Always, Sometimes, Never

I debated some Always, Sometimes, Never statements with my Geometry kids today. In groups, they had to choose the word that they thought went in the blank, as well as draw a picture to explain their choice.

Some example statements (taken straight out of our textbook):
Two planes ________ intersect in a line.
Lines ________ have endpoints.
Lines that are not parallel ________ intersect.
Two points _________ determine a line.

That last one created some interesting discussions, particularly in my last period. Many students wanted to put Sometimes in the blank. I didn’t look at the textbook’s answers, but I assume the authors wanted Always in the blank.

Why did so many students think Sometimes? Well, I think the statement was kind of confusing to them. What does it mean to “determine” a line? Does “a” line mean one line or does it many any line? I tried to resolve the matter by putting two random dots on the board and drawing a line through them. “Look, I can draw a line connecting any two points.” Not particularly convincing.

The students then told me to draw a line going through each of the points (parallel lines, for example). “See,” they told me, “there’s two lines, not a line.” I didn’t really know how to respond to that. I told them yes, I can draw different lines through each point, but only one line will connect them.

Well, I think I convinced them that any two points could be connected with a line, but we just left the Always, Sometimes, Never question unanswered. Which is okay. Of course, some kids insisted, “But what’s the answer?” and I replied, “Well, I think it’s Always, but I don’t think it’s totally clear.”

Perhaps the answer would have been less ambiguous if the original statement was Two points can _________ be connected with a line. But that statement seems way less powerful. So now I am intrigued by the word “determine”. I definitely think it’s important. It’s hard to explain to the kids what is meant by “determine” though.

One instructional difference I would have made during the activity was to require new people to be the writer and the speaker for each statement. In a couple groups, it was very obvious that two or three students were doing all the work while the others checked out, so some sort of rotation would have been smart.

I want to start the next class by playing Sarah Rubin’s Draw It game because some of the drawings I saw today were definitely off the mark, but that’s okay. Visualizing lines and planes and space can be tricky. I love seeing their eyes widen when they begin to “see” it.

High and Lows (Mostly Highs!) of the First Two Weeks

I survived my first two weeks as a high school math teacher! So many things have been running through my mind, but right now I’m going to make a list of things that are going well and things that need improvement. I just want to get it all out. I hope to blog more regularly from now on!

Things that are going well

• I love my school. It’s so, so, so good. My colleagues are incredibly supportive and amazingly talented. Our students care about their school and each other. I am very fortunate to be part of such a strong community.
• There are some very effective school-wide policies in place that administrators, teachers, and students are all on the same page about. I feel like this really promotes school pride and diminishes behavior problems.
• My Geometry and FST students are awesome kids. I am so impressed by them.
• Creating a classroom that values mistake making. This is a work in progress, but I’ve got a decent start.
• Establishing a classroom community where the kids feel comfortable talking to each other. Seniors are good with this (too good, actually), and I’m still working on Geometry kids.
• I have established some classroom routines! Phew. Thank you Andrew Stadel for Estimation180. It’s been a great way to start class every day. Similarly, ending class with an exit ticket lets students know that we work until the bell, as well as provides me with some great feedback.
• Using whiteboards (both big and small) has been an effective way to get students to share their thinking and to just get some students to write something down.
• I’ve done some deep activities, problems, tasks, or whatever you wanna call ’em that have produced good results.
  • Fawn’s Foxy Fives for Order of Operations
  • Visual Patterns
  • MAP “Interpreting Algebraic Expressions” formative assessment
  • GCF Speed Dating
  • Factor Craze (This was particularly powerful. I should write a blog post about it.)
  • Not a “deep” activity necessarily, but doing a stations review for the Geometry test was nice. Isolating specific skills seemed helpful to students. Gave them a clearer picture of what they knew or not.
• I am learning every day.
• I am finding time to exercise and cook dinner. (Sleep is another matter. Looks like I might pick up drinking coffee again…)

Thank you to all the inspiring teachers who share their wonderful ideas and activities so that I can use them. I stand on the shoulders of giants.

Things to improve

• Classroom management. Can you tell I’m a first year teacher?
• Similar to the first point, I struggle with engaging every student when I’m talking to the whole class. Group work is my strength: students discussing with each other with me floating around from group to group asking questions and guiding them along. In contrast, I feel like I’m not strong enough at whole-class lecturing and encouraging note-taking. I think I just need to be more strict about it. No talking when I’m talking. Pick up a pencil and write something down.
• Kids who are absent. And the kids who are just now switching into my class. How can I get them up to speed?
• Checking homework and going over answers. What a big ol’ unproductive time sink.
• Better hand-writing. I save my Smart Notebook documents and upload them to my class website for students to use as a reference. Neater hand-writing would be easier for kids to read and follow.

Have a lovely weekend, everyone! Here’s to a great year!

Exit Tickets and Group Work Norms

Ever experience kids packing up 5 minutes before the bell? I definitely did during student teaching, but half-way through, with support from my cooperating teacher and help from another math teacher, I decided to implement exit tickets. It definitely made a difference. Kids worked until the end of class when I handed out their exit ticket sheets and put the prompt up on the Smartboard. It was great. The other math teacher who did exit tickets used a weekly sheet with a space for each day’s answer, so dutifully followed her lead, but I didn’t really enjoy keeping track of the sheets for a whole week and it was a pain to pass them out at the end of each class. The kids complained that it took too long to get their sheets back so they didn’t have enough time to answer the question.

So anyway, this year I plan to just have a bunch of half-sheets of paper printed out for exit tickets. I am wasting more paper this way, which is a concern of mine, but it’ll have to do for now. On the back of the half-sheet there is a participation reflection. I’m  focusing on group work and creating healthy math culture in my classroom this year, so I want to remind the kids of our group work norms every day, and I want them to do some reflection on the day, hence the three questions on the back of the exit ticket.

I really like all of the norms I’ve decided to use, but unfortunately there are twelve of them, which is probably too many. I should try to shorten the list, but I don’t know which ones to give up. They’re all important to me!

Well, here’s the file with the exit ticket on the front and the reflection on the back. Nothing fancy, but check it out, and I’m interested in hearing your thoughts. Do you use exit tickets? How do you implement them, and what do you like about them? What classroom norms do you use? How do you get your students to think about your norms?

(I don’t have any word processing software on my computer, so I just use google docs for everything, but in the process of uploading my documents to scribd, the spacing gets a little weird, but you should still get the idea.)

View this document on Scribd

A Favorite Math Lesson Resource

I want to share one of my favorite resources for rich math lessons: the Mathematics Assessment Project, which is a collaboration between the University of Nottingham and UC Berkeley. I love the structure of their Formative Assessment Lessons. According to the website, these lessons are designed “to reveal and develop students’ conceptions, and misconceptions, of significant mathematical ideas and how these connect to their other knowledge”, as well as “to assess and develop students’ capacity to apply their mathematics flexibly to non-routine unstructured problems, both from the real world and within pure mathematics”.

When I taught middle school summer school, I used a couple of these lessons with great success. I love how they include matching and sorting activities that require the students to come up with their own mathematical justifications. When I tried these out in summer school, I had small classes and had the students do it individually, but this year I am definitely going to follow their suggested structure of using small groups because I want to see more mathematical discourse among the kids. I also want to do a better job of creating the final product and displaying the work on posters. In my summer school class, once the kids finished sorting they would just toss the cards in the trash (or worse, leave them scattered on tables or the floor for me to clean up). So this year I will definitely use the small group format and require the posters, which will hopefully get more students to explain their thoughts and reach a deeper understanding.

Conveniently, I was able to find a MAP lesson for each of the first four units in my Functions, Statistics, and Trigonometry (FST) class. (Students at my school have the option of the Geom–>Advanced Algebra–>Pre-Calc route, or the Geom–>Algebra 2–>FST route. So the curriculum for this class is comparable to the second half of an Algebra 2 class, I think. First year teacher here, so I can’t comment too much on “typical” math curriculums.)

Unit 1 – Our curriculum calls this first unit “Symbolic Manipulation” which is a decent description for it. It’s basically a review of Algebra topics they’ve previously encountered, such as the distributive property, combining like terms, and factoring. The MAP lesson I plan to use is Interpreting Algebraic Expressions. I like how this lesson helps students distinguish between expressions like (5n)^2 and 5n^2, and it also uses area diagrams that will come up again when I’m teaching factoring. I plan to try this lesson right away the first week of school, and it might be a fairly simple for this class content-wise, but that’s okay because it will also serve the purpose of introducing the students to this type of group work and what it’s like to share their thinking, which will probably be a very new experience for them.

Unit 2 – This one is all-things quadratics. The MAP lesson I want to use is Forming Quadratics. This is a great matching activity, and I like how it incorporates all of the different forms for quadratic functions and helps the students identify key features of the graphs.

Unit 3 – Polynomials. Not my favorite unit (I find long division, rational root theorem, etc to be somewhat tedious), but I can’t let that show. I think I will definitely emphasize SMP #6 – Make sense of problems and persevere in solving them during this unit. The MAP lesson here is Representing Polynomials, which is another great matching activity which connects what students already know about finding zeros to 3rd-degree polynomials and their graphs. The extension activity is also intriguing.

Unit 4 – Functions and Function Transformations. There are two awesome MAP lessons that could be incorporated here. The first one, Interpreting Distance-Time Graphs, was the one I used with my summer school class last summer (8th graders who would be taking Algebra 2 in the fall… so a very different group than the one I’m talking about now). Perhaps this one would be too simple for my FST kids, so the second option, Functions and Everyday Situations, might be a better fit. I like the open-endedness of Distance-Time, but I like how Everyday Situations has students translating between the algebra and the graph.

These lessons will be great formative assessments, and  I’m glad I can incorporate one into each unit. Hopefully you got a chance to check out the MAP website and lessons if you aren’t already using this great resource. If you’ve used these in your classroom before, what do you like about them? What about them is difficult to implement? Thanks in advance!

Creating math culture

Lately, I’ve been thinking about classroom culture, specifically math classroom culture. I want to create a culture of asking questions, learning from mistakes, sharing ideas, and justifying answers. Basically, I want my students to participate in rich mathematical discourse, and to do this, I am going to have to create an environment in which they feel safe enough to contribute.

One thing I tried while student teaching was to create a class mission statement. I had the students brainstorm ideas about what our class’s goals and purposes should be, and then we came up with some sort of summary to be the class mission. I didn’t do a great job with this activity and never referred to the mission statement again, but it was a small step in the right direction.

My cooperating teacher had group work norms posted on the wall, and I really liked that idea. I think they could be especially effective if we kept returning to them as class. My cooperating teacher suggested picking one to focus on every day, so I would announce one at the start of group work and tell the kids to keep it in mind as they worked. I didn’t do it every day, but I thought it helped the days when I did it. When I worked as an AVID tutor, I also began each tutorial session with an expectation.

So one thing I definitely want to put on my walls this year is a set of classroom norms. Some really good ones from my cooperating teacher’s wall were: “Reach consensus, not majority rule”, “Ask your group before you ask your teacher”, “Criticize ideas, not people”, and “Ask for reasons, not answers”. I’m definitely going to use those. They are similar to the ones Ilana Horn just posted on her awesome blog, Teaching Math Culture. Sarah Hagan at Math Equals Love also just posted the pdfs of the classroom norms she’s going to put on her wall, so check those out.

It’s also important to keep referring back to the norms. They aren’t just going to magically become part of my classroom, so, as I mentioned earlier, I want to pick one to have the kids focus on each day. We’ll see how that goes. If I’m really good, I might figure out some way to get the kids to reflect at the end of class on how they did with that norm.

How do you create classroom culture? Do you use norms? Which ones have worked in your classroom?