In the Ontario curriculum we have many expectations (standards) that tell us students are expected to, “Determine through investigation…” or at least contain the phrase “…through investigation…”. In fact, in every grade there are many expectations with these phrases. While these expectations are weaved throughout our curriculum, and are particularly noticeable throughout concepts that are new for students, the reality is that many teachers might not be familiar with what it looks like for students to determine something on their own. Probably in part because this was not how we experienced mathematics as students ourselves!

First of all, I believe the reason behind why investigating is included in our curriculum is an important conversation! I’ve shared this before, but maybe it will help explain why we want our students to investigate:

The chart shows 3 different teaching approaches and details for each (for more thoughts about the chart you might be interested in What does Day 1 Look Like). Hopefully you have made the connection between the Constructivist approach and the act of “determining through investigation.” Having our students construct their understanding can’t be overstated. For those students you have in your classroom that typically aren’t engaged, or who give up easily, or who typically struggle… this process of determining through investigation is the missing ingredient in their development. Skip this step and start with you explaining procedures, and you lose several students!

Traditionally, however, many teachers’ goal was to scaffold the learning. They believed that a gradual release of responsibilities would be the most helpful. Cathy Seeley in her book *Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem Solver* explains the issue clearly:

In the two pieces above Cathy explains the “upside-down teaching” approach. This is exactly the approach we believe our curriculum is suggesting when it says “determine through investigation,” and exactly the approach suggested here:

At the heart of this is the idea of “productive struggle”, we want our students actively constructing their own thinking. However, **I wonder if we could ever explain what “productive struggle” looks / feels like without ever experiencing it ourselves?** How might the following graphic help us reflect on our own understanding of “productive struggle” and “engagement”?

I think it would be a wonderful opportunity for us to share problems and tasks that allow for productive struggle, that have student reasoning as its goal, problems / tasks that fit into this “zone of optimal confusion”.

In the end, we know that these tasks, facilitated well, have the potential for deep learning because **the act of being confused, working through this confusion, then consolidating the learning effectively is how lasting learning happens!**

Let’s commit to sharing a sample, send a link to a problem / task that offers students to be confused and work through that confusion to deepen understanding. Let’s continue sharing so that we know what these ideals look like for ourselves, so we can experience them with our own students!

Hi Mark!

I loved this post! I have been trying to teach through exploration and discovery whenever possible with my special education 8th graders for several years now. One new set of lessons I started using this year came from the book “From Patterns to Algebra”. Here’s one of my posts on this: https://mathtechyblog.blogspot.com/2016/12/swdmathchat-from-patterns-to-algebra_17.html . They had hands-on constructivism-style opportunities to explore, notice and wonder with patterns, with some productive struggle along the way. After spring break I will be using one of my other favorite lessons on the Pythagorean Theorem. I’ve never blogged about it before, but this year I will be – stay tuned!

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Well said! This is something my group works hard to sell the teachers we work with on — instead of the “I-do-we-do-you-do,” model, try a “we-do-you-do-i-do” approach (where the “i do” is really more of a summarizing/pulling together of the ideas students have been investigating together). I also think you are absolutely right about teachers getting a chance to experience that productive struggle that we want for our students; as adults, it’s easy to forget what it feels like to really struggle with a hard problem that takes you a long time to make sense of, and when we forget that feeling (and the satisfaction of finally figuring something out), it’s hard to talk authentically about it to students. I honestly feel like some of the best PD includes teachers getting a chance to work on challenging mathematics together and debrief the experience.

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Great thoughts! I worked really hard this year to try to get my Algebra 1 students to experience productive struggle and get used to explaining their approaches. Here is a desmos activity I liked for introducing the best fit line:

https://teacher.desmos.com/activitybuilder/custom/57e563aa072703f509160cc2

And here are two of the learning tasks I used to introduce two-variable solution sets (linear equations and inequalities)

https://drive.google.com/open?id=0B-5elry4t9OOOW9tQmFjbUoxbnM

https://drive.google.com/open?id=0B-5elry4t9OOeVhBSTU4YTdsX00

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I do find the idea of “gradual release” to be useful for convincing teachers to differentiate the difficulty or complexity of math tasks for students. Teachers know that a key element in their literacy program is that the text must match the type of instruction or vice versa. And so, if a student is being asked to do a task independently, then that task should be at their independent level. This is equally true in Mathematics as in reading. It’s not that I’m advocating “I do – we do – you do” per se, so much as invoking the idea of differentiation that teachers are already comfortable with in literacy.

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Thanks for the response Karen. I find it difficult to use specific terms with other teachers because of just how much individual meaning those words have for each teacher. For example, “differentiated instruction” can have diverse or even contradictory meanings depending on who you are talking to. More on that here: https://buildingmathematicians.wordpress.com/2016/09/18/how-do-we-meet-the-needs-of-so-many-unique-students-in-a-mixed-ability-classroom

As for “gradual release” model, again, I’m not sure I know what you mean. To some, they mean providing the appropriate scaffolding all the time, making sure every student gets the right problems…. To do this, these teachers believe their role is to make sure all skills are in place before work with those skills can be used. In other words, they are teaching FOR problem solving. In direct contrast to this are teachers who attempt to teach THROUGH problem solving. More about that here: https://buildingmathematicians.wordpress.com/2016/06/09/teaching-approaches-what-does-day-1-look-like

Love to hear your thoughts about either link…

Cheers, Mark

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The trouble is that teachers feel pressured by common tests and exams (such as the VCE in Australia) to fall back to a skills (teach for test … or test and forget) approach.

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This is a problem! What are some things that are very important for our students to be able to do that aren’t directly measured on these tests? Forone, developing a student’s ability to reason requires us to be in situations where we are productively struggling to make sense of things. While there are no easy answers, I do think we need to find as many opportunities as possible for our students to be in the zone of optimal confusion followed by opportunities to consolidate.

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