Teaching Math Through Worked Examples
Do’s, Don’ts, and Discoveries by the Developer of You Teach You
When I first ran across John Sweller’s concept of worked examples fifteen years ago, it seemed as if a wish had suddenly been granted. I had tried everything to get math across to my 5th graders and middle schoolers over the previous two decades - from turning my classroom into “Math Land” with an electrical tape x-axis and y-axis on the square foot floor tiles and cubic foot boxes for 3D “construction projects” to writing hundreds of computer programs designed to help kids see mathematical causes and effects for themselves - and while my students enjoyed all of the activity and liked my class, very little of the math involved had ever stuck. I had begun to despair of ever solving the problem of teaching math to the point of mastery - and of even locating where the obstacles were - when suddenly, like a flash of light, Sweller’s concept revealed both why my Herculean efforts had failed and what I needed to do going forward: I had been flooding my students’ senses with a torrent of tenuously connected experiences and information; I needed to provide them with specific, sequential, and straightforward examples of the concepts and skills they were being asked to learn instead.
I immediately adopted a strategy of “working examples” at the board - going over sample problems with my students step-by-step as a group, and following up with related practice problems and detailed feedback afterward. And lo and behold, it worked!…for some of them. A larger percentage than ever before were able to latch on to concepts and demonstrate mastery of skills on various types of assessments - enough to convince me that worked examples were indeed the answer - but it was still nowhere near 100%. I tried to console myself with the thought that half of the students was better than none, but to no avail: the thought of never reaching the other half ate at me daily.
And then came a pandemic and remote learning.
In an instant - and to my horror - the half I had been reaching suddenly became unreachable too; working through examples on camera at a white board in my living room - without the give and take of the classroom, and with students passively watching at home (or not) - simply didn’t work.
Now my students - all of them - were learning nothing.
With the sickening realization that my only option now was to turn the job of learning completely over to them and hope for the best, I started creating self-instructional assignments they could check for themselves, each one based on a single pre-worked example. And to my surprise, even without me to explain them, they worked!…but only for the practice problems that were closely related to the examples. In a fit of frustration I found myself saying, “What am I supposed to do? Give them a worked example for every single practice problem?!”
Lightning struck.
I had time due to lockdown. I had the ability to distribute as many activities as I needed to with 1-to-1 Chromebooks and Google Classroom. And I had absolutely nothing to lose. I immediately started creating and assigning self-checking activities with a worked example for each and every practice problem.
And everything snapped into place. Instantly my inbox filled with messages from students saying it was the first time they had ever understood math - and from parents saying it was the first time they had ever found it so easy to help their children with math assignments.
By the end of that school year, I had seen more than enough to convince me: I was never going back to only working examples at the board with my students. I decided that when school opened again after lockdown, I would also be ready for them with a pre-worked example for every single thing they were being required to learn.
And when school opened again, I was. And at last they all understood precisely what they were being asked to do - and were actually able to do it. My wish for all of those years had finally been granted: all of my students, 100% of them, were now learning math - and their standardized test scores would prove it.
Since that time, I’ve identified what makes this specific method so successful, as well as how best to flesh it out and use it. I’m currently using these findings to create the You Teach You book series, but I’d also like to sum up my conclusions and recommendations here, for teachers and others who might be thinking of using You Teach You activities, as well as those who may be interested in creating activities of their own along these lines.
Worked Example Do’s and Don’ts:
For Teachers
Do: Use a 1:1 ratio of pre-worked examples to corresponding practice problems. Sweller’s concept of worked examples didn’t fail to start a revolution because educators didn’t apply it; it failed because those who did (like me) didn’t apply it enough. 1:1 is the correct ratio for beginners. Contrary to what might be expected, students don’t become overly reliant on the examples with this ratio - they only refer to them when they get stuck - and this precise ratio, and only this ratio, guarantees them a worked example directly related to every single problem they get stuck on.
Don’t: Feel the need to work through all of the worked examples with the students. Working through introductory examples is crucial, of course, but working through every single example is unnecessary - and possibly detrimental. Properly constructed worked examples are designed to be decipherable by the students themselves. Why? Because we learn best by figuring things out for ourselves, not by watching someone else “figure them out for us.” Why do we learn best this way? Because it’s far easier for us to remember what we ourselves did than what someone else did.
Do: Give students permanent access to the worked examples. Much of the power of worked examples comes from the fact that they stay put; students are free to study them at their leisure - and return to them later - without fear of them being erased or otherwise disappearing.
Don’t: Count on students to copy down worked examples perfectly from the board, or expect them to recall them accurately from memory. Asking beginners to copy down unfamiliar material correctly and neatly - with or without accompanying verbal explanations - is just setting them up for failure, and asking them to recall newly presented examples from memory alone is even worse.
Do: Give students access to fully worked solutions for all practice problems. This allows them to check their grasp of the material and track down mistakes and misconceptions in order to learn from them - and to do so without wasting time.
Don’t: Feel the need to go through all the solutions to the practice problems. Students are automatically interested in the solutions to the problems they’ve worked on, and will go through them carefully themselves. If they get stuck while examining a solution, they’ll come to you on their own - which is precisely what you want.
Do: Allow students to work through worked example activities at their own pace. Trying to get students to work at a pace faster than their own pace is futile by definition. And requiring them to work slower is just asking for boredom, disengagement, classroom distractions, and behavior problems.
Don’t: Limit the number of activities students may complete. Self-paced instruction requires that those who can push forward be permitted to do so. Anything less, again, is just asking for boredom, disengagement, classroom distractions, and behavior problems.
Do: Be the “sage at the side.” The pre-worked example strategy works best when there’s an expert in the room during independent work, offering explanations, inspiration, and assistance to individuals and small groups as needs - and teachable moments - arise.
Don’t: Feel the need to use worked example activities exclusively. The “sage on the stage” and the “guide at the side” strategies are critical too (and the super-efficiency of worked example activities leaves plenty of time for both).
Do: Allow students to repeat activities they’ve struggled with, ideally with some time in between. Here’s where I apply something I call the “Blue’s Clues Method” (named after the accidental discovery that repeating the same Blue’s Clues tv episode five days a week was far more educationally effective with young children than airing a different episode every day): simply have students repeat activities they’ve had difficulty with day after day until they can complete them correctly. There’s no use in a student trying to build on material they haven’t understood - and this approach promotes persistence and reassures them that they’re not being asked to build Rome in a day.
Don’t: Worry too much about mastery in the moment. Mastery occurs over time and math spirals back on itself; concepts that haven’t been fully mastered in the moment are certain to reappear. If a student feels the need to go back a lesson or two, they should, of course, be free to do so. If not, they should push forward. No worries: they’ll be sure to let you know if they get in over their heads.
For Creators
Do: Sequence worked example activities so they build from simple to complex concepts in an organized fashion, both within the activities themselves and between them. Nothing turns students off from math - and sows more anxiety and confusion - than instruction that jumps around in a seemingly random, disconnected way.
Don’t: Use written instructions if you can help it. Math is its own language with its own set of meaningful symbols - use this fact to create activities that communicate with as little verbiage as possible. Written instructions can be confusing or ambiguous (as any parent who’s tried to help their child with their math homework will tell you) - and math should not be inaccessible to students with reading difficulties.
Do: Use graphic representations, clearly but sparingly. The representational stage of mathematical reasoning - using drawings to stand for objects and amounts - is critical, but it should always be linked and lead to the ultimate goal of mathematical thought: abstract reasoning involving symbols and concepts alone. (Rule of thumb for incorporating a math drawing: if it can possibly be misinterpreted, leave it out.)
Don’t: Ignore algorithms and formulas. Graphic representations should be used to highlight the meanings behind algorithms and formulas, not to replace them. Math is far too complex to be approached by images alone (countless concepts, in fact, can’t be visualized at all); algorithms and formulas, properly understood, are what enable us to cope with this complexity.
Do: Make individual activities challenging. Students don’t need activities to be easy; they only need them to be do-able - and do-able and challenging is even better. Hit this sweet spot with your activities, and you’ll make them addictive, which is the best of all possible worlds.
Don’t: Make individual activities too hard. Each new activity should involve only a slight increase in complexity from the activity before it; a ladder with rungs too far apart cannot be easily climbed.
Do: Start each activity with what students already know and build from there. For example, don’t start addition instruction with addition; start it with counting.
Don’t: Assume perfect prior knowledge. Any background information required for an activity (formulas, definitions, etc.) should always be spelled out at the top of the activity itself, if possible; a workbench should always contain the proper tools for the task.
Do: Cover all possible situations and special cases. Scenarios that aren’t specifically covered are scenarios students won’t be prepared for.
Don’t: Overload the activities. The goal is to create a limited number of well-chosen problems filled with “learnable moments” - not to load up the lessons with “drill and kill.”
Do: Make sure the flow of each activity is obvious. Students should always know what they’re being asked to do and the order in which they’re being asked to do it. And when laying out a page, remember that in this part of the world we’ve been trained to read left to right, top to bottom.
Don’t: Get too “clever.” The goal is clear communication, not brilliance. What’s fun for the creator can be confusing - or worse - for the user.
And above all - keep things simple. Learning requires ample examples, but all the examples in the world won’t help if we let them or their contexts grow too complicated. (Considering these two critical factors, is it any wonder that most math textbooks and websites fail so spectacularly?) The sweet spot for beginners is many simple examples presented simply; the ideal is a 1:1 ratio of pre-worked examples to related practice problems, with both examples and problems gradually increasing in complexity over time, and with individual examples as clear-cut as they can possibly be. The relationship between the quantity and the simplicity of pre-worked examples and their presentation may be visualized like this:
The balance of quantity and simplicity indicated above is not only ideal, it has the potential to change math instruction forever (and will if we at YouTeachYou.org have our way). Why? Because such a balance, maintained continuously, permits students of all levels to access and conceptualize vast amounts of mathematical content to the point of mastery, and thus has the potential to turn math from something most people never learn to something they never forget.
John Sweller’s concept of worked examples was nothing short of revolutionary - let’s start applying it more extensively, and finally have the revolution to go along with it!
