Collaborative Teaching (10:00 - 10:50 and 11:00 - 11:50)
Powerpoint File: 6-4-15 Aaberg Collaborative Teaching
May the Odds Be Ever in the Back of Your Textbook (12:40 - 1:30 and 1:40 - 2:30 and 2:40 - 3:30)
Powerpoint File: 6-4-15 Odds Be Ever
Check out my magnum opus for the 2014-2015 school year, the summary of our special project trip to Omaha and Lincoln with 6 high school students.
On January 14, our math department participated in our second set of instructional rounds at SHS. We observed an applied communications (language arts) classroom, a marketing classroom, and a freshman English classroom being taught as a class within a class with an English teacher and a Special Education teacher exemplifying what strong team teaching could be.
The conversation after this set of rounds really got at the heart of what we are trying to do - take ideas from other teachers that we can incorporate into our own teaching practice. SHS Instr Rds Participant Walkthrough Form Version 2_0. When we visited classrooms on our first set of rounds in December, there was not sufficient space on the form we constructed for comments or to write what the observed teacher had up for objectives or a lesson agenda.
After this second set of rounds, it is also evident that we need to do rounds within our own department. We wish to examine the various aspects of lesson development. In particular, our department members want to see how a set of objectives is posed, how the selection of examples and tasks leads or does not lead to student engagement, and how an activity develops for an extended period of time. In our department of eight teachers, four of us have common planning time on our 'off' day, the day where we do not have our department common planning. For example, we have department common planning time for 90 minutes every Wednesday and Friday. Four of our eight math teachers, by pure luck of the master teaching schedule in our building, have 90 minutes of common planning time Tuesday and Thursday each week.
My next task is to re-tool our current form so it is useful for math teachers observing other math teachers. I need to make some adjustments that allow for us to carefully examine the mathematics and the teaching being observed. It is also important that all this fits on a single page. I am hoping to gain some ideas while in Phoenix with the other 2015 State Teachers of the Year this coming week.
I would love to get some feedback on our second version of the form we use while on instructional rounds at our school. To see the PDF of our form, please click the following link.
So far we have only had one set of formal instructional rounds. Our math department has common planning time as an administrative support from our administrators and counseling department, something for which we are eternally grateful. Our department went to visit two language arts classrooms and one science classroom during our first instructional rounds session. Below is a scanned image of the revisions made to the original form as a result of our conversation after rounds took place.
I collaborated with a social studies teacher at our alternative school while constructing the form. We drew from the works of both Charlotte Danielson and Robert Marzano. Our math department has spent some time together in our PLC (Professional Learning Community) meetings revising the form prior to its initial use.
What we have found with both novice and experienced teachers observing expert teachers is that it is really, really easy to be dazzled and get lost in the instruction and action in the classroom. Teachers often forget the reason why they came in the room in the first place. Filling out the form has three purposes:
Here's some background information about us. Our math department has nine teachers. One of these teachers spends her day at our alternative high school. The other eight of us are in the same building. One of our eight in the building is on a different floor, but roughly speaking, we are geographically located in the same area in our school building.
We have worked really hard to establish a culture that engages in cross observation on a consistent basis. Our math teachers know that, to improve their own instruction, they must learn from the instruction of others. For a year and a half, our department members have spent portions of class periods observing their peers in the act of teaching about once every two to three weeks. The frequency of the observations usually depends on how busy the teachers are, time of year, etc. But our conversations are always positive and lead back to supporting one another.
I have spent a lot of time visiting with Angela Mosier at Omaha Westside and followed the example set by Kristi Bundy at Ashland-Greenwood within our own state (Nebraska). They have established great cultures within their schools using this as an in-house professional development strategy. After observing the teaching of others, we send out a "positive blast" email to the observed teachers. This email highlights the positive actions and learning we observed in the classroom. Everyone involved learns something about teaching as well as learning in the classroom.
Our goal at SHS is to participate in instructional rounds on a once per month basis this spring.
Here's a video that explains different types of collaborative structures in a middle school setting. Administrative support is essential to creating a culture of collaboration and trust. Cross observation is featured as a professional development tool at the [2:05] time signature.
This post contains resources for the talk "Using Statistics in Mathematics Classes" given by Jason Vitosh (Falls City High School, Falls City, NE) and myself at the Midwest Regional Noyce Conference on Thursday, October 2 from 2:15 pm - 3:00 pm.
Click on the link below to access the presentation file containing resources, images, and links.
My elective math class has all grade levels represented within it. While not ideal, this is a feature of our scheduling system. So I have freshmen in Algebra I all the way up to seniors that have already completed Calculus AB. This poses a huge classroom differentiation challenge each Monday, Wednesday and Friday we hold class. Here is an instructional strategy I use that gets students writing about the mathematics they do in class.
Students worked on one of four things in the computer lab last Friday.
1. Construct 2013 Probe I Problem 7 diagram (2D)
2. Construct 2013 Probe I Problem 23 diagram (2D)
3. Construct 2013 Probe I Problem 11 diagram (3D)
4. Work on Alcumus problems independently
After we spent approximately 55 minutes in the lab, we returned to my classroom for a writing activity.
Here is the writing prompt I put on the board:
(Writing exercise on a separate sheet of paper to turn in to me)
Think about what you learned about the diagram or diagram(s) you built in Geogebra. Write a letter to the you of October 12. How did building diagrams in Geogebra help you understand the problem better?
I put eight minutes on the clock and informed students they would need to continuously write for the eight minute timeframe. Here are some samples of student writing from the activity. For convenience, I have inserted another copy of the problem. Immediately below each problem appears student writing associated with the problem.
The Students revealed their thinking about these math problems throughout their writing. While some students chose to concentrate on the construction process in Geogebra, others also revealed some of the mathematical structure they encountered while making the diagrams.
Writing samples from students that worked on Alcumus:
These writing samples revealed to me the depth of student thinking going on in the classroom. If I could have a superpower, I would be a mind reader. Then I wouldn't have to guess at what my students are thinking. Having the students write for an extended period of time gives me insight into how they are seeing the mathematics and gives me ideas on how I can help further their understanding and guide them as they struggle.
I collected these writings immediately after students completed them. I ran the pages through my ScanSnap scanner and converted them to a PDF for me to review later. I told students we would get these writings back out in a month's time, emphasizing the need for specificity on what they were working on and what they learned that day.
Going forward, these writings help me be more efficient with respect to differentiating classroom instruction. We don't need to be working on the exact same thing at the exact same time at the exact same pace for the students to engage in meaningful problem solving.
Here is a prime example why Twitter is a great collaboration resource for math teachers. Last night I was killing time while waiting for a haircut. Reading through some tweets, I noticed a chat going on with the hashtag #ggbchat. With some luck, I caught the very end of the session and posted a question about something that's been bugging me about Geogebra.
I use Geogebra to analyze student summative assessment data in my classes. I like to sort the data to guide me when deciding which students I should group together for class activities. Sorting the data inside a Geogebra spreadsheet would eliminate an extra step for me (specifically, entering the data into Excel and sorting prior to copying & pasting data into Geogebra). I would think the software should allow a user to select a list, right-click, and then be given the option to sort the column of data. Here's a solution to the problem I faced, compliments of Geogebra guru John Golden (@mathhombre).
John also followed up with an idea to create a "Sort" button using a script.
Here is a screencapture of John's suggestion for making a Sort button.
When we teach children to add, subtract, and multiply, we commonly see a vertical format used in school.
Then, the student learns division. Students usually see the division algorithm represented horizontally.
A student may not, beyond a worksheet with dozens of problems, be comfortable with a vertical format for division.
It almost feels strange to me to the type the problem vertically. Then comes fractions. We did the following exercise in a workshop I led today. I wanted participants to consider the question, “Does the size of a fraction bar matter?”
Below is an image of the problems we worked on in groups. Calculators were allowed. Spirited discussion and abundant disagreement followed.
As the back of the textbook often says, “Answers will vary.” The participants’ responses, split up by problem, appear below.
How can we resolve this conflict? Let’s type it into a reliable computation source, like Wolfram Alpha. It depends on how we type the input.
Above is one way to type the input. Below is another way to type the input.
Why the difference? Because multiplication is commutative, but division is not. 3 x 4 is the same as 4 x 3. But 3 divided by 4 does not yield the same result as 4 divided by 3.
In the original input, where I typed 3/4/1/2, the computer interpreted this as the product of 3 and the multiplicative inverses of 4, 1, and 2, like this:
In the second example, we see the computer associates the fractions ¾ and ½, treating the middle fraction bar as a grouping symbol.
The confusion for me stems from converting between vertical format and horizontal format. We teach students to follow order of operations, to divide in a horizontal statement as they encounter it from left to right. In school mathematics, we commonly write the fractions vertically when we mean to take the ratio between two ratios, specifically ¾ and ½ in this case.
Here is a state assessment practice item, taken from the Nebraska Department of Education website, that demonstrates this understanding of the vertical format when dividing fractions.
Let’s look at this Word document, specifically this problem. The author of the item intends for students to invert 7/3 and multiply. I worry a student may see this problem in a horizontal format as the work shown to the left, which gives a result which is obviously not an available choice, but if we are examining item reliability, I wonder if we should have the conversation about whether this expression should be typed in the following way:
Again, I’m just posing the question about the test item. I want to be careful in how I represent problems with my students. I want my own students to understand the context matters. I want my students to be procedurally fluent with fractions when they solve problems in the world. The conversation in this morning's workshop was great. Participants reflected on how what they say to their students while they teach the students can have unintended consequences on how students view procedures with fractions.
Here’s a link for a great blog entry on this topic: