In 1981, I moved from K-5 into high school. This transition felt almost like starting a new career. In many ways, what I had learned about pedagogy as an elementary school teacher still applied, but I needed to learn a lot more to grow into my new job. That’s when I came across *Geometry: A Guided Inquiry *(GGI), a way-ahead-of-its-time 1970 textbook by Chakerian, Crabill, and Stein. It was just what I needed, and had an enormous impact on me, both as a teacher and later on as a curriculum developer. In fact, of all the books I’ve seen in my 49 years in math education, this is probably the one that taught me the most, by far. In this post, I will try to explain why by listing some specifics.

**1. Group work helps student learning. **The front matter of the Instructors’ Edition explained the benefits of students working in collaborative groups. (Here, I have to trust my memory, because I only have a copy of the student book.) Students get a deeper understanding if they discuss the math with each other; and this setup allows the teacher to pay attention to (e.g.) eight groups rather than 32 students. The authors also argued that groups of four are optimal, because in groups of three one student is often left out, and groups of five are unwieldy.

**2. Tradition is not a good guide to the sequencing of topics. **Or at least, it should not prevent a teacher or curriculum developer to consider other options. For example, starting with definitions of points, lines, rays, etc. is a terribly boring start to a course. Or, the concept of deductive reasoning need not be introduced prior to doing interesting geometry. And so on. In particular, it verges on insanity to start with formal proofs of self-evident results, such as:

Midpoint Theorem: If M is the midpoint of AB, then AM = AB/2 and MB = AB/2

There is no quicker way to convince students that math is weird twilight zone where their teacher is an idiot, and yet they need to do what they say. (OK, I’m exaggerating, but the point of proof is to dispel doubt. The idea that obvious statements need proof is an advanced idea, completely inappropriate for the first few days of a 9th or 10th grade course.)

**3. Most students don’t learn things that they only see once. E**ach chapter of GGI is organized in three parts. *Central* is where the ideas are introduced. *Review *goes back over those ideas, using many interesting problems. And *Projects* include extensions which are not required for the book’s sequencing to work. At a certain point, I realized that forging ahead to the next Central while assigning Review problems as homework was a way to extend student exposure to the concepts. This was the seed of my idea about lagging homework. (Read about that here.)

**4. Guided inquiry provides the right balance between student discovery and direct instruction.** I can’t get into that here, but in short: neither is sufficient without the other. Students cannot hear answers to questions they do not have / students cannot discover all of math. The key to a healthy combination of discovery and direct instruction is the use of worthwhile problems that are both accessible and challenging, both before and after the key results are presented explicitly.

*Anchor problems and activities* help to introduce big ideas. Chapter 1 of GGI starts with the “burning tent” problem: a camper who happens to be carrying an empty pail near a straight river needs to run to the river, fill the pail with water, run to the tent, and put out the fire. What is the shortest path to accomplish this? Chapter 2 starts with the question: which polygons tile the plane? These lessons are engaging and accessible, and lead to many important and interesting ideas. Give me these openers any day over “the segment addition postulate” and the like. (I wrote about anchors in these posts: Mapping Out a Course and Sequencing.)

*Problems need not be sequenced in order of increasing difficulty.* This is countercultural, but effective. When students don’t know if the next problem is going to be easy or difficult, they are more likely to give it a shot. If the problems get harder and harder, many students will reach a point where they decide they can go no further.

*Practice need not be boring. *For example, the book had many entertaining problems in the form “what’s wrong with these?” which showed figures which violated one or another theorem.

*Answer-getting is not the point. *In fact, some answers are given right there in the margin, which allows students to check their understanding as they work. Most other answers are given at the end of each chapter.

**5. Inquiry and proof are not mutually exclusive. **Some geometry books prioritize inquiry at the expense of teaching proof. Others prioritize proofs at the expense of motivation and access. GGI keeps these in balance. Also, while Chakerian et al present both paragraph and two-column proofs, they make clear that the former is the standard in mathematics. If I remember correctly, in the Instructors’ Edition, they point out that students will be writing paragraphs for the rest of their lives. Two-column proofs, not so much.

Having absorbed all these ideas, I was ready to start developing curriculum myself, and to lead my department away from textbooks in our core classes. But even decades after using our own materials in geometry, we continued to use some of the problems from GGI.

I will forever be grateful to Chakerian, Crabill, and Stein. I was so lucky to have run into *Geometry: A Guided Inquiry* early in my high school career!

— Henri

PS: At the height of the math wars in the 1990’s, I was involved in a discussion with Chakerian. It ended well. Read about it here.

PPS: Chakerian, Crabill and Stein also wrote books for Algebra 1, Algebra 2, and Trigonometry. Those were not as successful as the geometry book, but I still got many good ideas from those, including the 10cm circle and a geometric approach to complex numbers.

]]>Early on, Johnson shares his goals as a teacher. Here are three good ones:

- I will give
*no*options for nonparticipation. - I will allow students to make mistakes without fear of failure or embarrassment.
- I will encourage student interaction during a portion of each class period.

After listing his eight goals, he adds: “I don’t remember a single day when I can honestly say that I have achieved all of them…” That’s evidence that the author is an actual teacher, not someone promoting the latest fad. In fact, the whole book confirms that: it consists of a mix of big picture ideas and specific implementation ideas about everything from homework, to the daily routine, to assessment, all of it stemming from one teacher’s experience.

The most useful part of the book, both in the 1980’s and now, is the chapter on “The Art of Questioning”, in which presents some great ideas about how to run a class discussion. He starts out by challenging what he calls the “one-on-one questioning method” (you ask a question, get the correct answer from a student, and move on, without knowing whether anyone else in the class understands what you’re talking about.) He then offers a list of 20 “try-to” principles of questioning. (Yes, twenty!) Here are three of the best:

- Try to avoid yes/no questions.
- Try to follow up student answers with “why?”
- Try to ask for reactions to a student answer.

Johnson’s overriding philosophy of class discussion is two-fold: on the one hand, the questions should get all students to engage, and on the other, student answers should reveal to the teacher the extent of student understanding. He discourages us from asking “Does everyone get this?” or similar questions, which rarely yield accurate feedback about what is really going on. He believes it is the teacher’s responsibility to figure out who gets it. This leads to his key insight: he needs answers from every single student. To achieve that, he has the whole class answer questions using the “paper-and-pencil method.” He asks a question, has all students write down their answers, and he walks around to see what they wrote. This way he can proceed with the lesson with full knowledge of who’s on board, whether he needs to backtrack, and so on. (Of course, there are other ways to get this sort of whole-class feedback, which I should probably discuss in a future post.)

To make this method more efficient, he lays out the student desks in a half-circle, with the teacher at the center — what he calls “a U-shape arrangement”. This means that no one is sitting in back, and the teacher can quickly see all the students’ work. Unfortunately, this is not optimal for student collaboration, so this is not a suggestion I would take up. Still, even in a collaborative classroom, teacher-led discussions play an important role. David R. Johnson’s techniques for genuine student engagement in class discussions are mostly excellent, and far superior to the standard pseudo-interactive lecture format, where a small number of vocal students finish the teacher’s sentences.

In fact, his ideas on questioning would make a great topic for a math department meeting, and for targeted visits from colleagues to your classroom. It appears the book is still in print, or is back in print, so you may want to get a copy.

Oh, one more thing. Johnson ends the book with 15 discussion starters, most of which address common student misconceptions. Here are three of them:

- Which is greater, x or -x?
- When is 1/x greater than x?
- Are some numbers greater than their squares?

— Henri

]]>Stepping back a bit, we may guess at cultural reasons for geometry’s decline. A friend who trains students for math competitions tells me that in those circles, it is well known that American students have much less grounding in geometry than their peers in other countries. If it is true, this does not surprise me. There may be a self-reinforcing mechanism at play: the very people who thrived in an educational program that privileges the manipulation of symbols become the educators of the next generation. The multi-generational retreat of geometry may also be symptomatic of a priority given to the pragmatic over the beautiful, which the students reflect back to us: “when will I ever use this?” Note that in *Catalyzing Change in High School Mathematics* (CCHSM) NCTM correctly argues that our goals in teaching math are many — not just preparation for college and career, which was the big emphasis of CCSSM. In any case, if it is pragmatism that led to the shrinking of geometry, it is a short-sighted pragmatism. In fact, thinking visually is important in further mathematics (including calculus!), in the sciences, in architecture, in the building trades, in design, in the arts, and no doubt in other fields.

In addition to the importance of visual thinking, there are more reasons to value geometry. It is the part of math where instead of breaking problems up into pieces, we look at the whole picture, literally and figuratively. It is a part of math with contributions from many cultures. It is a part of math that appeals to a different type of student (and, frankly, a different type of teacher!) It is the part of math where the connections to art and culture are the greatest. Thus, it is the part of math where the beauty of the subject is most apparent. It is the one part of math that “made sense” to so many of the people who tell us they are “not a math person”.

It is also the part of school math where students are introduced to proof, which of course is a core value in mathematics. However, in my view that is a less convincing argument for geometry. Reasoning and reflection about reasoning should occur at an appropriate level throughout K-12 math, and the CCSSM authors quite correctly embedded that in the excellent mathematical practices standards. In any case, the teaching of proof is an important topic which I hope to return to, but it is not the topic of this post.

Instead, I will suggest some of the geometric content I’d like to see.

- In elementary and middle school, I mostly like the content suggested by the CCSSM, but I would add
**geometric puzzles**to the mix. When I was a beginning teacher in the 1970’s, there were excellent geometry materials for the lower grades, some of which were developed by EDC in the Elementary Science Study (ESS) program. Those included activities with tangrams, pattern blocks, mirrors, and probably other things I did not see or do not remember. Alas, the ESS materials are out of print, and are not likely to see the light of day any time soon. In the 1980’s, I added to that genre with my pentomino and supertangram activities. (See my Geometric Puzzles pages for more on this.) Doing some work on geometric puzzles at an appropriate level is possible all the way from kindergarten to 12th grade, sometimes just to develop visual sense, and sometimes with implicit or explicit curricular connections. For example, in the lower grades, it ties in with the standards about “composing and decomposing shapes”. I was going to say that in high school puzzles could be used to explore the areas of similar figures, but I can’t find a CCSSM standard about this. (Could it be that this too is a topic that was a victim of geometry shrinkage?) - There are several mentions of
**symmetry**in the K-8 standards, with a specific standard about bilateral symmetry in 4th grade. There is plenty more that could be done in that arena: students could be introduced to rotational symmetry, to figures with multiple lines of symmetry, and to frieze and wallpaper symmetry. This can be explored for example by using pattern blocks, or geometry templates. I suggest some activities along these lines in my*Geometry Labs*book (free download), and I went into greater depth in a post-Algebra 2 elective (Space.) Symmetry reveals the deep connections between math and design in many cultures. It is highly interesting to many students, and not always the same students who enjoy symbol manipulation. The CCSSM emphasis on geometric transformations could have provided a unique opportunity to expand the place of symmetry in the secondary curriculum. **Tiling the plane**(tessellation) offers another geometry-rich context for exploration. It allows connections with symmetry, and can be used to introduce standard foundational geometry topics such as vertical angles, angles determined by parallels and transversals, the sum of the angles in a triangle or quadrilateral, angles in regular polygons, and isometries (known as rigid transformations in the CCSSM.) Find some tiling activities on my Web site.- In high school, I like much of the content of the traditional course, though not always how it is taught. I will return to this in a future post. Today, I will just say that the existence of GeoGebra and other
**interactive geometry software**makes it possible to teach the traditional content much better. This technology enhances learning in a variety of ways. Direct manipulation on a screen is an excellent environment to distinguish the essential properties of a figure from the particular way it appears on a static page. It makes it easy to generate conjectures, and to find counter-examples. It is what finally might help fulfill the promise of geometry as an appropriate arena for the teaching of proof. Finally, this technology is more accessible than ever, as it is now available in free software that will run on tablets as well as computers.

Even given the current version of math standards, the suggestions I have made up to this point are realistic, because they can be interpreted as ways to teach the existing standards effectively, and because most do not require much extra time. At least, they are realistic if you agree with the points I made in my last post. I will now go on to propose possible topics for 11th and 12th grade math electives. These topics probably should not be considered to be *really core*, even though I personally like them a lot more than much of what we usually teach in precalculus classes.

- School math has long underemphasized
**solid (3D) geometry**, a field with a very long history (Platonic and Archimedean solids,) beautiful and accessible results (Descartes’ theorem, Euler’s formula,) and multiple applications (chemistry, architecture, 3D printing.) This is another part of geometry made dramatically more accessible by technology. Absurdly, the CCSSM restricts 3D work to grades K-8. In high school, it appears only as a topic leading to the standard calculus problems. - We could do more advanced work with
**geometric transformations**: theorems about composition of isometries, connections with complex number arithmetic and very basic group theory, a more complete coverage of transformation matrices, and some introductory work with geometric transformations in three dimensions. (I taught all this in my Space elective and it provided an excellent opportunity for well-motivated more advanced work on proof.) **Fractals**, self-similarity, and fractional dimension. This ties in with similarity, obviously, as well as geometric sequences and series, logarithms, and computer programming. (I taught all this in my Infinity elective, and it was extremely popular with students.)

OK, I should stop. This turned out to be a longer post than I expected. You probably think it’s mostly a reflection of my own mathematical preferences, and of course you’re right. But that does not make it wrong! And in any case, I’m not the only one. I hope some day you’ll join me!

— Henri

PS: In this post, I not only quoted John Lennon at the end, but I re-used various bits from my article on the CCSSM, re-ordered, tweaked, expanded, and in some cases verbatim.

]]>There was a time in the distant past where solid geometry was a standard part of high school math in the US. That was before I became a teacher, perhaps even before I was born. (I myself was exposed to the basics of 3D geometry as a high school student in the French system back in the 1960’s.) There was a time more recently where the geometry of the conic sections was part of high school math in precalculus classes or in late chapters of geometry textbooks. As far as I can tell, that is no longer the case.

When I first started teaching high school, in the 1980’s, geometry was typically a one-year course, wedged between Algebra 1 and Algebra 2, with enough content to fill a fat textbook. Back then, algebra was taught to only some students, the ones who could pick it up quickly, and few students took calculus in high school. Thus there was plenty of geometry time for the college-bound. Still, as a department chair, I wanted all students to get a solid grounding in algebra, as it is necessary for any work in science, and of course in further math. That led me to move some Algebra 1 content into our Geometry course, which led to a reduction in the amount of geometry we offered. In other words, I participated in the very shrinkage of geometry that I decry. The importance of algebra is in part what is going on at the societal level, especially once you take into account the increasing numbers of students who take calculus in high school.

However, some algebra topics which are still widely taught should be eliminated. For example, the authors of the Common Core State Standards for Math (CCSSM) wisely removed solving absolute value equations and inequalities from the high school curriculum. Likewise, they eliminated arcane topics like synthetic division, Descartes’ rule of signs, and the rational roots theorem from the CCSSM. They probably should have gone further (see below!) but that’s a solid start.

But algebra is not the only time hog! There has been a substantial increase in the amount of statistics taught in high school math classes. Thus, something had to go, and geometry shrunk…

Teaching a lot of statistics in math class is not a good idea. Some work with data can enhance math education (for example binomial distribution, expected value, basic modeling) and should be taught. But concepts like standard deviation, regression, and correlation coefficients are based on a mathematical foundation that is completely outside the reach of most high school students, and in fact most high school teachers. (I certainly don’t understand the underlying mathematics.) This not only does not further our goals as math teachers, it actually works against what we usually try to do, which is teach for understanding. Instead, it forces reliance on black-box magical technology. (Using technology for line-fitting is quite different from simply using an electronic grapher, because we can and do thoroughly teach the mathematics of graphing — not so much the mathematics of regression.) Not only that, but to teach statistics effectively, students should collect their own data, which takes more time away from actual math. And finally, statistics is best taught with context and content, and therefore belongs in social studies and science classes.

Still, reducing the amount of statistics does not create enough space for more geometry. The reason is that there are just too many high school math standards. Most states have adopted the CCSSM, or something close to that. Given the governmental obsession with standardized testing, this forces a continuation of the mile-wide-inch-deep approach that has historically plagued high school math in the US. I know this, because I spent a supremely boring few weeks trying to sequence the high school CCSSM standards into a reasonable four-year sequence, and found that the only way to do it would be to race through everything.

When I compare the results of that thought experiment with the curriculum I developed and taught in my 30+ years as department chair, the biggest difference is that what worked in an actual classroom was to revisit the most important topics at greater and greater depth, in multiple representations, using a variety of learning tools. That can only be done if you focus on what I will call *really core* standards, and skip or deemphasize other topics. Which standards are *really core* is a good question. There are some obvious candidates: the distributive law, the Pythagorean theorem, the basic trig ratios, and so on. But figuring out where to draw the line between *really core* and merely interesting would require input from the whole community of math educators. As far as I know such input was not sought in the writing of the high school standards, and the standards are bloated. (Read my 20-page analysis of the high school CCSSM if you have time. I got nothing but rave reviews about it.)

Please don’t take this as an attack on the CCSSM authors. They made a huge and important contribution which moved the math education conversation forward on many fronts. Among other things, they brought focus and coherence to K-8 math, and got started on some good ideas for high school. But some time has passed, and I believe I’m within the professional consensus when I say there are too many standards in the high school program they recommend. Here is what NCTM has to say in *Catalyzing Change in High School Mathematics*:

Focusing on Essential Concepts is consistent with some concrete advice I offered on Pruning the Curriculum back in 2015. It would open up some time for better teaching, and yes, for more geometry. I will not give a complete list of standards I would remove — compiling such a list would take too much time I don’t have, and in any case no one would heed my suggestions.

Still, I hope I convinced you that it is possible to make some space for more geometry. In my next post, I explain why that would be desirable, and suggest what geometry I would like to see in grades K-12.

— Henri

]]>— Henri

by Shira Helft (@MsHelft) and Henri Picciotto (@hpicciotto)

Shira: Henri, thank you so much for inviting me to join you in this conversation. A lot of the thinking I’ve done already with story tables has been in a high school context. In today’s post, let’s consider what story tables might look like in middle school. More specifically, how might they be helpful as a way to understand solving equations and other pre-algebra skills?

Henri: As I explained in this post, I do not believe in “this is how you solve one-step equations, memorize these cases; this is how to solve two-step equations, etc.” I would rather have students learn some big concepts, and be in charge of applying those to problems of various levels of complexity. In other words, less memorizing, more thinking.

One worksheet that is useful early on is this one, which I’ve used to introduce the *cover-up method*, a good follow-up to the more important and more basic idea of solving by trial and error. You should take a look at it before reading on. Notice that I get to multiple-step equations very quickly. Also notice that in all of the equations, the unknown appears exactly once, so clearly things can and will get more complicated! Still, I think that set can be used to lead students to important insights.

Shira: I used the cover up method with my Algebra 1 students for the first time this year! A friend of mine, Anna Blinstein (@ablinstein), used those problems with her middle school students, and commented (on Twitter):

Really love this progression of problems from @hpicciotto for equation solving. They’ve resulted in conversations with students focused on strategy and key differences that students should notice when they solve equations, not mindless or memorized approaches.

This made us wonder – how might a story table illuminate these problems in a different way? Let’s look at the first equation, and set up a story table for it: Good math stories (or at least these ones) begin with a variable and follow the order of operations until you have built the entire story [3(x – 10)]. In this story, we know that the output should be 15.

Henri: Building on our past experience of trial and error, we decide to try 0, then 1 as the value for x:

Shira: Hmm…we are very far from the output we want but they do seem to be getting closer to 15. Let’s try 10 for the x and see what happens.

Henri: Better, but not great. But wait, what if we put 15 in the last column, where we want it, and work backwards from there?

Shira: Yes! The solution is x = 15! Many folks I have encountered have some way of thinking about “working backwards.” What I love about story tables is that they can both be a space for trial and error OR working backwards. They are also a helpful way to think of lots of other cool algebraic things, but we will leave that for another time. See if you can use story tables to help solve more equations! You can use trial and error in the table, or work backwards if you prefer.

Henri: Almost certainly, #3 will be problematic: 18/x + 12 = 15. How do we represent this situation in a story table, with x in the denominator? If no student has a good plan, I would suggest going from x to 18/x in two steps:

and thus:

Shira: Ooh! The solution is x = 6. The reciprocal here reminds me about how story tables help me dig into all of the nuances in algebraic symbols with my students. I can imagine this being a way to highlight the difference between 18/x and x/18.

Henri: Story tables will work well in problems where x appears only once. Students can set up their own tables as needed. But that is not the end of the road. After doing this for a while, the time is ripe for a discussion of the cover-up method, which is based on the same ideas, but is a bit less laborious (see page 1 of the worksheet.) Understanding “cover-up” makes it possible to generalize the key idea: each step should undo what was done, in reverse order. We work from the given expression to “x by itself” as students put it. This, in turn, is an insight which will be very useful when working on more complicated equations.

Shira: I love that with story tables there is very little memorization beyond order of operations and knowing what the symbols mean. In building the tables, we are also reviewing those skills, which I have found to be some of the biggest gatekeepers for students being successful at high school math. Henri, thanks for thinking with me about this.

Henri: Thanks for introducing me to story tables! Dear readers, if you try this approach, or find other ways to use story tables, let us know how it turns out!

]]>To address this problem, we distributed the content across three courses: right triangle trig in Math 2 (a course consisting mostly of geometry); law of sines, law of cosines, and intro to the unit circle in Math 3 (our version of Algebra 2); and trig functions in our one-term precalculus course, Functions. One benefit of this approach is that it allowed us to extend student exposure to this important topic beyond a single term, in fact, beyond a single year. (See this article for the general argument about extending exposure.) In each round you can go in depth into that year’s topic, and review the previous years’ work as needed. This ended up working really well.

Our most significant departures from tradition were right at the start when we first introduced trigonometry. Here are the key ingredients of our approach.

Our Math 2 students mostly understand slope. To build on what they know, we point out that there must be a relationship between the slope of a line, and the angle it makes with the (positive) x-axis. “Slope angles” can be used to solve problems such as “how tall is the flagpole?”

Answering the question would require knowing what slope corresponds to 39°. Fortunately, this can be figured out with the help of the ten-centimeter circle: the radius of the circle is 10 cm, providing the run, and the rise in this case is 8.1 cm or so, so the slope is approximately 0.81.

In fact, with the help of a straightedge, the 10-cm circle allows us to find the slope that corresponds to any given angle, and vice-versa. Note that there was no need to say “tangent”, or “trigonometry”. The idea is to introduce the concept first: many right triangle trig problems can be solved with this tool. When students understand it and know how to use it, it is time to reveal that there are keys on the calculator that can replace the 10-cm circle, that we’ve been talking about what is usually called the tangent function, and that our right triangles can be drawn every which way: the legs need not be horizontal or vertical if we use the formulation “opposite over adjacent” to replace slope.

We introduce “ratios involving the hypotenuse” a few months later, once again using the 10-cm circle, and completing the soh-cah-toa set, or as I prefer to put it: “soppy cadjy toad”.

The payoff of this approach is a strong visual understanding of the trig ratios, and an early preview of the unit circle. You should try it! Find more information, and downloadable PDFs on my Web site.

— Henri

]]>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!

Note: I follow up on this post with a conversation with Shira about using story tables in middle school or Algebra 1 for equation solving.

]]>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 arithmetic, 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

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