Here is an example:

Each column corresponds to a step in getting from *x* to 3|*x* – 4| – 6. Note that we’ve already improved on the usual approach to order of operations, which is usually discussed as a way to interpret the final expression. This table gives us insight as we go in the other direction, starting simple, and moving towards increasing complexity. As the words “story table” indicate, we are telling the story behind the expression. If that is all we got from this representation, it would be a lot. But that’s only scratching the surface: story tables help us deconstruct almost any function and get a deeper understanding of it. In this example, as we move from one column to the next, we can see the impact of each operation on moving and stretching the graph, and we can see that the symmetry of the graph starts in the third column (why?)

Shira and Taryn gave examples involving linear functions, quadratics, third degree polynomials, trig functions, exponentials, logarithms, and they challenged us to think of more. And they did it effectively: instead of doing a lot of talking, they gave us plenty of time to do our own explorations.

Note that the story table can be used in many ways, depending on which part of the table is revealed, and which part is left for the student to fill out. There is no way to spell all this out in this blog post. If Shira and Taryn ever write this up, which they should do as soon as possible, I will link to it.

For now, I will just say that *I learned more about teaching algebra in this one session than I had in any conference presentation in ages — perhaps ever*. This is because story tables are not just a good tool to teach a particular topic: they are a good tool to teach many, many topics. As Shira and Taryn put it, “when you can, use a knife”. There are lots of specialized kitchen tools: peelers, corers, slicers, pizza cutters, zesters, melon scoops, and so on. But a knife is a flexible tool, which can be used to do many things, in many situations, often replacing specialized, one-use tool. They suggest that as teachers, we should be judicious in selecting tools, and prioritize the ones with the widest range of applications, the *knives*.

I will use the rest of this post to discuss tool selection. If you’re familiar with my work as a teacher and curriculum developer, you know that I practice and promote a tool-rich pedagogy. In fact, that was part of my talk earlier in the day at the same conference! (More on the talk: Reaching the Full Range.)

In all my writings about tools, I may not have discussed a good strategy for tool selection, so here we go. Let’s start from Shira and Taryn’s advice. They gave an example of a *knife* : the rectangle model of multiplication. (I call it the rectangle model, not the area model, because for young children, it is an array of objects rather than a continuous area, but that’s another conversation.) That is indeed a good example, as it spans arithmetic, algebra, and calculus. Here are some other examples of multi-use tools:

- Electronic graphing, which these days is best exemplified by Desmos, and can be used from pre-algebra to calculus.
- Algebra manipulatives, especially the Lab Gear, which covers too many topics to list here, but check out my books.
- The geoboard, suitable to teach about slope, area, the Pythagorean theorem, and more.
- Function diagrams are an unfamiliar representation, and thus they are resisted by some teachers. Still, they are a powerful tool in understanding domain, range, composition, iteration, the chain rule, and more.

By all means prioritize such tools! And definitely add story tables to the list, as they are a truly brilliant and multifaceted tool.

However I don’t agree that we should limit ourselves to such a list. As one masters knife-level tools, there is nothing wrong with also adding specialized tools to one’s toolkit as one gets further along one’s career path. For example, I’ve used pattern blocks to introduce angles, geometric puzzles to illustrate scaling, the circular geoboard for inscribed angles, and the ten-centimeter circle for basic trig (for those last two, see my *Geometry Labs*.)

There are also electronic tools which on the one hand are fairly specialized, but on the other hand are so powerful they are definitely worth learning about. They are tools not just for the student, but also for teachers and curriculum developers. Here are three of those:

- Snap! for programming and more. I’ve used Snap!’s predecessors (Scratch, and all the way back to Logo) to introduce basic programming ideas, some fun turtle geometry concepts, and some deep math and computer science: fractals, recursion. I’ve also designed tools and games using accessible computational environments (See for example my games, coded in Snap! by Parisa Safa: Signed Number Arithmetic, and Complex Number Arithmetic. Slow to load, but worth the wait.)
- GeoGebra for geometry. Actually, GeoGebra also has graphing, spreadsheet, and a computer algebra system, all in one application. But it is mostly a phenomenal all-purpose tool for geometry, and a great environment to create worthwhile applets that zero in on specific concepts.
- Fathom for statistics and probability.

Actually, listing all these tools gets me back to Shira’s and Taryn’s advice: since you can’t learn them and use them all next week, prioritize! The only reason I have such a long list of tools in my repertoire is that I’m old, and I’ve had plenty of time to learn them, and to develop activities for them. But hey, if you’re planning on being a math teacher for a while, keep an open mind about learning new tools. More tools means a more varied classroom, more visual bridges to concepts, more student initiative and responsibility, multiple representations of the most important ideas, a better way to preview and review material, … I would recommend you start with story tables, but I don’t have materials for you. Let’s hope Shira and Taryn provide those soon!

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Over the decades, I have attended some great talks there, and this year was no exception. I will post some notes and reactions here, starting with two tech-oriented talks I attended.

Photomath is a free smart phone app which can “read” exercises (even hand-written ones), solve them instantly, and display one or more paths to the answer. John Martin and Gale Bach introduced us to its power, and to some of its limitations. Here is an example. I handwrote a system of equations, aimed the phone at it, and this is what I got:

Each step can be expanded to show more details. Scrolling down gets you to the solution, and then you are shown how to check whether the answer is correct, once again with the option to expand each step. And that’s not all! The app displays a graph of the two lines, and for some reason their x- and y-intercepts. Photomath can do many things: solve equations, simplify radicals, find derivatives and integrals, and so on.

John and Gale’s used a debate format to present two ways to respond to the existence of this new electronic tool: should we explain to the students they can only use such a tool effectively if they understand the underlying math? or should we take advantage of it to assign different, more interesting problems? In my view, we should do both, and I suspect John and Gale agree — the debate was merely a way to structure their presentation. Neither of them advocated banning Photomath altogether, probably because they realized that is impossible. Their session started an important conversation, one which has been delayed too long.

I was disappointed that the subsequent discussion focused on how to handle cheating, and on how students could use this tool to teach themselves how to carry out these manipulations. To me the more profound issues were the ones raised in the “debate”. We will need much more than one session at a conference to sort it out, but here are some initial thoughts, using an example.

Take basic linear equations, the subject of a lot of deadly drill in middle school. Students need to know what it means to solve an equation, but *they do not need to be able to solve super-complicated examples*. (Leave those to Photomath, or Wolfram Alpha, or GeoGebra, or…) But how do we teach the basic underlying concepts, you ask? One way is to solve a lot of equations mentally, perhaps using a number talks format. “If 3x = 18, what is x?” and increase the difficulty from there. (3x+2=20, 3x + 2 = 21 + 5x, and so on.) Another way, once the very basics have been established, is to ask questions like “create an equation whose solution is 6.” This is a good way to consolidate understandings about “doing the same thing to both sides”, and there are more correct answers than students. (See more ideas on this in my How To post from 2015.) And of course, there are word problems, modeling questions, and assorted applications, none of which can (yet) be solved by machine.

More generally, speed and accuracy in paper-pencil computational manipulations are no longer priorities in math education. Teaching for understanding is really the only game in town. Trying to figure out how to teach the same algorithmic material the same way as the technology keeps racing ahead becomes more obsolete every day. In my own lifetime, calculators displaced multidigit ariithmetic, scientific calculators replaced log and trig tables, graphing calculators superseded tedious graphing by hand, and we are now in the fourth phase of this revolution. Of course, this does not mean that we know what to do about the new state of affairs. Many questions remain, but they will not be answered by trying to find ways to continue business as usual. Let’s keep the conversation going!

Tim Erickson presented several activities where data is derived from a geometric situation. The geometry is explored in the real world, using rulers and protractors, then the data is displayed in Desmos, which is supremely easy to do. Looking at the resulting data points hopefully yields an insight. Given the nature of the examples, Tim (a self-described “statistics guy”) discouraged the use of regression, suggesting instead various strategies to help students interpret what they see in the graphs. He gave many great pointers on how to help students see and think about the numbers: ask for a prediction (left to right, will the points generally go down or up? what will happen for small, large, or extreme x or y? and so on.) He also encouraged us to discuss the effect of measurement errors, issues with the displayed domain and range, etc. Tim was a master teacher at work, alternating between talking to the whole group, and looking at our work and pursuing conversations with individuals.

For me, the key question here is whether the approach he shared is even appropriate in geometry. My aesthetic sense says no. Look at the damn figure, think about it, discuss it. Learn to think geometrically! Save data analysis and modeling for a statistics class, of course, but also for an algebra class. In fact, this style of lesson is sure to improve the teaching of algebra, and in my view that’s where it belongs. For example, one outstanding activity Tim shared was an exploration of how much vertical space a given paragraph requires if you change the margins and keep everything else (font size, etc.) the same. This turns out to be a great example of a (nearly) constant product. This activity, and many, many others can be found in Tim’s book, *Functions from Geometry*. Get your copy (and other great stuff from Tim) at eeps.com, and start using it in your algebra and precalculus classes!

More Asilomar notes and reactions.

— Henri

]]>— Henri

– by Rachel Chou

“We were disappointed to hear that the 6th grade wasn’t leveling. We had hoped that Monica would be on the same path as her older brother Brian.” This was a comment made to me in passing, by two parents new to the 6-12 independent school where I work. These two lovely parents, were already concerned about whether or not their 11-year-old daughter would be exposed to Calculus by the 11th grade. 6th grade had only begun two weeks prior. Their feelings and sentiments are normal though. Many parents at independent schools were themselves both serious and successful students. At any given point, if their children are not on a particular path to “be the best,” to be “ahead of everyone else,” they start to get nervous. These parents are well-meaning. They love their children, want the best for their children, and worry when a child gets “off-plan” on the way to this measure of success. While the parents that I interact with are almost always kind, well-meaning, and have their children’s best interests at heart, their concerns and desires are often short-sighted.

At my independent school, as a matter of philosophy and practice, we do not have students study a traditional Algebra 1 course in 7th grade followed by Geometry in 8th grade. I’m particularly unimpressed with middle schools that brag, “We get all 7th graders through Algebra 1!” Normal or even advanced adolescent cognitive development does not leave a child ready to deeply understand the topics in a traditional Algebra 1 sequence. When middle schools brag about this, it means that they are teaching a course titled Algebra 1, but that they are necessarily altering the content and cognitive demand of the tasks being included, in an effort to make the material accessible to younger students. Or worse, they are not altering the cognitive demand of the tasks being included, and instead are running a course that is predicated on teaching students to be docile memorizers of routines, but not mathematically thoughtful and powerful thinkers. I don’t want my own child in a class where she is learning to do Algebra. I want her in a class in which she is being exposed to meaningful learning experiences, which have an effect on her overall ability to deeply understand Algebra.

Absolutely no one choose a pediatrician because he brags, “I get all babies walking by 12 months old!” We accept that the normal age at which babies learn to walk is somewhere between 9 and 18 months of age, and the children who gain this skill on the latter end of the scale are no less athletic or physically capable children. We also certainly do not believe that when a child begins to walk is any reflection of the pediatrician caring for him.

Meanwhile, the parents who might think that their child is particularly advanced might choose a middle school because it is bragging that it gets all of its 7th graders thru Algebra 1; however, this is also very short-sighted. Remember that this course must be targeted at the average cognitive maturity of a 7th grader. If I thought my child was unusually precocious, such a course would necessarily defeat my purposes.

At our school, half of the incoming 9th grade students come from outside middle schools. Many of these middle schools do allow the acceleration mentioned above. We tell parents that we base placement on what a student knows and can do and not on the names of courses they may have taken. In practice, while 20 to 30 of our incoming ninth graders have taken a course called Geometry, we place only 2 – 4 of them in Honors Algebra 2 as freshmen. Parents often worry if their children have taken Geometry in middle school, that they will be “repeating” something in our Geometry class, and yet, exactly no one, has ever called my history colleagues asking if their children can skip 11th grade US History for the reason that “Their child studied US History in 8th grade.” We somehow accept that kids can look at the problems of history with a more thoughtful lens when they have reached an older age. The same is true for Geometry topics. As a matter of fact, students who have studied geometry in their 8th grade years report little to no advantage over their peers who haven’t. Often, they are at a disadvantage because their ability to handle algebraic abstraction is less well-developed specifically because they studied Algebra 1 when they were not yet cognitively mature enough to understand it.

It may sound somewhat contradictory that we allow advanced placement in our high school and not our middle school, but this is a thoughtful choice, not an accident. It is not our department’s belief that all students can acquire mathematical understandings or power at the exact same rate, but it is our belief that picking children out too early for advanced acceleration provides students with very little gain, and can lead to unwanted consequences. I offer the following analogy: The tallest child in the class in 6th grade is not always the tallest child in the class in 10th grade. (They haven’t gotten any shorter! Their friends just grew taller!) We accept and understand that children hit their physical growth spurts at very different ages. This is also true of cognitive growth spurts, though it is far less obvious to non-teachers. A child might be particularly mathematically precocious in sixth grade, and might be closer to the average by tenth grade. This child has not grown less capable or intelligent, but rather her friends have simply caught up! The point here is that there is a real danger in separating out children at a young age for active acceleration. They might handle the extra challenges in the 6th grade, but if they end up in a situation in which they are hitting Precalculus in the tenth grade, they are often not ready for the cognitive demands of the class. And all a school has succeeded in doing in this case is accelerating a child to a place of frustration. These students often report that they “loved math when they were little” but that “they no longer do now.” That can’t be our goal as parents or as educators.

Not surprisingly, when I speak to parents at open house events and inform them that it is common for 20 to 30 of our admitted students to seek an Algebra 2 placement, but only a small few actually are placed there, the parents immediately assume that their child will be one of these “select few.” This makes lots of sense. First, as parents we are biologically predisposed to believe in our kids and think they are generally awesome. Second, there is the issue of sample size. The average parent at my school has a sample size of 1 to 4. The parents with 3 or 4 kids will think their best math student is “incredibly brilliant” and needs more than what the grade-level curriculum has to offer. But the educator who is making placement decisions typically will have a sample size of about 2,000 students. Math educators have a far better ability to understand where a particular student fits in terms of their cognitive maturity as compared to their peers, and parents would be wise to heed the advice of the math educators caring for their children.

Back to the story of the two parents concerned about their 6th grade daughter’s eventual 11th grade math placement. Another point worth mentioning is the odd belief in this country that K-12 mathematics is a race toward studying calculus. First, there is so much interesting mathematics for kids to study that is not a pre-requisite for studying calculus. Students are typically quite drawn to discrete math topics such as probability, combinatorics, elementary graph theory, number theory, sorting algorithms, and the list goes on. If we view a traditionally taught calculus course as the only end goal of a K-12 math sequence, we might leave such interesting and mind-stretching topics out. Why? Instead of lobbying your local school or your private school to accelerate your child toward Calculus, consider advocating that math curriculums include more depth, more open-ended tasks, and more discrete math topics.

Second, and possibly conversely, why are we waiting so long to get to calculus exposure? Students need not have formally studied continuous math topics such as advanced trigonometry, exponential and logarithmic functions, polynomials and the like to appreciate calculus concepts. A thoughtful geometry teacher might guide her students to find a “slope-computing formula” (the derivative!) of the function which represents the top-half of a circle, by applying understandings of how a tangent line intersects the radius of a circle. Similarly, included in either a geometry course or an introductory programming course, students might write code to compute the area under curves by breaking the area into skinny slices. How many of us actually integrate by hand, unless we are ourselves teaching calculus, or training students for the next integration bee?

No child’s mathematical journey is the same. As teachers, in any single classroom, we are charged with furthering the mathematical growth of a wide variety of needs. The thoughtful teacher sets up her curriculum so that students can learn, grow, and deepen their mathematical thoughtfulness and creativity at varying levels within the same classroom. It is time to stop believing that the best thing to do for our children is to advocate that they advance through a sequence of math courses at an accelerated pace.

]]>– As I mentioned in my first catchphrase post, nothing works in all classroom situations, and teachers need to be eclectic. Too often, we are pressured to adopt one or another currently fashionable article of faith. Doing so rigidly is never a good idea. My advice: **trust your intuition, avoid dogma, be flexible, be kind**. Of course, your intuition will improve as you get more experience, but even if your intuition is “wrong”, at least it is yours. Your students deserve getting the real you, not a poor imitation. If you are true to all four of these admonitions, you are sure to get better at this job.

– Some teachers are reluctant to miss school, even when they are sick, or when they have an opportunity to attend a workshop or conference. They feel that even missing a day or two would be a betrayal of their students. To them, I say: **You are as important as your students.** Your getting healthy is of course in your students’ best interest: it will allow you to be at your best when you return. Valuing your own professional development may seem selfish, but if you have a chance of learning something useful to your teaching, keep in mind that it will help not only your current students, but the students in your future — probably many many more than are in your classes right now. And really, are you so great that missing you for a couple of days is going to permanently damage your students? Didn’t you manage to survive a number of less-than-perfect subs when you were their age?

– A balancing act faces all teachers: how much time should you spend grading? how much time should you spend planning? In the first year or two (or three) of one’s career it is difficult to think clearly about this. The main thing is to survive the day, the week, the semester. But maintaining an unrealistic workload is not sustainable in the long run: it will push you out of the profession sooner rather than later. If you want to stay in the classroom over the long run, you’ll need to balance the different parts of the job, and accept that perfection is not going to happen. (See my post on Growth Mindset for Teachers.) That said, here is a piece of advice from an old-timer: **When grading, you are working for one student. When planning, you are working for the whole class**. Keep that in mind when you are budgeting your time. Don’t grade more than you need to. Can students correct their own or each other’s work sometimes? When grading, do you really need to write a lot? Do your students read what you write? Do they heed what you write? I’m not suggesting you should be irresponsible, just that you should be efficient. I found that it was enough to circle mistakes on quizzes and tests, and to ask students to turn in quiz corrections. (I already know how to solve the problems: they are the ones who need to figure them out. And grading corrections is fast.)

Well, that’s all for now. If you have favorite catchphrases about our line of work, please share them in the comments!

— Henri

]]>In this post, I want to argue that while I agree with the fundamental underlying idea of a spiraled curriculum, there is such a thing as overdoing the spiral. I will end with specific recommendations for better spiraling.

Too much spiraling can lead to atomized, shallow learning. If there is too much jumping around between topics in a given week, or in a given homework assignment, it is difficult to get into any of the topics in depth. Extreme spiraling makes more sense in a shallow curriculum that prioritizes remembering micro-techniques. In a program that prioritizes understanding, you need to dedicate a substantial amount of time to the most important topics. This means approaching them in multiple representations, using various learning tools, and applying them in different contexts. This cannot be done if one is constantly switching among multiple topics.

In particular, in homework or class work, it is often useful to assign nonrandom sets of exercises, which are related, and build on each other. For example, “Find the distance from (*p, q*) to (0, 0) where *p* and *q* are whole numbers between 0 and 10.” (This assignment is taken from my *Geometry Labs.*) At first sight, this is unreasonable: there are 121 such points. But as students work on this and enter their answers on a grid, they start seeing that symmetry cuts that number way down. In fact, the distances for points that lie on the same line through the origin can easily be obtained as they are all multiples of the same number. (For example, on the 45° line, they’re all multiples of the square root of two.) Nonrandom sets of problems can deepen understanding, but they are not possible in an overly spiraled homework system.

The main problem with hyper-spiraling is the above-described impact on learning. But do not underestimate its impact on the teacher. For example, some spiraling advocates suggest homework schemes such as “half the exercises on today’s material, one quarter on last week, one quarter on basics.” Frankly, it is not fair to make such demands on already-overworked teachers. Complicated schemes along these lines take too much time and energy to implement, and must be re-invented every time one makes a change in textbook or sequencing. Those sorts of systems are likely to be abandoned after a while, except by teachers who do not value sleep.

Another problem for teachers is that it makes using a hyper-spiraled curriculum difficult to use, because it is difficult to find where a given concept or technique is taught. (In the case of *Algebra: Themes, Tools, Concepts* we tried to compensate for that by offering an Index of Selected Topics and Tools. We also included notes in the margin of the Teachers’ Edition: “What this Lesson is About”. But even with all that, a hyper-spiraled approach makes extreme and unrealistic demands on teachers’ planning time. In fact, some hyper-spiraled curricula lack even those organizational features. Without them, a teacher needs to spend the whole summer working through the curriculum in order to be ready to teach it. This can be fun if the curriculum is well designed (e.g. the Exeter curriculum), but no one should feel guilty if they’re not up to that level of workaholism.

So, you ask, what do I suggest? In the decades following the publication of my overly-spiraled book, I developed an approach to spiraling that:

- is unit-based, and allows for going in depth into each topic
- is easy to implement and does not make unrealistic demands on the teacher
- is transparent and does not hide what lessons are about (most of the time)

I have written a fair amount about this, under the heading *extending exposure*. The ingredients of this teacher-friendly approach are:

- Lagging homework and assessments
- Separating related topics
- Teaching two units at any one time (just two!)

Implementing these policies does not require more prep time, or more classroom time, and it creates a non-artificial, organic way to implement “constant forward motion, eternal review”. It helps all students with the benefits of spiraling, but without the possible disadvantages. You really should try it! Read an overview of this approach on my Web site: Reaching the Full Range

— Henri

]]>On this blog, the most popular post on extending exposure is Lagging Homework, and it links to other posts where I describe additional strategies (separating related topics, lagging assessments, and more.) If you haven’t read those posts, you should. The problem is, it does involve a bit of clicking around. Thus, I decided to combine all that information in a single longer article on my Web site.

However, before I do that, I need to share a few more related tidbits which hadn’t made it into those posts. Writing about those here will help when I’m ready to put it all together.

The article is written. Read it here: Reaching the Full Range.

A number of people, over the years, have told me that they don’t agree with my lagging homework system because they like to assign homework that prepares the students for the next day’s lesson. That, my friends, is not a disagreement! I love that idea.

Lagging homework is not a rigid system that requires homework to be assigned exactly one week (or day, or month) after the corresponding class work. My main point is that on most days, you should not assign homework based on the day’s lesson. A week’s delay, more or less, provides many advantages which I described in my original post, the main one being extended exposure to each topic. This in no way precludes homework that sets up the next day’s lesson, as long as it is not usually based on the day’s lesson. I described the characteristics of such preparatory problems in this post, based on Scott Farrand’s approach to in-class warm-ups.

In general, such problems are actually examples of long-lagged homework, based on ideas that were introduced the previous semester, or the previous year, or whenever. Such long lags can also be used for review (better than take precious class time for that). On the other hand, if preparing for the next day’s lesson requires completing homework about today’s lesson, and this needs to happen frequently, then I strongly discourage that as the collateral damage on some of your students is substantial. (Unfortunately, it is the students who suffer from this policy will get the blame.) Work based on the day’s lesson should be done in class, perhaps as warm-ups the next day.

I had the privilege of teaching in long periods for my whole career. One way I used what is sometimes known as a block schedule was to focus on two units at any one time. For example, here is the outline of semester 2 in a “Math 2″ class I taught before my retirement:

(My point is not to recommend this exact sequence, which depends on many

department-specific assumptions, but to use it as an example of what is possible.)

Here are some of the advantages of that approach.

- Any one day or week is more varied, which is helpful in keeping students interested and alert. Note in particular that we tried to match topics that are as unlike as possible.
- It takes roughly twice as many days to complete a unit. This is good for students who need that extra time.
- This takes nothing from students who pick up ideas quickly. In fact, they appreciate the variety.
- It makes it easier to balance challenging and accessible work: if you hit a difficult patch in one topic, you can ease up on the other one. More generally, you gain a lot of flexibility in your lesson planning.
- If your work on one topic hits a snag, you can emphasize the other topic while figuring out what to do.
- Perhaps most importantly, it carries a message to the students: you still need to know this when we’re working on something else.

In the long period, it is possible to hit both units in every class period, for example by introducing new ideas on one topic during the longer part of the period, and applying already-introduced ideas on the other topic in the remaining time. (Homework is typically on one or the other topic.) Can this approach, or a version of it, be used in traditional 50-minute classes? I don’t know. I am guessing that the answer is yes, but I have not tried it. It might involve, for example, focusing on each topic on alternate days, while the homework is on the other topic .

But, you ask, is this not confusing to students? Don’t they prefer focusing on one single topic? If they do, that is only because that is what they’re used to. At my school, this was a department-wide policy, and once they’re used to it, it does not even occur to them to question it. In teaching, the biggest obstacles to making changes are the cultural ones. The only way to tackle these obstacles is departmental collaboration, and a step-by-step approach: don’t make all the changes at once!

My friend is right: the authors of these op eds are not piano teachers, but they’re often not math educators either! Still, it is important to address their ideas, because they reflect a broad cultural consensus among many parents, administrators, students, and teachers. Some proponents of the “skills first” approach equate teaching for understanding to what they call “fuzzy math”, a flaky anything-goes sort of teaching, with no specific learning goals, no accountability, just feel-good teachers who allow students to wallow in their ignorance.

It behooves those of us who disagree with this caricature to clarify what we mean by understanding. That is the main purpose of this post.

To get a straw man out of the way, my position is that understanding cannot be divorced from the acquisition of skills. Without understanding, it is hard to develop an interest in the skills, or to retain them; without skills, understanding is out of reach. Good teaching requires a skillful weaving of those two strands. It is, in fact, like learning to play the piano! Skills are important, but it’s all about the music.

But on to the main point: what is understanding? This is a difficult question, and the true fact that experienced math teachers can “recognize it when they see it” is not a sufficient answer. Here is an attempt at spelling it out. **A student who understands a concept can:**

**Explain it.**For example, can they give a reason why 2(x+3) = 2x+6? Responding “it’s the distributive rule” is evidence that the student knows the name of the rule, but a better explanation might include numerical examples, or a figure using the area model, or a manipulative or visual representation. Therefore, we should routinely ask students to explain answers, verbally or in writing, even though many don’t enjoy doing that. It is a way for us to gauge their understanding, and thus improve our teaching, and more importantly, it is a way for them to go deeper and guarantee the ideas stick.**Reverse processes associated with it.**For example a student does not fully understand the distributive law if they cannot factor anything. More examples: can they create an equation whose multi-step solution is 4? Can they figure out an equation when given its graph? And so on. Reversibility is a both a test of understanding, and a way to improve understanding.**Flexibly use alternative approaches.**For example, for equation solving, in addition to the usual “do the same thing to both sides” for solving linear equations, students should be able to use the cover-up method, trial and error, graphs, tables, and technology. If they have this flexibility, they can decide on the best approach to solve a given equation, and moreover, they will have a better understanding of what equation solving actually is.**Navigate between multiple representations of it**. Famously, functions can be represented symbolically, or in tables, or in graphs. Making the connections between these three is a hallmark of understanding. I have found that a fourth representation (function diagrams) can also help deepen understanding, and be used to assess it. Multiple representations on the one hand offer different entry points that emphasize different aspects of functions, but making the connections between the representations is part and parcel of a deeper understanding.**Transfer it to different contexts**. For example, ideas about equivalent fractions are relevant in many contexts, such as similar figures and direct variation. Or, the Pythagorean theorem can be used to find the distance between two points, given their coordinates. If a student can only handle a concept in the form it was originally presented in class or in the textbook, then surely no one would claim they fully understand it.**Know when it does not apply**. When faced with an unfamiliar problem, students will tend to reach for familiar concepts, such as linear functions and proportional relationships. Sometimes, this makes sense, of course, but students need to be able to recognize situations where a given concept does*not*apply.

Clearly aiming for all this is a high bar, and it is tempting to just have students memorize some facts and techniques, and then test them to see if they remember those a few weeks later. (This is often how the “skills first” approach plays out.) But what good would that do? It would just add to the vast numbers who got A’s and B’s in secondary school math, went to college, and now tell us “they’re not math people”. Yes, teaching for understanding is ambitious, and it must be our goal for all the students.

Alas, there are obstacles. For one thing, understanding cannot be easily conferred by explanations. (A naive traditionalist once suggested that it was easy to teach about variables: patiently explain to the students that variables behave just like numbers! Would that it were that easy.) Moreover, understanding is not always valued by students, parents, and administrators, many of whom believe that everything would be so much more straightforward if we could just have the students memorize facts and algorithms. Finally, understanding is difficult to assess. (Actually, the list above is one way to improve assessments: each item on the list suggests possible avenues for authentic assessment. To reduce complaints that it’s “not fair” to assess students that way, such assessments can be ungraded. As the current jargon would have it: consider them formative assessments.) Being able to reproduce a memorized set of steps is a good test of memory and obedience. To test understanding, non-rote assessments are the most revealing.

The above list is also a tool in forward design. When planning a unit, ask yourself how you can incorporate reverse questions, alternate approaches, multiple representations, varied contexts, and so on. A tool-rich pedagogy is helpful, as different manipulative, technological, and paper-pencil tools provide a way to do this and avoid boring repetition. Of course, the implication of such planning is that it takes more time to teach any given concept. Because of the enormous pressure of coverage at all costs, it is generally necessary to take less time on less important topics, and approach the most important topics in as many ways as possible.

In any case, I hope you find this post helpful in your teaching, and also in your conversations with colleagues, administrators, parents, and students.

Good luck as you teach for understanding!

— Henri

[This post includes part of my Nothing Works article, updated and expanded.]

]]>I started thinking about this topic when I learned that Annie Fetter’s catchphrase “

And yet! I love coming up with my own catchphrases. For example, the point I made at the end of the previous paragraph is encapsulated in the catchphrase “

A friend told me that “Nothing woris” is self-referential, since it too doesn’t work universally: she claims that some things I believe in *do* work. (Alas, I’m unable to remember what she was referring to!) A related catchphrase, the motto for my Web site and for this blog, is “**There is no one way**” (as the Zen Buddhist said to the traffic cop.)

To elaborate: teachers sometimes look for the “best” way to teach something. That is the wrong question: for anything important, we should know *many ways* to teach it so as to have the needed flexibility when reaching a wall in a given class or with a given student. Thus the importance of learning tools and multiple representations, which make that possible.

My favorite among my own catchphrases may be “**Consta****nt ****for****ward**** motion. Eternal review.**” This is aspirational, as it is sometimes necessary to pause the forward motion, and there may not be sufficient time or resources for eternal review. Still, it is a great thing to aspire to. Forward motion is essential to keep a course interesting, especially to our strongest students. Review is essential if we want ideas and techniques to stick. Each requires the other to work well, and in combination, they make for *extended exposure*, a must for heterogeneous classes (i.e. all classes.) Striving for constant forward motion and eternal review is facilitated by such practices as lagging homework, separating related topics, test corrections, and no doubt other techniques.

Incorporating those changes, or any changes, into one’s teaching (or into departmental practice) is a long term project. This is captured in my catchphrase “**Fast is slow and slow is fast**”.

What I’m trying to say with this cryptic formulation is that if you try to make too many changes in a hurry, you may find that in fact the changes are superficial, and the underlying classroom realities are not affected. Or you may conclude that “it didn’t work” and go back to your old ways. If on the other hand you pace yourself, and make incremental changes one step at a time, you will find that on the one hand you reap immediate benefits, and on the other hand the changes will take root and become the new normal. Be the tortoise, not the hare. Even better if you do this in ongoing dialogue and collaboration with your colleagues.

You’ll notice that my catchphrases are largely aimed at teachers, not students. This is because I’m in general agreement with “**Nix the Tricks**” a great compendium of how to *avoid* catchphrases in our teaching. (Download the book, which was put together by Tina Cardone and the MTBoS.) Shortcuts like “cross-multiply” or “FOIL” usually obscure the underlying mathematics. They often reflect a cynical attitude: the students will never understand these concepts, so I’ll give them an easy-to-remember shortcut.

Nevertheless, I do occasionally use a catchphrase in my teaching. For example, in geometry I might be heard offering the hint: “When working with circles, you should listen to the radii”. It is a good hint, with substantial mathematical content, so it’s not really a trick that should be nixed. Overall, my stance is that** formulas and tricks should encapsulate understanding , not substitute for it**. That is the catchphrase I’ll end on.

[Three more catchphrases in this follow-up post.]

Feel free to share catchphrases you love or hate in the comments!

New York City April 5,4:30pm: I will presentGeometric Puzzlesat the Museum of Math Teachers’ Circle. Geometric puzzles are accessible to solvers of all ages, but they can also challenge even the most tenacious of solvers. Join math education author and consultant Henri Picciotto in an exploration of hands-on polyomino puzzles that involve area, perimeter, symmetry, congruence, and scaling — you’ll even participate in some collaborative pentomino research!New York City April 6,6:30pm: I will presentPlaying with Pentominoesat the Museum of Math Family Friday. Pentominoes are simple to create — just join five equal-sized squares together — but provide a host of classic challenges in the world of recreational mathematics. Discover them, play with them, and explore a variety of visual puzzles that span the whole range: from kindergarten to adult, from the most accessible to the most challenging, and from the meditative to the maddening.Atlanta April 11-15: I will attend theGathering for Gardner. I won’t be presenting, but I look forward to seeing old and new friends. I contributed a thematic cryptic crossword to the conference book.Washington, DC April 25 and 27 3:00pm: I will presentTaxicab Geometryfor the Math Teachers’ Circle sessions at the NCTM National Meeting (in the Networking Lounge). Many concepts depend on distance: the triangle inequality, the definition of a circle, the value of π, the properties of the perpendicular bisector, the geometry of the parabola, etc. In taxicab geometry, you can only move horizontally and vertically in the Cartesian plane, so distance is different from the usual “shortest path” definition. We will explore the implications of taxicab distance. There are no prerequisites, other than curiosity and a willingness to experiment on graph paper.Washington, DC April 26, 2pmandApril 27, 1pm, I will be at the Didax booth (#253 in the exhibit hall, NCTM National Meeting) for a 15-20 minute introduction to the Lab Gear. Participants will get a free sample!Washington, DC April 27 8:00am: I will present “Quadratic Equations and Functions Use Manipulatives and Technology for More Access and More Depth” at the NCTM National Meeting. (Convention Center 209 ABC) Algebra manipulatives provide an environment where students can make sense of two ways to solve quadratic equations: factoring and completing the square. Graphing technology allows students to link those approaches to quadratic functions. Using these tools and connecting these concepts makes the algebra come to life for all students. [I will give away manipulatives to the first 72 attendees.]If you’re an experienced Lab Gear user, I would love it if you would assist me during that session. Get in touch!

And from the comfort of your own home:

Anywhere, May 17, 5:00pm Pacific Time:Reaching the Full Range, a webinar. As everyone knows, students learn math at different rates. What should we do about it? I propose a two-prong strategy based on alliance with the strongest students, and support for the weakest. On the one hand, relatively easy-to-implement ways to insure constant forward motion and eternal review. On the other hand, a tool-based pedagogy (using manipulatives and technology) that supports multiple representations, and increases both access and challenge. Click here.

Whether you attend these events or not, you can find handouts and links on my Talks page.

- I’ll be presenting two summer workshops for teachers, at Menlo School in Silicon Valley:
**No Limits!**(Algebra 2, Trig, and Precalculus, with Rachel Chou, Aug 1-3), and**Visual Algebra**(grades 7-11, Aug 6-9.)- For more information about the workshops, visit my Web site.

*Info about registration and logistics*: Menlo School.

–Henri

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Start with definitions?No! Most students find it difficult to understand a definition for something they have no experience with. It is more effective to start with activities leading to concepts, and introduce vocabulary and notation when your students have some sense of what you’re talking about.

**Michael Pershan wrote**:

I also don’t want to introduce vocab before kids are ready for it, and there are often times when I introduce vocab when it comes up in class. When that happens, I like to launch the next class with that vocab and give kids a chance to use it during that next lesson.

I don’t disagree with what @hpicciotto says in this post either. I square the two by saying vocab probably shouldn’t be the start of the unit, but it works nicely for me when it’s the start of a lesson.

My heuristic is something like, I want the definition to be easy to understand, so I give the definition when it won’t be hard to comprehend. And that often (not always) requires some prior instruction. But I do like introducing vocab at the start of a lesson.

I don’t disagree with Michael. None of the guidelines I gave in that post should be interpreted as rules one cannot deviate from. The essence of what I was trying to get across is to avoid definitions if students cannot understand them. Michael’s system does not violate that principle. Starting a particular lesson with a definition, if students are ready for it, is not a problem at all.

As always when thinking about teaching, beware of dogma! Be eclectic, because nothing works, not even what I say on this blog!

**Mike Lawler gave a specific example**:

Teaching via definitions: I found it useful to introduce a formal definition of division to help my younger son understand fraction division initially (this video is from 4 years ago) :

https://www.youtube.com/watch?v=dC409YJ60mc

Then a few days later we did a more informal approach with snap cubes. Overall I thought this formal to informal approach was useful and helped him see fraction division in a few different ways:

https://www.youtube.com/watch?v=o33WPlC5Blw

You should definitely watch the videos. They provide a great example of starting with a definition, that worked. But let’s analyze this. The student, in this case, was indeed ready for a definition:

- He already knew that fractions are (or represent?) numbers.
- He already knew what a reciprocal is (not merely “flipping” the fraction, but the number by which you multiply to get 1, and from there he got to flipping)
- He already knew, that a division can be represented by a fraction.

Many students who are told to “invert and multiply” know none of this, and for them defining division this way would not carry a lot of meaning. Moreover, students at this level usually have an idea of what division means, and *it would be important to show that multiplying by the reciprocal is consistent with the meaning they already have in mind.*

So my approach might be to start with something students know (for example, “a divided by b” can be said “b times what equals a?) and from there find a way to get to “multiply by the reciprocal”. (That is my approach in this document.) However it is not easy to do in this case, and defining first, and then getting to a familiar meaning may well be preferable. In fact, that is very much the approach I use when defining complex number multiplication in high school.

To conclude: yes, a lesson can start with a definition, as long as the students know what you’re talking about, and will not instantly turn off. This does not invalidate my point. To return to the example I gave in my last post, in a bit more detail, compare these two approaches to introducing the tangent ratio.

**Standard**** approach**: “Today we’re starting trigonometry. Please take notes. In a right triangle, the ratio of the side opposite the angle to the side adjacent to the angle is called the tangent. (etc.)” This approach is likely to lead to eyes glazing over, to some anxiety induced by the word “trigonometry”, and to a worry about remembering which ratio is which. The latter is allayed by the strange incantation “soh-cah-toa”, but alas that does not throw much light on the topic.

**My suggested approach:** “As you know, for every angle a line makes with the x-axis, there is a slope. For a given slope, there is an angle with the x-axis. [More than one, but no need to dwell on that right now.] We’ll use this idea, a ruler, and the 10cm circle to solve some real-world problems.” This allows students to right away put the tangent ratio to use, without knowing its name. (See *Geometry Labs, *chapter 11. Do Lab 11.2 after introducing the 10cm circle, but before making tables as suggested in Lab 11.1. Free download.) Once they’re comfortable with the concept, you can tell them there’s a word for this, a notation, and a key on the calculator. And yes, at that point, you can do all that at the beginning of a lesson. And a few months later, you can introduce the sine and cosine in a similar way.

— Henri

PS: How to introduce trig, complex numbers, and matrices in Algebra 2 and Precalculus are among the topics Rachel Chou and I will address in our workshop **No Limits!** this summer. More info.