Report Card Time Machine

I am starting understand the term standards-based grading.  It’s one of those phrases I read in blogs and, assuming it was what it sounded like it was, just glossed over it.  (In the same way I got by for a long time assuming that penultimate meant better-than-ultimate – ha.)  I recently did my homework, which revealed an ever-expanding network of interrelated articles and posts – some of which said explicitly that if I didn’t already know what SBG was, I should stop reading the post and go read everything ever written by these other four people.  Thank you very much. One of the most exhaustive and inspiring posts was this one by Daniel Schneider (already well-lauded in the math blogoverse) about how changing the way you assess effects your whole practice.

Frank Noschese instigated all this reading with his short but excellent post about standards-based grading, The Spirit of SBG, exhorting us not to worry about making the transition to SBG, but to make small, sure inroads.  Recommendation number seven of ten (no trekkie pun intended) is: Assess what you value.  What do my assessments reveal about what I value?

My dad, the family genealogist, recently shared this: my Grandma Nancy’s seventh grade report card from 1939.  She lived that year in Ross County, Ohio and (just for that year) with her aunt – who happened to also be her teacher. What does this report say about what her school system valued?

Nancy's Report 1

Nancy's Report 2

There is a whole class devoted to geography.  She took arithmetic, not math.  The “Habits And Attitudes Desirable For Good Citizenship”, while a little quaint in its phrasing (Is dependable, Co-0perates with others, Keeps desk and floor clean), isn’t so different from items I have seen in the “conduct” section of modern report cards. Maybe what’s surprising is not that this looks so dated, but that it doesn’t look more dated.  My high school report card looked a little bit like this.  The reports I write to my students’ families much less so.

This makes me wonder about the next evolution of grading.  If I value the mathematical practices (perseverance, modeling, critiquing the reasoning of others, etc), how and when do I assess them?  The standards for these practices, despite being my favorite part of the Common Core, are mostly hidden in my classroom assessments.  I observe students having or not having them, I comment occasionally on them in general to the whole class or privately to a student or his/her parent (“I notice how precise Sophie is in the way she explains her mathematical ideas.  She helped the whole class understand why square numbers are the only ones with an odd number of factors.“)  My colleagues and I have tried a few times to include “Perseverance” and “Communication” as items on our report to families, only to re-phrase or discard them a year later because they didn’t quite capture what we meant, or were too easily misconstrued.

One obstacle was that we didn’t formally assess these practices within the classroom first.  Another is that the mathematical practices can only be observed in connection with mathematical content. The practices may be too abstract or too broad to be assessed independently.  Even so, what would it mean to be more transparent?  What would it look like for students (and ultimately their families) to know how their were progressing not only in their understanding of factors and multiples, but in their ability to, for example, look for and make use of structure?

Why Sketchpad Is Cool: Making Things Obvious

I am a little obsessed with Geometers’ Sketchpad. I haven’t used all its functionalities, but the basic principal is totally captivating: The constructions retain their features so you can manipulate them and watch how they behave. If you make two lines parallel, they will stay that way. If you construct two segments so they share an endpoint, they will be joined forever. On the other hand, anything not intentionally crafted is rendered a chaotic mess once you start moving objects around.

I redesigned part of my fifth grade geometry unit last year to include (I hoped) more problem solving and discovery. That experiment, which was mostly successful, is described in the post Detective Work. It inspired me to go further.

Why should introductory 2D geometry, arguably the most appealing and intuitive of all the topics I teach fifth grade, ever be didactic? One reason it has been in the past is the limitations of the tools. Most don’t really lend themselves to being messed with. Physical models (like the plastic polygons in Detective Work) are great for some activities, but not flexible. Flexible models like polystrips and pencil-and-paper constructions are all too taxing on the fine motor skills of 10 year olds. Enter Geometers’ Sketchpad. Sketchpad can model geometric relationships in a way that makes them intuitive and obvious. This year, I used it to teach almost the the entire fifth grade geometry unit.

  • Introducing the point, line segment and select tools. I asked students to construct and label polygons up to an octagon (or beyond). For some, this was new vocabulary. I told them they could manipulate, resize, and reshape them to their hearts content (helping to combat the misconception that these polygons always appear in their regular form). They liked watching the vertices ‘hang on’ to each other.Irregular Polygons
  • Classifying Polygons. I introduced the rotation tool, in which a line segment can be rotated any number of degrees around a marked point, and the parallel line tool, in which you can construct a line parallel to any other line or segment. I challenged them to make some of the ‘special’ triangles and quadrilaterals. “But,” I said, “they have to be indestructible. They can be flexible, but if you’re calling it an obtuse triangle, it has to stay an obtuse triangle, no matter what I do to it. A square has to retain its squareness.”

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(We had discussed the special qualities of these shapes – but briefly. I wanted them to have to ask themselves and their classmates – ‘How does a rectangle keep its rectangle-ness?’ ‘What qualities to I have to build in so it won’t collapse?’ ‘What needs to stay true?’) They worked on this for about a week. There were many different ways, it turned out, to make a parallelogram, and a rectangle. More than one intrepid young geometer came to tell me they were done, having made a handful of triangles and a square. ‘It’s also my rectangle, my parallelogram, my rhombus, my kite, and my trapezoid,’ they said. Nice.

  • Bonus: The Kite. I introduced the reflection tool on a whim, as some of them were asking about kites. (This took me a minute to consider: Can’t you make a kite with just the rotation tool? Or parallel lines? Can you? Not without knowing anything about angle sums.) So I taught them how to reflect objects over a line segment and how to hide objects from view rather than delete them.

These are some of the constructions with their “hidden” components showing

  • The Rule of Angle Sums. The next week, we talked about what made a polygon regular and I challenged them to construct all the regular polygons they could by rotating a line segment. At what angle should they rotate it? They had two choices: to guess and check or measure plastic versions of regular polygons with angle rulers/protractors. They didn’t all choose to use these tools, so some of them still have no experience with them. (I’m still not sure if this is ok. Probably.) The students who used guess-and-check had to interpret the results of using the ‘wrong’ angle – ‘What does it mean when the line segments cross at the end?’ ‘What does it mean if they never reach each other?’ ‘How do I adjust my angle? And by how much?’ Eventually, I gave everyone recording sheets and asked them to keep track of the angle sum. “What’s happening here?” I asked. “Can anyone explain it?” Disclosure: Prior to this moment, I myself had never been asked to explain this. I actually wanted to know.

Regular Polygons

  • The class made two important discoveries: One, the internal angle of a septagon is not 128.5, even though that looked about right on everyone’s sketch. In fact, it is a tiny bit off. The angle sum is 900, not 899.5 (which came from 7 x 128.5, the most common estimate). A few students backward-engineered a perfect septagon by dividing 900 by 7. The fact that it is a repeating decimal reassured them (‘We couldn’t have estimated that.’) Two, the angle sum increases by 180 degrees when you add a side because to turn a square into a pentagon, for example, you’d simply add a vertex to the middle of one of the sides of the square. This increases the number of sides by one and it creates one brand new straight angle. You can alter the shape of the new pentagon and distribute those 180 degrees among some of the other angles, making it look more properly pentagonal. This fact continues to blow my mind. Thank you, class of 2020!
Turning a 360 degree quadrilateral into a 540 degree pentagon by adding a vertex.  Genius.

Turning a 360 degree quadrilateral into a 540 degree pentagon by adding a vertex. Genius.

  • Designs. The students went on to make, among other things, cyclic and dihedral figures with rotation and reflection and to make semi-regular tessellations – some pictured below. When I asked them to give me feedback about the year a few weeks ago, they almost universally reported that this unit was their favorite part.

Ciro SymmetryEthan Tessellation

Reflection Symmetry

Tessellation - Hanna

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Faulty Logic: Math Wars

If you haven’t seen it yet, two professors (one philosophy, one math ed) just published an opinion piece in the NY Times online blog, decrying the state of the reform movement in math education.  These reformers, they warn, are trying to get away with not teaching the standard American algorithms for computation.  These upstarts and their “numerical reasoning” may destroy us all.

“At stake in the math wars is the value of a “reform” strategy for teaching math that, over the past 25 years, has taken American schools by storm. Today the emphasis of most math instruction is on — to use the new lingo — numerical reasoning. This is in contrast with a more traditional focus on understanding and mastery of the most efficient mathematical algorithms.”

It may have been a mistake to go on to browse the comments.  The top comment right now reads,

“I am a mathematician, and I am indeed outraged and baffled at the move against teaching the standard algorithms in grade school…”


“Anyone who would propose that one can teach “mathematics” without the standard algorithms is simply innocent of any notion of what mathematics is. He or she is proposing to “educate” students in a subject while withholding from them the definitions and meanings of the nouns and verbs one uses in thinking and talking about mathematics.”

Oh dear.

The authors argue that the value of standard algorithms is that “algorithm-based calculation involves an important kind of thinking.”  In order to know whether you have mastered the algorithm, they say, you need to understand it on more than a procedural level.  I think all math teachers would agree with that.  However, using algorithms doesn’t produce such understanding understanding any more than repeatedly transposing sheets of music from one key to another would produce the ability to make music (see A Mathematician’s Lament for more on this wonderful analogy).

So how do we get students to understand the algorithms they use? (And by algorithm, the authors don’t seem to mean ‘method’ but the much more specific Standard American Algorithm, or The Algorithm, as if it is self-evident precisely which methods the real ones are). The reformers, they say,

“insist that the point of math classes should be to get children to reason independently, and in their own styles, about numbers and numerical concepts. The standard algorithms should be avoided because, reformists claim, mastering them is a merely mechanical exercise that threatens individual growth.”

The reform math teacher’s criticism about a traditional approach is actually not that The Algorithm threatens individual growth, but that it  doesn’t produce any growth.  Students don’t come to understand something because they heard someone else describe what they understand. They will absorb and remember what they actually are asked to do, not what they’re asked to watch or listen to. That’s the real foundation of an inquiry-based classroom.  If students actually make math – that is, grapple with numbers and operations in an authentic way – then they will be able to interpret and make sense of the math that others do, including the standard methods.

The biggest mistake the authors make, though, is when they say that the standard algorithms

“are the most elegant and powerful methods for specific operations… Our best representations of connections among mathematical concepts.”

That’s wrong in at least three ways.  First, the fact that these methods are the most common doesn’t make them most elegant and powerful.  That’s like saying that fast food is superior because its so widely available. These methods are the ones that prevailed and flourished in our country during a completely different time, when people had no alternatives but to do computation to the nth decimal number by hand.

Second, other countries and cultures use different algorithms for basic computation and those methods are no less elegant.

Finally, and most importantly, the standard American algorithms are hugely inelegant in many cases.  Use the standard algorithm to multiply 999 x 123.  It’s ridiculous.  Any fifth grader could tell you its a better idea to multiply 1000 x 123 and take away one group of 123.

Maybe this opinion piece is a gift – a throwing down of the gauntlet for math educators.  I challenge all of us to respond to this.  I invite you to read the opinion piece and try – in the comments.  50 words or less. Without profanity.  Unless absolutely necessary.

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Computer Multiplication

I recently saw a video demonstration of “Egyptian Multiplication” in which the presenter described how both Egyptians and modern computers multiply using binary.  It was presented like this:

  1. Write the problem down horizontally.
  2. Below the first factor, list the value of each place in binary (1, 2, 4, 8… etc).
  3. Circle the ones that compose your first number.  If your number is 22, for example, you would circle 16, 4 and 2.  All numbers can be written this way – or in any other base – but the neat thing about base 2 is that each place can only count once (or none) – so you never need to double-circle any number).
  4. Below the second factor, you create a doubling list, starting with the number itself. (If the number were 5, your list would read 5, 10, 20, 40… etc).
  5. Circle the numbers in the second list that are across from numbers that were circled in the first (below).
  6. The sum of those circled numbers is the product.

This is how computers multiply? This video was made in the 80’s.  I am certain that now, if not then, there is more to it.  Regardless, I like this because I talk a lot about multiplication strategies in my 5th grade classroom, modeling how multiplication works and what it means. There is so much that is mathematically powerful in asking and answering the questions, Why did you do it that way? How do you know it works? Ideally, this helps kids flexibly choose the method that’s most appropriate rather than always resorting to the standard American algorithm, which is most adult’s go-to but often the least efficient way. Take the case of 98 x 26, for example.

Of course, kids get creative with multiplying in this way if they understand what multiplication means – that it’s shorthand for making groups.

I asked myself about this binary multiplication the same questions I’m constantly asking my students: Why does it work?  The video wasn’t meant to be transparent, opting instead for the ‘wow’ factor of a cool math trick.*

So what’s the relationship of this doubling method to more familiar ways of multiplying? Are there numbers this doesn’t work for? This seems ridiculously efficient and easy – is there some reason we don’t all multiply this way instead? Maybe it gets unwieldy if you use certain factors.

I saw that the connection between numbers that are across from each other is the key to the question of why this method works.  In 22 x 5, for example, the pairs are 1-5, 2-10, 4-20… Looking at it, the second column is just groups of 5.  First one, then two, then four, etc… doubling.  Since every number can be composed in binary, then you can figure out how much 22 groups of 5 is by taking the sum of 16 groups, 4 groups, and 2 groups.

Isn’t decomposing numbers into smaller groups and then multiplying just what we do when we multiply in our own number system?  How is this any different? I started to experiment by using this method with base 10 instead of base 2.  I tried it (below), listing our own place values (1, 10, 100…) below the first factor and below the second, multiplying the factor by ten as opposed to 2.

There are two problems.  First of all, unlike binary, each place can be occupied many times over – you can have 2 ones and 2 tens, for example, rather than just one of each.  This muddies the method and adds steps.  Above, I had to circle some numbers multiple times.  Second, the size of the places are so far from each other, it sometimes doesn’t help at all to use the method.

If you want to multiply something by 67, for example, you are only using two of the places – the ones and the tens – and then you have to go through all the trouble of multiple copies of each value.  Binary is so elegant because even small numbers are spread across many places.  You could use base 3 for this method without too much trouble, but most of us are not as facile with tripling as we are with doubling, so it’s harder.

You can use this same binary method for dividing, incidentally, with a slight reversal of the steps.  I’m wondering if I can or should present this 5th graders.

* An editorial note: These days, I physically recoil when I hear educators refer to a math ‘trick.’ It’s not magic.  It’s a language that everyone is equally entitled to understand. A mathematical method is not worth the paper it’s written on if you don’t understand why it works.
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Detective Work

I have a set of translucent plastic polygons that I use with my 5th graders.  They come with the Connected Math unit “Shapes and Designs.”  If you’re not familiar with it, the program is full of good problems – but this particular unit is underwhelming.  It includes the kinds of activities no one would continue doing if the teacher and school suddenly vaporized (measure this, put these into groups… a collection of unconnected 30-minute tasks).  I’ve been tinkering with ways to use these polygons – a set of mostly triangles and quadrilaterals – in a way that is harder. Harder in a good way.  Whole ladder of abstraction way.

Last year, I elected to scrap several of the usual lessons in favor of what I decided to call Detective Work  –  students would spend over a week creating visual proofs for angles in polygons.  The classes were lively and students worked out some of the most essential and important elements of the the unit without much input from me.  I was impressed with their thinking and energy.  (This, more than anything else, seems like a sign of a good lesson: that I am impressed with their work.  Conversely, if I’m feeling disappointed in their work, it’s usually my own bad planning.)

Here is what we did and what I think made a difference:

  • I kept the shapes out longer.  We spent several classes categorizing and sorting.  The students were able to get familiar enough with them that they could use them as tools.  I always forget to account for this adjustment phase – I hand out manipulatives and expect that everyone will be as ready to use them as I am – an adult who has essentially graduated from fifth grade with flying colors every year for a dozen years in a row.
  • The challenge was simple: Prove the measure of any of the angles in any of the shapes without measuring (estimates are helpful but don’t count).  As a hint, I told them that they could use multiple copies of the same shape if they wanted to.
  • I didn’t help much after that.  Starting with the simplest polygons in the set (like the equilateral triangle which is called “shape A”), they tiled them around a point.  Since three make a straight angle, each one must be 60 degrees, they reasoned.  They moved on to more polygons –  the various rhombuses and parallelograms, the three different  trapezoids.  They started to use the polygons they knew to fill in the ones they didn’t: “This one is a B and V, so it’s 90 + 30… 120 degrees!
  • I let this go on way longer than planned because they were enjoying themselves and discovering important things along the way. The parallelograms have two pairs of equal angles.  You can use the shapes build up from an angle you know (like 90) or subtract away from one you know (like 180).  They became very good at estimating the sizes of angles.
  • It was intrinsically differentiated. Students chose polygons that were appropriately difficult for them, and they had the option to do as many or as few as they wanted. Students reshuffled themselves into the groups and partnerships of their choice, sometimes choosing to work alone.
  • Solutions were public and communal.  There was a real audience for the finished proofs (so much more persuasive than my vague urgings to be legible because I said so). We posted them on the bulletin board as we went along, and students compared and consulted with each other.  There were at least five different methods for the regular octagon.

It was a week well spent. I’ll try it again this year.

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Big Plans

Yesterday, inspired by Sam Shah’s new school planner and riding the first wave of back-to-school delight, I skipped over the details of which program he used (Adobe something – whatever), rolled up my sleeves, and created a new planner on Adobe Illustrator.  I’ve made my own planner before, and am good at bending word processing programs to my will (below), but was interested to see what I might be able to do with more sophisticated software.  Also, I’ve been curious for a while.  What’s with these Adobe Programs? What do they do that’s so great?

Old Planner

Apparently, they do it all.  It’s overwhelming, actually, to be able to make literally ANY design you can think of.  I’d gotten used to having to work around the limitations of Word and Pages.  It took me over an hour, but I learned to use enough of the tools to make what I need.  I especially like the live paint bucket tool, which is brilliant.  I can now successfully make straight and curved lines.  And I found the warp, twirl, crystalize, and pucker tools – which are addictive in a lets-go-crazy psychedelic way. Only after these efforts and newfound design skills did I realize I’d used Illustrator instead of InDesign – and had to do a funky PDF work-around to increase the number of pages from 2 to 82. (I’m a perennial directions-skipper. I frequently over-estimate my ability to do things myself.  Ikea shelves? Making risotto? Growing tomatoes? How hard can it be?)

My favorite innovation of the 2012-2013 planner (below), besides the doodle around the words “Scrap Paper”, is having just one area for each prep rather than one for each class.  Since I teach two sections of each class, I have often made planners with spaces for each.  Since I almost always teach both sections on the same day, the second section’s area always winds up empty, with all my notes and reminders front-loaded in the first section’s box.  This waste of space bothers me more aesthetically than practically, but it’s remedied here.  There’s one area on Monday for 5th grade, for example, though I teach 5th grade twice that day.  We’ll see how it goes.

New Planner

And the cover, with my cats on the back, because they remind me to be nice, and a crystallize/warp/twirl tool creation behind them because Adobe Illustrator lets you do stuff like that to a rectangle.


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What do we do when we do math?

I started thinking about this while planning a math workshop for 5th grade parents last year.  I’ve been hosting these twice a year for long time now and they’re always slightly different.  The main messages are the same:

    • Math is accessible and fun.
    • Computation is just a tiny part of a rich world of numbers and is not an end in itself.
    • Struggle is the heart of problem solving; resist the urge to interrupt your child to show her the “easy” way.
    • Don’t ever boast about how bad at math you are, or your family is, or how little you understand your child’s homework.

Each year, depending on what I’m thinking about, and what parents seem to want or need, these mornings are a little different.  Last fall I started wondering about the verb we use with math.  Why do we do math (and science)? Other do’s: we do laundry, do exercises, we do card tricks.  We don’t do English or history – those have their own verbs: read, write, discuss, analyze. We make art and music.  Why? And what is the thing we do when we do math?

Doing is for things that already exist.  The only contribution you can make is to be obedient to the routine – ie, doing chores.  Making is for new things.  Even though the artistic process is full of borrowing (techniques, themes, and materials), we think of its product as unique, so we say making art, not doing art.  Doing math implies that math exists already and the best we can do is to not screw it up.  Many adults seem to feel this way – that is, weirdly cowed by the whole discipline of mathematics – like it’s looking down on them, waiting for them to show weakness.

In my experience, when people say, “do the math” what they actually mean is, “compute.”  Or, more precisely, they mean: do the steps you were taught.  Also, do them quickly and get the same answer that a calculator would get.  It’s no wonder most people remember math class as being totally joyless (and by extension, think anyone teaching the subject must be a little off – maybe brilliant, maybe just defective). Most computing is more paint-by-numbers than math.  Is this dog, who can bark out the correct answers to basic arithmetic, doing math?

Problem-solving is creative, not imitative, and it deserves an appropriately active verb.  When a kid, through her own (often non-linear) efforts, realizes how something is true (ie, any whole number times an even number will always be even), she’s created a new idea, a new model.  It doesn’t matter that someone else has seen it, proved it, published about it, because it feels like she made it.  And didn’t she?

Students make sense of things, and make connections. I told the assembled 5th grade parents last year that we should set our sights higher than just doing the steps – what Jo Boaler calls “intellectual obedience,” in which the teacher offers a (great) explanation and lightbulbs go off around the room.  Explaining feels so good – so productive.  But I don’t really want my students to do as I do.  I want them to have their own mathematical experience – wondering, predicting, testing, comparing, sketching, discussing, modeling, proving, and – ultimately – making a new idea, then another and another.

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