to self…NEVER complain about a “rough” day at school again…)

If you have the means, get to next year’s EARCOS Teacher’s Conference, if only to be inspired for the rest of your professional life by the stories of people like these!

Important Websites to Check Out


Education Changemakers

Kim Foundation

Christophe Galfard’s Website




Google Virtual Reality Academy at EARCOS 2017 (Part 1)

Greetings from Sutera Harbour, Kota Kinabalu, and the EARCOS 2017 Teacher's Conference!
Greetings from Sutera Harbour, Kota Kinabalu, and the EARCOS 2017 Teacher’s Conference!

What began as a relatively subdued morning of training that included interesting but expected icebreakers soon turned into a day in which our eyes were popping out of our skulls here in friendly Kota Kinabalu, Malaysian Borneo at the EARCOS 2017 Teacher’s Conference.

Welcome to the Google Virtual Reality Academy with Jay Atwood from the EdTechTeam!

Atwood primed the educator-laced hall with a sequence of Four Corners discussions aimed at gauging our preexisting perceptions of virtual reality, then raced through the history of this long developing technology, from it’s relatively primitive beginnings in military and NASA simulators, to the Oculus Rift and Google Cardboard of today.

Then came the foray into the unknown for many of us.

It all began with a box…a little CARDBOARD box…

IMG_9449 (1)
Such an innocent box.

Once the Cardboard was out of the BOX, we popped our smart phones in and started to explore, and the pro-tips from some of the more experienced teachers began to drop.

*Note: At the behest of Atwood, we had all loaded the following apps already onto our phones: Google Expeditions, Google Cardboard, Google Streetview, YouTube, and Google Cardboard Camera.

Richard from Jakarta "Intercultural" School gave some fantastic advice for how to strengthen and adolescent-proof these humble Cardboards...
One teacher gave some fantastic advice for how to strengthen and adolescent-proof these humble Cardboards…with popsicle sticks and clear plastic “grease guards”! Also, fall risk (phone may fall out – if you have not yet realised the purpose of the rubber band.)
Putting the virtual reality into Cardboard.

When I loaded up the Google Cardboard app, slipped my iPhone 5 into the flap, and put the head set on…this was the scene I initially saw that Google has created as a demo, albeit 3D due to the split lenses.

*Spoiler alert!?!?

False alarm…I have not given away any of the best details of this first demo virtual reality ride. Check it out yourself!

Being teachers, we did not just “tinker” with this tech…we recorded our findings in a tinkering Google Doc!

Sketchnotes of this half of the Google VR Academy Session

Check in later today for part 2 of this post in which I will discuss how Google Expeditions, Google Cardboard Camera, and other apps work with the Google Cardboard apparatus. I’m also going to give my basic conclusions about what I loved about Google Virtual Reality, potential places for development, possible ways to use the technology in classrooms now, and questions I have!

EARCOS (East Asian Regional Conference of Schools) 2017 Conference Blog Updates Coming

Just a note to all my readers. I will be attending this year’s EARCOS Conference, “Connecting Global Minds”, in Kota Kinabulu, Malaysia from March 29 to April 1, 2017.

Possible meetings I (plan) to attend are as follows:

  • ETC Google Virtual Reality Academy with Jay Atwood
  • Writing and Mathematics – It Can Be Done with Jessica Balli and Patrick Callahan
  • Step Into the Lab: Gamification, Differentiation, and Documentation, Oh, My!Supporting Teachers from a Positive Discipline Perspective withEric Schoonard and Mark Crowell
  • Drumming Activities as Metaphors in Math and Science with Martin Robinson
  • Priming the Brain for Learning with Julian Thornbury
  • Learning Vocabulary Through Drama and Games with Hamorn Lau
  • Forest School Supporting the Development of Social Skills with Karen Killeen 
  • Redesigning the Learning Experience (Design Process) with Wesley Przybylski
  • Making Time to Create with Kim Cofino
  • Failing Well – Cultivating Growth Mindset, Self Directed Learners and Self Paced Learning with Yojin Chung
  • Engaging Young Writers – Strategies That Work with Alis Gorcea and Emily Bevington
  • The Neurobiology of Learning withRhonda Wildeman and Darrell Sharp
  • Moods Before Mindsets with Jacob Humes

As part of my PD agreement with my school, I will be trying to blog about a few of these meetings, particularly ones that involve a tech element. Please check the blog on those days for updates!