Some Thoughts & Questions About Differentiation (Part II)

This is a response to a question posed by Paul Hawking in a previous post on differentiation.

Have you considered how you would differentiate for those who are significantly behind the rest of the class, either in prerequisite knowledge or skills? … I’m curious as to your thoughts about providing meaningful learning and avoiding frustration for the bottom students.

In my previous post I mentioned, “A general rule of thumb I have come to is that differentiation should move upwards, not downwards.”  By differentiating downwards I am making judgments about what a student or group of students can or cannot do, something I believe is a very slippery slope.  In general, I think that different approaches for that situation may be a safer bet.  Such approaches include lesson planning which create tasks with access points for students at all levels, or structuring group-work and assigning status such that all students know they have skills to help others as well as something to learn from others.  Nonetheless, there are cases where I have differentiated downwards.  This is my working rule of thumb for such differentiation (which of course is subject to change as I change and improve my pedagogy).

If I am to differentiate downwards I need to ensure that 1) students who receive extra supports are still held to the same standards as all other students, and 2) I am not usurping learning experiences from students who do not need extra supports.

Allow me to provide an example of how I have attempted this.

In the past I have written that I believe that there is a chain of events that lead to deep understanding.  It goes a little something like this:

Students begin by having a basic idea of a concept, teachers blow their mind by guiding the students to something new, thus making them confused (and often frustrated), but as students push through the confusion they develop a deeper understanding.  The problem is that not all students move from confusion to a deeper understanding at the same time.  I often see that many students are emerging from the confusion while others are still frustrated and confused.  In this situation, many students need time to let their understanding seep in by utilizing their knowledge while other students still need focused support and encouragement to push through the confusion.

When this situation occurs, I find it valuable to hold the materials constant and differentiate by individual teacher support.  To do this, I assign a relatively procedural partner-task and tape a solution key to the white board, telling students that they can use one another to answer most questions they have, and use the solution key to answer the question of “is this right” (after all, that is the most common question of students who are entering into a deeper understanding).  I help them through valuable questions when they arise, but with the down time (and there is a lot of it) I pull a chair up to the group of students who is clearly still struggling through the confusion and give them the bulk of my attention.

I justify this extra attention to the students who need it most because I believe that student-teacher time does not have a clearcut, direct relationship to student understanding, and that there exist times when students benefit more from the freedom to work with their peers.  When students develop a deeper understanding they need to flex their mental muscles in a low pressure situation.  In these moments I can focus on ensuring all students reach that deeper understanding without detracting from the learning experiences for those who already have.

Some Thoughts & Questions About Differentiation (Part I)

Differentiation is a concept which I have spent considerable time thinking about. Given the heterogeneity of the class I teach (and probably the class most teachers teach) it seems that a one size fits all approach to education always ends up excluding someone.   Differentiation is something I’ve wondered so much about that it’s almost become a buzz word in my mind. I feel as though I’ve asked professors over and over, “how do you differentiate?” and over and over get a politician’s response.

So this is not a post on how to differentiate; this is a post on my working thoughts on differentiation. These thoughts are all subject to change as I gain more experience through teaching and interacting with other teachers. This post represents a catalog of my dabblings in differentiation (a sampler of my stumblings?) – not a step by step guide.

Recently I’ve come to the idea that the question “how do you differentiate?” is the wrong question to ask (or at least coming in the wrong sequence). The essential question should be “what do you differentiate?” By differentiating instruction I am changing something for some group of students. To do this I have to ask myself “what can I change and what do I need to keep the same?” What follows are three cases of differentiation, all of which change a different facet of learning.

Case 1: Differentiation in Scaffolding

These are two versions of the unit exam (sshhh SBGers) my CT and I  gave for the modeling unit:

Version A – Scaffolded

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Version B – Unscaffolded

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In giving this test we wanted to test students on their ability to use regressions to model real world data.  Both versions of the test hold the students to this standard and both versions require students to complete the same task.  The only difference in the two tests is that one of them asks the students to do each task in a series of sub-questions while the other gives the students a prompt and requires them to organize their information and present it themselves.  By differentiating in this way we are holding the content constant while varying the scaffolds in place for the students.

Case 2: Differentiation in Difficulty

These are two versions of an in class pairs-check we gave during the conics unit (apologies for scribd’s incompatibility with tables – the worksheets were 2 pages each):

Version A – Straightforward

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Version B – Connecting different mathematical domains*

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The goal of this lesson was to give students practice working with their partner to graph conic sections.  With that in mind, I designed these two worksheets so that they both are focused on the specific goal in this unit, but the second asks students to take an extra step in connecting that ability with previously learned concepts (a.k.a. transfer – a skill I believe is an undervalued, but quite important).  By differentiating in this way I held the learning objectives constant while varying the difficulty of the task.

Case 3: Differentiation in Content

When we gave an algebra II assessment the students average score was just under 40%.  We’ve also noticed significant gaps in the students algebra abilities.  We have used some techniques of simultaneously teaching algebra and precalculus, but we decided most students could benefit from a two-week algebra II refresher / boot camp.  But of course there were about fifteen students who had great scores on the algebra II benchmark.  They clearly would not benefit from the review.  In this case, we differentiated by content.  While most students participated in mini-lectures, quizzes, and pairs-work around algebra II concepts, about 15% of the class focused on SAT test prep/skills practice/miscellaneous challenges.  They still completed all the homework, but did not actively participate in the review.  By differentiating this way I held the pace constant while varying the content of study.

Every time I differentiate I have to question my beliefs as a teacher.  I believe some situations allow for students to work at different paces, work on different skills, learn skills in different ways, or a multitude of other variations.  A general rule of thumb I have come to is that differentiation should move upwards, not downwards.  When writing the modeling test, I took away scaffolds to make it harder.  When writing the pairs-check I added domain to increase the difficulty.  When reviewing Algebra II, I assigned a different task to a minority of the students who had mastered the planned task.  I am still holding all students to the high standards I set, but I am altering the standards in ways which create more of a challenge, not less.

This is what I am thinking right now.  I await the day where I look back at this and laugh at my own ignorance.

*If you graph all of these functions on a single sheet of paper you get a sail boat floating in the water under a moon.

Random Thoughts About Life

I’ve been planning to write a post for a long time, but there has always been something else to focus on.  Who am I kidding though, that’s not an excuse – there is always no matter what something to focus on other than blogging.  But I should get this out there.

This post was supposed to be on differentiation, but it is clearly not.  I still plan on posting about my experience with differentiation soon, but I still need to figure out exactly what to say and how to say it in a reasonable amount of spacetime.

This post is a classroom reflection, a grad school reflection, and a life reflection.

Part I: Teaching and Learning

Teaching and learning are really two heads of the same coin.  When I think of all the things I’ve learned – all the the skills and knowledge I’ve acquired in my path towards becoming a teacher I am reminded of this image:

Replace size with knowledge and you have an adequate description of learning to teach(Image courtesy of Dave Jarvis and made available under the Creative Commons Attribution 3.0 Unported license.)

When I decided to become a teacher I maybe had a Mercury’s volume of knowledge about what it means to teach.  After minoring in education I had a Mars’s volume of knowledge (this is when I first convinced myself I was ready to teach).  I went to graduate school and have extended my understanding of how to teach considerably (maybe to Earth’s volume).  Now I really do feel ready to take on my own classroom and feel as though I can be successful.  Perhaps the biggest development from graduate school is my ability to see the whole picture and how amateur and unskilled I really am.  I’m not saying I’m a bad teacher – I actually feel pretty confident about teaching right now; I just know that I have a long way to go before I will be able to do all the things I want to.

One thing I realized recently is that when I began graduate school I felt like a student who sometimes taught a high school class.  Today I feel like a high school teacher who occasionally goes to graduate school classes.  I think that’s a good thing.

Part II: The future

The future is still such an ominous cloud.  When I am not in the classroom (the only time that I feel like I am living in the present) all I can think about is the future.  That’s probably why this post a drifting stream of consciousness rather than a focused, well-developed collection of thoughts about differentiation.  Am I worried about the future?  Yes.  Should I be? Probably not.  As I’ve told a number of interviewers at different schools, I feel as though I have spent the last four years working towards a specific goal.  I am now at the gate and ready to begin a long future of teaching.  There are a few more hurtles hurdles along the way, but I will get there.  I have two more weeks of mathematics instruction at the high school I work at and eight more weeks of graduate school before I am done.  I would love to know what my future holds for me, but I suppose that will come in time.

When it comes to the point I want to make I find that there is always someone who can make the point better than I can.  In this case, it’s Bob Dylan.

Teaching to the Lowest Common Denominator

When I was young I remember riding on the freeway with my mom.  Whenever we would come to a hill she would complain about all the cars slowing down.  It is logical that to maintain the same speed going up hill you would have to apply more gas than you would going on flat ground.  Even as kid this made sense to me.  I wondered why it seemed like none of the other drivers knew that.  It wasn’t until much later that I realized most of the other drivers probably did know that, but it fell on the few drivers who didn’t to slow down the rest of traffic.  After all, when driving on the freeway people can only move as fast as the slowest wall of cars.

I just got home from a string of college visits with my class.  It was stressful, exhausting, and one of the most interesting learning experiences I’ve had this year.  For the most part the kids were amazing.  They had a blast seeing the different colleges, they were super helpful, and they were overall great people to be around.  But of course there were a few who had to push boundaries; students who would wander off if given the opportunity and who would try to sneak out of their rooms at night or arrange co-ed rooms.  As a result, the teachers had to enter cold-war mode and lay down the law.  After all, we can only provide as much trust as we would for the least trustworthy student (not that I have a student in mind, but by the well-ordering theorem there does exist a “least trustworthy student”).

But why can’t we allow the students to wander off?  Why can’t we trust them to be responsible?  Why do we have to monitor them at all times and ensure no sketchy behavior occurs?  Well it’s simple really – we can only grant the students as much freedom as would be allowed by the most controlling parents.

This dilemma permeates the classroom too.  I hated high school because we’d spend a week learning about what it took me a day to understand.  Now as a teacher I fear that I will fall in to this same trap – that my tasks will only be useful to those who need them most.  That my instruction will be lost on the students who have done the work to get the concept quickly and are ready to move on.  There’s ways around it – through differentiation or groupworthy tasks and complex instruction.  But it’s another constant internal battle to fight.

And what about every other battle of the same type?  How can I grant students freedom knowing that a minority will abuse it?  How do I let students explore the mathematical world knowing that some will choose not to?  How can a provide my students with choice knowing that there will be some who consciously choose to make the wrong ones?

Assessment, Limits, & the Farmer’s Fence

There is a common math problem that is given to students studying geometry.  It asks them to find the biggest area possible with a given length of fence.  After exploring different shapes, the students generally come to the conclusion that a circular fence provides the biggest area.  There is a good account of this activity in Jo Boaler’s What’s Math Got to Do with It.  I know for a fact that all of my students have seen the problem at least once.

Fast forward to precalculus.  My students are studying limits.  I have given assessments where they are asked to compute limits at infinity and at a point, algebraically compute limits of rational functions, and explore discontinuities.  But that’s not enough.  I want them to be able to make arguments about limits.  I want my students to be able to use limits to solve the paradoxes that stumped the Greeks and answer fundamental questions about the infinite and the infinitesimal.

I present to you, the farmer & the fence problem:

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TL/DR version: A farmer notices that adding more sides to the fence results in a bigger area.  He concludes that adding infinite sides will result in an infinitely large area.

I talked with the English teacher and got advice on how to structure an essay prompt.  I had the format mirror that of the ACT argument essays.  In short, I asked the students to use what they have learned about limits to disprove the farmer’s logic and show the faulty conclusion.

What I got blew me away…

Some students put the data points into a calculator and modeled the function to show that it had an asymptote.  Some students showed that as you add more sides the shape begins to resemble a circle so the limit of the area would be the area of a circle (side note - did you know an infinite sided polygon is called an Apeirogon?).  Some students calculated the area for more and more sided shapes and concluded that, while the area does increase, the rate at which the area increases, decreases.  Students defined functions and used formal limit notation to illustrate their point.  They included diagrams of graphs, sequences of polygons, and just about every other image you could imagine.

My favorite introduction was, “Just because a farmer can count sheep, does that make him a mathematician?”

But as I graded the essays I started to notice a trend.  Many of the students who usually score low on procedural work wrote brilliant essays and got perfect scores (which was amazing).  Many of the high academic status students wrote the most mediocre essays where they answered the question with no justification.  They got a mediocre score which reflected their mediocre work (which was amazing in a different sense).

So here is my question: how can I say that any assessment I give is valid if the format in which I give it determines the academic success of my students.  I’ve always known that assessment can only give me a vague idea of what students might know, but this essay has really shaken the foundation of how accurate I think my assessments are.  Are my perceptions of my students’ academic capabilities merely a commentary on my preferred form of assessment?

In short, how do I ensure that I am assessing my student’s mathematical ability rather than my ability to make valid assessments?

Looking on the Bright Side

This has been a tough week – there is no questioning it.  I knew it’d be a tough week going in, but I had no idea how bad it would be.  If I limit my scope to grad school and teaching, there is PACT (the Performance Assessment for California Teachers – A.K.A. the most tedious, obnoxious, and time-consuming assignment of all time).  There is the continuous dribble of assignments, essays, and reflections for the grad program.  Then there’s the grading and the assessing and the clerical work for the classes I teach.  Add a killer sore throat and that frustrating sickness that makes you feel like you’d be all better if you could just get that cough just right but no, it always makes it worse – and there you have my week

So imagine going through this week gathering video footage that you know you’ll have to analyze for hours.  Imagine watching the video of failed lesson after failed lesson, punctuated by the time-stamp of throat clearing every five seconds, and my raspy voice that is almost as painful to listen to as it is to make.  Imagine sifting through this educational vomit hunting for a morsel of good teaching and you will know what I have to look forward to for the next month.

But this isn’t a post to complain (well maybe just a little bit).  This is a post to make a comment about teaching.  I don’t think I could phrase it any better than Helen Keller did:

Character cannot be developed in ease and quiet. Only through experience of trial and suffering can the soul be strengthened, ambition inspired, and success achieved.

As a grad student, I have been infused with this false idea about teaching.  I’ve been deluded into thinking that every lesson should be awesome; that students should be left with a love of learning; that teaching is about perfect lesson plans, perfect learning environments, and all around perfection.  When I sit in class, I get to witness the best of the best.  But that is not where I am, nor where I will be any time soon.  That is a goal to strive for, not a pedagogy to compare myself to.  I may be there eventually, but there will be a lot of terrible lessons and painful classes on my journey.

But there is a light at the end of the tunnel.  It came after a particularly painful morning with little help from my students and a number of awkward silences.  It was a comment my CT made to me yesterday afternoon.  He told me:

Zach – that lesson really flopped.  But you stuck with it and forced the students to learn.  You didn’t let their attitude define the class.  At the end of the day they may be frustrated, but at least they all know how to compute limits of rational functions at infinity – and that was your goal going in.

The fact is that I know I have a long way to go.  Sometimes teaching is painful – sometimes teaching isn’t fun – sometimes at the end of the day I just want to bury my head in my hands, close my eyes, and turn off my mind.  But as long as I can say the students learned something I know that I have achieved the bare minimum of my goals… and sometimes that’s all I can ask for.

My Grading Policy 2.0

After having my first draft torn apart so viciously that I no longer felt comfortable having my name associated with it in the first place, I present to you draft #2 of my grading plan.  While I am sure it is not perfect, I hope it represents an improvement over draft #1.  The biggest changes are that I’ve tried to take out as much of the educationese as possible and give straight details about how grades will be made without the history of my thoughts on the purpose of mathematics education as a whole and yadda yadda yadda.

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Begin the destruction…

[note: I'm sure there are still minor errors in places.  I decided not to make the grammar and spelling perfect because I am pretty sure this letter will be destroyed like my previous one was]