Write this down…

I’m wondering….again.

What is the purpose of note-taking?? (Specifically, in math class; still more specifically, in MS/HS math classes, where note-taking is more common, suddenly the more appropriate pedagogy.) Why are students expected to take notes? How essential are they, really, to learning? How much of it is about compliance (or The Way it Has Always Been) and how much about active, student-generated sense-making? Are notes primarily for recall of facts and rules, to practice processes and prepare for tests, or can notes aid in the construction of conceptual/relational understanding? What does it mean to “take notes”, and are some options better than others? Which ones? Why? Are there times when note-taking is appropriate and beneficial for the learner, and other times it is not? If so, when and why?  Is there a difference between taking notes and making notes?

But before that, I noticed…..

I’ve heard teachers (including myself) repeatedly remind their students to take notes (Get out your journals! Write this down!) only to lament their lack of use.  Like you, I’ve seen student notes range from dutifully copied examples to partial and chaotic scratches on a random page to nothing at all. I realize students do not, after all, magically know how to take or use notes just because they reach middle school. I’m aware that copying examples/filling in blanks ≠ understanding what the heck is going on, let alone why. I know some students are able to mimic processes well enough to maintain their status as the “smart” kids…and you know what happens to everyone else. I notice a lot of popularity with ISN’s and more recently, sketch or doodle-notes, both claiming to be improvements on “traditional” note-taking routines.  There is a lot that intrigues me here, but I am cautious.  Maybe even confused.

Here’s why.  Much (all?) of the ISN and doodle/sketch-note materials I’ve seen via my brief ‘research’ online are teacher-generated, limiting student interaction to following directions/filling in blanks, keeping students passive and unburdened by any need to make sense of ideas. Happy and busy ≠ engaged in content and thinking critically; what risks are being taken, what fabulous mistakes are being made? How will there be any WTF or AHA moments?

My gut-feeling is that note-taking actually plays a more significant role in the culture of our classrooms that we realize, impacting everything from mindset to equity to assessment. The parameters and expectations we set communicate what and who we value in our classrooms and defines who takes the active or passive role, teacher or learner.

Mind you, I’m no expert on this subject; my teaching skewed toward less notes, more tasks, and my knowledge is limited.  Note-taking does not seem to be a blog-worthy topic; I’m suggesting it should be.  Surely I am not the only person asking these questions!  There are plenty of blogs, mostly positive, about ISN’s, which seem to at least have the potential to include more than rules, examples, and definitions.  I’m imagining how powerful post-exploration student-generated sketch notes would be…

I just want to push hard on some assumptions about notes and note-taking, start a conversation, ask more questions, and gain some insights.  My gut is tellling me this is another component in education in need of a transformation.  And to go eat a peanut butter cookie, but I am trying to ignore that part.

A Picture is Worth…..Everything.

I spent about an hour yesterday playing with geometry tiles.  I was:

  1.  Fully engaged and enjoying myself.
  2. Experiencing OMG, WTF, and AHA moments.  (AKA actively learning.)

The inspiration for my mathematical play time was this photo from  Sara Vanderwerf’s recent post  on Stand and Talks, a bold routine she uses to increase the number of students talking about math.   Really great stuff and worth your time to read it through.

IMG_2094But back to the photo, which is her example of using a Stand and Talk to introduce a new idea, in this case, the Pythagorean Theroem.   My eyes perk up (is that a thing?) because I have been pondering about this very topic.  Specifically, wondering about what activities/tasks would place students in the active role of sense-making.

The first thing that popped into my head was….wait, trapezoids?  (Thank you, Sara!) Then I started some noticin’ and wonderin’ from my teacher-y perspective and also imagining/guessing what students might say.  The more I noticed and wondered, the more I realized how FULL OF MATH IDEAS this one photo is.

I find this exciting.  Really, really exciting.  Can you imagine starting here, with this one little image, asking students what they notice and wonder, and then letting them run with their curiosity?  Isn’t it joyful to know in your heart of hearts that this is so much BETTER than, “Hey kids, here’s a little something called the PT, copy it in your journal/onto your pre-made “doodle” notes, plus copy these three examples of how to get the right answer, and do this worksheet”? How superior this single photo and all its potential are to “I do, we do, you do” and /or anything else that keeps students in the passive role?

It didn’t take me long to drag out my tub of tiles and start messing around with them.  TOTALLY what I would want my students to ask beg to do.

The questions I decided to guide my play were:

First, can a proportional, scaled-up version of each polygon tile (that I have) be built with only that particular tile?
If so, why? Is there a pattern?
If not, why not?

Second, can any three similar polygons be used to make any type of triangular “hole”? Why or why not? Which ones make right triangles? Why? Is there a pattern??


I started with squares.  Don’t know why, maybe because that’s what is already familiar to me.  No surprises, the areas are the square numbers, 1, 4, 9, 16…..

So I moved on to the equilateral triangles, wondering/kinda expecting that I will make triangular numbers.  Imagine my surprise IMG_2119(my OMG moment) to find SQUARE NUMBERS AGAIN!  Holy shit!  NOT what I was expecting!  Now I feel compelled to keep exploring.  COMPELLED.  (Imagine your students feeling compelled!!)

I quickly confirmed that the next in the series takes 16 triangles, and moved on to IMG_2125the rhombus.  YES, the areas are square numbers AGAIN!  Which gets me reconsidering my understanding of square numbers and forming conjectureish questions:  Can square numbers be made out of any polygon, not just squares? These are all regular; what if they are irregular?

I’ve been reading Tracy Zager’s fabulous book Becoming the Math Teacher You Wish You Had.   She starts Chapter 7, Mathematicians Ask Questions, with a quote from Peter Hilton: “Computation involves going from a question to an answer.  Mathematics involves going from an answer to a question.”  Brilliant distinction.  Looks like I’m living Chapter 7.  Know I want this for my students!

Remember, one photo.

I finally get to the trapezoids, which are more challenging to tile and require some analysis.   I managed to construct one, but the number of tiles it took (7) was a red flag for me, so I  examined it closely to verify if it was proportional or not.  It wasn’t, so I kept playing, keeping theseIMG_2126 two beliefs in my mind: a proportional trapezoid was possible, and it could be made out of 4 tiles.  And lo and behold….

TA DA!  “Square” Numbers!  (Whew!)



By now, I have even more questions.  By now, I have reviewed and applied my prior understanding of proportionality and similar figures, without ever touching a worksheet.  By now, I’ve taken risks, persevered, reasoned, looked for patterns, used vocabulary, formed conjectures, and justified.  All from looking at one picture and allowing myself both time and pleasure to play and be curious….and I haven’t even begun exploring that thing with the triangular “hole”.   Yet.

Finally, the hexagons.  And….What?!?  You can’t even make a similar hex without using some non-hex friends!  WTF!

New question!  WHY NOT?





Week 1, Late.

I was going to jump on board the MTBoS 2017 Blogging Initiative .  I was planning to take a deep breath, close my eyes, and plunge in.  But for some unfathomable reason, I read “submit by midnight Sat., Jan. 7” as “submit by midnight Sun., Jan. 8th”.   Oops, I missed that first boat; fortunately, no one is going to dock points!  I’m determined to not let Week 2 drift by, but a part of my brain is still on Week 1’s theme, “Favorites”.

My brainstorm for Week 1 included a task from Visual Mathematics Course II and the folks at The Math Learning Center.

Original Version, Submarine Task

Its annual use qualified it as a “favorite”, although over the years I used it in various ways.  An assignment, a formative assessment, a group task, a summative assessment.   I always required diagrams and equations that supported each other, and eventually figured out to omit the question to allow students to focus first on understanding the situation.  That would be Version 2 (with all verbs in agreement as well.)

In the spirit of You Can Always Add here’s Version 3:

The Submarine Task, Version 3

A submarine cruises in an ocean. First it dives down, then climbs up, dives again, and finally climbs up.

Before you read my ideas, what would you do with this version?

What I think I would do, feel free to poke holes:

First, do some Noticing and Wondering  Ask students to make a sketch of what is going on.  Have students suggest not only what questions they could ask, but also what information they would need to answer their questions.  (BTW, typically the question in V2 is, “Where does the sub end?” I am curious what, if anything, will be different for V3.)   Then, allowing private think-time before working in small groups, give them this:

Which of these sets values fit this situation, which do not, and why?   Use visuals and equations to explore each list and be prepared to justify your reasoning.  Use values exactly as they appear, and in the order given.

-200, 150, 115, 180, 100

                                     -200 -150 -115, -180, -100

                                      -200, -150, 115, -180, 100

                                     -200, 150, -115, 180, -100 

                                      -200, 150, 115, -180, -100 

Compare what this task is asking of students, and what the others versions ask.  What do you notice?  What are your thoughts on the lists?

Continuing with my ideas:  Perhaps give the lists to groups on strips of paper so they can move them around a sort them.   I am dying to know which lists students accept and which (if any) they reject; should be an interesting discussion!  In my mind, V3 would be appropriately placed after some reasoning  and conjecturing about adding and subtracting integers, during a time when there is still room for questioning and sense-making, and before students practice fluency.

Since they have done most of the heavy lifting already, end with this:

1.  Chose one set of numbers to answer the questions you asked.  Be sure to include ALL of your work.

2.  Use a second set, including your work.  

3.  Compare the strategies:  How are they the same?  Different?  Which one is “better” for you and why?  (Do not describe what you DID; your work should already clearly show your steps!)  

A possible sequal to  V3, although probably not immediatly:

Which of these lists of values fit the Submarine Situation?  Explore with diagrams and equations and be prepared with viable arguments. 

                        -142.5, -157.8, 315.25 , -273.0, 198.75

                         -3,127, -1098, 4105, -3627, 2503

                          218.5, 105.6, 162.4, 298.3, 57.7

                          -410, 119.5, -338.26, 937.01, -705.635

                           -5/6, -3/2, 1/4, -2/3, 5/12

For each situation that DOES fit, answer the questions you previously asked. For each situation that does NOT fit, you may change ONE NUMBER so that it does work. Justify your choice.

And a couple of Reflections, if you’re into that sort of thing:

  1.  Understanding why subtracting a negative value results in an increase to a higher value is often perplexing.  Why do you think this kind of calculation exists if it feels so awkward?

        2.  Consider these two questions:

                       What is the distance between the highest and lowest elevations?

                      What is the difference between the highest and lowest elevations?

Would your answers to these two questions be the same?  Why or why not?  Would your work to find these answers look the same or not?  Explain.

If you made it all the way through this post, THANK YOU!  I would appreciate feedback on any or all of these areas in the comment section:

  • The value (or lack of value) of this post, with specific, non-judgemental suggestions for improving it and/or my blog.
  • Strengths you see or improvements needed in the task and lesson suggestions. What would you do differently and why?
  • Actually use Version 3 and/or some of the additional materials and let me know how it went!

…equals (write answer here).

When students are shown how to “do” math and are asked to perform copious amounts of computational answer-getting, they quickly and unfortunately develop the idea that math (and learning math) is primarily about right and wrong answers, void of context and steeped in mystery.  And the correct location for the answer is right after the equal sign. In such a setting, students who say 5×6 “is the same as” 6×5 get their statement affirmed and handed some new vocabulary to memorize: commutative property.

What does that statement even mean?  In context, 5 groups of 6 objects is different than 6 groups of 5 objects. That is, the pictures look different, although the total is the same. Is that it, end of lesson?  In some classrooms, yes.  But I think there is risk here by glossing over why and caring only about what or how, and that risk is losing the kids who are trying (desperately) to make sense out of math, especially when we assume they easily move between concrete and abstract. The pictures look different.  I don’t get it.  

You could use area models/congruent rectangles to try to convince them: See? 5×6 is the same as 6×5! Same dimensions! Same area! Same rectangle! Ta-da!

Wouldn’t it be amazing if school encouraged kids to be dubious and curious?  Wouln’t it be glorious if they asked, What about 21+ 9? Or 60 ÷ 2?, Or any of the infinite number of expressions that also equal 30? Are some expressions “more” equal than others? What exactly is going on here?

When people like Kristin  ask 3rd graders to sketch pictures to match two similar stories, then discusses with them whether or not order matters in symbolic notation–while remaining genuinely interested in and openly curious about their thinking– they are helping young students develop significantly different ideas about math and learning. In this setting, students know that the teacher respects and values their reasoning; together, they are all making sense of an idea of equality that reaches beyond “is” or “write the answer here” in a way that addresses misconceptions and helps them build a bridge between abstract and concrete.

While we’re on the subject of equality, what about these?

4/5 = 0.8
5 – 8 = 5 + (-8)
-12 – (-7) = -12 + 7 = 7 + (-12) = 7 – 12
a÷ x/y = a • y/x
-3/4 = 3/-4 = -(3/4)
-4(9) = 4(-9)
3(n + 7) = 3n + 21
-(a + b) = -a – b
x^0 = 1
Etc., including any formula or rule “given” to students.

If I want my students to understand equality deeply so they can use it purposefully, then I’ve got to do more than tell them that these expressions are equal, show how to change from one to the other, and then test that they can. And I’m beginning to understand that it is also a disservice to impose my reasons why. The sense-making needs to be theirs, not mine; they are completely capable of figuring out why these equations in particular are worthy of their attention.

“Same value” is just not convincing enough for me any more.

Math Dreams

I’m not kidding.  I seriously dreamt recently about a variation on the Spiral of Theodorus , woke up and thought, hey, that might be a way for students to create some math— specifically, the well-known theorem attributed to Pythagoras.

I’m not trying to dis Pythagorus here; I’m just trying to figure out how to create opportunities for students to arrive at the theorem WITHOUT just showing them “how” and having them practice finding missing lengths umpteen times, with some “real world” applications thrown in involving a leaning ladder or a shadow and a tree.  Maybe that’s why my brain came up with something while I was sleeping.

For some reason, people tend to like the PT, maybe because it feels magical,  maybe because it’s an easy equation to memorize and recall and sound smart (unless you are The Scarecrow).

It certainly isn’t due to dutifully copying how-to examples from the board (or a book or a video or website), memorizing the steps without ever knowing/wondering/asking what it all means.  (I’m describing my school-aged self here, but I know you know/teach people who are just like this.)


There are thankfully ample visual and symbolic proofs available that help students make some sense of the theorem (usually after they have been introduced to it), so that may be part of the attraction.   Its cool factor seems to be sufficient enough to make it memorable.  (Oh, that’s what it all means?  Cool.)

Just imagine the way studens would feel if they were empowered to discover it themselves!

Mind you, I don’t have research-based data or definitive lesson plans that guarantee deriving the PT.  I just have a fleeting dream about a seed of an idea, and wonder if there is something worthwhile here.  I know it needs considerable fleshing out and trial runs with real life teachers and students.  Hint hint.

This is not part of the dream, but I would begin by having students use graph paper to make as many squares with whole number areas as they possibly can.  (This idea is not original to me.  If and when I find the source, I will include it.)  Challenge them to make all areas from 1-10 (or even higher), label the side lengths, and justify the areas.  (Dammit,  I can’t figure out areas of 3, 6, and 7!!)  This is a genuinely engaging activity, and depending on learning goals, there are lots of connections to similar figures, parallel and perpendicular slopes, and similfying radicals.  For me, the essential learning this activity offers is a concrete, visual understanding of the relationship between the area of a square and the length of its sides, between “square” numbers and their “roots”, even the not-so-perfect ones.


In my dream, I made a right isosceles triangle with legs of 1 unit, then used the hypotenuse as the leg of the next right triangle.  This is where the Spiral of Pat differs from that of Theodorus– each right triangle in my spiral is isosceles.  His goal was (according to Wikipedia) to “prove that all the of the square roots of non-square integers from 3 to 17 are irrational.”  My goal is for students to look for patterns and make and test conjectures.  All the while honing their skills and fortifying their understanding of squares and roots.

When I tried my spiral out during daylight hours, I saw potential.  Do you?

The Spiral of Pat

What do you think students will notice and wonder about?  What conjectures do you think students will make?  Do you think they will arrive at the “right” one?  Why or why not?  Would you also have students test this relationship with other polygons?  What about cubic units?  What would you do with this?


PS.  If you have a way of helping me find areas of 3, 6, and 7 WITHOUT actually telling/showing me, I’d appreciate getting un-stuck.

What’s the Big Idea?

I’ve been thinking about a more effective way (for me) to teach proportionality (to 7th graders). I never have wanted to reduce it, like so many textbooks and videos and apps do, to a series of how-to lessons and repetitive just-do-it-this-way practice. You know what I mean; Lesson 1: How to write a ratio. Lesson 2: How to write a unit rate. Lesson 3. How to write and solve proportions: Method1: cross products Method 2: equivalent fractions. Lesson 4: How to… make everyone think learning math is only about following directions to get the right answer. Yuck.

In spite of my best efforts to teach for understanding, I think know I’ve missed the mark. Fortunately, I’ve had lots of glorious time lately to read incredible blogs and do some heavy self-reflecting. Here’s where I stand currently regarding teaching proportionality:

WHAT IF… I build a series of lessons (primarily exploratory activities and worthwhile tasks, discussion and reflection, but also instruction and practice) that explore the Big Essential Idea of Relationship. Have all lessons contain this common, recurring thread: proportionality is about a consistent relationship between two things that change together (co-vary). This idea needs to be explicitly noticed and talked about from multiple perspectives– in rates, in probability, in percent change, in similar figures, in scale drawings, in proportional contexts, graphs and slope, tables, equations, in formulas– and in experiencing all these things, students develop a solid understanding of relationship, a foundational algebra concept, make oodles of connections, and all the while develop a meaningful set of skills.

Just wondering, how do you address the relationship concept in your classes? I’ve got some ideas brewing for a series of posts.

BTW, the other two 7th grade BEIs are, IMO, Equality and Operation.

Seriously, I want to know.

[There’s a lengthy update to this post, published 1/5/17.]

Huston, we have an inconsistency.

In a slope ratio, the dependent variable is always compared to the independent, y:x or more commonly, y/x. If you’re making a right triangle on a graph to determine the slope, this y to x comparison needs to remain intact: changes in y are vertical: changes in x are horizontal. You can train kiddos to always move vertically first to draw one leg of the triangle, then horizontally, always to the right. Yes, I know this is a trick to figure out the right slope, but bear with me, I’m trying to get to understanding.


You can also train your students to create a table that lists the independent variable in first column, dependent in the second. Very easy to then grab the ordered pairs straight from the table and plot those points. The independent variable axis is always horizontal, you tell them. The first value in the ordered pair is the independent variable, which means to first move (from zero) horizontally. Followed by the 2nd value, and the 2nd move, which is vertical. This gets confusing to students because for plotting points, its horizontal, vertical, but graphing slope is vertical, horizontal.

If you are lucky, they wonder why.

I’ve been wondering why for a long time, too, and I’m stuck. I figure this is a safe place to admit that. What do you say to a curious student who is trying to keep track of (x,y) and y/x and asks, “Why is it different?” I want to be able to give them a better answer than, “Its convention.” I really do want to know why the ratio is written dependent/independent, and not the other way around.  It’s inconsistent.

Here’s my thinking so far: For the sake of sanity argument, let’s agree that the placement of independent and dependent variables on the X and Y axes, respectively, are convention. Also, that slope is a ratio that describes the steepness of a line; comparing point A to point B, point B is both higher and to the right of point A.


If a table lists the coordinates in this order (ind. var., dep. var.), then plotting points on the graph requires that same order, using the appropriate axis. First distance from zero horizontally, then distance from zero vertically. As long as a person understands which variable is which, and which variable goes with which axis, then the table could be reversed (dep. var., ind. var.) and the plotting would be accurate as long as the same order is followed: first vertical distance from zero, then horizontal distance. The key is consistency with the order.  Rather than memorizing convention, you need to apply an understanding of the co-variance of independent and dependent variables and make sense of horizontal and vertical changes.  Perhaps it is just convention (you tell me!) that guides us to always list the independent variable first in tables and coordinates.

Yet when I experiment with the order within the ratio, inverting it to x/y, I get the same line. Example: Suppose y/x = 3/2. That means x/y = 2/3. These look like completely different slopes. But they are not, because both the values and the variables are inverted. So, to correctly graph a y/x = 3/2, even if I have inverted it and written x/y = 2/3, everything will be super groovy AS LONG AS I understand which variable is which and which variable gets a vertical or horizontal move. Moving first horizontally a distance of 2, then vertically a distance of 3, creates the same slope as before. Huh. The x/y ratio fits nicely with (x,y).


So why do we use y/x instead? Does it all really boil down to convention? I’m sure there are people who can explain the reasoning for all this to me– please do, it’s bugging me. What do you say to your students when they notice the inconsistencies? What lessons and activities do you use to get at the heart of this?

UPDATE:  Do we use y/x instead of x/y because we want to associate a higher ratio with a steeper line? That only works for y/x.



In his 1976 article , the late Richard Skemp identifies and explains two meanings for “understanding”, two divergent interpretations of the same word that undoubtably influence everything from pedagogy to policy. (Go here  to read David Wees’ thoughtful post about Instrumental and Relational Understanding.) These two meanings co-exist simultaneously with two significantly different interpretations of what math is.

Does this become a self-perpetuating, chicken-egg circle? Which came first? Surely our beliefs about what math is rise out of our own experiences as math learners. If we adhere to those same beliefs as educators, we will teach in the same way in which we were taught, developing in turn the same beliefs in a new generation of learners. The only break in the cycle is when someone, teacher or learner, has an opportunity to experience learning math in an alternative way.  And I think this experience needs to be profound enough to at the very least allow you to see that choices exist.

I was fortunate enough to have several such experiences. Here’s my story: When my own children were nearing adulthood, I decided I wanted to become a teacher. I had no clue about what I would teach, but began taking general classes at the local community college that were required to get into a graduate education program. This also meant taking some math classes, and I tested into a level of math about equal to high school Algebra 1.  The extent of my prior math education was limited to one year each Algebra and Geometry in high school.  No math classes in college. Not required, not interested.  Yet I had been an “A” student in math because I was able to successfully imitate what the teacher was doing and did all my assignments, memorizing and performing routine processes and formulas. Nothing more was expected of me. I did not hate it, but I did not love it, either.

So there I was, repeating the algebra content I last ended with, which is when I had the first of a series of eye-opening experiences about math and learning math. One of my instructors was a retired high school teacher, and although he did quite a bit of direct instruction, he also got us talking about WHY, not just how. (Cue heavenly voices.) The moment I recognized that there was so much more to math than imitate-memorize-regurgitate, the moment I realized that math was both beautiful and interesting, was the exact moment I decided to become a math teacher.

I continued devouring math classes (through the first term of calculus) until it was time to begin my grad program. I was no longer satisfied with instrumental understanding, I wanted to really understand relationally (although I did not call it that) so that I could pass that understanding on to my future, oh-so-lucky students. Along the way, I also got a Middle School Math certificate.  Through my math and education coursework,  I found out that the best learning is learning with others, not next to them. Glorious. That you can (and should) revise your work as you gain new insights, that you can follow more than one solution path, that math ideas are wonderfully interconnected. In Skemp’s words, I was now “able to consider the alternative goals of instrumental and relational understanding on their merits” and choose.

           To make an informed choice of this kind implies awareness of the distinction, and relational understanding of the mathematics itself. So nothing else but relational understanding can ever be adequate for a teacher. (R. Skemp)

I totally stole this image from David Wees because I think it is awesome. If I committed a blogging  no-no, I’m sure someone will let me know.

How successful have I been in my endeavors to teach relationally? That’s a whole other post (or three). What do I think relational teaching and learning looks like? I’m working on figuring that out. That’s a whole blog.

Thank Goodness

As a brand new member of the Global Math Department (you folks are AWESOME!), I recently received my first newsletter. One link led to another, and I eventually found myself reading Richard Skemp’s article here on Relational Understanding and Instrumental Understanding.  If you have not had the pleasure yet, go! Now!

What initially jumped out at me and slapped me in the face was the date it was published: 1976! Apparently his concerns for the isolating nature of teaching and the lack of conversation around reform was not unfounded, since seriously, not enough has changed in the last 40 years.  Too bad he is not around to see what is happening now, how teachers are connecting in amazing ways through blogs and tweets and MTBoS and GMD and heaven knows what else. Dan Meyer recently marveled at what is taking place.

Is it possible that technology, which has so radically altered the way we live and work, will be the catalyst to radically alter education? I hope so.