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.):

Inspiration Post

For all those who need this today.