Inquiry Based Math Strategies

During our half day of Personalized Professional Learning, I hosted a workshop on inquiry-based math strategies, but not everyone who wanted to attend could attend… so I thought I’d recap the workshop here for those of you who could not make it – and for those of you at different schools who might be interested in this topic as well.

The structure of the workshop was very hands on, so in the absence of you being able to actually engage with the materials and manipulatives, I will provide a combination of notes, photos, questions and reflections that will hopefully allow you to engage in some of the same ideas, just in a different way.

Tuning in – What do already know?

Think about or jot down your current understanding of each of the inquiry-based math strategies listed below:

  • math time capsule
  • open ended centers
  • magic question
  • open-ended questions
  • number talks
  • math congress
  • visible thinking routines
  • inquiry cycle

If you have a thorough understanding of each of these strategies, you probably do not need to read on. If you think your current understanding has room to grow, read on!

Open-Ended Centers

I’ve already written a post about open-ended math centers and how they work in our early years classrooms. During the workshop today, each group had a bin with the three essential ingredients of an open-ended math center: manipulatives, writing utensils, and a placemat/whiteboard.

Here are some pictures of how teachers tested out a few open-ended math centers:

math workshop 3 math workshop 2 math workshop 1

The Magic Question

I’ve also written about my favourite inquiry questionWhat do you notice? In the workshop we looked at how this question can be used for math specifically.

Take a look at this multiplication chart. What do YOU notice?

Open-Ended Questions

Answer this question: Compare the following fractions using < > or =

1/4   ____  1/2

Now answer this question:

What is the same as a half?

Reflect on the difference between answering the first and second question. What are the benefits of asking open-questions in math?

Number Talk

Take a look at the following image. How many dots are there?

How did YOU figure it out? Here is a picture of all the different ways the participants of the workshop figured it out.

math workshop 8

Math Congress

Step 1 – Present the problem: A sports store has a number of bicycles and tricycles. There are 60 wheels in total. How many of each kind of bike could there be?

Step 2 – Work towards solving the problem. Markers and chart paper work best!

math workshop 6 math workshop 5 math workshop 4

Step 3 – Share discoveries and strategies with fellow mathematicians. Make sure fellow mathematicians are invited to ask questions, make connections, comments and conjectures!

Visible Thinking Routines:

Use the Visible Thinking Routine “Claim, Support, Question” to share some of your thinking about decimals.


There are also many other Visible Thinking Routines that are helpful in approaching math in an inquiry based way!

Inquiry Cycle:

Kath Murdoch’s inquiry cycle is a great way to make any math more inquiry-based.

A CCSS math standard: Know and apply the properties of integer exponents to generate equivalent numerical expressions.

What do YOU already know about this?

What do YOU need to find out about this?

How could YOU find out about this?

Math Time Capsule:

Now, think about or jot down your understanding of each of the inquiry-based math strategies listed below. A math time capsule is a great way to show growth and progress in math – whether it’s over the course of a unit, a year… or even of the course of a workshop!

  • math time capsule
  • open ended centers
  • magic question
  • open-ended questions
  • number talks
  • math congress
  • visible thinking routines
  • inquiry cycle

How did your understanding of these strategies grow and change?

In the actual workshop, after each strategy, we took some time to discuss how the strategy could be applied/adapted to different content and different age levels. Too often when we are looking at strategies we are focused on the actual strategy within to confines of the example that is used. This leads to the conclusion that “That doesn’t work for the grade/content that I teach”. Instead, I challenged the participants in the workshop – and I challenge you in the same way – to focus on the essence of each strategy and how that same approach can be used in different ways, for different ages and for different strands of math.

Here are a few examples of how the same strategy can be adapted for different content and different ages:

Math time capsules – In Grade 5 you might give students the summative task on the first day and then again on the last day to show all of the growth and progress they experienced. But in KG, you may conference with a student and voice/video record everything they know about shapes, and then record them again at the end of a unit to capture growth in their understanding.

Magic question –  In KG you might show a ten frame and ask “What do you notice?”. In Grade 2 you might show a hundreds chart and ask “What do you notice?”. In Grade 4 you might show a multiplication chart and ask “What do you notice?”.

Inquiry cycle – In Grade 1 you may use the inquiry cycle to structure a whole class inquiry into measurement. What do we know about measuring objects? What do we want to know about measuring objects? How can we find out more about measuring objects? In Grade 6 you might use the inquiry cycle to structure self-directed, personal inquiries towards calculating volume of 3-d shapes. What do I already know about finding volume of 3-D shapes? What do I still need to find out? How can go about that?

The possibilities are endless. If you focus on the “why” a strategy is effective and “how” a strategy helps foster thinking and exploration… then the “whats” become infinite! I also shared this google doc with some of my favourite inquiry-based math resources (books, blogs and Tweeters!) Feel free to have a look!

What are your favourite inquiry-based math strategies?

Open-Ended Math Centers

At our school, we strongly believe in the benefits of inquiry, exploration and play based learning – for all of our students, but especially our youngest students in KG (kindergarten). One of the best strategies our teachers implement are open-ended math centers.

Here are a few reasons why we love open-ended math centers:

  • open-ended math centers have no start or finish, which means there are never students who are ‘done early’ and never students who need to ‘finish their work’
  • open-ended math centers allow students of different abilities to self-differentiate and explore the math concepts and skills they are developmentally ready for
  • open-ended math centers allow students to construct their own meaning, collaborate with their peers and engage in authentic conversations about math
  • open-ended math centers allow teachers to observe and collect assessment data in a non-threatening, non-stressful environment.

Take a look at some of our open-ended math centers in action. What do you notice?

image image image image image image image image image image image image

How many of the following Common Core State Standards for Kindergarten Math are being explored?


How many of these Common Core State Standard Mathematical Practices are being developed?


Here is a sneak peak into how we plan for open-ended math centers:

Manipulatives Writing Tools Boards/Placemats Teacher Questions/Prompts CCSS
Dot cards (p.34 guide for effective in kindergarten) white boards markers, pencils white boards How many dots are there?

Which has more? Has less?

Any ( peoples, farm animals, cubes) white board markets, pencils white boards, papers, dot cards, stampers, two circle placemats How many are in this circle? How many are in that circle? Which group has more? Less? How could we make it equal? 6
number line, counters white boards markers, papers white boards, papers How can you show me this number? Can you show me a number bigger than this number? Less than this number? 6
counters white boards /white board markers ten frames What number did you build? How many more do you need to make ___? How do we make ex: 11 ? 3
shape blocks pencil, markers paper What do you notice? What are you drawing? Tell me about that shape? How are these shapes the same? 3
building blocks

number cards

playdo How do the numbers look different? How do they look different?
Choose two numbers. One of them is a lot more than the other. What are they and how could you write them?

We are always growing in our own understanding of math centers and play-based learning, so we would love your feedback about our open-ended math centres. We would also love to hear about and see what early math learning looks like in your classroom! 

Learning Time Capsules – shifting the focus from achievement to progress

Here is an example of how one of our Grade 4 teachers is shifting his students’ focus from achievement to progress through the use of a math “time capsule”.

Diagnostic: This teacher looked at all the big concepts the Common Core outlined for fractions in Grade 4 and created open-ended questions to allow students to show what they already knew or thought they knew about each big idea. Students were encouraged to be risk-takers and try every question!

Grade 4 Open Fractions

The teacher then tracked students’ prior knowledge on an excel sheet. This allowed him to plan full group, small group, guided and individual math inquiries based on needs.

Formative: After a few weeks of inquiring into these fraction concepts, the teacher gave back the same task and highlighted questions that students were required to try (based on the concepts that had been learned over the past few weeks in class). Green meant they showed competent understanding the first time they tried the question (during the diagnostic), but could still show extended understanding if they added to it. Pink meant they had not previously attempted it or showed a developing understanding and would need to add or change their answer. Students were encouraged again to be risk-takers and try the questions that were not highlighted, as their new knowledge and understanding might help them figure out concepts that had not yet been explored as a class.

Grade 4 Fraction formative

The teacher then added this formative data to the excel sheet to show the progress each student had made in each area, who was ready for a challenge and who needed more support.

Students were also given the chance to reflect their own understanding of the concepts learned in class so far and indicate which areas they were feeling confident in and which they wish to work on more. The teacher also filled in the same feedback sheet which highlighted his perspective on what the student did well and what they could still practice. This feedback was shared with parents along with recommendations for support at home.

Stars and Wishes Template

Summative: At the end of the Unit, the teacher gave the same task back and the students were instructed try every question in order to show their final knowledge and understanding. Again the questions were colour coded so students knew which of their answers showed a competent understanding and which answers needed to be added to, changed or attempted. Prior to handing out the time capsule, the class came up with a student generated rubric for each question, indicating what would show a competent or extended answer. Students had access to both an electronic and paper copy of the rubric to help them understand how to be successful at each question

After completing the time capsule, students completed the Visible Thinking Routine “I used to think… Now I think” to reflect on how their thinking about fractions changed from the beginning of the unit to the end of the unit. The time capsule, self-assessment rubric and meta-cognitive reflection were all sent home so students could share their progress with their parents.

Used to think now i think

When the focus is on achievement, students have no choice but to compare their achievement to the achievement of others. But when you place the importance on progress, students focus on how their knowledge and understanding grows and changes over time. Each time the students added to or changed their time capsule it was a visual representation of how their knowledge and understanding had grown and changed. Each student felt successful in his or her own way because they could see the progress they made over the course of the unit.

How do you help your students focus on progress and growth?

Teachers as Risk-Takers

I love the teachers I work with. They are passionate, professional, open-minded, reflective… and my absolute favourite thing about them … they are risk-takers!

As this year wraps up we’ve been taking lots of professional development time to reflect, rethink and refine our PYP practices. We’ve reflected on the transdisciplinarity of our programme. We’ve reflected on the way we collaboratively plan for inquiry. We’ve reflected on our PYP learning environments. We’ve reflected on how our own thinking about teaching and learning has changed. As a result of these reflections, I’ve had some inspiring conversations with passionate teachers who are interested in trying something new next year.

Here are a few of those ideas:

Math and Literacy Integration

One of our seasoned Grade 4 teachers came into my office after our PD on transdisciplinarity. The teams had just finished identifying the Common Core literacy and math standards that were either essential to a Unit of Inquiry, could enhance a Unit of Inquiry or could be taught through the context of a Unit of Inquiy. Then they built Scope and Sequence documents through this transdisciplinary lens.

Grade 4 Math Scope and Sequence

This Grade 4 teacher shared his wondering with me, “I wonder how much literacy and math would be integrated accidentally… authentically… if we allowed students to inquire into each Unit and then took notes of what reading, writing and mathematical skills popped up, out of the necessity of their inquiries?”

Brilliant! If literacy and math – as disciplines – are just ways for learning about and communicating about life then surely if students were truly learning about life (Units of Inquiry) literacy and math skills would be necessary! So we’ve hatched a scheme which will start in September. We are going to let the students inquire into the Unit of Inquiry and retroactively see which Common Core literacy and math standards came up accidentally…authentically. Instead of using the pre-planned Scope and Sequence, at the end of each week we are going to look at the raw Common Core documents and highlight the concepts and skills that were needed for students’ inquiries. At the end of the 6 week unit we will reflect and see if a sufficient amount of literacy and math standards came up. From here we decide whether or not this model is beneficial to continue with during Unit 2.


One of our new and enthusiastic Grade 1 teachers scheduled a meeting with me to reflect on her year and receive feedback about how to grow for next year. Through our conversation we stumbled on the topic of Student-Led Conferences and the current procedures for student portfolios at our school. Throughout the year, teachers collect a variety of work samples, then they have their students reflect on them and finally place them in a construction paper folder to share with parents.


This Grade 1 teacher said it would be great to try building ongoing e-portfolios throughout the year. Brilliant! Let’s give it a try! So we’ve decided that she will be the e-portfolio pioneer and try it out with her class next year. She is currently inquiring into different formats and thinking about the procedures that will be needed to make this new system a success. Throughout the year we will meet to discuss the successes, challenges and discoveries along the way.

Inquiry Based Scheduling

Usually, homeroom teachers at our school chunk their classroom time into three categories. Unit of Inquiry, math and literacy.

PYP Schedule

Upon wrapping up this school year and beginning to think about next year, a different enthusiastic Grade 1 teacher (we have many enthusiastic Grade 1 teachers!) wanted to stretch herself and knock down the boundaries between UOI, math and literacy. She wants to create a more open class schedule that allows her to plan her time in response to students’ questions and inquiries. Brilliant! She is going to give it a try and then together we can reflect and discuss the benefits and limitations of this type of schedule!

I’m optimistic about all three initiatives and so honoured to be working with such visionaries who are comfortable taking risks and veering off the beaten path. If all goes well and these initiatives prove beneficial to student learning, we will share our discoveries with the larger staff in hopes of inspiring these changes on a larger scale… and maybe even inspiring more risk-takers!

Transdisciplinarity. (It’s a word…I think)


The Golden Goose of the PYP.

Every PYP teacher’s dream…

“Wouldn’t it be amazing to spend the whole day on our Unit of Inquiry”

“How great would it be to have no stand-alones?”

“I wish all my math and literacy was authentically integrated!”

This past year, our school adopted the Common Core State Standards for Literacy and Math and we have spent the year trying our best to integrate the standards into our PYP.

We survived… and we did a pretty darn good job!

Now, as we begin to wrap up this year and think about next year, our teachers were chomping at the bit to “transdisciplinary-ify” (that ones definitely not a word) our language and math standards further. So we used one of our half day PD sessions to inquire into transdisciplinary learning. 

First our staff did a growing definition to tune into what they thought transdisciplinary learning was all about.

3 minutes to write your own definition on a post-it:

post it

5 minutes to combine your definition with a partner onto a recipe card:

recipe card

10 minutes to construct a collective definition with your whole teaching team on a sheet of blank paper:


After giving each team a chance to share, we looked at what they PYP says about transdisciplinary learning:

pyp says

We used this video as a provocation to get teachers thinking about the endless possibilities of transdisciplinary learning:


Then we unpacked 3 levels of transdisciplinary connections:

Level 1:
What concepts, knowledge or skills are essential in order to:
– understand the central idea
– inquire into the lines of inquiry
– complete the summative

Level 2:
What concepts, knowledge or skills can enhance

Level 3:
What concepts, knowledge or skills can be taught
within the context of your Unit of Inquiry?

Finally we gave teams the rest of the afternoon to look at the Common Core State Standards with fresh eyes, through the lens of transdisciplinarity.

The result was amazing! All of our teams ended the day with a more transdisciplinary math and literacy scope and sequence. Some teams even ended up with no stand-alones!

Hopefully these documents will help to guide our thinking more next year about how math and literacy can serve the units of inquiry. It will be interesting to come back and look at these documents at the end of next year and reflect on how we can refine them again to further enhance the opportunities for transdisciplinary learning.

As we often say to our students…once you’re done, you’ve just begun!