Steve Hare

If I could have been granted one wish as a 5th grade math teacher years ago, I would have pushed for two: I needed to get inside the heads of my students to see why they were getting stuck with what seemed to me to be very simple problems, and I needed them to check the reasonableness of their answers before committing to them. Not a day passed without students submitting answers indicating that I had failed to get across the material, failed to get them to see if their answers made sense, or both.

One day, after metaphorically banging my head on the chalkboard for the umpteenth time, a solution popped out. “Who Wants to Be a Millionaire?” was the most popular show in the US at the time, and its climactic catch phrase “Final answer?” was everywhere. I suddenly envisioned a cardboard dropbox marked “Final Answer?” where students could anonymously submit answers to specific math problems - along with their work - on folded slips of paper. Such a box would add drama to the practice portion of each lesson, especially when I revealed the correct answers after a simulated drumroll. More importantly, the collected slips of paper would give me a glimpse into what students actually had in their heads, and the question on the front of the box would remind them to see if their answers made sense before submitting them. 

I suspected the “Final Answer?” box would give me detailed insight into students’ thinking and help them develop habits of self-reflection, and I was right. (Things like formal assessments and “exit tickets” weren’t enough to give me the kind of anonymous moment-by-moment feedback I needed.)

I had no idea, however, that the “Final Answer?” box would change my life.

It may sound crazy - and I would have thought it crazy earlier in my career too - but I now know it to be fact: almost 

 can master math given the right materials/instruction.

Prior to the “Final Answer?” box, if you had pried open 

 head, you would have found that I didn’t 

 believe that all of my students could learn math. I had seen too many bizarre and seemingly random “solutions” to what should have been easy problems over the years to be convinced that math could ever be understood by all.

The “Final Answer?” box showed me in an instant that I was wrong. It happened on a typical day after a typical question. As I started going through the answers the students had dropped in the slot, I sighed - yet again - at how many of them were wrong. But as I unfolded the last one, I got a shock I’ll never forget: half of the class may have gotten the wrong answer - 

. The ones who had gotten stuck had all run into the exact same roadblock! “Remove the roadblock,” I thought, “and they should all get the next question right!” I did …and they did! I had chills as the students cheered their own perfect performance.

 that didn’t make sense to the kids; it was 

. This would have depressed me to no end in the past, but at the same moment this problem revealed itself, a means of finding the solution did too: the “Final Answer?” box! The kids never tired of it, it was teaching them to think deeply about their answers, and it could teach 

 where they were getting stuck and how to “unstick” them!

Over the next few years, the “Final Answer?” box taught me far more about teaching basic math than my math training ever had. It taught me that if I didn’t have students total up more than two addends at a time when practicing multi-digit addition, many of them would mistakenly conclude that the regrouping digits were always 1’s, and would develop no understanding of the place value involved. It taught me that a series of multiplication problems like 526 x 3, 526 x 33, and 526 x 333 was far more effective for introducing the concept of place-holding zeros than if I had them solve, say, 753 x 3, 892 x 34, and 615 x 348. It taught me that if I didn’t include zeros in the answers (quotients) of the long division problems I asked students to solve, many of them would get stuck when zeros 

 appear and settle on answers like 57 remainder 8 when the real answer was 507 remainder 8 or 5,007 remainder 8 - even though they knew these answers were unreasonable! The “Final Answer?” box taught me all of these things and hundreds more besides. I’m now systematically incorporating everything I’ve learned over the years into the example-based 

But far more importantly than any of these things, the “Final Answer?” box taught me that almost 

. Math is logical. Kids understand logic (because it’s logical!). 

When math is presented logically - in sequence, in stages, in detail, and with an eye toward sticking points and special cases - there’s literally 

FFBLOG

The Best Math Teacher I Ever Had Was a Cardboard Box

Do’s, Don’ts, and Discoveries by the Developer of

When I first ran across John Sweller’s concept of 

 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 

-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

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 

 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 

 activities, as well as those who may be interested in creating activities of their own along these lines. 

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 

. 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 

a worked example directly related to every single problem they get stuck on

 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.” 

 do we learn best this way? Because it’s far easier for us to remember what we ourselves did than what someone else did.

Give students permanent access to the worked examples.

 Much of the power of worked examples comes from the fact that they 

; students are free to study them at their leisure - and return to them later - without fear of them being erased or otherwise disappearing. 

 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.

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.

 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.

 Allow students to work through worked example activities at their own pace.

 Trying to get students to work at a pace 

 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.

 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.

 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.

 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).

 Allow students to repeat activities they’ve struggled with, ideally with some time in between.

 Here’s where I apply something I call the “

Method” (named after the accidental discovery that repeating the same 

 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.

 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.

 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.

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.

 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.)

 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.

 Students don’t need activities to be easy; they only need them to be do-able - and do-able and 

 is even better. Hit this sweet spot with your activities, and you’ll make them addictive, which is the best of all possible worlds.

 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.

 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.

 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.

 Cover all possible situations and special cases.

 Scenarios that aren’t specifically covered are scenarios students won’t be prepared for.

 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.”

 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.

 The goal is clear communication, not brilliance. What’s fun for the creator can be confusing - or worse - for the user.

 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 

 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 

 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!

Teaching Math Through Worked Examples

[Disclaimer: serious misbehavior in the classroom is a serious issue and should be dealt with accordingly. This article is not about such misbehavior, but rather everyday, ordinary “goofing around.”]

A few weeks ago, two of my most hard-working and conscientious students were seated next to each other in the computer lab. Both of them had completed the coding assignment we were working on. I was helping other students debug their computer programs on the other side of the room, and was unaware that these two had finished their work – and now had nothing to do. Seasoned teachers already know what happened next: escalating tomfoolery. One of them typed a bunch of gibberish on the other one’s screen. The other one turned off the first one’s monitor. The first one pulled the plug on the other’s CPU. The other stole the first one’s mouse. Typical seventh grade misbehavior, of course – but this time from two very unexpected sources.

What caused these two “good kids” to suddenly “go bad”?

I should have told everyone to occupy themselves with something (appropriate) online if they finished early. Since I didn’t, the two cherubs in question followed my lack of directions perfectly - and did nothing. And got bored. And tried to cure their boredom as seventh graders tend to do, by 1) trying to make each other laugh, 2) getting on each other’s nerves, and 3) both.

A few years ago, as I was observing another math classroom, I watched a kid clown around day after day instead of working on the week-long project he was supposed to be working on. This kid was ordinarily a high-level math student, and I had found his behavior to be exemplary in other classes, so the misbehavior in this case interested me. After he was ejected from the class one day on account of his increasingly disruptive antics, I had the chance (in the detention room) to ask for his side of the story. Fighting back tears, he informed me that he had no problem doing the math and had been trying to work on the project, but that he didn’t understand the point of the project or its directions. (This last part I could readily believe; none of the students understood them.) He also told me that he had been hurt by the teacher repeatedly accusing him of laziness, especially since, up to this point, he had considered the teacher to be a friend.

These situations illustrate a law that I’ve observed countless times over my 30-year teaching career, a law that should be right up there with Newton’s law of universal gravitation and Bernoulli’s law of fluid dynamics: Kids don’t goof off because they’re bad; they goof off because they’re bored, baffled, or browbeaten.

How can we make sure they aren’t bored or baffled? (I’m assuming we all know how to avoid browbeating them.) By keeping all instruction within the Goldilocks zone (or flow channel), the mental region where tasks aren’t too hard (confusing) or too easy (boring), but instead are continually just right. I’ve illustrated this zone below (by modifying an image from my article 

Now before we look at how to achieve this goal, let me talk about what I’m 

 talking about. I’m not talking about busy work (word searches, crossword puzzles, mazes, etc., etc., etc.). Schools are littered with this useless crap, because 1.) kids will happily do it, 2.) it’s not confusing, 3.) a kid can complete it incorrectly and it doesn’t matter, and, most importantly 4.) it gobbles up huge amounts of class time. Kids are fine with “working” on this nonsense because it runs out the clock and beats the other more mystifying or soul-crushing work the teacher could be giving them instead. Busy work is excluded from this discussion not because kids don’t learn anything from it (they don’t), or because it doesn’t empower them or make them feel better about themselves in any way (it doesn’t), but because it doesn’t prevent behavior problems. Why not? Because kids aren’t stupid; they know when they’re being condescended to – and they resent it just like we do.

Instruction within the Goldilocks zone prevents behavior problems because it requires and rewards sustained focus, and because it makes kids feel good about themselves, as all genuine accomplishments do. When students feel themselves actually growing and getting smarter – I mean, for real – they're more inclined to do their work, and less inclined to goof around or seek attention.

So how do we get there? By using the classic 1-2-3 punch of “see it, do it, check it.” Each learning task we assign must include an instantly understandable and fully worked-out example to refer to, a related practice problem, and a key containing the fully worked-out solution to the practice problem. (Stay tuned for forthcoming activities along these lines!) By working their way through sequences of related and slightly more complex tasks of this type, and by gradually progressing through “levels” of related and slightly more advanced topics, students will discover more reasons to apply themselves in school and take their learning seriously.

Why Students Goof Off in Class (Hint: It's Not Because They're Bad)

Think for a moment about how a kid learns to play a video game. Gameplay begins with an understandable situation in which the kid learns skills to solve specific challenges, by trial and error. Successes are immediately apparent, as are failures – and the reasons for both are obvious.

Gameplay progresses until the kid has mastered the skills – and the situation is no longer challenging. The kid bumps up a level, to a new, but related, situation. New skills must be mastered, new challenges must be met. In time, the kid masters these skills and meets the new challenges, then gets rewarded

Why? Because the successful application of a hard-to-acquire skill to a hard-to-beat challenge provides the brain with a hit of dopamine. This hit feels great, so more are sought. But there’s a problem: the old challenge is no longer a challenge. New challenges must be sought, along with the new skills required to meet them. The new challenges can’t be 

 challenging, though; when they are, we simply give up. And they can’t be too easy either, or we become bored and find other things to do.

Game developers know this, of course, so they purposely design their games to stay within what psychologist Mihály Csikszentmihályi termed the 

, the zone where there’s a constant and addictive balance between skill and challenge levels. When we manage to stay within this zone, we enter into the brain state Csikszentmihályi called 

, a state of complete absorption that produces a deep sense of enjoyment – and even alters our perception of time; hence kids playing video games for hours on end with no letup.

You could also call this channel “the Goldilocks zone,” for obvious reasons. It’s illustrated below.

Well, for one thing, it explains why most kids don’t learn it. Too often, kids are given math tasks that are too hard or confusing (so they give up, and don’t learn anything) or too easy (so they get bored and zone out – or misbehave – and 

It also provides us with 7 guidelines for successful math instruction:

 of math in order to master it. They’ll never get there if they don’t find learning it to be at least slightly addictive.

Thanks to Frogger, Fortnite and Final Fantasy, we now know how to make that happen.

Why Kids Master Insanely Hard Video Games but Struggle with Basic Math

If you ever want proof that the way you teach something determines whether or not students actually learn it, try teaching a room full of kids to say supercalifragilisticexpialidocious backwards (as Julie Andrews does during the famous song from Walt Disney’s 

First, teach them badly, by simply writing the backwards version on the board:

If your students are like mine, when you ask them to read it - even by “sounding it out” - most of them will freeze up and draw a blank. Some, of course, will 

 to pronounce it, but they too will soon give up. In my 30 years as a teacher, exactly zero students have been able to read it correctly on their own the first time through - despite the fact that it has a regular English spelling.

Now draw the following on the board and ask them to try it again:

Most or all of the students will try it this time, and some will actually succeed (it’s pronounced doh-sh

-al-i-ek-spee-is-tik-fraj-ee-kal-ee-roo-p

). Pronounce it yourself several times and give the students some time to practice, and pretty soon they’ll 

 be saying it correctly. (Then, for an added challenge, try getting them to 

Imagine: a 0% success rate quickly converted to a 100% success rate by simply breaking the task into pieces. What’s the explanation for such different rates of success? It’s all based on the way our working memory works - or should I say, 

. As George A Miller revealed in his famous 1956 paper 

, there appears to be a limit to how many items we can hold in our working memories at one time, and that limit is about seven. A 33-letter, 14-syllable nonsense word simply doesn’t stand a chance.

But that’s only when we try to pronounce it in one go. If we apply our innate abilities to work with 

 of items (with no more than 7 items per group as a general rule), we can bypass our working memory limits and enable our minds to work with practically limitless amounts of complex material. By turning dociousaliexpeisticfragicalirupus into seven two-syllable groupings and stringing them together, the load on working memory is reduced and our natural ability to work with groups kicks in - and the correct pronunciation becomes a piece of cake. Notice that the number of 

 is key here, as we’re teaching kids to 

 it, the approach would work just as well in this case; the longest grouping - “docious” - is exactly 7 letters long.

Scientists call the individual groupings of information “chunks,” and the act of breaking complex information into sub-groups and stringing them together into coherent groups “chunking”. There is evidence of both all around us. Consider phone numbers; instead of trying to memorize a number like 18423891224 we break it into a single-digit country code (1), a 3-digit area code (842), a 3-digit exchange code (389), and a 4-digit subscriber number (1224), producing the much more easily remembered 1-842-389-1224. (Seven digit telephone numbers were actually 

 to conform to Miller’s “magical number seven,” in order to make them easy to remember.) Zip codes, as well, contain easy-to-remember “chunks” now that they’ve created the extended “Zip+4” codes. Notice what the Postal Service 

 do when they created them: they didn’t just tack on 4 extra digits to create 9-digit Zip codes.

Interestingly, the Sherman Brothers, the songwriters who wrote 

, intuitively used chunking when they created the backward spelling. As you may have noticed, if you actually try to read the word backwards, the result doesn’t even come close to dociousaliexpeisticfragicalirupus; the actual backwards spelling is virtually unpronounceable. To navigate this dilemma, the Shermans broke the word into chunks and simply reversed their order - with the exception of “super,” which they “fake reversed” to “rupus,” which sounded funnier.

Chunking is a powerful  memory device, of course, but to think of it as 

 a memory device would be to miss its larger implications; chunking tells us how our brains master complex skills, namely by breaking them into sub-skills, mastering those sub-skills, and then putting everything back together.

Take the complex skill of playing softball. In order to master it, you first have to master sub-skills involving how to hold a bat, how to run the bases, how to throw the ball, how to field a grounder, and so forth. Then, on the field, you simply put all of those sub-skills together. If you fail to master any of the sub-skills - actually 

 them - you won’t be a good softball player; a player who can do everything except field a grounder will most likely spend a good deal of time “riding the pine.”

This is no different in principle from our dociousaliexpeisticfragicalirupus example; if you can pronounce every part of the word correctly except for “expe” (ek-spee), you're not pronouncing it correctly at all.

Of course, this is all very obvious, to the point of not needing to be said. So why don’t we teach math this way? Why aren’t students mastering math sub-skills, and putting them together to master the math skills themselves?

Because we haven’t been aiming for mastery at all! If I had a dime for every time I’ve heard an educator say, in private, that we’re really just “exposing them” to math; you don’t just “expose” a kid to softball - and then put them in the game. (There are subjects - like music, art, and computer science - where exposure to the subject may be sufficient; math just isn’t one of them.) How do I know math teachers in general aren’t shooting for mastery? Because they’re not employing the “see it, do it, check it” learning strategy that mastery requires. Students aren’t being given enough worked-out examples to refer to, practice problems related to those worked-out examples, and worked-out solutions or other immediate feedback so that they can check their work and nail down each individual concept. Isn’t that how you master new skills?

Imagine the complex math skills our students 

 master if they first mastered the “chunks” that math is made of. Imagine the 

 skills they could master if they mastered those math skills!

A supercalifragilisticexpealidocious future where all students know math backwards and forward is entirely possible - so long as we build it one dociousaliexpeisticfragicalirupus chunk at a time.

How to Say Supercalifragilisticexpialidocious Backwards: A Key to Learning Math

I am looking at the very first lesson in a popular and widely-used 7th grade online math textbook right now. The lesson involves ordering rational numbers and is spread out over 6 pages.

The introductory section includes an open-ended activity in which students make number lines on the floor to solve numerous self-generated math problems with a partner. This follows an introductory activity for the 

 that requires students to plot 41 points on a coordinate grid and then create a “dot-to-dot drawing” of their own design with an additional 20 points.

There are 51 problems in the actual lesson, ranging from basic tasks involving absolute values to extremely difficult word problems involving scientific tests on the moon, measuring the corrective powers of contact lenses (in 

, whatever they are), summit elevations of volcanoes, and the freezing points of various liquids. This does not include the 27 problems of various difficulty levels on the workbook pages designed to accompany the lesson. There are answers to

none of this in the text itself; there are answers to the odd-numbered problems in the back of the book, but even then there are no explanations of how the individual problems were solved. Instead, there are two “self-assessments,” one for “skills and concepts” and one for “problem-solving.” Both of the self-assessments invite students to rate their own understanding of the “success criteria” in their “journals,” but include no correct answers or explanations to help them do so. The three “success criteria” are listed in small type back on page one - I know because I 

. They are, “I can graph rational numbers on a number line,” “I can find the absolute value of a rational number,” and “I can use a number line to compare rational numbers.” None of the practice problems in the lesson explicitly involve graphing rational numbers on a number line or using a number line to compare rational numbers. It would be interesting to see how students rate their ability to do something they haven’t been explicitly asked to do.

There are only four examples provided over the course of this 6-page lesson (!), one of which requires reading three full paragraphs of text (you know, because none of our kids have reading difficulties). There 

 a one-line definition of rational numbers on page one of the lesson, but there are no explicit examples or non-examples to indicate what a rational number is or isn’t.

There will be roughly 179 more days this year just like it.

The sad truth is that math textbooks don’t teach math: they assault you with it. Unexplained, uncheckable, unrelenting math, math, math.

This shouldn’t be surprising, though. Textbook publishers aren’t rewarded for teaching math; they’re rewarded for selling textbooks. Who cares if the kids are learning anything so long as sales are up? Can we all just finally admit that textbook publishing is a 

 these snake oil salesmen, not buying books 

 them. To appeal to as many teachers, administrators, and school boards as possible, publishers fill textbooks (and supplementary materials) with truckloads of content – despite the fact that this produces 

on the part of the students. That they still do this in 2023 is inexcusable. We’ve known about the very real limits of human cognition since at least 1956, when Harvard’s George A. Miller famously published 

, one of the most cited papers in the history of psychology.

The fact that the limits of attention span and working memory haven’t been baked into instructional design since then is a failure of positively epic proportions.

And then, to add insult to injury, textbook authors don’t lift a finger to help the students learn the content they’ve dumped between the covers. Almost no examples. Very few answers. Almost no explanations.

And most of us know it. If you’re like me, you remember writing down any-old-answers during textbook lessons or in workbooks because you had no idea what you were supposed to be doing (not enough examples) and no way of knowing if you were doing the work correctly anyway (no keys to check your work, and no explanations to help you learn from your mistakes).

And if you’re like most of us teachers, you long ago gave up trying to check all of the students’ textbook and workbook work yourself because 1) it’s mathematically impossible, and 2) it’s too depressing. There’s no worse feeling than checking long division worksheets and realizing that, with no examples or key to refer to, most of the students got most of the problems wrong - and just practiced how 

 change, because textbook publishing is Big Business

The publisher of the book I just referred to, Cengage Learning, has $1.5 billion in annual revenues, and is one of the five publishing companies that control 80% of the $13.9 billion textbook market and is accused of being a cartel. No publisher is ever going to come close to admitting that all of their expensive books, and workbooks, and worksheets, and supplemental games, and e-games, and e-resources, and e-

complete and total waste of precious educational funds

Because they don’t work. As you can see in the 

 below, the US is in the bottom third for industrialized nations despite our extensive

Because they unnecessarily complicate a subject that ought to be a pleasure to learn.

But it’s even worse than that. Textbooks make kids feel stupid, and kids who feel stupid in math class start giving up on math itself.

We are literally teaching our kids to hate math, and spending a fortune doing so.

Now this would be an absolute tragedy if we didn’t have alternatives.

Math is learned one problem at a time – and 

 one problem at a time. Contrast that with the 119+ problems students are barraged with in the very first lesson of the year in the example above. All you need is an example problem (which can involve manipulatives, drawings, or symbols depending on the age or level of the student), a similar problem to try, and access to the worked-out solution or solutions to the problem you tried. Once you’ve finished your work and checked it, you need a related and slightly more complex example and practice problem to try, along with the worked-out solution or solutions to 

 problem. Then it’s just keep ‘em coming.

This “see it, do it, check it” strategy is 

 of us learn. It’s low-stress, devastatingly effective, and as simple as Lego instructions.

It’s how math should actually be taught.

Anybody who tells you differently doesn’t understand how kids learn - or is trying to sell you something.

Why Math Textbooks Don’t Work and Never Will

Diane Hare

Gene has been involved in education for over thirty years. He has served as coach, teacher, counselor, administrator, and consultant. He is the author of 

. Dr. Tavernetti holds a core belief about children and adults: Given the right environment and proper support, everyone can improve and succeed. We, at FactFreaks, share that belief.

Do your students struggle with story problems? Are you searching for a new strategy or trick? I'm sure that you've run across the multitudes of various strategies that claim to teach students how to solve story (also known as word or application) problems. What is disenchanting about the many strategies, tips, and tricks, many of which are found in textbooks, is that the authors provide these strategies as an ala carte menu of offerings as if there were no cohesive agreement regarding problem solving.

But that does seem to be the case. Most subjects, with the exception of math, have an agreed upon framework or protocol for solving problems, with only small variations in the various disciplines. In content areas other than math, students are taught to use some form of the scientific method to solve problems:

The good news is that this methodology is so common it is constantly observed in our everyday life! Think about the last time you went to your doctor with a malady. Your interaction with the doctor likely followed this structure. The doctor asks, “What is wrong?” (Statement of the Problem) Next, “Have you or anyone in your family had this, or something like this, before?" (Identify the problem) "What were you doing differently when you first noticed this?” (Determine what is already known) After several questions like this, the doctor may say, “I have seen something like this before” or “Let me do some research on that." (Generate a hypothesis) From the collected data, a diagnosis is made, a treatment is recommended (Conduct an experiment), and a follow-up appointment is made (Review the data).

The most time consuming, and I argue, the most important components of the process are the first two steps; identifying and gaining knowledge about the problem. A student should continually re-visit their initial questions, which identified the problem and determined what was already known, multiple times throughout the process to stay focused on getting the correct answer. 

Use of the scientific method is ubiquitous in our daily lives and is, in fact, a largely intuitive process.  We observe a crying baby. We ask when has this happened before? Maybe the diaper is dirty, so we change the diaper but the baby is still crying. Maybe she is hungry, but when offered food she won’t eat. Imagine that the crying is due to pain but we decide to stop asking questions and just put her in her crib and hope she falls asleep. The first two steps are so crucial, that if we stop gathering data and skip to the wrong conclusion without generating a hypothesis,

This process continued correctly - identify the problem, determine what is known, generate a hypothesis, conduct an experiment, and review data - is simply common sense. We use it everywhere except when we teach students to solve story problems. To be fair, I have 

about teachers using this framework, but in my twenty years as an instructional coach I have never 

 it. Instead of teaching students a process to identify the problem, instead of teaching them a structured systematic process to reflect on whether they have encountered similar problems, we observe teachers directing students to use disparate strategies that lack the coherence of the problem-solving framework. Strategies such as finding the numbers in the problem, looking for clue words or phrases such as “all together”, “remain”, “how many groups?”, are strategies that may work for some problems but will fail for others.

These disparate strategies do not lead to deeper understanding of the problems by the students, and they rob teachers of the opportunity to gather discrete formative data that could guide further instruction. If a student is unable to solve a problem, 

we need to find out where they are getting stuck

Blog

Updates & Upgrades

Guest Spotlight

Features