Friday, June 23, 2017

Advice for Getting Started with SBG

One of our teacher ed grads emailed one of my colleagues in search of SBG resources for a pre-algebra course he's developing for next year. He wrote:
When I had you for ____, you graded our assessments using a form of standards based grading. I remember receiving a paper back that listed what standards I had mastered and which ones I still needed to work on. I have wanted to try using a standards based grading system ever since I saw it in your classroom. Would you have any resources that you would be willing to share with me?
My colleague replied with several tips and resources, including Matt Townsley's growing list of scholarly articles SBG. He also Cc'd me, and while I was drafting my own response I realized it might be better to write it as a blog post. So here it is, for what it's worth: my list of suggestions and resources for getting started with SBG in the math classroom.

Hi ___,

I'm glad to hear you are thinking of making the move to SBG. I've been using it for some time now, and I'll never go back. Doug Reeves' comparison of traditional vs SBG principles (from Ahead of the Curve, p. 130) does a fair job of summarizing the reasons why:

 As you move to SBG, there are (at least) three important things to think about in advance:
  1. Which SBG Learning Targets (Objectives, Goals) will you use? 
  2. How will you manage the grades? (And what rubric will you use?)
  3. How will you manage reassessment?
Learning Targets: Early on, I had used lists of "I can” statements to organize the SBG assessments in my course, but I've moved away from that practice. There are a lot of reasons for that (see previous link), but one I've noticed recently is the fact that my students don't seem to use "I can" statements to work out what they're supposed to be learning anyway. Instead, they look at examples from assessment tasks I've given previously on a particular target (or those that were discussed in class) and say, "Oh, that's what I have to be able to."

So I've taken a simpler approach. I disaggregate my course content along two dimensions: content domains vs. skill areas.

For example, this summer I used SBG in my intermediate algebra course. I used eight content domains:

  00 = Functions
  01 = Linear Functions
  02 = Linear Systems, 
  03 = Exponential Functions
  04 = Logarithmic Functions
  05 = Polynomial Functions
  06 = Quadratic Functions
  07 = Rational & Radical Functions*
          *light coverage, so I lumped them together

and three skills areas:

  (a) Procedural fluency,
  (b) Concepts, Connections, and Representations, and
  (c) Modeling & Problem Solving.

to form a grid of 24 distinct target areas (e.g., Target 02(a): Linear Systems--Procedural Fluency).
Every graded assessment task was linked to one of those target areas. (Technically, "target" is not the right term, but it's the one I tend to use with students.) This has greatly simplified my life: I no longer have to agonize over the best way to cluster the dozens upon dozens of course objectives into a manageable-yet-meaningful set of assessment targets. Not everyone does it this way, but I've used it in several different settings now, and it works for me.

Managing Grades: You'll need to find yourself a gradebook option that lets you track reassessments and record scores by target instead of by assignment. Some find it easiest to do this with an old pencil & paper gradebook. There are some dedicated SBG gradebooks out there (e.g., JumpRope). Personally, I use a spreadsheet-based SBG gradebook that I've been developing bit-by-bit over the last few years. It's at a point now where it tracks reassessments and generates automatic grade reports from the grades entered to date, which I print out for each student after each major assessment. Here's a sample from this summer:

Because grades are recorded by target, not by assignment, I make sure each question on each assessment is clearly aligned with one or more learning targets, and I record separate rubric-scores for each target. To help students remember what sort of evidence is expected for each skill-area, I include a short statement to that effect under each Target label. When I grade the assessment, I evaluate the body of evidence the student has provided (i.e, their work on the tasks associated with a given target) and assign a single rubric-score for the relevant target(s) based on that evidence. I'll discuss my rubric next, but first here is an example from this summer showing how I set up my assessments.

Evaluation Rubric: As a rule of thumb, it's best to use a rubric with three or four levels. Anything beyond that and we are at risk of making fine-grained judgements that offer little in terms of increased accuracy and resolution. I use the following four-point rubric to evaluate students' work:
  P = Proficient,
  NP = Near Proficient,
  MP = Mixed Proficiency (or Making Progress), and
  B = Beginning.

I also use two other codes:
  I = Insufficient Evidence (I cannot determine the level of proficiency based on the work shown),
  X = Student was absent or did not complete the assessment.

Reassessment Options: Reassessment is an integral part of the SBG philosophy, but it's also one of the biggest challenges to implement.

In general, I have students complete written reassessment tasks, although I remind them that some reassessment is automatic as targets generally appear on several different assessment tasks over time.

Reassessment tasks, listed by target.
I prepare a set of parallel tasks for each target (not one per assessment) and save them as separate ready-to-print files labeled by both target and assessment (see screenclip). That way I can see at a glance whether I have an reassessment ready for a given target, and I can keep track of where the reassessment falls in the scope of the course. I print these on demand on tan or grey paper (to distinguish them from regular assessments), and only after a student has shown me they're ready. I grade the task and will go over it with a student afterwards if necessary, but I do not return reassessment tasks to students so I can keep reusing them.

There's a nice article (.pdf) by Rick Wormelli that discusses the reassessment in an SBG setting. It concludes with practical tips for managing "redos" in the classroom. I highly recommend it, and would add one more tip: Some reassessments may be done as oral interviews. If a student made a small mistake and can demonstrate understanding by convincingly explaining why it was incorrect, I will generally change their score without requiring a written reassessment.

I hope some of that helps as you work out the details on your own SBG implementation. I hope you'll share what you come up with!

Monday, February 6, 2017

MIA2017 - Do you have a growth mindset?

We share how the math teachers in our hall
empowered kids by focusing on growth mindset.
(Based in part on Boaler's book Mathematical Mindsets)

Our presentation was part of GVSU's
2017 Math in Action Conference
Saturday, Feb. 25, 2017

Session Resources:






Saturday, February 27, 2016

MIA2016 - Coaching Principles in the Mathematics Classroom

Molly Carter, Jamie Stuart, and I presented the following session at the 2016 Math in Action conference. Our talk was called "Promoting Collaboration During Problem Solving: Coaching Principles in Action". Here are our slides and electronic versions of the session handouts. Thank you for attending! 

Our Slides:

Session Resources:

Collaboration and Problem Solving Guide
Levels of Discourse chart

Author: Jamie Stuart (2015) Source: Principles to Actions (NCTM, 2014)
Image source:

Wednesday, February 24, 2016

Best Teaching Practices, by GVSU's Math Student Teachers

Over the years, our math student teachers have shared their best ideas and most memorable experiences from the mathematics classroom. We prompt them by asking what they have seen or done that will stick with them or have a lasting impact on their practice. I thought I would share it here in the hope that others may find something useful.

Feel free to submit your own ideas or experiences in the Comments.

Wednesday, October 28, 2015

TMWYK What if you

Me: Hey, why did you choose 7-2=5 for this stretch your thinking question? Is it because you just did that one on #5?

No! It's because 7 is my favorite number. And then the other one had to be 2 to make 5.

Oh, I see. Well I have a challenge for you then. I say you can use your favorite number 7, but you have to put it in the other box. What would go in the first box then? But that's probably too hard for you, right?


He counted on his figures, and a moment later said, "it's 14." Then added, "wait, that's not right.... Is it?"

You tell me. Let's finish the rest of the homework (other side). But when you know you have the right answer to this one, (_)-7=5, you can tell me. But I don't want guesses. You tell me what it is when you know it's right. And then you can tell me how you were so sure. I can't wait to find out how you think about this one!

He came back later and said in no uncertain terms, "it's 12, and I can prove it!"

He held up 12 fingers by flashing 5 & 5 and then 2. That's 12, he said, and to take away 7, you can take away the 2 first, and then 5 more. What's left is the other 5. So 12-7=5, ha!

That's awesome Adam, I totally get that argument. Nice job, buddy!


Thursday, September 10, 2015

Transformations vs. Order of Operations

The following question was raised by one of the work groups in class today: Why is the order of performing transformations different than the order of operations?

We are studying function transformations like these:
From College Algebra by Coburn & Herdlick

Our explorations in class have supported our book's claim that function transformations must be applied in the following order:
  1. horizontal shifts
  2. reflections
  3. stretches/compressions
  4. vertical shifts
Does that sequence conflict with the order of operations? What a great question! Let’s explore it using the absolute value function f(x) = |x| as our parent function and using
g(x) = -2|x-1| + 5.
to define the intended transformation. Consider what happens to a point on the graph of y = |x| under this transformation. Let's use the point (5,5). Where does it end up after the transformation?

To find out, we evaluate f(5)= -2|5-1| + 8. This requires the following sequence of calculations:

  1.     5-1 = 4.         that’s the x-1 piece; there’s the horizontal shift*
  2.     |4| = 4.           that’s |x-1|; we have just applied the parent function, |x|.
  3.     -2*4 = 8.       that’s -2|x-1|; there’s the reflection (-) and vertical stretch (by 2).
  4.     8+5 = 13.      that’s -2|x-1|+5; there’s the vertical shift.
*But why is it a rightward shift? That's for another post.
Has the order of operations been maintained?

It is probably easiest to see if we use GEMA rather than PEMDAS to track the order of operations. They reflect the same underlying order of operations, but GEMA seems to produce fewer order of operations misconceptions (sounds like a PhD thesis topic to me!)

GEMA = Grouping symbols first, then Exponents, then Multiplication (and Division, from left to right), and finally Addition (and Subtraction, from left to right).

Now let’s step through GEMA:
G: Grouping symbols. The absolute value bars a type of grouping symbol (so are parentheses and brackets, square root symbols, and even the horizontal line that separates the numerator and denominator in a fraction). First, we work on the expression inside the grouping symbols (absolute value bars). There is only one operation to do in there: subtract 1 (step 1). Now we apply the absolute value bars (step 2), at which point the Grouping symbols are gone and we move on to….

E: Exponents. No exponents to deal with this time. Move on to….

M: Multiply (or Divide): With the || bars gone, the function now reads: f(4) = -2*4 + 5. We multiply by -2 next. This creates the reflection (step 3a) and stretch (3b).

A: Add (or Subtract): Only one thing left to do! (step 4).
Conclusion: It appears the transformations sequence is consistent with the order of operations.

I'm convinced. Are you?

Monday, February 23, 2015

Giving Effective Feedback

I had a nice discussion with my assessment committee colleagues today. Afterwards, at the request of one of my colleagues, I shared a few resources about effective feedback. I decided to kill two birds worth one in a handbasket by posting them on my blog, too.

The first resource that came to mind is this article by Grant Wiggins (2012): Seven keys to Effective Feedback.

I also like this article (from the same September 2012 issue of Ed Leadership) by Fisher & Frey (2012): Making Time for Feedback. It offers practical feedback strategies, including this gem: it can be counterproductive to mark every mistake a student makes.

Actually, the collection of abstracts suggests the entire Sept 2012 issue may be a treasure trove of excellent articles on feedback. I'll have to check out the rest when I have more time.

Finally, I encourage anyone looking for a more in depth look at feedback to check out the first chapter of Classroom Instruction that Works (2nd ed.):