TMWYK What if you

Here's an exchange I had with my almost-7-year-old where I use "What If" to stretch his thinking a bit further:

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?

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?

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.

Source: eatoneducationalinsights.edublogs.org

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.

https://s-media-cache-ak0.pinimg.com/736x/2b/de/d3/2bded3e02dfc2f5d55ab162b296baf43.jpg

MIA2015 - Facilitating Growth through SBG

How can the use of standards based grading support a growth mindset in students?

Presented at GVSU's Math In Action Conference 

Saturday, Feb. 21, 2015 

by Dr. Pamela Wells and Dr. Jon Hasenbank

(Session E6, 1:20-2:20 pm)
See below for slides and resources.

AMTE 2015: Using Standards Based Grading with PSTs

This page hosts the materials for the presentation by Jon Hasenbank and Pamela Wells on the use of standards based grading in math courses for future teachers (presented at AMTE 2015).

AMTE 2015: Supporting Growth Through Cognitive Coaching

This page hosts the materials for the presentation by Profs Coffey, Gerson, and Hasenbank on the use of Cognitive Coaching(SM) for preservice teacher field supervision (presented at AMTE 2015).

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Coaching Information and Resources:

• Cognitive Coaching official home: thinkingcollaborative.com
• Dave Coffey's posts on Cognitive Coaching
• Jon Hasenbank's posts on Cognitive Coaching 
• An unofficial wiki on coaching: mynalyssa.wikispaces.com/Cognitive+Coaching 
• An wiki-adaptation of the official reflecting conversation map.

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 AMTE 2015 Slides:

Finding the "Hidden Wows"

I was working with some middle school teachers-to-be who are noticing some of the aspects of algebra we tend to take for granted after years of practice and application, and they are noticing how difficult it can be to anticipate the strategies and struggles of students who are just learning algebra.
 
One way I get around that is to try to hold myself still with a problem for a bit and look for the hidden connections. With kids, I might tell them we're looking for the Hidden Wows.

Suppose I am preparing a lesson for sixth graders to introduce problems of this form:

Solve: ax = b

 My planning might start out sounding a bit like this:

SBG Indicators - Your Experience May Vary

Uh-huh...

Learning targets are hard to write.

Here's one (of 27) that I used in F'13:

D.3 I can calculate, work flexibly with, and demonstrate understanding of statistical measures of center and spread for numerical data, including: mean, median, MAD, and IQR.

If you tease that target apart, you realize it contains 12 distinct skills: {calculate, work flexibly with, and understand} x {mean, median, MAD, and IQR}.

That's a problem: such a complex target is very difficult to assess. Do they have to show all parts on a single task for a proficient score? Can they piece it together over several tasks? If so, how do we keep track?

So I swore off using complex targets for my Su'13 College Algebra course: