Wednesday, 16 December 2015

Strategies for Creating Enquiry-Based Maths Learning

What effective strategies can we use with enquiry-based maths?

I'm the first to admit that my journey into enquiry-based maths learning over the years has involved many trips and stumbles along the way.  

Sometimes strategies for a particular maths unit that proved so engaging and successful with one class can start to fall flat with the next year's class. So, as teachers we need to be constantly re-evaluating and reflecting on what is happening with each cohort of children we have - and then take that further to each child. 

Enquiry-based learning has the student at the centre driving the learning that will take place.  

I've noticed that if I try to steer a group of children down a journey of discovering I saw children last year experiencing successfully, then it is not as successful.  By doing this, I am placing unfair expectations and also taking away the essence of enquiry-based learning by steering them into directions they perhaps haven't a strong curiosity in finding out.  Letting go and allowing children to have more ownership of the directions they want to take is key. 

To help instill student-driven maths enquiries, strategies are needed in the classroom to help them create a framework of where to go and how to share what they are learning.

I experiment a lot and try out different strategies for each maths unit and reflect on what makes them successful or not so successful.  I always ensure I ask the children to reflect with me and we discuss how helpful or unhelpful each strategy was.  

Is there a recipe or general formula to create enquiry-based maths units that are student-driven?

Below are some of the general ingredients I tend to use for maths units; I find selecting one or more of these per maths unit helps to create an enquiry-based maths learning environment framework.  

The list isn't complete- just those that have popped into my mind now.

If you have other strategies, why not post them in the comments section below for myself and other teachers to try out?  :)

Beginning a maths unit strategies:

° Begin a unit with an open-ended provocation.  

This arouses children's curiosity and gets them wondering....wondering...wondering.  If it's a successful provocation, these initial questions can form the basis of the maths unit.  Giving children time to explore their generated questions and then share their findings with their peers makes the learning student-owned and raises engagement. 

To differentiate to different levels, grouping children and giving different provocations to think about and then peer-share as a whole class can be an effective alternative. 

Pros - tunes children into the topic, sparks curiosity, forms a pre-assessment

° Introduce and unpack the central idea with the class.  

Sharing, discussing, wondering about a central idea can be a very powerful strategy to help arouse curiosity and to also make the unit student-owned.

When unpacking a central idea, I sometimes say things like:

- To become experts on this central idea, what do we need to discover?

- If we are to find out everything there is to know about this central idea, what do we find out?

- What sort of activities could we do to enquire into this central idea? 
   (Sometimes, children come up with amazing ideas that I would never have
    thought of!)

Example of unpacking our central idea:

Pros- arouses curiosity, makes the unit student rather than teacher owned, presents to children
        what they will be learning about

° Use the PYP key concepts to create questions we want to find out about.

Either share the central idea and then have children generate enquiry questions based on PYP concepts OR simply write the maths topic (eg, angles) on the board and brainstorm PYP key concept questions that could help us enquire into that maths topic. 

Link: Why use the PYP key concepts in maths?

Pros - ensures deep higher-order thinking enquiries will take place, helps children to make 
         connections, find the relevance of learning about it and discover the WHYS of maths 
         instead of just the HOWS

° A 5-10 minute pre-assessment

Being able to group children into ability groups early on is crucial for differentiating to each child's levels and needs.  To create a pre-assessment, I sometimes create a few short skills-based questions for children to show whether they know how to solve or not.  I then often use an open-ended question such as:

How do we use XXX in real life situations?

(This question is powerful.  It starts tuning children into the idea that there is a real purpose to what we are about to learn which is pivotal to engagement. After everyone has finished, we share these ideas as a whole class to help us deepen our understanding of the usefulness of what we are about to enquire into)

I think a pre-assessment should only be 5 or 10 minutes maximum long.  I don't want to discourage children early on and perhaps create a negative feeling towards that maths topic.

Whenever we do I pre-assessment, we always begin by reviewing the meaning of the prefix 'pre-' and then I always say:

- If you know how to answer these questions, that's great.  If you don't know how to answer these questions, that's great too.  Remember that the purpose of this pre-assessment is to help me find out how I can best help you. 

A short pre-assessment also helps me to quickly group children and helps me to start thinking about which skills or conceptual understandings will be needed to be introduced to which children. 

Pros - group children into ability groups quickly to provide effective differentiation 

° KWL Chart

A KWL chart can be a really effective tool to make a maths unit student-driven. The K part can serve as a very interesting pre-assessment as well as tuning each child into the maths topic. The W part makes the learning student-owned and the L part serves as a great reflection.  

Using a KLW chart can be successful either individually, in partners or in small groups (a short pre-assessment can be useful in grouping kids before they create a group KWL chart).  Sometimes I even have the whole class create a KWL chart together to create our maths unit and as a means of sharing our understandings together.

Pros - student-owned, ensures reflection on learning, forms a useful pre-assessment - formative AND summative

During the maths unit:

° Differentiated group activities / learning centres etc

Usually teacher-created, but sometimes children are encouraged to create their own learning activities to do - or for another group to do. When children create their own learning activities, higher order, critical and creative thinking are developed.

Pros - collaborative learning, differentiating

° Maths Wonder Wall

Encourage children to add wonderings to our maths wonder wall in the room for themselves or others to find answers to throughout the entire unit. We sometimes use these at the beginning of a maths period or at the end as a reflective discussion.
As the unit develops, if we have a lot of unanswered questions on our wonder wall, an entire maths period will be devoted to finding and sharing the answers. The children love these days as they again area able to take ownership of their own learning and feel proud when they share their discoveries.

Pros - student-owned, fosters student curiosity, broadens learning

° Have children create questions or problems for their peers to solve

Creating word problems in particular is an effective strategy towards the end of a unit for children to apply their understandings and to further solidify their understanding of WHY we are learning this - how do we use this in real life?

Pros - student-owned, fosters student curiosity, broadens learning,  critical, creative and higher-
         order thinking

° Continually refer back to our central idea  

   Ask- What have we learnt about our central idea today?
        - What have we already discovered about our central idea?
        - Which questions do we feel we have good understanding of now?
        - What could we do next to find out more?

Pros- reflection on learning and understanding, reminding of the enquiry framework and our
        purpose / goals

° Encourage children to constantly be learning with different classmates. 
Often like-minded, but sometimes mixed ability groupings are really successful especially with open-ended tasks. Keeping the learning environment flexible is key to optimising student-peer teaching and learning.

Pros - broadens learning, develops collaboration skills, peer teaching / learning

° Open-ended activities
Open-ended activities allow for deep, creative, critical and reflective learning to take place.  Every child is helped to feel successful as a mathematician and such interesting mathematical thoughts are created and shared with them.

This book is a bible of amazing open-ended tasks and all presented at different levels:

(It's unfortunately tricky to buy outside Australia, but you can at  and they will post overseas)

Pros - differentiating, giving opportunities for student-owned thinking, developing responsibility
         for own learning, higher-order, critical and creative thinking, informal formative 

° Provide three levels of questions 

Using the data screen or google docs, I always create 3 columns.  The first column is 'easier', middle column 'bit more challenging' and the third column 'more challenging'.


( The above might look like a 'drill n kill'.  However, the way it works is the children have 3 or 4 minutes to answer as many as they can but the goal isn't to answer as many questions as possible. The aim is to use these questions to help us understand the two big questions above. They are just a tool for discovering)

Children choose which column level they will solve. They can also flip between columns too if they want. 

This helps children feel a sense of ownership towards their own learning as they get to choose which questions to solve.  It also allows for differentiating all those different levels in the classroom. 

Additionally, it provides as a valuable informal assessment taking mental notes of which level of questions each child is selecting to do.  

Incidentally, children are amazing at the levels they choose.  Over the years, I might have a one or two students who choose easier levels as a 'lazy' option, but after a quick conference discussing with them how these are an opportunity for themselves to develop, that 'laziness' always stops and they start selecting more appropriate levels for where they are at. 

Another great advantage to this strategy, is that children often gain a marvellous sense of pride when they begin realising they can now solve problems in more challenging columns than they could at the beginning of the unit.  They are seeing their own learning development and take a lot of pride in that.

This strategy is also great for catering to children who are strong at that particular maths strand.  The 'Ive finished!' usually heard by stronger mathematicians isn't heard much because they have problems designed to help them think at a higher level.  If children do happen to finish a column of questions, they soon understand they should then attempt questions in another column as this will help them further develop their understandings (even if they do questions in an 'easier' column)

Link: Three column levels problem solving- subtracting decimals

Pros - differentiating, developing responsibility  for own learning, informal formative

° Children create own enquiry-planner
Perhaps using a KWL chart or simply selecting one or two PYP key concept generated questions, children are given time to find out and share their findings. Having children share their findings by creating hands-on activities for others in the class to do develops creative, critical and reflective thinking.  Plus, they LOVE it and are so enthusiastic.

Pros - student-owned, differentiating, developing sense of responsibility for own learning,

         formative & summative assessment

° Reflective formative / summative assessment

I used to use this strategy as a summative, but now am seeing how much more powerful it is to use this as a formative AND a summative.  After every 2 or 3 maths periods, children are given 10-15 minutes to reflect on what they have learnt during the unit.  

This gives each child the important time to collect and deepen their thoughts as they reflect on what they have learnt.  It also helps the children to take their maths learning to a more conceptual understanding as they reflect on the central idea.

Every child feels successful as a mathematician and feels proud of their learning regardless of their 'level'.


End of maths unit:

° Complete formative /summative reflection

At the end of the unit, children are given extra time to add their understandings to their formative /summative reflections where they show what they understand about our central idea.  

Additionally, they use the 'I used to think......Now I know.......' strategy to help them deepen their own reflections:


° Share with peers and then with parents

After reflecting, groups or partners form and the children share and discuss what they recorded.   After receiving some feedback from me, they then share these with their parents.  Since the assessment is a positive experience, parents have little choice but to celebrate with their child what they have achieved rather than negatively focusing on 'mistakes' made in a traditional maths test. This discussion with their parents helps reinforce in the child that they are successful mathematicians!

What strategies do you find successful for enquiry-based maths learning?  

It would be great if you posted below, so other teachers and myself could build our repoitures! :)


Links to assessments and reasons why:

Positive Maths Summatives (Fractions / Decimals / % Samples)

Open-Ended & Enquiry-Based Summative (Data Handling)

Meaningful Maths Summatives (Base 10 / Place Value Samples)

Every Child Feeling Successful in a Summative (Probability samples)

When a Formative Becomes a Summative (Measuring Time Samples)


  1. Hi Graeme....loved your post! Such great strategies and tools to enable and facilitate authentic maths inquiries in the classroom. Choice, differentiation, student questioning, reflections, feedback, conceptual understandings, authentic's all there! Some great ideas for me to introduce in my classroom when school returns after the holiday break. I have attended some great PD recently that has also provided me with some great insight and practical examples on how to use the thinking routines outlined in Making Thinking Visible (Ritchhart, Church, Morrison). Each strategy having the potential to be used as a pre-, formative or summative assessment at any stage of the unit (depending on the concept and understanding being targeted). Again, thanks for sharing snapshots of your practice!

  2. Fantastic post! Thank-you for sharing these ideas.


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