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?

Desmos rocks my world

Last night I took the batteries out of my TI (83 plus!) graphing calculator to use in my bike lights. Sorry not sorry. Priorities.

In case you haven’t already discovered this fantastic resource, there is an online graphing calculator (and so much more) called desmos. Use it once and you will never want to use your old TI again. It’s easy to zoom in and out. It’s IN COLOR. If you are graphing multiple functions, you can make each one a different color. You can create sliders. And it is all FREE. Also be sure to check out all of the beautiful artwork while you’re there.

I won’t be ditching the TI for good because using it is a major part of my school’s current FST (aka the second half of Algebra 2) curriculum, which is fine. Students need to learn how to use TIs because right now they are the accepted technology for tests, both in the classroom and for standardized tests like the ACT and SAT.

Desmos is definitely worth incorporating into the classroom though. I used desmos with great success last summer when teaching summer school, and I think it’s great for doing investigations and creating visuals. That tiny, pixelated TI screen seems rather clunky and out-dated next to desmos, where students can really “see” the graphs and play around with them more easily. When I personally do math, I always use desmos if my laptop is with me. (Being significantly lighter than my laptop, the TI is more likely to be in my backpack on any given day.)

So I plan to use both desmos and the TIs in my classroom this fall. The TIs will be our go-to use-every-day type of calculator, but I’ll pull out the laptops as much as possible to use desmos for graphing investigations. I also hope the kids will come to appreciate desmos and start to use it at home or when they come to the math resource room during study hall. From summer school, I already have desmos investigations made up for quadratic functions and rational functions, but they were made hastily and need some improvement and some updating to more closely match my school’s FST curriculum. I’m excited! Now I just have to do some work and make these plans actually happen.

How do you incorporate desmos into your classroom?

Here are some examples I’ve found on the MTBoS:

Fawn’s Des-man which inspired the desmos team to create this awesome version of the project

Bob Lochel’s Desmos Filing Cabinet

Formative Assessment Brain Dump

Today I enjoyed another get together with my FST (kind of like the 2nd half of an Algebra 2 class) co-worker. We had a good discussion on formative assessment and how to grade it, and many thoughts and ideas are still running through my mind, but here’s a brain dump of where I am so far.

School policy requires each class to grade 25% on “Effort” and 75% on “Knowledge and Skills”. Pretty much as a whole, the math department uses homework and quizlets (your typical formative assessment short quiz) to make up that 25% Effort grade.

Last year my co-worker graded homework for completion and quizlets for correctness, but he was having trouble reconciling the fact that something graded for correctness was going into the “Effort” grade. So he proposed that the kids take the quizlets like normal, he grades them like normal, but if the kids make corrections then they get 100%.

At first, I didn’t really like the idea, but now as I’m typing this I’m kind of warming up to it. Well, let me back up. First, I’ve done lots of reading on Standards-Based Grading and am intrigued by it, so I personally don’t really like the 25% Effort thing in general, but I have to accept it and move on. Likewise, I don’t really want to bother grading homework. I want great math to happen during class so that there’s no need to dole out the typical “page 155 #1-27 odd” homework assignments. If I feel like the kids need more practice (or if some individual students ask for it), then I can give some homework problems, but otherwise I’m not really interested in seeing a bunch of kids copy off each other every day just to get their completion grade.

But anyway, my co-worker and I want to basically have the same set up because our kids get shuffled at semester, so I’ll play along with grading homework for completion. No big deal.

Now, regarding the quizlets, like I said, at first I didn’t like the idea of kids blowing off their quizlets and then copying the correct answers for 100%. So I told my co-worker that although their effort grade will be higher from that easy 100%, I worry that their actual effort will go down because they won’t really care about being prepared for the quizlet if they know they can just correct it and get 100%. Basically, I want them to take the quizlets seriously, and I worry that they won’t if they know they can just correct it.

But, as I’m typing this, I am opening up to the idea. If I can create the expectation that they come prepared for the quizlet, and I continually emphasize it’s importance as an indicator of what they know and don’t know, then it’s very possible that they will take the quizlet seriously despite the “easy” grading of it. In fact, maybe the “easy” grading of it will take some pressure off of them and really encourage them to make corrections and learn from their mistakes. And that’s the most important thing about formative assessment, right? If they can identify their errors and learn from their mistakes then they’re doing exactly what I want, so why not give them 100%?

Grading is weird.