AHA #3. If it’s broken, throw it out.

I’ve been reading Alfie Kohn’s book The Schools Our Children Deserve; it is part of the required reading for a course I’m taking. Just before summer got crazy busy with Other Stuff, I had read through Part 1: “Tougher Standards versus Better Education”. I admit I do not feel intrinsically motivated to write thoughtful any reflections on Chapter 5 (Getting School Reform Wrong) or Chapter 6 (Getting Improvement Wrong); while they contain additional compelling evidence for what is currently amiss in eduction and with ed reform, I am already long on board. In fact, even though I took a un-planned break from the course, I’ve continued to think/read/journal/blog/reflect in order to keep moving myself towards insights about teaching and learning.

Recently, I decided to take up Kohn’s book again, looking forward to “Part 2”, in which Pooh Kohn offers solutions. Chapter 7: Starting From Scratch.

Well, beam me up Scottie if this chapter isn’t fully in sync with my current thinking and wondering.  Suddenly, I feel motivated. Huh.

What I’ve been wondering about in recent weeks stems from a conversation I had with Jackie, a middle school math teacher friend. I volunteer regularly in her room and we meet to talk almost as regularly about pedagogy and moving student learning forward and other Important Things. My role in part is to support her implementation of several strategies that are intended to create an active learning culture in which students make sense of math.

She noticed that in spite of the on-going, long term math PD that has been invested in K-12 teachers in her district and in spite of her efforts to increase student talk and focus on student reasoning, learning remains for the most part passive and feels glacially slow. Mindsets cling to answer-getting, compliance, and performance, about speed and status. In many ways, what she wants happening in her classroom differs from what is actually happening.

Been there, done that.

Neither she nor I are implying that their previous teachers are at fault! Nor are we blaming the students! (I just heard again today that The Reason students don’t learn is that they “just don’t try hard enough”.  I disagree.  I may need to blog about that.) At this point, this is merely an observation that her desired change seems too minimal. Its frustrating. We know there is more work to do, more than we anticipated.  Persevere.


Every worthwhile observation should be followed by a genuine question. It’s how we learn, how we find solutions.  What I wonder is, exactly what is going on in schools and classrooms (including mine) that impedes or prevents moving from a performance culture to a learning culture?

If we only make changes- even great ones– to some of the things we do while maintaining others, the “old-school” culture prevails. This is a big AHA for me.  Most teachers seeking to improve practice consider what to do differently or what to add in, but it is equally important is to identify what needs to be (gasp) thrown out.  In October 19’s post, I listed several widely accepted, rarely scrutinized practices that I believe undermine honest efforts to transform teaching and learning. Alfie Kohn provides a similar list and includes justification.  For both of us, grades top the list. Teachers, administrators, parents, even students, should make their own lists. This is where conversations need to begin; this is where priorities get identified, appropriate long-term goals get made, resources located, and plans developed . This is where meaningful change- change that moves education forward–stands a chance.

I recommend reading Kohn’s book. In the meantime, here’s some of my additional take-aways from Chapter 7:

  • You can’t make people want to do something, like learn. You might offer rewards and threaten punishments (grades, points, stickers, perks, honor roll, a different track, detention, praise, admonishments, etc.) but you get compliance or defiance, not motivation. Being extrinsically motivated to get the reward or avoid the punishment is not the same as being intrinsically motivated to learn. Intrinsic motivation comes naturally and heightens engagement, but can be extinguished or nourished, depending on what takes place in the learning environment.
  • The purpose of school, the reasons for educating people, tend to be debated in rather polarized terms. This or that.  Kohn writes, “It’s more a continuum than an either-or, but the point on that continuum we identify as ideal makes all the difference.”  A lot of this and a little that.  For individual educators, teams, schools, and communities, identifying where that point is located will help clarify which practices support or impede genuine learning.
  • In “traditional” school cultures, there is a disconnect between long term goals (those ideals shared by most educators and parents) and short term goals (those skills, mandated and tested) and practices within classrooms. We say we want one thing, and we do and/or allow something else. (Is that the problem Jackie and I are facing??) If our long term goals are for students to be curious, to be intrinsically motivated, active learners, to listen empathetically, to construct viable arguments and critique the reasoning of others, to persevere, to make sense of complicated ideas, and so on (all valuable life skills, by the way), but we put more energy into short term goals like being sure they can correctly add and subtract rational numbers, then it is possible– even probable— that the long term goals will never be met.
  • If, on the other hand, we first identify that ideal place on the continuum, clarify our long-term goals, and then teach in such a way that always supports these goals, then both long and short term goals will be met. Frequently meeting and reflecting on our long term goals and evaluating what is happening in our learning communities is a must.  Remember, some practices may need to be updated, included, or even discarded to be sure long term goals are supported.
  • Achievement should not ever be a long term goal. Low achievement is not the real issue (although it is popular to say so), disengagement is. Its not that students are not “trying” hard enough; they just don’t see the point.  We should be wondering why.  The performance/compliance/accountability culture that is touted as the “solution” is actually a massive obstacle to achievement; focusing on grades and points and data and test scores in order to improve learning simply backfires.  John Nichols said, “Teaching requires the consent of the students, and discontent will not be chased away by the exercise of power.”  (That quote is in Kohn’s book.)
  •  In contrast, high achievement results naturally when both long and short term goals are met by capitalizing on intrinsic motivation and student interest. Common sense tells us that all people learn best when they are interested in what they are learning about; schools can and should be putting their efforts into creating and sustaining learning environments for students and their teachers.

If you have not already done so, please check out  MARK CHUBB’S informative post about how his district set and went about meeting long term learningculture goals and what happened to student achievement.

Where is the “ideal” place on the continuum?  For you, for your school, for your students.  Why?

What practices and policies do you use or see used in your school or district that are intended to create and sustain an active learning culture? How effective are they?

What else is going on that undermines these efforts and perpetuates fixed mindsets, passive learning, and disengagement?

What’s on your list? Why?  Who will you share it with?

…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.

Aha #2: Learning Communities

When I started teaching, I had an vision for what I considered an ideal culture in my classroom.  Every year, I tried to make one professional goal support this ideal.  My vision was of a Culture of Learning.  Not the kind of learning that is mostly about memorizing and skill performance, but the kind of learning that is mostly about exploring ideas and constructing understanding; the kind of learning that helps develop critical thinkers.   I envisioned students working closely on interesting and worthwhile math tasks, asking questions, listening empathetically, taking risks, justifying their reasoning, learning from and with each other and having glorious aha moments that moved their learning forward.  I consciously chose to use a proficiency grading policy from the get-go (and stuck to it in spite of being the only teacher using it) because I wanted my students to grow as learners and know that they were.  Ideally, they would monitor their own learning, be willing to revise their thinking and their work, and learn from mistakes.  Ideally.

I felt it was my responsibility to make my vision happen.  I still do.  The reality was, many of these things actually did happen to some degree, but never to my satisfaction.  The various moves I tried did not ever seem to make a big enough difference.  At times, it was difficult to not feel like a failure.  Each year, I would try again, because I know beyond a shadow of doubt that the culture in a classroom matters.

This post is not about woulda-coulda-shoulda regrets.  Or about blame.  It is about trying to make sense of a culture in which learning flourishes, and part of that process involves figuring out and examining what hinders, undermines, or flat-out prevents it.    Grades.  Worksheets.  Right and wrong answers.  Performance culture.  Testing.    Compliance.  Grades.  (I said that already?  Oops.)  Government mandates.  Time.  Status Quo.  “Ability” leveling.  Homework.  Tradition.  Myths and misconceptions.  Fixed mindsets.  Data overload.  Just to name a few, of course.

In an attempt to be succinct, here’s what reflecting on my experiences and efforts has revealed to me so far about culture:

AHA #2a:  In order for a teacher to improve her daily practice, in order for her to develop and sustain a classroom learning culture,  she needs to be working in a learning community.  That is, the thriving learning culture we desire for our students needs to begin with a thriving learning culture for their teachers.  In order for teachers to learn, they need a safe and supportive learning community that is willing to talk about and examine practices honestly and critically, to make time to find and use excellent resources, to implement ideas, ask questions, collaborate, make mistakes, revise, and reflect, reflect, reflect.   While fabulous online communities can and do support the learning of individuals, I am not aware of any real impact on a school’s culture.

Please read Mark Chubb’s post * on how his district made this happen.  Notice the non-performance goal, the commitment to time, and investment in people.  If we expect/want our students to be active learners, then we’d better desire and demand it for ourselves.

AHA #2b:  You can’t simply “add” in a few pedagogical moves or latest research-based ideas and expect significant change to happen, even over time.  Even if you get training and/or implement excellent ideas well, there simply is no magic bullet.  Figuring out what you personally and you as a community need to STOP altogether or significantly ALTER — and understanding why and figuring out how–are equally important as adding in the good stuff.  Less feeling like your’re making shit up and trying to survive and more intentional learning and professional growth.  See Aha #2a.

What kind of community do you work in?  What kind of community do your students work in?  Who is surviving and who is thriving?  Why?

* Mark Chubb is my latest blog crush.  Go.  Totally worth reading everything he has to say.  Not to mention that delightful photo gracing the top.  (Other crushes I’ve had are Dan Meyer, Christopher Danielson, Fawn Nguyen, and Bree Pickford-Murray.  There are many amazing other bloggers I follow and am thrilled each time a new post shows up in my reader, but these five I have gone back and read every one of their posts.)

Aha! Aha! Aha?

During my time reflecting on teaching and learning these past 15 months or so,  I have arrived at some significant insights.  Significant for me, at least.  These insights came from reading as well as my journaling.   Sometimes I got there on my own and then stumbled on a post or two that echoed my very thoughts, usually with greather elequence.  Better yet, backed with reasearch.  Other times, what I read got me revisiting and questioning my beliefs and pushed me to grow.  I should and may blog about each insight, but for now, I want to just summarize the biggies.

Aha #1.  Grading, no matter how you slice it, is horribly detrimental to learning.  Some systems more so than others, but they all boil down to judgement handed down by someone who is not the learner, someone in a position of power.  Even when a system is intended to communicate learning, it is received as judgement.  Grades (points, percentages, levels, etc)  do not inspire or motivate, at least not instricially.  They teach compliance, which is not the same as responsibility.  They generate status in classrooms, schools, and communities.  They develop fixed mindsets and negative beliefs about self and learning and school.  They open doors of opportunity for some and close them for others.  I can’t even say they do more harm than good, because there simply is no good.

I say this understanding that most teachers are genuinely interesting in being fair, in doing what is right.  They find or create or use a system that makes sense to them and look for ways to make it efficient and meaningful.   I say this understanding that grading is deeply, so very deeply entrenched in the Institution of School that it is rarely questioned, rarely examined honestly and openly, and incredibly resistant to change.

Yet change is desperately needed.  We need to reject grading and adopt practices that support and foster learning.    I say this with very little to offer of what to do instead, because this is largely uncharted waters.  (Hence my aha with a question mark.) Yet I also say this with absolute conviction.  Grading is broken (always has been), and we need admit that and throw it out, not try to fix it.  Changing how we assess student learning (NOT the students themselves!) requires us to ask why we need to so do in the first place. I think the conversation needs to start there. For me, everything we (teachers, admin, parents) do including assessment should promote learning. Inspire learning. Deepen learning. Celebrate learning. For. Every. Student.

I believe the solution lies with involving students.   The Art of Learning, if you want to call it that, includes metacognition and self-reflection.  Anybody, any age, any where, knows whether or not they are learning, whether or not they understand, where their strengths are and where growth can happen. Frequent student led conferences with teachers and peers, written and verbal reflections, peer and teacher feedback, formative assessments, opportunities to revisit and revise,  and portfolios are all potential components, I think.  I’m certain there’s more.

Its going to require a stronger role from the learner and a more supportive role from the teacher.  It’s going to take effort and patience and flexibility.   Change is always difficult, but the difficulty of the task (and this one is really complex) should not be a deterrent and  is certainly is not a valid reason to maintain “tradition”.  After all, we are talking about the education of our youth, the adults of tomorrow.  Perseverance is mandatory; they’re worth it.

This turned out to be a longer post than I anticipated;  I guess I’m more passionate about this than I realized.   So I’ll close with a quote from David Wees’  latest post:

The goal of teaching though is not to generate specific student performances. The goal of teaching is to produce long-term changes in what students know and can do. While we study performances in classes and use these to make short-term decisions about what to with our students, we should also systematically compare these short-term performances with the long-term changes in student performances that then correspond to their learning.


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.

Making Sense of “Same”

Monday, Jackie’s classes wrote updated conjectures about what they thought it meant for ratios to be equal. The language varied a bit from class to class, but they generally looked something like this:

Two ratios are equal if one is scaled up or down to get the other one,  or when simplified, are the same ratio.

Jackie then introduced the word proportional, subbing it in for the word “equal” in their statement.

That’s not too shabby. Its taken quite a while to get to this point, but we’re counting on (hoping?) this investment in conceptual understanding to pay off in the long run!

Tuesday, students dove into investigating “same” shapes by making rectangles with areas of 24 square units. There was lots of debate as they looked at how their shapes were the same and different. So far, they have:

If two shapes are the same kind of shape, have the same area and same dimensions, they are congruent.

Since they were only looking at rectangles, there was no mention of angles in their conjecture. We asked them to compare the bases to heights because we wanted them using their recently written statement about proportional ratios.  They were somewhat surprised to found out that even congruent rectangles, when drawn with different orientations on paper (and therefore with different bases and heights), do not have the same base to height ratios. Their home assignment (because we ran out of time– WOW it took them so long to make those rectangles!) was to choose one of their rectangles, and make another that is not congruent but does have the same base to height ratio. I saw a lot of puzzlement on their faces, but by reminding them that they just said congruent rectangles have the same area, clouds parted (well, for some of them) as they realized the restriction on area was lifted.

(I’ve got to start remembering to snap photos of student work, img_1804 if only to insert and break up all this text!  How’s this alternative?)

My biggest concern was not enough discussion participation by all students; we’re really pushing sharing reasoning, and the pace of a whole class discussion feels sluggish to me.  Probably to most of the students, too.  The ones with Hermione Syndrome seem like the only ones getting anything out of a whole-class discussion. I realize that many other students may be actively tuned into the discussion, but are not comfortable with raising their hand to voice their thoughts. I know this is true because when we pose a question to the whole class, *crickets*, yet when we then say, “Turn and talk to your neighbor about…..” suddenly we see kids talking and gesturing with their hands. So cool to listen in!

Jackie is realistically expecting that not every students will follow through with the rectangle assignment. Will there be too few, making discussion and moving forward awkward? Her plan is for them to use their proportional rectangles to wrestle with defining a different kind of “same” (similarity). She predicts/hopes their conjecture about similar figures will include something about proportional ratios, and is going to push students to use their sketches to “see” and justify proportionality in the linear measurements. Students who need it can expand into wondering about the growth in area.

Then in small groups, they’re going to look at three sets of triangles to refine their notions of similarity. During prep, Jackie and I had a lengthy discussion about angles. They matter, but we don’t want measuring them to distract from the main focus on proportional ratios. We think/hope these sets will address the congruent angles issue pretty well on an intuitive, visual level.



On Ratios, Fractions, and Equality

The glossary of the math book lying about in Jackie’s room has this gem:



Jackie and I want students making sense out of this statement (without ever subjecting them to it.) What does it mean to state that two ratios are equal? How do you know they are equal? What do you mean by equal? We recently had them look at some images, determine whether or not the ratios they saw in them were equal, and explain why. It was interesting, to say the least, and they were surprisingly challenged, especially on the ‘splainin part. They’ve also been developing some conjectures about what they believe is true about equal ratios. All this has got me pondering (more than usual) about fractions and ratios, how these ideas are the same, and maybe more importantly, how they are not.

Fractions can be considered ratios, if viewed as part-to-whole comparisons: I have 3 out of 4 squares of a chocolate bar. Each number in the ratio quantifies a separate pieces of information: the 3 is an amount/number/value, an adjective-ish word/symbol that states how many parts I have, and the number 4 quantifies how many parts are were the whole candy bar before I snitched some. Conversely not all ratios are fractions because they could compare parts to parts, this to that, etc.

When we extend the ratio point of view, we find the size/value of the part depends on the size/ value of the whole unit, which can be an-y-thing. Take 3/4. For one whole hour, the part is 45 minutes; for one whole dollar, the part is 75¢; for one whole yard, the part is 27 inches; for one whole deck of cards, the part is 39 cards; for one whole 8oz candy bar, the part is 6 oz.  That’s probably overkill on the examples, but the point is that the independent/dependent variables thing belongs to ratios, not fractions.

Yet the entire fraction itself is also a stand-alone quantity: I have 3/4 of a candy bar. When the same ratios from above are abstracted to a quantity, each are located in the exact same place on a single number line. One point to rule them all. So, 45/60 = .75/1.00 = 27/36 = 39/52 = 6/8 because they each represent the same idea (3/4 of one whole). This “sameness” can be demonstrated by ‘simplifying’ or ‘reducing’ each value to 3/4. Both simplifying and scaling up require you to shift back into fractions-as-ratios mode (although no one mentions it when they are showing you “how”.)


The most perplexing thing so far about is these layers of meaning– fractions as ratio and quantity. (There’s division, too, but I’m not getting into that here.) It would not surprise me that this is the source of confusion and anxiety whenever the f-word gets mentioned. (Fractions. What were you thinking?)

Ratios by contrast are not placed on a single number line. You need two lines to represent both quantities in the relationship; for every 3 parts (one axis) there is a whole of 4 (the other axis). There are an infinite number of points possible, located somewhere between the two axes.  Put enough of ’em there and a pattern emerges. If you define rates as a comparison of two different units, like time and distance, they are always relational, never a single quantity/fraction. Oh, wait, they become a quantity, or are treated as such, as soon as you generalize the relationship into an algebraic equation!  Drat. (Is this what is meant by ‘constant of proportionality’?) No wonder kids get confused. Is this a ratio or a fraction or division or what? Yes.

Let’s model equality. Consider the candy bar. Suppose the 3 remaining parts are shared evenly among 3 people, each person receiving one. Yet if a same-sized candy bar contains twice as many parts, each person would get two parts. They are getting exactly the same amount of candy in both cases. Therefore 3/4 and 6/8 are (still) equal because they are the same amount. This justifies the idea of equal quantities. The size of the whole did not change (it remained constant), which is why we can scale fractions (as ratios) up and down to get common denominators for comparison or addition/subtraction purposes. img_0684Notice though that to get 8 parts, each original part in the model had to be divided. However, when you write this process symbolically, it translates bizarrely to multiplying both values by 2. Is this a WTOther F-word moment for anyone else but me? I mean, if I give you one when you asked for two, and to solve the conflict I break one into two parts, wouldn’t you be kinda miffed?


Now look at another candy bar model that’s a part : whole comparison, 3 parts to 4 total.  Scaling up by doubling the 3 shaded parts as well as the 4 total parts (which totally matches the symbolize version) I get…..3/4 of TWO candy bars, and all the pieces are still the same size!!  WTF again!

Here’s what so significant and what we want our students to truly understand–ratios are about relationships between two quantities, not about a single quantity. Even thought the amount of candy changed, the relationship between did not. There’s still 3 parts for every 4 parts. So now 3/4 = 6/8 for an ENTIRELY DIFFERENT REASON. And you get your two pieces after all.

When we asked the 7th graders studying proportionality why the ratios 3/4 and 6/8 are equal, they said things like, “…because 3×2 = 6 and 4×2 = 8.” They’re calling on their prior experiences with fractions (nothing new to learn here!) but are they thinking about quantity or relationship? I suspect it’s quantity, and I wonder which candy bar model would they choose to represent their idea of “equal” ratios, or if they can identify the scale factor in either model. Others simply said, “….because they are the same number” so at least we know they’re thinking about. In any case, students seem pretty sure that ratios are just fractions and that equal means same value and the same-old reasoning about fractions applies.  So let’s thrown them a curve ball:


Really? 3/4 and 6/8 are equal? Are you sure?