Publié dans Aha moments/ moments Aha, Conversations

Convergent or Divergent problem solving (#mtbos30 – 23)

Robert Kaplinsky wrote a blog post that really made me think.  Please read it first, then come back here for my thoughts.

The Difference Between Open-Ended and Open-Middled Problems

Ok so now that you read it (if you’re like me, you probably skipped reading it), here are my thoughts.

I am a passionate lover of math ever since I was a kid. Until I started teaching, math was all about getting the solution. For me it was like a puzzle. For others it was torture.

When I first saw Dan Meyer’s TEDtalk years ago (can you imagine it was filmed in 2010!), I knew like most of us that there was something there.

You know you’ve baited the hook right?

His 3-acts-math approach let the students ask the question, dont give them facts until they ask for them, let them work together, let them be in a state of productive struggling and have a discours with the whole classe at the end.  Different strategies emerged.  Sometimes math fights broke out.  The solution wasn’t always the same as the real world solution.  It was great!  The teacher and the students could concentrate on how they got to the solution as opposed to the solution itself.

For me that is a great way to engage students in a productive struggle based on the fact they deeply wanted to find out « how much time will it take to fill the tank! »  It was very much a low-floor/high ceiling problem. Every kid has an opinion or a gut feeling of the solution or an idea about the problem. And every kid could participate in the classroom discourse.

For me this type of problem is an open-middle problem.  There is always several ways to think or go about the problem.  It creates great class discourse since multiple strategies converge to the solution of the problem. It allows the teacher to assess student’s thinking and push the learning further and deeper.


Then, I met Marian Small.  I am lucky enough to meet her with facilitators of the 12 French school boards of Ontario.   She introduced us to open-ended problems.  They way she made us think was amazing.  She is a master at coming up with problems that start at a same point, with an entry point for every student but diverge to different solutions at different levels of complexity and learning at the end of the lesson.  To her, it’s ok to live with students being at different levels of understanding since as individuals we are all at different levels all the time.

In her open-ended approach, the classroom discourse is very present and needed.  Different ways of thinking are presented.  Different solutions are offered, patterns seem to emerge out these differences and resemblances.  Math fights break out!

The math is no longer about the solution, it’s about the deep thinking and deep learning going on.  It doesn’t need to be about real world problems either.  There could be no context at all and still be a deep learning experience.

For example, take linking cubes.  You have a group of red cubes and twice as many yellow cubes.  How many reds and yellows do you have?

Another example:  Two perpendicular lines cross at the point (10,5).  What could be the slope of each line?

She has 2 books full of these.

Here is an audio excerpt of Marian Small talking about creativity and mathematically interesting problems.

I want kids to love to learn and to love to think!

Marian Small states that open-ended problems should meet the goal of differentiation in order to meet the needs of all students. In order for this to be achieved the teacher needs to create or develop a single question that is inclusive, allowing students to approach the problem using different approaches and methods. The divergent question posed should allow for students at different stages of mathematical development to benefit from the problem that is solved, because students can solve it the way that makes sense to them.

Converge or Diverge?

I was pedagogically torn!  So, as teacher should I converge to a solution or diverge to different thinking with my students?  Should I do converge sometimes and diverge other times?  If so, when should I converge or diverge?

I have decided that a mix of both are important.  But as a whole, we have to start letting the students do the thinking.  We should be orchestrating the conversations to help them discover (not cover) our curriculum and push deep learning further along.

I’ll take a great example that came out from a Dan Meyer and Andrew Stadel 3-acts-math.  The cup stacking problem.   The way the problem was shown, the solutions would converge to a single solution.

Then came Alex Overwijk!  He opened up this question by not showing a way of stacking the cups.  He just asked his students « What do you notice? What do you wonder? »

cup stacking Alex Overwijk

His way of asking the question really opened the problem to be open-ended!  Multiple solutions could be possible.  Divergence coming out of a Convergent problem!  Look at the possibilities!

I remember a tweet from someone that opened it up even more!  Imagine the possible ways of stacking these cups?

I saw this first hand in a classroom where the teacher had class of 16 students from grade 9 remedial level right up to grade 10 academic kids. The teacher presented the cup stacking problem.  It was amazing to see that everyone in the class started with the same problem but they quickly diverged to different ways of seeing the problem, all based by there proximal zone of development.  The class discours at the end of the lesson was rich and complex and every student’s thinking was pushed further than their solution.

Make problems more open!  Make them low floor but raise that ceiling as much as you can!

Raise the ceiling!


Found this list of blog posts about cup stacking! 

Leave a comment. Laissez un commentaire

Publié dans 3 acts math

Mon jardin – 3 acts math (#mtbos30 – 22)

Voici une autre situation d’apprentissage en 3 actes.  J’aimerais bien avoir des commentaires si jamais vous l’utilisez dans votre salle de classe. Qu’est-ce qui fonctionne bien? Qu’est-ce qui peux être amélioré?

For my English readers, let me know if you want this translated.

Acte 1

Que remarques-tu? Quelles questions te poses-tu?

Comment disposerais-tu les briques pour avoir le plus d’espace de jardin possible? Combien de sac de terre est-ce que j’aurais de besoin?

Acte 2

Acte 3

Je me suis servi de 5 sacs de terre!



Si on double le montant de brique, qu’arriverait à la surface du jardin?  Combien de sac de terre aurais-je besoin?

Leave a comment. Laissez un commentaire.

Publié dans Conversations, Liens

Being the facilitator you wanted when you were a teacher (#mtbos30 – 21)

When I became a facilitator for my board I am very glad I had some great people to help me figure out how to accompany a group of teachers.  Marc Goulet and Celeste Harvey, both seasoned facilitators in my board, guided me thru my first projects with teachers.

I was fortunate to participate to a talk from Lucy West at our provincial math conference.  There was a follow up session with her for a discussion about facilitating.  I asked how could I become a better facilitator.  She said something I will never forget.

Who coaches the coaches?  We take great teachers out of classes and expect them to become great coaches!

I was very thankful to have mentors. As a provincial math facilitator now, my goal is to be a mentor and collaborating collegue to all the facilitators of the French school boards of Ontario.

A document, created by a team led by Cathy Bruce, really helped me become the facilitator that I would of wanted as a teacher.

Facilitator Brochurefacilitator

There is a lot of great thinking in there. I like the explanation of the difference between facilitation and presentation.  There is also a great section on paradoxes on being a facilitator.   For example:

Focus on the big stuff, yet, narrow in on the details.

For my French readers, here is the same document translated.

Faciliter l’apprentissage en mathématiquesfacilitateur

This is a must read for any facilitator that wants to become the one they wanted as a teacher.

Publié dans Aha moments/ moments Aha, Conversations

Math Wars (#mtbos30 – 20)

Since the start of my journey in the pedagogy of mathematics, there has always been a war between two ways of teaching;  inquiry based teaching and procedural based teaching.  As a new teacher back then, I was so envious of teachers that had their prep binders with lesson 1, lesson 1.1, lesson 1.2, test, lesson 2, … etc.   Every lesson had an example or two, a counter example and then practice exercices.  Unit 1 had 5 to 6 lessons then a test and sometimes a quiz half away.  Oh and lets not forget the « pop quiz » that were planned.  I had a hard time prepping like that.  Sometimes I was teaching them and I knew they got the big idea of the lesson but I was plowing thru because I felt I had too.  It was surreal.

It all changed when I decided to be myself.  I decided to listen more to my students and let them do the work.  I got the first smartboard and document camera in my class.  All of a sudden, the students voice was heard.  I would take the « challenge » question at the end of each unit in the manual and display it at the start of the unit.  They worked in teams to solve it.  When some great thinking was popping up, I would display it and ask that each student tell me why it was interesting.  My document camera was mostly used to celebrate mistakes.  When I saw an « awesome mistake » I would slap it on the screen and let every student try to figure out the mistake.  I would wait, and wait and wait till the very last light bub flick on and ask that last student to show me the mistake and a possible solution.  By the end, I had students bring me their mistake and ask that we all « figure out » their mistake.

That was 7 years ago.  I was still very much unit based teaching but I was happier, my students were happier and their success was remarquable.  In 3 years, my EQAO (provincial testing) scores for my applied grade 9 math class climbed from 5% to 77% of my class reaching the norm.

A big thing for me was making sure each student was ready for their tests. (yes I was still using pencil-paper tests)  The day before each tests, I would give them an exit ticket.  I could tell who was ready and who wasn’t.  Those who were ready would write their tests, the others worked with me on figuring out the exit ticket and discussing where they went wrong.  The next day they would do their tests while the others worked on a challenge task together that they would share with the others the next day.

Then I read about inquiry based teaching and putting my students in productive struggling situations.  I had to try it!  I was amazed at seeing how hard the students were working and collaborating together.

I became a math facilitator for my board, and now I’m a math facilitator for the CFORP, an organisation working for the ministry of education to help professional development across the province.

This is when I met people like Marian Small, Dan Meyer, Alex Overwijk, Andrew Stadel, Robert Kaplinsky and Graham Fletcher.  They helped me evolve my pedagogy to another level.

That is when I discovered that the math wars were alive and well.  Petitions were going around to make the schools go « back to basics » while others were saying « never teach times tables ».  Parents were also divided.  This « new math » was weird for them.

These articles really opened my eyes to this math war that was happening.

Canada’s math teachers should get back to basics, report says

Back to Basics: Mastering the fundamentals of mathematics

Why the war over math is distracting and futile

At the NCTM in San Francisco I participated to Matthew Larson’s talk.  I was amazed that this math war has been the same since the beginning of math teaching in america!  Here is an ignite that summarizes his message.


Here is another talk from Dr. Daniel Ansari’s at the Canadian Education Association (CEA) Neurosymposium 2015;  What is the Best Way for Children to Learn Math?

My take away from all this is that we should be developing deep math knowledge thru productive struggle and collaboration.  We have to develop patient problem solvers.


Publié dans Conférences, Conversations

AFEMO – AGA 2016 (#mtbos30 – 19)

Aujourd’hui ce tenait l’AGA de l’AFEMO (l’Association Francophone pour l’Enseignement des Mathématiques en Ontario). Nous étions rassemblé de Timmins à Chatham à Ottawa de façon hybride.  Les membres se sont rendus à des sites où un responsable de site de l’AFEMO les accueillaient.  Tout les sites étaient connecter par l’entremise de Google Hangout on Air.  Ceci veux dire que les membres ne pouvant pas se rendre à un site, pouvais suivre le flux en direct sur la chaîne YouTube de l’AFEMO

Suite à l’AGA, les membres ont eu le choix de suivre une présentation de Céline Renault-Charrette pour l’élémentaire et de Jérôme Proulx pour le secondaire.  Les présentations seront accessible sur la plateforme des membres très bientôt. 

L’AFEMO a maintenant un site publique et une plateforme des membres.  Le site publique contient entre-autres les 6 premiers InforMATHeur; magazine offrant des problèmes vedettes, des entrevues avec des experts et pleins d’autres rubriques intéressantes.  Les prochaines éditions seront disponibles pour les membres seulement sur la plateforme des membres. 

Si vous n’êtes pas membre, vous n’avez qu’aller vous inscrire sur la plateforme.  Vous aurez accès à des ressources réservées aux membres, aux actes de congrès, aux documents officiels et aux événement des membres tel que le congrès 2016. 

Cette année le thème de ce congrès: Penser mathématiques c’est critique!  L’inscription commencera très bientôt sur la plateforme des membres. 

Vive l’AFEMO!  Et au plaisir de se revoir en octobre. 

Publié dans Conversations

Technology in math education (#mtbos30 – 18)

As a math facilitator, I am often asked about using technology in the mathematics classroom.  Most of these requests are centered on what app should be used to teach a certain concept.  I actually had a teacher ask me: « What app can teach my students algebra skills. »  Then they start searching for that magical app that can teach a specific concept.  The result of this search is a lot of drill type questions with often a timer so you get more points or stars or apples if you do the math faster.  Don’t take me wrong, I’m not totally against drill, but if that is all that you do, you rob the students of the underlying math behind the concept.  Lots of schools sell the idea that they are a One-to-One school.  Every kid gets a computer or ipad.  The result is 30 kids looking at a screen.  For me, this is not how math education should look like.

MJ_laptop (37)

That said, there are great apps the help the discovery of concepts (Dragon box), that help students make sense of a concept (Desmos), apps the let you check your work (Photomath) and even apps that read your written math expressions (Mathpix).  All these great apps can be used in the wrong way.  As a teacher you have to ask yourself:  Does the technology help my students to collaborate, to communicate and to think mathematically?

For that to happen, I think that a Two-to-One approach is necessary.  For instance, I’ve tried the Polygraph game on the teacher side of Desmos with a One-to-One approach.  I was very strange to see 25 kids typing away on their screens.  Some were not trying, some were writing non appropriate things and some we genuinely interested.  The teacher was doing his best to walk around and help students or trying to notice great thinking to use as a consolidation discussion afterwards.  I realized one thing… these kids have to start talking!  Although they were linked, they were not collaborating or even learning that much.

With a Tw0-to-One approach, I saw kids talking to each other to ask a really good question to their linked team.  I saw kids discussing and figuring out together the question they were asked from the other team.  I saw the teacher being able to catch great thinking and use it as a teachable moment to generate richer discussions in other teams.

When I was thinking about this blog post, I fell on this image on Twitter.


It summarized very well my thinking.  I wanted to share with you the source of this slide.  After a shout-out to the Twittersphere, I found this ignite session from Micheal Fenton.

How are you going to flip technology in your classroom to the right side of that list?

Publié dans Conversations, Lectures

Gérer les discussions! (#mtbos30 – 17)

Dans mon évolution de la pédagogie des mathématiques, je m’aperçois que l’enseignant est un facilitateur de discussions et d’idées.

  • Il faut savoir quand donner de l’information, sans trop en donner.
  • Il faut savoir quand intervenir sans donné la solution immédiatement pour pouvoir permettre l’élève de fournir l’effort cognitif.  J’aime bien l’expression en anglais « productive struggle ».
  • Il faut savoir brancher une conversation mathématique avec un autre.  Il faut avoir de la patience pour que les mots sortent un peu tout croche de la bouche de l’élève.  Ce balbutiement est une phase importante de sa communication et de sa compréhension d’un concept.
  • Il faut savoir juger le temps qu’on laisse à l’erreur.
  • Il faut célébrer les erreurs pour que les élèves se sentent à l’aise d’en faire et de s’en servir pour construire leur compréhension.
  • Il faut offrir des tâches qui vont susciter des discussions, voir même des débats mathématiques!

Jai fait plusieurs liens quand j’ai lu le livre « 5 practices for orchestrating productive mathematics discussions ».img_0890

Les 5 pratiques sont:

  1. Anticiper
  2. Monitorer
  3. Choisir
  4. Séquencer
  5. Faire des liens



Lorsqu’on donne une tâche, on doit anticiper les solutions et les processus que les élèves prendront pour faire la tâche.  Il est donc important de faire la tâche sois-même ou encore mieux, avec des collègues.


Pendant la tâche, il faut bien monitorer les élèves afin de faire attention spécialement à leur raisonnement mathématique.  Quelles stratégies utilisent-ils?


Il faut maintenant choisir les travaux des élèves en fonctions des stratégies utilisées et des concepts ou grandes idées que l’enseignant veut faire ressortir.


Il ne faut pas seulement choisir 2 ou 3 traces d’élève mais bien les organiser pour faire ressortir l’élément clé recherché.

Faire des liens

Finalement, l’enseignant aide l’élève à faire des liens entre sa solution et la solution d’un autre élève ou même avec des concepts et des grandes idées déjà vu jusqu’à maintenant.


Ce n’est pas évident de mettre tout ça en pratique, mais c’est faisable!  Ça se pratique, si on est voulant de prendre des risques et d’accepter de travailler avec d’autre afin de collaborer à des pistes de solutions!