Why Limit Opportunity?

You know how sometimes TV shows begin at some point in the middle of a story arc, right at the point of high drama (a door opens to aliens, the heroine at the brink of death with no escape in sight….) and then on the screen you see “8 hours earlier…” or “one week ago” and you’re taken abruptly back to the calm beginning, still knowing where its all going to lead?
Ya know what I mean here?

I’m going to do that now.

IMG_2416                             Point of high drama in my day: WTF!

Far,  far too many hours ago….*.

Recently, colleague “Jackie” and I looked at the opening lesson of her new curriculum, in which students review HOW to change a ratio/fraction into a decimal.**  The “real world” application included is so faux its not even funny. Seriously, if you want to know which wrench (with fractional measurements) fits which bolt (with decimal measurements) on your bike, you are not going to grab pencil and paper to set up proportions!  You’re not even going to use a calculator and divide.  You are just going to try them until you find one that works. OK, maybe a little funny, in a sad sort of way.

Students are next prompted to change a repeating decimal into a ratio by GIVING THEM step by step instructions showing HOW, using algebra (Cue wah-wah sounds.) No connection AT ALL to the strategies just reviewed, no reason to do this other than to comply.  What?!

After we finished gnashing our teeth and pulling out our hair, we started thinking about how to approach this content differently. That is, how to generate a need to convert repeating decimals and create a headache around the process that would have kids begging for some aspirin. All while resting the responsibility of sense-making on the shoulders of students.  Where.  It.  Belongs.

We decided we’ll begin with a Which One Doesn’t Belong? to activate some prior knowledge and vocabulary.


The thought is, through some discussion and probing questions, students could arrive at these questions:
What are rational numbers?
Which of these are rational? Why?
Does every ratio/fraction have an equivalent decimal version? (Why/Why not?)
Does every decimal have an equivalent ratio/fraction version? (Why/Why not?)  What about that repeating decimal…?


Next, we’ll give them some time to wrestle with converting repeating decimals, then when they ask for salvation, show them The Aforementioned Algebra Process in its entirety, without explanation, and have them work in small groups to 1) identify what is happening, 2) ask questions and 3) make sense of it.

Commercial break and time passes. Jackie and I part with our vague plan and our fingers crossed, and I sit down to think about it some more. Because vague does not sit well with me. Naturally, I end up overthinking it all evening and again the next day, which is my problem with trying to make a silk purse out of a sow’s ear. I also spent some time thinking about how I would incorporate Socrative, a tech teaching tool totally  new to me filled with potential that I am dying to try. (More on that another time.  Maybe.)

At some point I remember that I am only going to be in Jackie’s room for one short day and that this lesson is going to take several. I’m over-creating for my minor role. To focus, I decide to make the WODB above, and just to make sure I understand The Algebra Process and to anticipate student difficulties/misconceptions, I give it a go.



I’ve intentionally included alatta steps so students can (hopefully) dust off algebra skills and increase the chance of sense-making.  I also used 1/3, since they may already be familiar with this equivalence.




If you give a mouse a cookie give an 8th grader a process to make sense of, they’re going to want to try it out on another repeating decimal. Well, at least that’s what I wanted to do. Maybe they will, too.

IMG_2418As I finished, I remembered that Jackie mentioned something about the process always involving 9’s. Now I see why.

Does this mean, I wonder, that EVERY repeating decimal’s fraction version has a 9 (or 99 or 999, depending) in the denominator? Let’s find out!

I also see an opportunity for students to notice the pattern and make a conjecture. An opportunity that would have passed me and students by had I never attempted to makeover this lesson because the student sensmaking in it is nonexistent.

Do you feel how close we are getting to that opening drama here? Truthfully, I was really enjoying where this was going; my childhood math experiences did not include this type of exploration, and it is FUN.  Seeing an opportunity I did not know was there is exciting.  I imagine students might think this pattern is just another neat-o/mysterious math trick and stop there. Unless you insist they test their conjecture….

And now, a word from our sponsor.
This entire explorative experience and the inevitable WTF moment will never happen if students are merely asked to imitate ad nauseam a process they don’t understand, followed by a test and a grade. If the culture of a classroom (and its supporting curriculum) revolves around “standard” algorithms and “right”answers instead of noticing and wondering, curiosity and perplexity, student-centered sense-making, and celebrations of WTF and AHA moments, then our students are being robbed of opportunities to see the beauty and humanness of math,  are being denied a chance to know they are mathematically capable, and are less likely to grow into curious and creative people who can develop viable arguments and critique the reasoning of others.  Life skills, for sure.

Where was I?  Ah, yes, testing a conjecture.

Which I did. And ended up with 0.99999…. = 1.  A fabulous WTF moment, I must say.

Notice the purpose of converting repeating decimals shifted from performing a rote process (booooring) to students uncovering something Big (exilerating).  It does not matter that you will probably not be able to resolve their angst over this issue; in fact, it is OK to discuss a bit, argue a bit, consider a bit, and then leave this perplexing moment…a bit unresolved. (If you google it, you’ll find a lot of arguing.  Infinity is difficult to nail down.)  It is sufficient for students to learn that it is in excatly these kinds of moments where humans need to make sense of something that does not make sense that new ideas are born and learning happens.  Zero. Place value. Fractions. Negative numbers. Irrational numbers. Imaginary numbers, for heaven’s sake!  Infinity (and beyond).

You get the idea.



* Due to the fact that I am trying to help Jackie make over a less-than-satisfactory “new” curriculum; see my previous post about finally understanding why having an exemplary curriculum is a much better situation.

** The presentation of this topic is so very rote and unexciting it will do a great job of keeping kids hating math.


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?






Its been so long since I’ve blogged that I had to look up my password.

One of the many reasons for my little hiatus is that there are plenty of UH-MAZING blogs out there– that people actually follow and read– making what I think and wonder and write about rather…superfluous.

For example, Ilana Horn’s insightful, intelligent, inspirational blog.  Informed and intriguing.   I recently started following, and back-read several posts she has written on status in the classroom, in part because I believe know status is deterimental to equitable learning yet is created and deeply ingrained and even actively perpetuated in the Old School system/institution of teaching and learning.  She explains it all  very clearly in a series of posts:

Status: The Social Organization of “Smartness”

Seeing Status in the Classroom

What Does it Mean to be Smart in Mathematics

Recognizing Smartness and Addressing Status in the Classroom

This morning (since I kind-of overdid it hauling bark dust yesterday), I decided to chill a bit and create a list of competencies I value in my classroom.  Not besides “fast calculations and right answers”, but instead of.  A definite and requisite shift in classroom currency if one is striving to achieve an active and equitable learning culture.

In no particular order….

Students in the role of sense-makers.
Connections between mathematical ideas.
Connections between representations and models.
Clear communication of thinking (the WHY), even if incomplete or unsure.
Active and intentional listening to all peers.
Multiple strategies and solution paths.
Gaining insights by making mistakes.
Willingness to revise thinking and understanding.
Great respect for the value of every person, their learning, and the strengths they already have.
Genuine Questions and Wonderings.
Collaboration in learning as a community.
Flexible thinking.
Creative thinking.
Visual/alternative representations of reasoning and ideas.
Connections between multiple representations.
Connections between different strategies.
AHA and WTF* moments.
Active awareness and regulation of learning.
Attention to reasonableness of solutions (yours and others’).
Private time to think (and respecting it).
Critique of thinking, reasoning (not people).
Critical and deep thinking.
Understanding the thinking of others, even when it differs from your own.
Respectful disagreement.
Respect for (and celebration of) strengths and strategies that differ from one’s own.
Genuine/legitimate peer support in learning.
Growth mindset.
Engagement and involvement.
Willingness to start even if you are not sure.
Equitable collaboration.
Consideration of ideas other than your own.
Ability and willingness to adjust your reasoning/opinion and change your mind.
Learning from peers.

Wow, that list is a lot longer than I expected.  Which would you add, revise, or omit?  Why?

Here’s what I might do with such a list.  At the beginning of the school year, cut it up and have students in small groups sort them into 2-5 or so categories, their choice.  Sorting activities are a worthwhile way to get kids talking to each other, voicing opinions, making choices.  Listen in, because you’re finding out about them, too.  Notice common choices as well as different ones.  Ask groups to explain their categories to you.

Then, as a whole class, share and discuss.  Ask them to notice things.  I have NO IDEA what will happen here, but I’m wondering if anyone will notice that “right answers”, “smart”, “good grades”, “fast thinking” and those types of competencies typically over-valued (and detrimental to learning) are MISSING.  So are generic behavior-type rules, like arrive on time, do your homework, pay attention….Will they notice the focus on inclusiveness and learning instead of on first and fastest?  Will they identify with some of them?  I’m really curious about how kids will sort these and what they will say!  Finally (if there are enough common themes?), use their input to develop a SHORT list of classroom norms that recognize and support these valuable competencies.



*Probably should change this to WTH What the Heck, or HIW Hold it, What!? Or some such thing more socially appropriate, right?

WTF moments are not moments of frustration, though.  They are moments of realizing something is amiss, some reasoning, intuition, or process is not going the way you expected, or the solution make no sense.  Disequilibrium and perplexity reside here.  In a sense, these moments are insights, too, a realization that an adjustment is needed;  understanding WHY one path works and the other does not paves the way to the bigger insight (AHA!) and gains in learning and understanding.  When students people share their thinking, they tend to leave the WTF moments out and share only what worked, saving face and strengthening the currency of “right” answers.  However, in a healthy, inclusive culture of learning, WTF moments are valued as an important and natural part of the learning process, worthy of sharing, even celebrating!   “First, we thought….because….then we. saw…realized…tried….because….figured out….learned….”. Even “First we tried….because…not working……and now we wonder….not sure….have some questions…..”

…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.