I’ve noticed that opinions are split on this question with some people calling the image above a task and others calling it a lesson. In my opinion, unless an image like this includes a description of how the teachers and students will interact with the image, it’s a task. (Aside: There’s too much variety in how lessons are described to have a very clear definition of lesson in this post, so I’ll have to save that for a future post, but this blog post by Annie Forest on different lesson structures is a great read).

One might be able to imagine how you would use this task with students and form a lesson plan based on a task, but without some insight into the intended use of a task there is enormous variation in how any particular task might be used.

This is not just pedantry. There is significant evidence that how one teaches matters and that there is far more to teaching than just putting tasks in front of kids. As a profession, if we are to have any hope of solving the problem of communicating nuance about teaching with each other, we should at least start with being clear about how we use some basic professional terms like task and lesson.

Here are at least six problems that often make inquiry-based lessons fail.

Some Problems

Students have too much information to process when attempting to solve a problem which can quickly overwhelm their working memory.

When given a new problem type, students do not always have all of the prerequisite knowledge necessary to approach the new problem successfully, leading them to need to make leaps of logic larger than they are able.

When students are sharing their ideas and strategies for solving a problem, other students are either not really listening or hearing what they want to hear rather than what is actually being said.

Students sometimes focus on the short-term “How do we solve this type of problem?” rather than “What mathematical principle can I generalize to be able to solve other problems?”

Alternatively, students attempt to generalize and run out of working memory since they often have to hold both their solution and the generalizations from their solution simultaneously in their heads in order to generalize.

The goals and/or structure of inquiry-based lessons are often unclear. When students need to remember the goal of a lesson and the structure of an activity in order to be successful, they have less working memory available to actually be able to focus on the task at hand – the problem they are working on.

Students often use “means-end analysis” when problem solving which means they tend to focus less on what process they are using (and improving that process) and more on what answer will get them out of needing to continue problem solving.

Some solutions Note: The solutions described below assume that your lesson structure is still inquiry-based. It may be that the best solution to whatever goal you have for the day is not to use inquiry at all, but have students study worked examples instead.

There are a variety of solutions to problems 1, 2, 4, and 6 which are all based on problem-structure and task selection.

For example, in the instructional routine Connecting Representations, the goal of any given task is not to come up with answers to problems but to name connections between two representation types. Students are focused on a small amount of information at any given time, which is given students in a deliberately staggered way in order to reduce the amount of information to process all at once. Connections between representation types are also easier to generalize than solutions to problems.

A sample Connecting Representation task

To improve the odds that students can generalize from their problem-solving experience, make generalizations a focus of any whole group discussion following the problem-solving time and have students reflect (in writing, perhaps in a regular journal) on what they learned today that they think they may be able to use in the future. Sentence prompts like “Today I learned to pay attention to … because …” can scaffold these reflections for students.

Another suggestion is to select tasks for which students can use a lot of what they already know to solve the problem and only have to make small leaps. We want to balance our students’ use of their long-term memory to aid them in solving problems with their working memory to make the small new leaps or connections necessary. A good rule of thumb is that if students need to make 3 or more small discoveries or new connections in the course of solving any particular problem, they probably won’t.

We’ve also noticed in our work that it can be helpful to distribute problem solving tasks both over time and over a group of people.

Distributing a problem solving task over time means giving out portions of the problem to learners in smaller chunks rather than all at once. For example, it can be really helpful to give students some independent time to first consider a problem on their own before working with a partner. Or you can divide a more complex problem solving task into smaller pieces so that students can “chunk” their earlier work into their later, more complex solutions. The Math Forum’s “Notice and Wonder” protocol is one example of this principle being used in practice.

Distributing a problem solving task over a group of people means giving learners deliberate access to each other as resources while they are problem solving.

In Peter Liljedahl‘s “Building Thinking Classrooms” work, this looks like students working at vertical whiteboards in small groups or with a partner, during which I have often noticed groups borrowing ideas from each other.

An example of the use of vertical nonpermanent surfaces, shared with permission from Michael Pruner.

In our work with instructional routines (with lots of help from Grace Kelemanik and Amy Lucenta) this means making problem solving sessions short and/or interrupting an unsuccessful problem solving session with whole class opportunities for students to share observations and ideas. In order to improve the odds that students actually listen to and understand each other (and solve problem #3 listed above), we have students first share a strategy/idea while we or a student points, then another student restates a strategy/idea while we either continue to point, another student points, or we annotate the strategy using color/symbols/small amounts of text. We have also found it helpful to press students to provide complete explanations, especially when there are missing details/jumps in logic in their explanations.

To get a sense of how this supports students, try watching the following two videos to see how the use of restating and annotation makes a huge difference in your own clarity around a strategy being shared.

Student sharing a strategy: no gesturing or annotation

Student sharing a strategy: with gesturing and annotation

In terms of the 5th problem with inquiry, around the goals and structures of a lesson being unclear, I’ve written extensively here about how instructional routines support students (and their teachers) in minimizing unnecessary extraneous cognitive load focused on “what am i doing next?” If you don’t have time to read that other post, tl;dr: routines free up working memory by allowing students to delegate questions about what their role is, what they are doing next, and why they are doing it, to their long-term memory.

Conclusion:

In many schools around the world, learning how to use inquiry-focused lessons in mathematics class is a focus of the school or mathematics department. However, inquiry-focused lessons come with their own set of challenges, raised above.

I have some proposed solutions to those challenges listed above, but I’d love to hear what other people are doing to tackle the same problems or what other problems people have noticed occur when they try to implement inquiry-focused lessons.