How to do SBG, better, next time!

Wordle of this semester's SBG Targets

Over the last few semesters, I have begun implementing a standards-based grading approach in my classes. My first real attempt was over the summer in a small intermediate algebra class. The results were very positive, but I had fewer than 12 students in the class for most of the semester. Plus it was summer. Could SBG stand up to the challenges of an academic year? Thinking the benefits were worth it, I decided to go for it.

It was worth it -- and I'll talk about why another time -- but I also learned a few things along the way. As the semester is drawing to a close, and I'm grading a lot of student work as they seek to fill in "holes" in the SBG gradebook, I thought it was prime time to reflect on what I would do differently next time: hence, this post.

My Intermediate Algebra Targets (Su13)

I used these targets as part of a my standards-based grading effort in a summer offering of our Mth110 course (Intermediate Algebra) at GVSU.

They are not perfect. Some changes I would make for next time include:

• Reducing the overlap between certain pairs of targets (Target 1.2 and Target 1.14 are examples of that).
• Splitting some targets that merge two concepts or skills in one target (Targets 1.5 and 1.11 are two examples of that).

But overall these worked well and were well-received by my students.

Feel free to use or modify for your own non-commercial educational purposes. For all other purposes, please contact me and let me know what you have in mind.

Intermediate Algebra Learning Targets

I don't know this...yet!

I've written before about my experiences with Standards Based Grading and why it feels so right. See "Show me what you can do" and "I can show you more than that".

Yesterday I gave a quiz on measurement and data. During the quiz, I noticed a few students really seemed to be struggling with some questions (evidence: prolonged periods of time spent staring at a blank spot on their paper).

So interrupted the class with the following announcement:

Remember, one of the great things about this SBG grading system is that you can always turn in more evidence later. So if you don't know how to do something yet, it really is okay to just write "I don't know yet." We'll keep working on it, and then you can turn in stronger evidence when you're ready." 

The release from the students was immediate. Tense shoulders relaxed. One student said out loud how much better that made her feel. It was a memorable moment.

How Math Saved Bedtime

(This post was inspired by @Trianglemancsd's blog Talking Math with Kids.)

The bedtime routine at our house begins at 7:30. We have a chart showing the steps: take a vitamin, go potty, brush teeth, put PJs on, read three books, go to sleep. Lately, the middle three steps have evolved into a serious power struggle with my four year old. He whines, complains, drags his feet, says he's too tired... and the resulting cajoling amounts to leading a horse to water and finding that it stubbornly will not take a drink.

From https://www.babysleepsite.com/wp-content/uploads/2008/12/coverears.jpg

After a particularly frustrating night, I happened to notice the potty / brush his teeth / change into PJs sequence had taken 24 minutes. This gave me an idea. I went downstairs and found a piece of black construction paper and a white crayon. I quickly constructed the axes for a simple bar chart, extending the vertical scale up to an optimistic 25 minutes, and sketched the horizontal scale that would hold the days of the week.

Phronesis? (Using Tools Strategically)

I was preparing a lesson on representing data with dot plots and box plots. The goal was to help students recognize the pros & cons of each type of visual display, among other things (like simply making sure everyone knew how to construct one).

But it turned into something more powerful, at least for a moment, and I thought I would share that moment with you via this post.

The lesson also built off the previous session where we had used "MAD Minute" type tasks to generate data. You know, like these:

 

Lots of kids are good at those. I was good at them. Many of my preservice elementary teacher students said they loved them too. "Oh man, I loved doing these!" But you didn't love them if you weren't good at them. And we had that discussion: For some kids, these high-stakes timed worksheets were sources of extreme anxiety.

I'm not grading this

I asked my students to turn in a draft of the Cheesecake Task last week. But when I sat down this weekend to prepare to write feedback on their tasks, I hit a snag. Simply put, the work was not good, but their self-evaluations were off the charts high. How could this be?

 Before I started inking comments, I decided to sort the stack into two piles:

Pile one: Almost got it, needs minimal feedback. 
Pile two: Needs a lot of work (and lots of feedback).

Pile one had 5 papers in it. Pile two had 19.

Ugh. 

A Planning Post

I have had many conversations with preservice teachers about what it looks like when an experienced teacher plans a lesson. I suspect we all approach it differently, but I had a really nice lesson the other day that I wanted to understand better. So I decided to kill two birds with one post: in writing up my planning efforts, hopefully I can help some of my newest colleagues understand something about lesson planning while also coming to a better understanding of why that particular lesson "worked". So, here's what lesson planning sometimes looks like for me:

...why rubrics? (part 1)

Why I use rubrics, #1: Rubrics help me to focus on proficiencies, not deficits, and they support my efforts to give feedback that can be used to improve future performances.

In earlier posts, I discussed the "In the cups" performance task and my use of a proficiency-based assessment system in my intermediate algebra course. In this post, I'll share an example to illustrate how the use of an evidence-based rubric has supported my implementation.

I can show you more than that

In a previous post, I talked about my grading system based on the mantra, "Show me what you can do." Here's a quick example of a student's response to a quiz item that shows why this proficiency-based approach feels so right:

One of my learning targets was:

___1.5. I can solve linear systems and represent the solution symbolically and graphically.

I stated that objective at the top of the quiz, along with several others, then gave the following task:

The solutions to the following system of equations are provided. Show that you can use the elimination and substitution methods (use each one once) to solve these problems.

{y = 3x+6
{2x + 4y = -4              solution is (-2,0)  

{7.5x - y = 10
{15x - 4y = 10            solution is (2,5)

Show me what you can do

I have a new mantra this semester, reflecting a new (for me) way of thinking about my assessment and evaluation. The mantra is basically: "Show me what you can do." It's a big shift from my early assessment systems.

"Show off!"

What it Means to Solve (Again): Finding Wormholes

In two previous posts, I have explored what it means to solve linear inequalities and what it means to solve quadratic equations. The latter post describes a classroom moment in which my students and I contemplate what it means to find and visualize the (complex) solutions of a quadratic equation that has no real solutions. I wrote:

"A picture formed in my mind of an invisible, ethereal, wormhole-style thread binding the two curves. How could we represent the functions so that the "intersection" would be visible?"

This post documents my post-class exploration of that issue.

What (else) it means to solve

In class yesterday, I had an opportunity to reinforce what it means to solve equations and inequalities (a connection I've discussed in a previous post). My students and I were working on a task that required that we verify that x = 2 and x = 0 are the two solutions of the quadratic equation -2x^2 + 4x + 1 = 1. How can we do that? A chorus of replies: "plug them in and check." 

"Right," I said, "We need to see if we have found both (all) of the values that make the equation true."

Learner #1 raised his hand and asked, "Doesn't it also mean finding the points where the parabola intersects the line y = 1?" I know from a previous conversation (described in this previous post) that this was how he learned to solve quadratics in high school: enter the parabola into Y1, enter the other expression into Y2, and use the CALC -> INTERSECT to solve.

Created with Geogebra

So I was happy to affirm the connection.

A short time later, in the same lesson, we were exploring the function f(x) = (x-3)^2 + 5. We had determined that because the parabola opened upward and had its vertex at (3,5), so there must not be any solutions to the quadratic equation f(x) = 1. Furthermore, we agreed, the equation f(x) = 5 would have just one solution (at the vertex) and f(x) = k for any k > 5 would have two real solutions.

Created with Geogebra

Then Learner #2 offered: "So if we ever find that there's no solution, we can conclude that the curves do not touch, right?" It was a nice extension of Learner #1's remark.

But it depends a bit on what you mean by "no solution," doesn't it? The situation is exactly analogous to the 1st grader who asks the question, "Teacher, what's 1 minus 3?" to which the teacher response helpfully, "Honey, you can't do 1 minus 3." I like to imagine a precocious 1st grader reply: "Yeah, maybe you can't, but I can! It's negative 2!"

In the Cups

Our department has a shared e-folder where we can share tasks we develop for use in specific courses. I've used it a lot this summer to support my planning for the intermediate algebra course I'm teaching. The activities there range from routine to inspiring--and both types have been useful!--which reminds of a quote I saw recently: Good teachers borrow, great teachers steal.

So I have been drawing heavily on the excellent activities there. Some tasks I use almost verbatim, and others I modify to suit my own purposes. One of the activities I found was a nice math modeling activity called "In the Cups". It is designed to support students with creating and using linear models to make predictions and solve problems.

The original document presents the following data to students:

 

Number of Cups	Height of stack (in cm)
1	7
2	7.6
3	8.2
4	8.7
5	9.2
7	10.1
8	10.6

The task proceeds to ask students to predict the height of 6-cup, 10-cup, and 25-cup stacks, and eventually invites students to find a model expressing the stack height as a function of the number of cups in the stack.

I noticed that the data in the table do not quite conform to a perfectly linear relationship--notice that the unit rate of change varies from 0.6cm to 0.5cm per cup--which got me thinking. I wondered if the author had introduced some random variation into the stack heights presented in the table in attempt to make the task seem more authentic. Or perhaps he or she had really measured some cups and the data were authentic approximations. Either way, I liked it.

I started measuring a stack of dixie cups I had in my office, then I realized that by doing the measuring myself, I was about to rob my students them of an opportunity to do some critical thinking. So I decided to replace the table of values with a photograph. 

That one picture (of 27 cups) contains all of the information needed to model the relationship, and it invites students to consider questions about the accuracy of the approximation. Replacing the table with the photograph felt like a more authentic way to present the task to my students.

What's with that title anyway?

Let me say a bit about the title of this new blog of mine.

My teaching and scholarship have always centered around striving for understanding, and so I settled on the title "...do we understand" as I reflected on the types of questions I anticipate exploring here. These are the questions that interest me as a teacher, as a researcher, and as a human being.

So I expect I'll be exploring questions like:

• Do we understand the content we are teaching and learning? How? What? Why?
• Do we understand our students' ways of thinking? How? What? Why?
• If not, how can we find out?  

What it means to solve

"Do we understand" was on my mind as I was grading intermediate algebra quizzes over lunch today. In particular, I wondered: do my students really understand what it means to solve an equation or inequality? It was clear enough that they knew how to do it, but do they know what they are accomplishing in the process?

One of the learning targets I have identified for my students reads:

I can explain what it means to solve equations, inequalities, and systems, and I can use this knowledge to check my answers for reasonableness and correctness. 

I'll be honest: I really love that learning target. It was inspired based on one of the Common Core State Standards for Mathematics, specifically, one of the Grade 6 Expressions and Equations standards:

CCSS-6.EE.5 [Students will] understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? ... (emphasis added)

That concept is fundamental to the work we do in algebra. Consider the wide array of situations where the sole directive is "solve". Come to think of it, one of my old doctoral advisors once described "solve" as one of the four universal directives of algebra homework sets. (Can you come up with the other three? Answers appear at the end of the post.)

In the case of "solve", the meaning is universal and boils down to "find the set of [thing(s)] that make this [thing] true". Consider: 

• "Solve 3x+4=7" means "Find the values of x that make the equation true."
• "Solve {3x + y = 4, x - 2y = -1}" means "find the ordered pairs that simultaneously make both equations true."
• "Solve y'(x) - y(x) = 0" means "find the family of functions that make the differential equation true."
• "Solve Ax = λx" means "find the real numbers λ and corresponding vectors x, called the eigenvalues and eigenvectors of A, that make the matrix equation true."
• "Solve Ax = 0" means "describe the set of vectors x, called the null space of A, that make the matrix equation true."  

In with the old, out with the new

I've been tinkering with Tablet PC's lately.

What's a Tablet PC, you ask? I'm talking about a fully functional laptop computer that has a touch screen interface on top of the traditional keyboard and trackpad, and one that folds or flips flat so the user can write on the screen as if it was a notepad.

My old tablet is an HP Elitebook 2730p. I've been using it for years as my primary computing device. In my office, it spends most of its time docked, so I get a full size keyboard and monitor. But I also regularly used it for writing papers, teaching classes, presenting at conferences, creating mathcasts, and taking notes during meetings. It came home from work in my backpack every evening and returned with me every morning. Once, during the summer, I even left it on top of my car on my way to work. We made it about halfway there before it caught the wind at 45mph, did a tumbling backflip off the roof of my vehicle, and bellyflopped onto the asphalt behind me. I hit the brakes and managed to retrieve it before anyone ran it over -- and would you believe it, the thing booted right up!

Suffice it to say, I love my HP Elitebook. But its getting old and slow. I'll need a new one soon.

So I'm testing out a contender: the Dell XPS 12 Convertible Ultrabook, pictured above. I decided to see how it would compare with my old HP Elitebook for creating Mathcasts.