Layers Of Differentiated Assessment In Math Class

The York Region District School Board recently posted a YouTube video describing the “Comprehensive Math Program”.  Watch all 4.5 minutes of it. It is worth it.

There are a number of important ideas, concepts, and vision pieces that I can take away from this video.  I am also certain that there are some important ideas, concepts, and vision pieces that still need to reveal themselves to me as I continue to develop as an educator.

One idea I gleaned from the YRDSB video is the importance of building assessment (for, as, and of) into the very planning of the learning process. The entire comprehensive math program is built from the foundation of knowing our students first and providing students with opportunities for regular ‘check-ins’ so that students can monitor what they deeply ‘know’ and can successfully ‘do’, over a learning arc.  As students monitor their own learning, so too can teachers monitor the learning of their students.   Assessment then becomes interdependent between student, student, and teacher; not simply dependent on the teacher.

Layers of Assessment (Conversations, Observations, and Products)

Actively engaging in conversations, as a member of a math learning community, is an important setting for students to develop as a mathematician.  By engaging in conversation with peers and the teacher, the student is encouraged to make their thinking visible.  In doing so, the student is placed in a safe zone where their thinking is open to being questioned.  The benefit being that the student then has the opportunity to justify or perhaps adjust their thinking.  Digital platforms like Learning Management System discussion forums, social medial like Twitter, or student blogs can add an important level of differentiation for our learners, especially when the assessment need is to have students engage in conversation.

A student or teacher walking around the class watching and listening to responses to meaningful math questions, puzzles or games, are both engaged in another important layer of assessment:  Observation.  These observations can, but don’t necessarily have to, lead to conversations.  The benefit here is that through observation, student thinking can be seen in its raw form; in the moment.  These important glimpses into student thinking, as they work to make sense of how new math ideas match with what they already know, are essential to the assessment process.  Digital platforms like EduCreations or VoiceThread can help the teacher capture those important but at times elusive opportunities to observe a student in a class of thirty.   This was discussed in more detail by Dan Meyer, here.

Products provide another important glimpse into what students can do. Tickets out the door, tests, projects, or assignments are familiar assessment tools and in general are completed independently by students.  Providing students access to multiple modes of responding such as paper-pencil, web 2.0 tools, or manipulatives can provide an important level of differentiation. These artifacts of student thinking provide yet another important component of how we as educators can develop a broader understanding of our learners.

Engaging in only one of these forms of assessment provides a narrow view of our learners, a view which needs to be more panoramic.  By engaging in the different layers of assessment ranging from observations, conversations, and products, and differentiating with technology, we as educators can get to know our learners a little bit better.

How else have you differentiated for assessment in your classroom?



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Math Talk Community. It’s more than just talk, it’s about addressing status and curbing the fixed mindset

Teaching math through problem solving or inquiry needs a math-talk rich community of learners.  A community of learners that is at ease with asking clarifying questions, advancing personal conjectures and awaiting feedback from peers.  This is an idea I have advanced elsewhere.

At that point in my thinking, I believed developing a math talk community was essential so that students could freely talk about the math they are learning about.  I now believe it is deeper than that. Developing norms around a ‘math-talk community’ is more than just generating discussion.  It’s about addressing the ‘math smarts’ status in the classroom and a fixed mindset towards math learning (via Helen Hindle). A mindset that stunts the learner because of an ingrained belief that you are either born smart in math or not, and can’t change anything about it.

My recent thinking regarding the necessity for a rich math-talk community has been influenced by Education Professor Ilana Horn, from Vanderbilt University.  Her research explains that perceived “smartness” in math is a status issue that leaves students feeling that only those who are ‘smart’ in math can understand math. Illana Horn argues, unfortunately, that most math classrooms honour the student who can quickly compute and not those who might need more time to think because they are trying to deeply understand how math ideas work and fit together. By making it normal to ‘ask questions’ during math learning, students begin to engage with math ideas as a sense maker.  A mathematician.

The desire to share this learning was to provide students with broader and richer experiences with math, which include but go beyond computation.  Students need to experience what it is like to be a mathematician.  They need opportunity and time for conjecture, sense making, and proving their thinking.

In this post, I will share the process my colleague Ashleigh Mcintosh and I used to develop math-norms for her grade 4 class, to address this ‘status’ and fixed mindset issue. The process was heavily influenced by Illana Horn’s research.  For now, a math-norm will be defined as a co-constructed and mutually accepted code of student-to-student interaction, and student-to-teacher interaction.

Please note that I am offering a process, not a procedure.

Step 1:  Chart Intuitive Ideas

Talk to students about what makes them feel “happy face” and “sad face” when talking with a partner. The happy face and sad face choice was intentional because we believed it would be general enough to let students provide a more open and perhaps personal interpretation of how they feel.  The words ‘happy’ or ‘sad’ might be too limiting.

Provide students an opportunity to talk in a group of three to share their thinking.  Then, ask students to share what their partners talked about.  Having a student report on their partner’s ideas takes the pressure of sharing ‘what you think’ away and replaces it with an accountability to listen to your partner and develop your own ideas.  In time, I have noticed that students replace the “my partner shared…” prompt to a “my partner and I discussed…” prompt.

Chart thinking on a T-Chart.  Slowly guide conversation towards how it feels when talking about math. The idea is to build onto students intuitive ideas of how it feels like when there is an equitable exchange of ideas.

Below is the chart of ideas developed by Ashleigh’s grade 4 class.


Step 2:  Record Synthesized Math Norms

Group students in pairs to discuss what they deem important from the co-constructed list and re-write it as a statement instead of an observation.

Next, call on students to share what their partner talked about as being important.  Chart the ideas on poster paper.  Keep in mind that this is intended to be an initial list of norms that the class can add to as they become more experienced within a math-talk rich community.

Below are some of the initial norms Ashleigh’s grade 4 class developed.


Step 3: What do you need to work on?

Again, students talk with a partner to discuss what their own needs might be when working in a group.  Refocus the discussion and have students share what they would like to work on, and what they wish a partner might do to help them learn.  Chart and record new norms if needed.

Teacher then shares a norm they want to work on.

Ashleigh and I, for example, shared with the class that we as teachers need to work on Helping by asking questions because sometimes we might accidentally take learning opportunities away from our students by giving answers instead of asking questions.  To emphasize our reasons, Ashleigh and I constantly referred back to our class constructed T-chart (from step 1).  This helped the class and teachers understand how we could help promote discussion and develop a growth mindset in the math classroom.

Step 4:   Try it out.

In our case Ashleigh and I devised, in advance, a meaningful fraction question that we would share back to the class in a 3-part lesson format.  We then let students work on the problem.  As a team, Ashleigh and I walked around the classroom to support students not only with the math thinking but also how to interact with each other using the norms we co-created.

There was a definite buzz to their work as the students attempted to interact with their math partners in a way that they and the class believed would help them learn better.

In one corner of the class Ashleigh and I overheard a student exclaim, “this question is hard!” and their partner quietly explain, “But that’s the point.  Feeling stuck is normal so let’s just try to figure it out.”

All Ashleigh and I could do was smile.

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A Probability and Number Sense Math Game – Please contribute

Ontario educator Stepan Pruchnicky aka @stepanpruch sent out a request calling for math games (perhaps using cards or dice) that might be helpful for a grade 5 classroom learning about Numeracy / Probability topics.

If you know of any, then kindly leave a link or description in the comments section, or reply directly to Stepan using the twitter link above.

Here is my contribution which comes courtesy of Dr. Cathy Bruce and Louisa Chan.

River Crossing:

Here is the Twitter discussion I had with Dr. Bruce and Louisa.

Based on this brief Twitter discussion, I believe the River Crossing Game is played as follows.

Materials:  8.5 x 11 paper, 20 ‘counters’, and two 6-sided dice.

Prepare game board:

1)  Get a sheet of 8.5 x 11 paper and fold in half, lengthwise.

2)  Each player draws 10 ‘docks’ on their half of the page.

3)  Place one counter on each dock.

4)  Each player numbers their 10 docks by picking any number from 1 to 12.

river crossing

(Photo courtesy of @louisa_cee)

Goal of game:   To take turns rolling two 6-sided dice.  Add the value of your roll and if you have a dock with that number, then you can remove your boat (counter) from the dock. First one to have all their docks clear, wins.

The game, as described above, will likely lead to fierce competition, disagreements, and developing strategies for winning.  I am wondering, however, if this game might be more of a grade 6,7, or 8 type of ‘probability topic’, as it relates to the Ontario Curriculum.

Can it be revised to work with the grade 5 expectation of “…determine and represent all the possible outcomes in a simple probability experiment…”?  For sure.  In the end however, if you view curriculum expectations as a floor instead of a ceiling, then don’t worry.

How I would revise “River Crossing” to use 1 die (D) or 1 dice

Students could construct a D for each game of River Crossing they make.  Imagine multi-sided dice like:

Roll Your Color

(Photo courtesy of BEV Norton)

Students could construct the game D by selecting a plastic or wooden 3-D shape from your math work room.

Wooden Blocks
(Photo courtesy of MacAttck)

And, using post-it notes, put numbers on each side as it corresponds to the created River Crossing game board.

Want another level of probability gaming added to it?

Perhaps have each student role a 6-sided D, add them up, and then write the total on a post-it note, to be place on the selected D shape.  Imagine the ‘agony’ of your dock number not being rolled, even if the game hasn’t started.

Perhaps a revised rule needs to be created for this event.  For example, if your dock number is not represented on the D, then maybe you get to ‘change’ your dock number.  Is this fair?  Can a player disregard their undesirable dock number to only then select the number that shows up most frequently on the co-constructed game D?  Perhaps your opponent can pawn off one of their undesirable dock numbers to you so that they themselves can pick a new dock number? Either way, a good discussion will ensue because of this problem.

Want to up the number sense difficulty?  Use more 6 sided dice or have students multiply the two rolled numbers instead of adding.

How I would revise “River Crossing” to include a colour spinner

Instead of ‘numbering your docks’ you could ‘colour code your docks’.

Pull up your favourite colour spinner, web-tool.  A quick search brought me to


In the change spinner option you can construct your game spinner.  For example, before game is played, the two players can roll two 6-sided dice and add the rolled values together, to construct the size ‘value’ of each colour of the spinner.

For the spinner example above, I quickly constructed it like this:

spinner change

Maybe in this game, players can preselect what colours are permitted to colour your dock with. That way the event of a colour not being one of the possible events on the spinner, will not be an issue.

How I would revise “River Crossing” the concept

Maybe it is the Marvel Universe comic book collector kid, I used to be, talking right now.

But perhaps Spider-Man and Wolverine decide to practice together to hone their respective abilities.  Note that they aren’t fighting…

The player docks could then be replaced by “Spider-Man” and “Wolverine” specific moves.

Everything else stays the same.

Just ideas.

Don’t like this concept because you are uneasy with potential not appropriate for school arguments?  As with all things, adjust accordingly to suit your students’ needs.

In certain settings, the idea of students using cards or dice might even need to be a topic for discussion/permission.

Back to Stepan…

I am sure you have your own math game ideas, or math games you have played or borrowed from someone else.  If so, then kindly add a link or description in the comments section so that we can all benefit.

Thanks for reading.

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Balancing Knowing Math Basics with Developing Mathematical Thinking

So, what does it mean to think mathematically?

Hindsight being 20/20, I understand now that I used to equate “mathematical thinking” with whether or not a student could follow a set of prearranged steps (algorithm) I synthesized for them.  For instance, as an educator in my first few years of teaching, I believed that if a student could perform the long division algorithm, then they knew “how to divide”.  Additionally, if a student could then read a word problem and correctly choose to complete long division to solve the problem, then I believed they really got it.

Want to do a quick check of your students’ grasp of division based on their proficiency with the long division algorithm?  Ask them what they are counting at each step of the algorithm.

Consider 257 divided by 4.  To solve, one student might suggest that we first need to count how many times 4 can go into 2.  A different student suggests that we first need to consider separating 200 pieces into 4 equal groups, then count how many pieces there are in a group.

Which student has a more developed sense of division and relationships between numbers?   

As I spent more time reflecting on my math practice, with the help of  mentors and professional resources aimed at enhancing math practice, I began to realize that I was only providing a very narrow opportunity for students to develop their mathematical thinking by focusing on math computation ‘basics’ that I found useful.  I did not provide students with a broader opportunity to make connections between math ideas or by extension an opportunity to develop as a mathematician.

In Ontario, we have a math curriculum document that outlines what students essentially need to ‘know’ and ‘do’.  An important part of this curriculum is a description of seven math process expectations that can help students ‘know the math’, deeply.  Here is an excerpt listing the process expectations, found on page 11 of the Ontario mathematics curriculum document.

math process expectations

Based on this list of mathematical processes, the narrow opportunity I used to provide students with to learn math was essentially through the lens of selecting tools and computational strategies.  This learning experience I provided for students “fit” into my main instructional model which was predominantly lecture style teaching where I ‘showed’ students how to solve problems.

At that early period of my teaching I felt that if a student could read a ‘word problem’ and solve it by applying an algorithm I taught them, then I was teaching through ‘problem solving’.  In time I realized, “teaching though problem solving is not the same as solving word problems”.

What I try my best to do now is provide broader opportunities for students to engage and interact with math.  Opportunities where students problem solve, develop their intuitive math thinking, share and prove their thinking to a partner, listen to the ideas of others within a large group setting, reflect on the validity of strategies, and try to make connections between math strands or ideas.

In my opinion, mathematical thinking is more than applying an operational strategy.  It is at the very least being able to think through a problem and account for additional information needed to make sense of the question.   Mathematical thinking involves students devising a plan to investigate the problem, reflecting upon the “reasonableness”of a solution, then communicating and proving to an audience why they are right.

By providing students an opportunity to interact with math through the process expectations listed above, then perhaps students can do more than learn math content and instead, more importantly, develop as mathematicians.

It might be time to replace ‘teaching math’ with an ‘inspiring math learning’ mindset.

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3 degrees of separation to Dean Shareski

I trace myself back to Dean Shareski through my former Vice Principal, Brian Harrison.  In a recent post, Dean assigned the following piece of homework for a few of the bloggers he follows.  Here is the post.

Brian nominated me to share a bit about myself by telling 11 random things about me, and to answer some questions.  Here are some things you may not know about me.

  1. Telling people about the stuff I do is rather difficult for me.
  2. To help pay my way through university I worked in the mail room of an insurance agency by day and was a DJ at night.
  3. My wife was also my girlfriend in high school.
  4. I like going through the process of reinventing the wheel.
  5. For 3 months I lived in Trinidad & Tobago.
  6. Gemini
  7. I train in Goju-Ryu Karate – Meibukan – and am currently a 3rd degree blackbelt.
  8. John Stockton is my favourite basketball player.
  9. If my dad had his way I would have been named:  Giovanni.
  10. Learning a lot in my new role as a math coach – both as teacher and coach –  and I have my amazing colleagues to thank for this.
  11. Love it when my 5 y.o. or 3 y.o. son call for me when they wake-up in the middle of the night. I typically go running to them.

Here are the questions Brian posed to me:

  1. Who is your favourite superhero?  Spider-Man. He isn’t the strongest hero but with ingenuity and smarts, he gets the job done.
  2. What is the most interesting place you have visited? Philippines
  3. IOS or Android?  IOS
  4. Would you rather be a hammer or a nail-Why? Tough one… Hammer.  Potential for usefulness of nail remains after hammer has left. Hammer has at least 2 uses to it.
  5. What was your first part time job? Record Store clerk – yes vinyl records.
  6. Left on a desert island, what 3 books do you take with you? Musashi, Photo Album, and Surviving Deserted Islands for Dummies.
  7. When do you usually write your blog posts? Whenever I get the inspiration.  During the week and then I try to post on the weekend (a little late with this one).
  8. Pizza- thin crust or regular?  Deep Dish – Giordano’s Pizza in Chicago
  9. What was the topic of your first blog post? Basically I am nervous about blogging and I have Dean Shareski and Brian Harrison to thank for it.
  10. Did you ever own an 8-Track cassette? No.
  11. Lennon or McCartney? McCartney

If you want to play along too, then I have some prompts for you to respond to.  But please share 11 things about yourself first.  Looking forward to your responses.

  1. Who is the better Captain:  Jean Luc Picard or Han Solo?
  2. Where in the world would you like most to visit?
  3. Your “go-to” dish to cook when trying to impress someone.
  4. Coffee, tea, or hot chocolate?
  5. What is your funniest ‘teaching’ memory?
  6. The last book you read was…
  7. What was your first part-time job (borrowing from Brian)?
  8. Favourite Seinfeld quote?
  9. Do you have a nickname and can you share it?
  10. Eggs over easy or scrambled?
  11. What is your favourite web tool or app?
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PISA 2012 Results: What does it mean for your students?

In 2011, the school I presently work at developed a learning focus on enhancing math practice.  Through open, collegial, respectful, but at times difficult discussions, we teachers recognized a need to learn more about how inquiry or problem based learning could be used to enhance our math pedagogy (philosophy of teaching and learning).  This learning focus was chosen because we believed it would help address our students’ needs.  Fast forward to December 2013, when the Programme for International Student Assessment (PISA) released its results on student learning, specifically on math in Canada.

PISA reported that Canada’s ranking in math moved from 10th, to 13th out of 65 in the world.  Media responses on Canada’s move resulted in some claiming that ‘discovery learning’ and lack of teaching math “basics” were to blame.

Ultimately, I feel it is important for educators viewing the PISA results and media interpretations of the results, to do so through the lens of how does this impact the needs of the students in your school?

Before the PISA results were released, my colleagues and I realized that students at our school were adept at solving questions that required them to pick and apply an appropriate math computation.  However, we as a staff also realized that students at our school needed more learning opportunities to enhance their ability to reason, prove and communicate their thinking, especially during open-ended ‘problem solving’ questions.  We also identified supplementing our math practice with inquiry, as a teacher need.  These remain student and teacher needs at my school.

My Principal, Rob Dixon, shared his opinion with the entire staff about math learning in our school, especially in light of the concern around PISA 2012 results.

…it is important for us to balance our teaching to include inquiry as well as time to build basic skills. 
As well, properly designed inquiry promotes excellence, imagination and creativity for each student to soar and achieve well beyond the limitations a text book and drill and kill teaching promote. 
This is where the teacher is the difference maker, the knowledgable, skilled facilitator of planned, integrated and differentiated inquiry practices that bring math to life for all students. Very exciting stuff!!

This I find relevant for my students and their needs.

Can we learn from the PISA report and those who reference it?  Of course.  For example, in an attempt to address the growing call for a return to the ‘basics’ in math, Dr. Cathy Bruce (Trent University Professor, Faculty of Education), who has been reviewing the PISA 2012 results for Canada explains:

YRDSB Principal, Brian Harrison, shares in his blog that “PISA results confirm that the students who perform best in math have teachers who are well trained, both initially and over the course of their careers.”

These results are more in-line with the needs of the students and teachers at my school.  On the one hand, the students in my school need to develop their ability to problem solve, prove their thinking and communicate it effectively.  On the other hand, my colleagues and I have realized that we need to supplement our math program with tasks that challenge student thinking and encourage students to reason, justify and communicate their thinking effectively.  And that this professional learning should be on-going.

I am sure there is more we can learn.

What are some of the things you take away from the PISA 2012 results, as it relates to the needs of the students in your classroom or school?

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Doing a 3-part lesson does not mean you are doing math inquiry

Back in February 2013, I published a post ‘Thoughts on math inquiry…subject to change after I click publish‘. In that post I listed a few ideas I had with respect to math inquiry. One of the ideas I expressed was this idea that “Doing a 3-part lesson does not mean you are doing inquiry”.

I think this was related to my observation that most teachers attempting math inquiry for the first time, become preoccupied with the 3-part lesson framework and the inquiry question you might ask students.  I know this was the case with me.  As a result, little attention was being paid to important steps needed to sustain the inquiry process, such as developing a math-talk learning environment.  In other words, you can ask the best inquiry question, but if students are not accustomed to showing, proving and reflecting upon their thinking, then inquiry becomes difficult. This is something I have discussed, elsewhere.

To use the image of an iceberg.  A teacher new to inquiry, walking into the classroom of a skilled math inquiry teacher, might notice the inquiry question and 3-part lesson framework standing out prominently amid a sea of math pedagogy or practice.

iceberg metaphor 1

But is there more to math inquiry?

If we were to redirect our gaze to other parts of that inquiry iceberg, then one might observe that there is perhaps more to it.

iceberg metaphor 2

With this visual in mind, I wonder…

…what else helps to contribute to the sustainability of the math problem solving or inquiry process?

…what other elements can be used (in addition to 3-part lesson) to contribute to a comprehensive math program?

…what important components of the problem solving or math inquiry process go unnoticed but are nevertheless important to learn about and develop?

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