A blog for sharing my math interests on the web, to post new materials for elementary, secondary and teacher ed, and vent mathematical steam when needed.
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It was disconcerting recently to realize that my grading policy is still dysfunctional. As much as I've moved towards standards-based grading for content, end result evaluation, and student-chosen exemplars, there's vestiges of teacher control and obedience.

With my spectacular grad class this summer (see here and here), there's still an attendance policy. I felt a conflict because going by my own policy would mean giving a grade or two below A to students who exceeded my goals for the course. I asked the students how they felt about it. All inservice teachers, they asked "were the goals met?"

I was lucky in that this was about an A, A- or B+. No heart-rending decisions. But in another situation, it easily could be.

In the end I did the right thing. But now I'm considering how much coercion I want in my syllabi. I make class worthwhile, as much as I can. (This class's most frequent student evaluation comment was wanting more classes to explore what we did in more depth.) Do I have to try to make my students come to class? Is it coercion or support? You know, to help them make the "right" decision.

As it is, I get requests to have more due dates, and require things from them more frequently. Do they need it? What can I do instead, with which I can live? Help!

It's a little hard to share this because he speaks so nicely about our class. But my realization is that it's really his work that he's talking about.

Our end of semester assignment is one I cribbed from Maria Anderson at her college math/tech camp. She asked us each to do a Little Big. That is, a little presentation that gets at the big ideas. I ask the teachers to make it available over the internet, as I see that as an important professional participation tool. Slideshare, Prezi, Google presentations or even a PowerPoint file shared via box.net. Bill is the first person to share by webcam, and the result seemed pretty powerful to me. He gave permission to share it more broadly, so you get to see what you think.

Bill is an experienced teacher who speaks his mind directly and is very free to share what he does and listen to others. Follow him on Twitter at @breenw77.

Post a comment here or tweet him directly with your response.

A recent blogging assignment in my grad class was this: Blog: Your choice. What about your thinking or practice do you want to share with the world at large. This can be a record of something you’ve done, a particular activity to share, work from this course to share, or an opinion piece on an issue of the day (Khan Academy, standardized testing, Michigan education funding, ...)

They wrote interesting pieces, and I love hearing teachers' voices about that which they care most. Four of the teachers have public blogs, so I'll point to theirs with this post. Leave them a comment, encourage them to keep blogging! Those who were blogging on Blackboard, I'll quote more extensively. Post a comment here for them on how they should export and continue their writing!

Bill: Credo about what it takes to teach mathematics. Also see: Bill's take on the 3 Act Story (I'm quite interested in this as a structure. Here's a short urli.st of blogposts I've found on it.)

Eric: Textbooks. "The secret to being a lazy math teacher? Good textbooks!..." Good hook!

Melissa: only assignment she missed all course! Plenty to read at her blog, though. See: Her description of her lesson planning process. Ted: quick take on the public's perception of teachers and Summer Vacation.

Also see: his outstanding concept map for linear intercepts. (Not a blogpost, but wowser.)

Amy: Traditional Teaching
Taking this class has really got me thinking about the way that math is traditionally taught, and the way that I teach math. It has opened my eyes to the importance of teaching students how to problem solve and think critically. This has caused me to feel uncomfortable in my classroom for the past week or so. I feel that I want to make some changes, but yet it seems so overwhelming. The math department I work in is very concentrated on "everybody doing the same thing." This includes assessments and lessons. Also, the time that it takes to develop new tasks for student also is a daunting task.

So, as I reflect about changes I want to make for next year. I am thinking about making small changes. I think my first focus is going to be on formative and summative asssessments. Providing more opportunities for mastery rather than completion. I would also like to incorporate reflection on a daily basis to get students thinking about their problem solving and being able to put their thoughts into words.

In the future, I would like to work towards having more discovery and problem solving activities related to the concepts in my classes.

Erin: the switch from Michigan's High School Content to the Common Core State Standards
I have to complain for a minute about the switch from Michigan standards to the new Common Core standards. I have no problem with switching to common standards, I have no problem with the content of the standards, and I don't even have a problem with standardized tests based on the standards. My problem is this...

We are supposed to begin teaching to the new standards this year with testing based on the new standards to begin in 2 years, however, in the meantime, we are still being tested on the Michigan standards. That doesn't make any sense. The standards really are different in some ways and we are in the process of designing our curriculum based on the new standards. We have to be careful for the next two years that we also teach the HSCEs becuase that will be on the MME.

Someone tell me why we are rolling out the new and testing the old. Is testing really that important that we can't miss 2 years? Perhaps at least we can remove some of the consequences in the meantime so that we can develop a coherent curriculum without fear of the government taking over our schools.

Monica: Differentiation
When I took my Curriculum Development Class, we focused on Understanding by Design and Differentiated Instruction. Prior to that class, I had believed that differentiated instruction was synonymous with individualized instruction. And it wasn’t until the last 4 weeks or so that we started to talk more and more about what DI was, and how to adapt lessons and differentiate them.

First, some things I learned about DI:

Quality differentiation begins with a growth mindset, moves to student-teacher connections, and evolves to community. In a growth mindset students persist in the face of setbacks and see their effort as a pathway to mastery. Students embrace challenges, learn from criticism, and are able to find inspiration and motivation in others’ success. Rather than plateauing with skills and knowledge, students with a growth mindset reach higher levels of achievement.

Quality DI is rooted in meaningful curriculum (not fluff!)

DI is guided by on-going assessment which is used not for grades, but for instructional planning and providing feedback.

and DI addresses students’ readiness, their interest, and their preferred method of learning.

Different methods of differentiating instruction include using

Choice Boards--like a tic tac toe, where you have 9 activities listed, one in each box, and have students choose whichever 3 activities they’d like to do, as long as they make a tic-tac-toe. These activities should be rooted in the same learning objective, but address different learning types (multiple intelligences).

Cubes or Think Dots--cubes would be using a net for a cube and having one question on each of the six sides. Think dots follow the same idea, but the students would roll a die or number cube, and then do the problem underneath the number they rolled on a worksheet (the worksheet would have 1 through 6 on the top, and the 6 problems listed under it—rather than making the cubes, now you just make a worksheet, and use a number cube).

Sternberg’s Tri-Mind--list three different sets of directions to address the same objective. One way of addressing the objective would be analytical, one way would be practical, and another would be creative. There are tests, similar to multiple intelligence tests that students can take to see of the three they prefer.

You can also have two similar worksheets, where one is more advanced and the other is more basic, depending on the level of the student. Students don’t know they have different leveled activities, but this is a great way to address the “just-right” problems for students on a case-by-case basis. You can make more than a basic and advanced, setting up four or five levels, but remember, differentiation is not individualizing.

In this class, I wish we would have learned earlier what differentiation was all about, and how to differentiate activities. This was most beneficial part of the class for me, and I wonder why “stuff like this” hasn’t been around longer. I feel that DI addresses issues that have been around in schools longer than solutions have, and this is something that should be included in all undergraduate teacher prep classes now (which I’m hoping it is)!

Monica's digital decimal differentiation designs are available at Scribd or by email from me. There's quite a bit of work done on tic-tac-toe, tri-mind and cubes.

Closing Thought
Powerful stuff when teachers start sharing on Twitter or writing for sharing.

So why don't you join the conversation, you?

Photo credits: Search Engine People Blog, Cliff1066 @ Flickr

Went to get on the elevator at work, and like a bad stereotype of the absent-minded prof almost said "Oh! Excuse me" to this roll of carpet. I thought, 'I should take a picture of that!' Got my student Ted to pose with it, and I'm asking: any questions?

Ted said, "I thought that's what that was about."

Here are more photos. The label probably has the answer to many questions that could be asked. The original image is on the left, a shot of the edge of the roll, and then the shot of the label. Ted is 6 feet precisely, so perfect for the occasion.

One further thing I'm wondering is that when the carpet is laid out it looks like it's actually in squares instead of from rolls. So now I'm wondering if that's an illusion or ...

In our spring grad class we've spent the week looking at assessment and instruction in the context of quadratics. The pedagogical side of class was spent talking about Skemp, making a assessment concept map on Mindmeister, a rare mindmapping tool that allows realtime collaboration...

...a questioning framework (shared in this old blogpost), a video of a teacher leading a lesson on solving quadratics (from the new to me resource of Inside Mathematics, with videos, coaching, lessons and problems), sharing a variety of articles on assessment, and watching Shawn Cornally's TEDx talk. The assessment articles included one new to me, "Using Assessment for Effective Learning," Clare Lee, Mathematics Teaching, Jan 2001. That led me to two books (that are available online through our library), Language for Learning Mathematics, by Lee, and Learning to Teach Mathematics in the Secondary School, in which she has several chapters. Shawn's talk is worthy of a post of it's own... really has me thinking.

Between the classes, the teachers tried to find quadratic data of their own. One speculated about pulse rates in fight or flight situations (concave down quadratic?) but couldn't find a good set of data. Another found a neat class of problems about water draining in a quadratic pattern. Sadly, my remaining physics doesn't let me know why that would be. One of the students did a great little think aloud using Jing (click for the short video; I don't know how to embed someone else's screencast) with a picture of a record dirt bike jump.

He did a quadratic fit to this using GeoGebra. The picture does not seem to be from perpendicular to the plane of the jump, which raised the nice question: is a parabola seen from askew still a parabola? (The other nice tip he had was to use SmartNotebook for within the screencast... interesting.)

To me, one sign that students are genuinely investigating is that they get to questions that I have to think about or don't know the answer to.

With the focus being on deepening their understanding, I asked the teachers to choose from these investigations, with the addition of the water draining.

1) Galileo Galilei (essentially that means his father was named Galileo, too; so he’s Galileo Jr.) conclusively disproved Aristotle’s idea that heavier things fall faster. He took Aristotle to task for never figuring out a good way to test it. Next he wanted to study how and if speed changed during a fall. But it was too fast for his available tech. So he devised several clever ways to slow it down. One way was to roll a ball down an inclined plane rather than dropping it. (Why would this have the same information as falling?) He did one experiment by building bumps on the ramp and spacing them until the sound of the clicking over them was periodic. That’s hard to replicate in class. But another experiment was to time a rolling ball down a plane, and then find a point where it took half or a quarter of the time. (Read more at http://bit.ly/lwcK1h but after you experiment.) Try that out, take some measurements and discuss what you find. How can you get a variety of data to look for patterns?

2) There’s a famous relationship between quadratics and second differences in the output, if the inputs havea constant difference. Choose a couple of different quadratic functions and experiment, generating and organizing data.

a. What is the relationship?
b. Why is the relationship?
c. How does the relationship connect to either the standard form or vertex form of a parabola?

3) We often care about the roots of quadratic functions. Is it easier to solve for the roots a function in standard or vertex form? Why? What would the quadratic formula look like for vertex form? Could you use that to find or derive the quadratic formula for standard form?

4) Not much for visual learners here. Can you devise a pictorial growth pattern that has a quadratic relationship between input and output? Well probably it’s easy!

But there must be more interesting patterns than that. Can you make a pattern with a ≠ 1? b and c ≠ 0? How would you visualize first and second differences in such a pattern? Would the function for your pattern have roots? What would they mean?

I felt choice was important, especially for experienced students who already knew a lot about quadratic functions. They did an excellent job with their choices, looking for something new to do. Here are some results of their investigations. Afterwards, they came together and shared the results and had a great discourse.

(1) Not to gender stereotype, but the boys were rolling things even before they finished reading the first question. The first attempt was rolling a tennis ball on about a -1/3 slope. They got this data at right. Barely any difference between linear and quadratic. They were surprised by the small size of the a value. They thought it was quite interesting that this was a situation with distance as an independent variable and time as the dependent. (I thought that was cool but hadn't noticed or expected that.) They noticed that if they didn't know to get a quadratic, they probably would have stopped at linear, because r=.985 is clearly good enough.

I asked if they could slow it down at all. So they immediately changed the slope and switched to a hot wheels car that would run truer.

An in between occasion, before (0,0) was included.

This interesting question came up: is averaging the data and finding the regression curve any different from finding the regression curve on all the data? (For this one it turned out the same.) One teacher included (0,0) as a point, which raised the question should the other one using all the data use one (0,0) or three? Should they add (0,0) - because if there was some experimental effect it would not be in a theoretical point like that. How can you tell from the data that the car is speeding up? How did Galileo notice that falling wasn't uniform?

Seems to me that real investigations always raise interesting questions. Canned investigations raise many fewer. The big question came up: how did Galileo know that rolling was essentially like falling? While this was the most dramatic investigation, each of the others raised great questions and connections, too.

(2) Only one student looked into differences, with some consultation. She quickly found a pattern relating the second difference to a. But I put up the following GeoGebra sketch. (File here, or as a webpage.) The teacher checked her work because of the 4.5 2nd difference. She realized it was because of the delta, and set to work finding the relationship between the 2nd difference, a, and delta. Excellent math detective work. We discussed the new applications of difference equations because of computers, and the weird connection with derivatives. They noticed that b and c had nothing to do with the second difference, and how that made sense if the second difference detected quadratic behavior.

(3) This was the one I thought was the least interesting, but the teachers found several cool bits. First of all, they did it by rewriting the vertex form into the standard form, and then used the quadratic formula to get a vertex form quadratic equation. But then they liked it well enough that they wondered why they had never seen it before. It was noticed that the work they did identifying the standard form with the vertex form gave an equation for the x-location of the vertex for the standard form, -b/2a.
I mentioned that I would never have thought of doing it the way they did, and they started wondering what I meant. Another teacher saw a possibility, and solved directly from the vertex form, getting another quadratic equation that was even simpler.

(4) Two groups tried this. They captured their work in photo, so it's probably just best to show.

The teacher on the left was surprised that there was a 3-dimensional way to think about it. The teachers on the right were surprised that there was a non-3-dimensional way to think about it. The rightside teachers liked the idea of colorcoding from the leftside, and made an adjustment.

Left as an exercise for the blog-reader what the function rules are for these sequences. The discussion revolved around trying to make the pattern understandable to a teacher that finds it very hard to visualize. The unanswered question was: what would roots mean in this situation?

Our closing reflection was on what elements of the investigation allowed them to deepen understanding. They brought up the closing discourse as well as collaboration, the manipulatives, the new questions, and trying to collect their own data.

Pretty good lessons to draw, and some quality mathematics.

Yeah! I have rarely been so excited to get an email. "We just pulled your Graphing Story out of the oven!" The only downside is that my name is prominent, when I really just incited other people to make them. But I'm still geeked. Huge props to Dan Meyer and BuzzMath for doing this.

First, the Balloon. Filmmaker - Anna Minnebo, Balloonist - Gregg Minnebo.

Second, The Towers of Hanoi. Stacker/Graphist - Eric Thuemmel; Camera - Monica Leneway. In the original he just got the fourth stack complete on the 15th second, but it just barely gets cut off here. He was quick.

Still to come: four simultaneous tower builders... can understand why the graph for that is a special problem.

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.

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:

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!

Dan Meyer, purveyor of pedagogical principles and master of mathematical memes, has popularized or coined the idea of #anyqs on twitter. A math teacher will post a photo (or video) and ask, "Any Questions?"

Here's my photos:

Post questions in the comments or I'll record any twitter questions I see.