A Few Problems of My Own

Some of my favorite assessment problems from this semester in College Algebra. Each problem is labeled with the standards it mainly addresses. The main function families for our course are:

• Q - quadratics
• P - polynomials degree 3 and higher
• R - rational functions
• E - exponential
• L - logarithmic

And then there's a category of function concepts and special functions.

For each function family, there's four repeating standards:

1. Basics - vocabulary, characteristics
2. Representation - being able to move amongst table, graph and equation
3. Symbolic - traditional algebraic skills, solving and simplifying.
4. Context - solving in application. Being able to mathematize situations.

I like having a structure, as students getting used to the Standards Based Grading is one of the big hurdles in the course. That's why all the problems are labeled with standards, too.

These problems are different than the ones I use in class or for homework, with more scaffolding, and often reminiscent of things we've tried in groups. They also choose problems on which they're ready to be assessed. They can write up longer responses at home to turn in on problems that they do not do in class.

This first one's not mine. They read a Glen Waddell post on the Exeter method, so I thought it was a nice connection to put on an assessment.  The element of the cup really made clear which students understood what the height function was saying.

We did a fun activity in class on sound frequencies that you can hear at different ages (due to hearing loss/damage) in class when we were talking about decibels. Frequency is not logarithmic, of course, but I love using a topic as arcane as logarithms to make sense of something they've heard of (get it?) as much as decibels.

One of their favorite application problems in the semester was using exponentials to model weekend movie grosses, which are roughly exponential decay. But of course the sum of geometric sequences is also a transformation of an exponential. This problem unfortunately highlighted careful reading or the lack of it, which is not super-useful, so that some people tried an A*k^x model. But it did show the people who were making sense of that not working, and the people who blindly accepted the results. Desmos reliers (as opposed to Desmos-only-if-my-calculator-can't-do-it) did better on this.

This problem was another great one for function notation. The difference between evaluation and solving was crystal clear. And it was such a nice pattern for the people who correctly interpreted the problem that there was a good payoff.

The top problem here is also not mine, of course, coming from WODB.ca. Well, originally. The task of making a function to look like it is a great representation prompt. Students who made sense of marble slides were really able to strut their stuff.

 It was the second and third problems that were really interesting to me here. They were open middle-like in the variety of different methods people used to solve them. Recursive rules, which we never used in class, really, excellent table use, some regression... such a nice mix.

I liked it enough that I wrote a follow up for the last SBAR opportunity. (AKA the final.)

One thing that we never addressed directly that we got to the last time I taught this course is that the sum of a certain degree polynomial sequence gives the next degree. Much like the differences give the next lower degree. I LOVE that structure.

For the special function types they only had to demonstrate one of the four kind of standards that they did for the main function families. Both of these offered a lot of opportunities for sense making. In particular the normal distribution question highlighted whether people understood mean and standard deviation as descriptors.

If you have feedback on writing assessment questions for SBARs, I'd love to hear it. Whether it's modifying these or a whole different direction. Here's all my assessments from the semester in a Google folder, if that could be of help or interest.

PS. Thanks Ann and #MTBoS30 - I've now blogged more in May than I did all January to April.

Two Final Problems

Trig Problem 2

For my preservice high school teachers' "final" (really a last Standards Based Grading opportunity), there were two problems that while similar in many respects were quite different in results. All of the problems were listed by one standard, but typically could be used for other standards. It's the student's responsibility to describe what standards they are demonstrating, though I will help if it demonstrates something well that they need.

Trig Problem 2. (Standard: Law of Sines, Law of Cosines and applications)

Figure out some of the missing information in the diagram.



The pictures were made in GeoGebra, which I highly recommend for mathematical image creation, as well as more active uses.




Geometry Problem 1. (Standard Lines: parallel, perpendicular, properties of angles)

Find more angles.

Geometry Problem 1

Similarities: visual, finding connections, geometry, students have previously done and been assessed on similar problems.

Have to love easy-to-draw memes.

Differences:  throughout the semester students saw trigonometry as something difficult, and had much less confidence on them.  Students were very successful with the angles problem, able to find all the angles, and be able to justify their results. Why vertical angles are congruent, why there are 180º in a triangle, etc. On the "trig" they quickly resorted to visual inference (like the angles at A were all 60º), supposition, and ignored contradictions (such as finding that the length of CD was less than 6 units), and did almost no extension to other standards from circle geometry.

It was fascinating to read their work, and I wish we had more class time to look at the results. It felt like direct confirmation of the Van Hiele levels, and convicted me that as much time as we devoted to trigonometry, I need to find more ways to increase their experience.  While I thought the circle diagram was more subtle, I didn't realize the great difference in how students would see it. Only one student realized CD must be 6 units, which is the entry to me for many of the possible values that can be determined.

SBG Resources

From Rainbowcatz @ Flickr

I was gathering Standards Based Grading (SBG) resources for a colleague and thought that would be worth sharing.  Maybe this should be a LiveBinder? There's a definite math focus to my selections below, those it's not strict. Many people refer to SBG as Standards Based Assessment and Reporting (SBAR), which gets the whole 'grade' idea right out.

People: (Name links to Twitter)

• Jason Buell - SBG resources that includes the SBG galas.
• Shawn Cornally - collected resources at his blog
• Riley Lark - SBG blogposts
• Frank Noschese - SBG blogposts

Fundamentals and Further:

• Ed Leadership article: Seven Reasons for Standards-Based Grading by Patricia Scriffiny
• SBG Beginners Wiki, which also has math skills lists , started by Elissa Miller
• SBG page at the MSmathwiki.
• Blue Harvest - Shawn's SBG software
• ActiveGrade - Riley's SBG software

Twitter discussion

• #sbar - find more SBG folk, or people trying it in your discipline, by a Twitter search.
• #sbarbook - book group that 'meets' weekly for discussion about a particular book on assessment. Doesn't look like this semester's book is very engaging, though.

For completeness sake, here's my 2 (so far) SBG posts. Hey, this makes 3! If there are more examples of SBG in college, especially college math, please help me find them.

Grading: SBG and U

Math Monster
by Mister Awesome @ Flickr

Standards Based Grading to me is the idea that the teacher lays out what students are responsible for demonstrating ahead of teaching, and students have a long period during which to demonstrate them, possibly up until grades are finalized.  And students have multiple opportunities to demonstrate.  (This is a Part II to the previous grading post.)

Other people describe it better and more thoroughly.  Especially Sam Shah and Shawn Cornally.  Also please check out the beginner's wiki started Elissa Miller and the SBG gala hosted by Matt Townsley. (Note that you could be interacting with these outstanding professionals on Twitter: @samjshah, @thinkthankthunk, @misscalcul8, @mctownsley) Frank Noschese is thinking about it powerfully, too, in physics, but I haven't had the chance to interact with him about it.

I will say that I've only used it with preservice teachers so far, but they were mostly an appreciative audience for it, and would like to see it in their content classes.  I will be doing it in my content classes, starting with a graduate calculus class in the fall, but we're so pinched for math educators right now that I don't get to teach any straight content courses.

The preservice teachers have been helpful for improving my practice of it with their feedback.  If you're making the change, I'd encourage you to discuss it with your students, give your reasons, and involve them in the process.  I was only going to do it through in class assessments and similar things in office hours, but I added an SBG option to portfolio submissions and added an interview option for office hours.  The biggest remaining thing is how to communicate it better at the outset, with which the resources in the second paragraph will help.

Another Speedbump Classic

The most powerful concept to the shift has been giving the students a clearer purpose on the assessments: to demonstrate what you understand by communicating your thinking.  Much of the emphasis on the right answer is gone, as is the expectation that test questions will be trivial repeats of tasks already done.  Not that my tests were like that lately (have to go back over 20 years for one of those), but it was a bone of contention with students.  Now it makes (more) sense to them that they couldn't show understanding on a question like that.  I've had a few students reject a problem because they knew how to do it already.  (That's not the majority, but some day...)

It's different from K-12 use because in the university we see the students so much less. We give up class time for independent work outside of class, which minimizes time for summative assessment.  I struggled to provide multiple assessment points.  Put lots of former standards on assessments as choice, and polled students as to what previous standards they wanted on.  My standards were much broader than they would be in a content course, as math ed classes wind up covering things like "all of high school mathematics."  So I made my standards pretty broad, but we looked at examples of more focused grade level standards.  In the future as I reuse, I'll try to add some of those specifics as ways to demonstrate the broad standards.  I also let them know that the final grade would take into account which standards we had covered and assessed in class.  Some of the content I don't set until the preassessment is in, so it's hard to know ahead of the semester.

Here's the policy on my middle school math syllabus.

Standards Based Grading: SBG is a relatively new way to assess students that seeks to get a higher correlation between grade and understanding. On each of the objectives below, you will have opportunities to demonstrate your understanding. These objectives are a bit broader than you would expect in a secondary classroom, since we are seeing content from three years of schooling. In a secondary classroom, the teacher identifies the standard demonstrated, but in this preservice teacher preparation course you will also be trying to identify which of your work is evidence of which standard.

Scores do not mean an answer is right/wrong, but are meant to reflect how much understanding was demonstrated. It is possible to demonstrate good understanding of a concept without even finishing a particular problem. The score for each category is the average of the 2 highest scores. If there is only one score it is discounted by 1; a single A becomes a B, etc. You can reassess on specific objectives during office hours or at arranged times.

A+ complete understanding and can extend on your own
A complete understanding, can apply when appropriate
B some small difficulty applying or missing a small point of understanding
C significant difficulty in application or missing a major point of understanding
D mechanical application of ideas without understanding
F little to no understanding or evidence of understanding

Mathematical Content Objectives

A. Number: representations and operational concepts

1. Integers
2. Operations on integers
3. Rational numbers: fractions
4. Operations on fractions
5. Rational numbers: decimals
6. Operations on decimals

B. Algebra: representation, operations and modeling

1. Patterns: recognizing and generalizing
2. Variable: as unknown and changing quantities
3. Linear and exponential relationships

C. Geometry

1. Similar figures and proportional reasoning
2. 2-D figures: characteristics and sorting
3. 3-D figures: characteristics and sorting
4. 3-D representation

After a messy fall semester of trying to run parallel SBG and traditional, and a messy winter semester of struggling with full implementation, I'm very happy I came down this road.  I have four basic goals for my grading:

From Comically Vintage
Don't be a Dodo!

• fair - reassessment helps this.
• measures real understanding - move away from non-problems helps this.
• not fear or anxiety inducing - students said this was a big improvement.
• measures where the student is at the end of the course - clear improvement.

While I didn't get a lot of out of classroom reassessment until the end of the semester, I did get people using the in class assessments to reassess.  Students were more responsible for their own marks than ever before, and rather than tracking grades, they were attending to objectives. Broad over-generalized objectives, but I had to start someplace!

I strongly recommend you consider SBG, whether you be K-12 or 13-19.  If you do, let's talk!

Grading: Road to SBG

From the excellent
comicallyvintage.tumblr.com

So, my son and I (X=11) have been discussing starting sentences with "so."  It appears that it may be genetic.  In real life I'm quite awkward in conversation and transitions are especially difficult, and I think "so" is an all purpose connector.  Leads to something that fits or to a change in topic.

Of course, transitions being awkward is something that I think is true in general.  How to get students to transition from what they've known to something that may be completely new is the crux of Standards Based Grading, or more wholesome and holistic ways to evaluate as a whole.  Back When I Was a Worse Teacher (tm) I could sometimes be heard to joke (sortofjoke) that I wished I could offer my students a deal: take a C and leave the class, stay and take the class for a chance - no guarantee - at better.  I was known to be an easy grader though, and I wanted it to be the case that if students made a genuine effort that they would get at least a C.  I also use to jokingly propose a Survivor style grading system where we vote people out of class starting at the third week.  First out fail, but get the rest of the semester off.  Thought the tests could be like challenges where multiple people can earn immunity.

Strange Brew
my current favorite comic strip
does my favorite movie line to quote?

It was a joke!  The point of the joke was to get people to think of why they were in the class, and the grade might not be the point.

Working with Dave Coffey, who is our local assessment guru, I saw him use a portfolio to assess our Math for Middle School class, and we have long used a portfolio for our student teacher assistants.  In our introduction to mathematical reasoning class there's a summative assessment called the proof portfolio.  One time I was teaching it, at the end of the course one of the students asked: "I'm so much better than I was at the beginning of the semester - why does that writing count in my grade?"  Whoa. My only response was that THAT was a very good question.

I realized that more of my grading had less to do with where the student was at the end of the semester than I had wanted to think.  Mostly because of the reason I hate the most: that's the way it's always been done.  Furthermore, I realized I hadn't ever really thought about what I wanted grading to achieve, let alone whether it was accomplishing the job.  It's usually pretty quick for a room full of students to agree on some characteristics of good grading:

• fair
• measures real understanding
• not fear or anxiety inducing

It usually takes some discussion to get to:

• measures where the student is at the end of the course

My portfolios shifted to feedback only up until the end of the semester, and then the last time they are graded.  The grade is about half on completion (as a percentage) and half on exemplars.  The students choose the exemplars.  In early turn ins, the students ask for feedback on what they want to know, and I share what the grade would have been, and give feedback on their issues.  The would-have-been grade allows me to identify big issues that the students don't seem to recognize.  I collect at the 1/3 and 2/3 point, though some students want more frequently.  (So they can be more responsible.)  The final grading is relatively easy though visually daunting; how many boxes of grading?  The final exemplars need annotation about what makes them exemplars, and have been the best way for me to evaluate problem solving and communication.

What are we assessing?
brad_holt @ Flickr

So what about the content? That was hard for me.  How to evaluate content in a way that allowed for improvement right up to the end.  Getting involved in twitter around the same time I was reading more and more educational blogs exposed me to standards based grading.  Especially Sam Shah and Shawn Cornally.  They supported with good stories and honest difficulties, links to several educators trying or mastering the practice, and good resources, like the SBG wiki.  I definitely wanted my preservice teachers to be exposed to it, and felt like it was the missing piece of the puzzle.  Or at least better.  In the fall I tried running parallel systems - wow, was that a bad idea.  This past semester I went full out, and did better.  Learned a lot for the fall.

Next time: what I did, how it went, and how I'll adjust.