Math Inquiry or Problem Based Learning: Plan to be less helpful

(Photo courtesy of Flickr user: .michael.newman.)

This is a follow up to my preparing for math inquiry blog entry.  In that post I explain that an important place to start, when preparing for math inquiry, is to focus on developing the math talk community.  Or attending to the classroom culture, as the National Council of Teachers of Mathematics (NCTM) explain in a recent research brief (January, 2013).  By extension this post focuses on what the teacher can do to help contribute to the healthy development of an inquiry or problem based learning environment, within a math context.

With the goal of implementing inquiry in the classroom, the first major and daunting change I made in my math practice was to REPOSITION the textbook in my planning framework from ‘program’ to ‘resource’.  In other words, I placed more responsibility on myself – as the teacher – to make conscious decisions about where the math learning needed to go, based on the needs of my learners, and how I believe the curriculum could seamlessly link together.  This was done in place of following a textbook from one unit to the next.

I have learned that as the textbook is repositioned from program to resource, so to the teacher must reposition themselves in the classroom from ‘math sage’ to ‘math guide and activator’.  To paraphrase math researcher Fosnot, a math program should ‘not strive to fix math thinking, but instead work to develop the math thinker’.

I believe this idea tries to address what math researcher, Dan Meyer, suggests are 5 ways you know you are doing math reasoning wrong (Math class needs a makeover). Specifically, students demonstrate:

I have observed that when I made a few specific but small changes to my teaching practice, on the scale of my daily interactions with students and their learning, I have noted significant gains in addressing the 5 points listed above.

What did I do as a teacher?

In short, I planned to be less helpful.

What I mean by this is, I made a purposeful and strategic effort to:

  • plan to present math topics as contextual problems;
  • plan to let learners mull over these shared problems with partners then individually;
  • plan to promote peer to peer discussion so as to push student thinking and not, rescue the student from thinking by ‘telling’ them the strategy to use; and
  • plan to highlight student thinking during the “reflect and connect” stage of the 3-part lesson.  This is something educator Thomas Ro has suggested is often an unfortunately overlooked process embedded within the 3-Part lesson.

An important change I needed to make as a teacher was to adjust how I interacted with students during the problem solving stage.  What I mean by this is some teachers new to ‘inquiry’ often inadvertently project a sense of “sorry but I can’t tell you right now” or “just think about it some more and we will look at the answer later” to the student.  My feeling is that this does not contribute to the spirit of inquiry we are trying to develop within our classrooms.

Instead I have found it useful to project a sense of “Let’s figure this out together” to the student.  A strategy I employ to help promote the spirit of inquiry within the classroom learning community is to use ‘talk moves’ so as to avoid ‘tells’.  Tells are general comments made by the teacher which essentially relieves the student from the duty of developing their own intuitive math thinking or strategy.  An example of a tell might be a comment made by a teacher to a student like: “for this question you need to add the 3 apples in one basket and the 2 apples in the other”.

A ‘talk move’, on the other hand, elicits dialogue and quietly encourage students to ‘show’ their thinking.  A talk move can be as simple as, “How does their thinking match with yours?” I have noticed that this question elicits much conversation, especially once students have become comfortable with the problem based or inquiry community.  The work of Chapin, O’Connor, and Anderson (2003), Classroom Discussions:  Using math talk to help students learn, have informed my practice.  Here is an adapted version of their work, create by the YRDSB Math Team (Five Talk Moves), which explain 5 high yield talk move strategies.  They are:  revoicing, paraphrasing, agree or disagree & why?, prompting for further participation, and using wait time.

Are there any other important ways we might ‘plan’ to be less helpful for the benefit of our students?

What else should we consider when planning for inquiry or problem based learning?

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Preparing for Math Inquiry or Problem Based Learning: Where do I start? How do I start?

Inquiry or Problem based learning in math is not new.  It is the math pedagogy advanced by the Ontario Ministry of Education (2005) and is based on the belief that “…by learning [math] through problem solving, students are given numerous opportunities to connect mathematical ideas and to develop conceptual understanding” (Ontario Ministry of Education: 2005, p. 11).

If you are reading this post, then you probably already buy into this.  You might also be starting your math transformation by locating websites to help you generate the “mother of all inquiry questions”.


tweet making questions tough

Generating your own inquiry question is a skill that needs to be developed, over time.

If I had to suggest an alternative starting point in preparing for math inquiry or problem based learning, then it would be to instead focus on developing the math talk community, within your classroom.  Specifically, each student’s ability to defend, and prove their own thinking, while at the same time questioning and reflecting on the math thinking of others.  This being done with a view to enhancing the understanding of math concepts, for each member of the talk community.


There are many resources that already contain well crafted inquiry questions and could be used as a starting point for familiarizing oneself with how an inquiry question is framed.  Fosnot’s math kits, websites like math researcher Dan Meyer’s 101 questions, and some textbook problems contain excellent inquiry contexts for students to work through.  Math inquiry questions or engaging math contexts tend to satisfy the following criteria:

  • The solution is not immediately obvious.
  • The problem provides a learning situation related to a key concept or big idea.
  • The context of the problem is meaningful to students.
  • There may be more than one solution.
  • The problem promotes the use of one or more strategies.
  • The situation requires decision making above and beyond the choosing of a mathematical operation.
  • The solution time is reasonable.
  • The situation may encourage collaboration in seeking solutions

(Guide to Effective Instruction in Mathematics, vol. 2, p. 26)

Simply taking a question from any of these resources and using it within your classroom is fine.  Don’t feel like you are only doing inquiry if you make your own question.

Here are some ideas that might help a reader consider how to start preparing for math inquiry by developing a math talk community.

Develop math norms of working with a partner

Provide a common experience for learners where they work with a partner to problem solve a given math context.  By engaging with the task in a problem based way, students develop a sense of what solving math problems with a partner “looks like”, “sounds like”, and “feels like”.  Students also start getting into the habit of using their intuitive math sense to develop new understanding.  From there, have students develop class norms around working with a math partner and record their ideas on chart paper.


(Norm exemplar courtesy of Carrie Byer)

Develop question starters

Develop question starters with students that help them probe into their partner’s thinking, in  constructive ways.  Record these question starters on chart paper.  This provides students with a useful framework for questioning that ultimately gets the class community into the habit of ‘showing and proving their thinking’.  Over time, you will notice students devising their own questions, and an increase in peer to peer discussion about their math thinking.


(Question prompts courtesy of Ashleigh McIntosh)

Institute regular math meetings (math congress)

According to the Ontario Math Curriculum (2005) document, “One of the best opportunities for students to reflect is immediately after they have completed an investigation, when the teacher brings students together to share and analyse their solutions” (p. 14).  In this larger group setting, “Students then share strategies, defend the procedures they used, justify their answers, and clarify any misunderstandings they may have had” (Ontario Ministry of Education: 2005, p. 14).  To help sustain this important aspect of developing the math community, it would be important to create math meeting (or math congress – Fosnot) norms, like in the case of working with a partner.  Again, record the norms on chart paper and post in the classroom for students to review prior to math meetings.


(Math meeting norms courtesy of Carrie Byer)

Keep these norms and question starters posted throughout the year, and revisit them regularly – even if you feel the students “have it”.


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Math inquiry and planning – Where does the curriculum fit?

Radial Perspective
(Photo courtesy of Flickr user: Mark_twisT)

How we view the math curriculum influences how we plan for learning in our math classes.

If the curriculum is viewed as content to be covered, then it is understandable that there will be a desire to focus on the teaching of algorithms.  I say this because speed of covering content becomes a major concern of a math program with this view of the curricululm.  Of all the math skills available to the student, RECALL is deemed the most valuable.  In this regard, math lessons tend to be chunked into discrete units and delivered successively.

I cannot speak for anyone else, but my early math teaching experience might have at one time reflected this mindset.

Under a content driven view of the curriculum, I used to:

  • Refer to the math textbook when planning my lessons
  • Do a diagnostic to diagnose what students know and more importantly don’t know.
  • Find pages and numbers of homework that students could work on quietly after I did my lesson. Questions are essentially the same so students are engaged in practice of a similar type of problem.
  • The next lesson would often be the ‘next page’ in the textbook.  Followed by a note to be copied where I highlighted the steps and algorithm used to solve a specific problem.
  • Assessment was essentially 2 or 3 quizzes marked out of 20 or 30, and then a unit test typically marked out of 50.  A nice 3 week math unit.

dr jeff discuss twitter

If, however, the curriculum is viewed as contexts to explore, then attention is given to learning of math processes and concepts.  In this view of the curriculum, exploration and development of concepts, not recall, is key.  Of all the math skills available to a student:  questioning, problem solving, proving, communicating and reflecting hold common ground and are all important. This ‘inquiry’ view of the curriculum then lead to math lessons that unfold and link from one unit to the next.  Math is presented not so much as a group of discrete and separate units, but instead as a ‘whole’ where making connections from one math concept or context is encouraged and takes place.

Under this ‘inquiry’ view of the curriculum, what I try to do now is:

  • Refer to my math curriculum document and develop an understanding of where my students, with respect to math learning, need to traverse.
  • Instead of diagnosing math understanding, I attempt to create an environment where the students and I ACTIVATE math thinking.  This is usually done using an overarching math inquiry question, a common context as it were, where students work with partners to share knowledge and engage with the problem using their intuitive math understanding.
  • Questions for homework are developed from problems students had about the math task they worked on, while with partners. In this regard, the math homework questions students work on are not prefabricated.
  • The next lesson often involves a problem that focuses on an area students showed difficulty with the other day.  This is then usually followed by a math congress and a highlighting of what we have learned from our common context.  On a BYOD note, I can get an immediate sense of ‘what students know’ the night before school because they post their homework electronically, on our learning management system (LMS).
  • Assessment is more ‘small scale’ and frequent.  Initially I don’t give levels or marks because it is all about activating knowledge and developing intuitive thinking about math.  In my opinion, guess and check, and getting things wrong, is an important part of learning in this process and needs to be honoured.

It took me a fair bit of learning and work to get to this point, however.  With a lot of help from math mentors.  I am certain I still have more to learn.  A reality of this whole process is that math inquiry requires solid planning and assessment because you are anticipating and guiding student responses to questions instead of, like what I used to do, anticipating when the unit will be completed.

How do you plan or prepare for math inquiry in your classroom?

How has your thinking about math and the learning of it changed over the years?

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What Bruce Lee might say about the Math Learner

Yes, I am talking about Bruce Lee, Jeet Kune Do founder and star of martial arts classics such as Enter the Dragon.

Bruce Lee 'Game of Death' action figure
(Photo courtesy of Flickr user: comigirl)

But what does Bruce Lee have to do with math learning?

To me, it can be summed up in the following line taken from the opening scene of Enter the Dragon.

[Learning] is like a finger pointing away to the moon…don’t look at the finger or you will miss all of that heavenly glory…

This critique of a martial arts student, made by Bruce Lee in the 1970s, is in my opinion a relevant critique of the math learner in classrooms, today. Students engaged in watching a finger pointing at a textbook or algorithm, and not necessarily looking past it to wonder and make sense of the math.  In this case, students are typically engaged in recalling predetermined steps to responding to math questions, instead of:

  • Problem solving;
  • reasoning and proving;
  • reflecting;
  • selecting tools and computational strategies;
  • connecting; and
  • representing or communicating their thinking.

(Ontario Mathematics Curriculum – Grades 1-8, 2005, pg. 11)

math context or content

So, the challenge becomes, can we open our math teaching styles to include opportunity for students to explore math in context, and not just the isolated parts we point to?

Can we push student thinking, without rescuing the student from thinking?

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My class is ready for BYOD, but how do I start?

Instagram iPhone Photo School Desk

Willingness to create a new norm within the classroom such as introducing digital devices to students, can lead to unpredicted outcomes.  On the one hand, it can be unpredicted in the awesome sense like when a student downloads a ‘dice’ app to help calculate the experimental probability of rolling a seven on two, 6-sided dice.  On the other hand, it could also be unpredicted in the “oops…I should gone over this with them” sense too.

This post is a follow up to the BYOD reflection I provided early in the week. My hope is to shed light on some areas where the ‘devil could be in the details’ when starting the BYOD process.  Keep in mind, that the reflections I provide were generated out of a need for a specific group of students, familiar with a specific type of class culture.  Therefore a reader wanting to apply the ideas or strategies outlined below, should determine whether or not the ideas presented would be appropriate for the existing culture and needs within their own classroom.

1)  Create a culture of using digital devices in your classroom, before you go 1:1 through the use of BYOD.

With a new group of students, it usually takes me about 2 months to construct a culture of using digital devices before beginning BYOD. But, if I have students with existing BYOD experience, then it is sooner than 2 months.

The idea here is for students and teachers to get into good habits of using devices to access, collaborate on, and modify classroom tasks.  A suggestion here is to find and start using a Learning Management System (LMS) in your classroom, before you roll out BYOD.  Here is a list of 20 popular Learning Management Systems, created by Jeff Dunn. By having a good LMS in place, supported of course by sound pedagogy, learning no longer becomes space or time dependent.  In extreme cases, for example, a student could be on a family trip but still be able to collaborate with peers, complete and submit their work, and access missed class content.  In the end, parents are able to see the benefits of using digital devices in the classroom.  Once you have helped parents understand this, then asking permission for students to bring their digital devices to school becomes easy.

2)  Have a comprehensive information letter to send to parents

Take the time to create a letter that outlines your rationale in having students bring devices to class.  Parents need to buy into this because they will be the deciding factor as to whether or not students will be allowed to bring their devices to school.  Make the time to share how their kids will be using the devices to benefit learning.  You owe that much to them.

3)  Many voices – same message.

As part of creating a culture around the use of digital devices, we must also create a culture around the appropriate use of digital devices.  There is no need to create a new code of behaviour when using digital devices.  Instead, work with administrators, other classroom teachers, Police (community liaison officer), Digital Literacy Resource teachers (consultants), lunchtime supervisors, and caretakers (read:  adults in the school) to share with students how existing school and societal expectations for behaviour are apropos when using digital devices.

In the end, let students know that when using devices we want them to be safe, respectful, and productive.  

4)  Be proactive, not reactive when it comes to appropriate use of devices.

I believe students have a heightened sense of what is considered fair play.  So have them be part of the creation of norms around the use of devices in school.  Here are some norms my students and I have created to address the above.

  • Be safe, respectful and productive when using your device.
  • When not using your device, be safe and place your device screen down on the table to hide any private text messages.
  • When speaking to someone, show attentive listening and mutual respect by removing ear phones from ears, even if the music is turned off.
  • What may be okay with Mr. Aniceto, may not be okay with another adult in the school.  Listen to the adult first and then talk to Mr. Aniceto later about the appropriate use of digital devices.

Provide plenty of reminders when you first begin BYOD and continue to remind the class of the norms after.

If you are in a primary or early junior classroom, then you may want to create norms around the safe storage of digital devices during recess, lunch, physical education, and so on.

5)   Hit the ground running.

Use the devices productively in the classroom the day the students bring their device.  That way all the work you and the class have done in constructing a culture of using digital devices, carries over smoothly.

6)  Share the wealth with other teachers.

BYOD has many learning benefits, so share the possibilities.  Especially with teachers who teach your class on rotary.  It can be as simple as offering to scan and post a handout on the class LMS, or updating homework on an online agenda.  It can also be as exciting as sharing with the French teacher that students have the capabilities to record, edit and share French videos, starring them.  BYOD is something that you should try to built in your school, just not in your classroom.

After I first introduced BYOD at my school, four classes went BYOD the following year.  All intermediate grades (7-8) and one junior grade (split 4/5 class).  I hope more classes will add to this, next year.

7)  Check back often.  

From my experience, quality of work does not immediately increase as students begin working digitally. If anything, I have observed a slight decline in the quality of work, when students first engage with using digital devices for learning. So take the time to help generate good habits.  Checking back often shows students that you are a willing learner and participant in the BYOD environment.  By doing so, you as the teacher have the opportunity to model the social and academic norms you and the class have constructed.  This goes a long way.

These are some of the important areas I feel a teacher starting the BYOD process, may want to consider.  Are there any that I am missing?  If so, please leave a comment below.

Starting a BYOD classroom culture is definitely an exciting prospect but it is made better by sharing the benefits with your colleagues.

The question now is, “What do I do now that I have started?”  That is a question best served in a new post.

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Reflections on the BYOD Classroom and Digital Citizenship


Photo courtesy of Flickr user jennip98

If you are reading this, then I hope you will be willing to share your Bring Your Own Device (BYOD) expertise with my new friend, Anita (@anitaasimpson), by leaving a comment.  She will be implementing a BYOD classroom in March and was hoping to get some advice to make the journey exciting but smooth, so to speak. For those unfamiliar with BYOD, it is a classroom environment where students are encouraged to bring their personal digital device (smartphone, laptop, tablet, and so on), to assist them with learning and sharing.

This initial blog post on BYOD will therefore be written with Anita in mind (and teachers considering or beginning the process of implementing BYOD).  The theme of this post will hopefully be one that provide opportunity for reflection, instead of simple advice.

Why do you want to go BYOD?

Your response to this simple yet loaded question could be a first hint as to whether or not creating a 1:1 classroom environment, is right for you.  If your responses fall in the realm of:  “I want to use technology to engage my students”, then perhaps you are not quite ready for BYOD.  In my humble opinion, pedagogy must lead the need for BYOD.  Students should already be engaged in class and BYOD allows for improved flow, sharing, collaboration, and efficiency when completing tasks.

In my case, I chose to go with the BYOD route to create a 1:1 device classroom because signing out the laptop cart and waiting for the students to login took way too much learning time away.  I also steadily noticed that I was needing the laptops everyday so students could use web 2.0 tools and our Learning Management System (LMS) – Moodle & Edmodo – to complete our tasks.  Tasks which often required students access to tools that allow them to collaborate at school or at home, to receive teacher feedback in a timely fashion, access the internet, and use modifiable documents or links that I have provided to them on our LMS.

We needed BYOD.

BYOD, is just part of how I teach everyday.  It is not an event.

What are your expectations of students when they interact with school, digitally? A question of digital citizenship.

In my classroom we have a saying that we follow with respect to BYOD, which is “Let’s be safe and productive.”  However, as a school we also adhere to the Tribes agreements of:  Mutual Respect, Attentive Listening, Appreciations, and The Right to Pass.  With the help of my Principal and Vice Principal.   Students, parents, and teachers have been made aware that school and classroom expectations are consequently extended to the digital realm too.  There is no threshold between the physical and digital classroom.

Digital presence in the classroom is not the Wild West.  It is ‘a part’ of, not ‘apart’ from, the classroom. How will you share this understanding with students and parents?  A recent post by @gcouros is useful when reflecting on this aspect of BYOD.

More BYOD reflections to come.  Hopefully you can help add to them by leaving a comment for Anita.

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Terra Incognita: An exploration into math inquiry

Antique map 53

(Photo courtesy of:  Changhua Coast Conservation Action)

I haven’t been teaching for very long, but for as long as I have been teaching (10 years), literacy has been the focus of the school board I have worked in.  Owing to this important and sustained focus on literacy, a level of organizational sustainability has been achieved with respect to literacy, specifically as it pertains to pedagogy.  So much so, that a push to achieve similar results in mathematics has become a growing trend in the school I work at and the network of schools, my school belongs to.

On a large scale, implementation of math inquiry is uncharted territories – Terra Incognita – as the early map makers called it.  These uncharted territories were often depicted being inhabited by fantastical sea monsters or dragons, as made popular by the Lennox Globe and the inscription “Here be Dragons” that describe these mysterious areas.  The anxiety of traveling over uncharted territories is evident, but doing so over a terrain such as math, which has its own palpable anxieties related to it, can be unsettling to many.

The thing with uncharted areas is that, if one person made it through, no matter how efficient, good, bad, or indifferent…that often became THE path to follow because, “Hey they survived, and there is nothing wrong with them.”  This can be similarly said of math instruction.  As I reflect back to my early teaching days, I realize now that I often replicated the same outdated methods, practices, and tools that were already out of sync with my needs of tomorrow, when I was a student.  Let alone the needs of my students as I helped to prepare them for their tomorrow.

This is a feeling that I believe may be shared by other teachers and parents.  It can be particularly evident in the way certain die-hard math ‘hallmarks’ of believed math mastery are defended such as the long division algorithm, or the “Dracula Must Suck Blood” method of long division.  Fortunately, this is not shared by our students.  Do I then, not owe it to them as an education professional to provide opportunity for students to walk a wider path on their journey through mathematics?  Why should I force students onto narrow (algorithm) paths that blinker all the wonderful surrounding scenery of the mathematics landscape as Fosnot and Dolk describe?

In a previous blog post, I listed several lessons I have learned as I made my way through murky inquiry waters.  The first lesson I shared was that:  “Inquiry seems difficult.  Until you start.”  Perhaps I should have included the word “very” to the first sentence.  This is why I blog.  On one hand I want to help share my experiences, my dos and don’ts, as it were in hopes that it might make the task of engaging in math inquiry, just a bit easier for others.  On the other hand I in turn hope to receive nudges on to a better math inquiry path by others who are traveling with me; whether in front, beside or just starting.

Terra Incognita…here be dragons…

Let’s work together to shed more light upon these uncharted math areas.

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