Monthly Archives: May 2015

Letter to the Arizona Superintendent

[The Arizona Superintendent is going to be in town tonight doing a town-hall sort of event to hear feedback from the public on issues in education. Her office is also accepting feedback online during this time, so I decided to write in regarding the standards in Arizona, which I know she is against. I would also like to attend the event tonight, but I have no desire to speak up. I thought writing in was my best option to make my opinion known.]

[We adopted the Common Core Standards when they were rolled out, and then changed to the Arizona College and Career Readiness Standards, which are basically the CCSS with a different name and a 4th year of high school math standards added.]

I am a high school math teacher and I am writing in favor of the AZCCRS or Common Core State Standards. I have not been a teacher long, so I was completing my teacher certification courses when the standards were in the process of being accepted and implemented by states across the country. I attended a 3-day workshop being led by one of the people responsible for writing the math standards – a mathematics professor who used to be the head of the department at the University of Arizona. He emphasized that there are a few big advantages to a nationwide set of standards, with which I agree wholeheartedly.

First, for students who transfer between schools, a nationwide set of standards means that we can know more reliably what these students have learned in previous grade levels, even if they did not attend schools in the same state. As teachers, it was extremely difficult for us to work with students from other states, as we had to take into account some pretty significant differences between these learning expectations.

(I do realize that standards and curriculum are not the same thing, and I am not in favor of a nationwide curriculum. I want to maintain the freedom to make the best choices in my classroom for my students. I also want us to raise the bar for students in Arizona, which I believe these standards do.)

Second, a nationwide set of standards like this actually takes some of the power (and funding) away from the textbook companies. At my school, we are moving away from using textbooks altogether. We can do this because we can now network with people all across the country to get ideas and resources for lessons for every standard. The pool of people we can collaborate with is now so much wider than it was with individual state standards.

As far as the standards themselves go, I do not believe they are perfect. However, I do believe they are a significant improvement over what we had before, especially in the state of Arizona. The standards have been written in such a way that we need to re-define what mathematics education looks like at all schooling levels – a change that has been a long time coming. We can no longer teach kids “rules” and “tricks” at the elementary level and allow them to rely on those to get them through high school. We must now focus on teaching for understanding, not just memorization. The CCSSM were written with a focus on coherence – the idea that math is not a series of rules to be memorized, but a logical body of interconnected ideas. As we turn our attention away from memorization and toward a focus on true understanding of mathematical ideas, we are really teaching our students about the beauty of the entire discipline of mathematics. And this focus on understanding is what will allow our students to truly be successful in upper-level math courses.

If students do not understand math and just memorize it, eventually they will reach a point in their education where that memorization just can’t help them anymore. For many students, this happens in high school. For some, it happens in their early college courses. It is inevitable – if students do not understand the relationships between ideas in math, eventually they will reach a point where they cannot go further. The CCSSM have actually been written with attention to this coherence between ideas and grade levels so that teachers will have to teach differently, which in turn will help us enable our students to be more successful.

Furthermore, the CCSSM include the 8 Standards for Mathematical Practice, which are a complete game-changer for math teachers. We now have a specific list of habits of mind that allow students to be successful in mathematics. We know what to look for in our students, and we can speak specifically about what they can improve. These habits of mind are arguably more important than the content standards. In fact, the Mathematical Practices are not just math skills – they are life skills. Perseverance in problem-solving, defending your own reasoning and evaluating that of others, being precise in your communication, analyzing and using structures and patterns – these are things that people do every day in their adult lives, and they are valuable skills to teach our students.

Do I believe that the Common Core State Standards (or AZCCRS) are the silver bullet that will fix our educational system? Of course not. The troubles that plague our educational system in this country are legion and to assume a single thing will fix them is infantile at best.

Do I believe that keeping the AZCCRS is a step in the right direction? Yes. We need to raise our standards for our students in Arizona. We need to make sure we are working toward the goal that our students are on the same level as those in the top states. We need a set of standards that is written with an intentional focus on coherence and rigor.

Furthermore, after spending the last few years implementing the AZCCRS, which are based on the Common Core Standards, it would be incredibly frustrating for us as teachers to be told that we are changing the standards again, not to mention expensive for the state of Arizona. We are already burdened by low salaries, high student-teacher ratios, and increasing levels of frustration and burnout at all levels. Please do not subject us to another round of change so soon. Please respect our need for a framework in which we can do our jobs, and let us do them.

Please do not do away with the AZCCRS. Give us time to learn how to teach them effectively, give our students time to learn and understand math concepts in way they have never been asked to before, and evaluate the effectiveness of the standards a few years down the road. No change happens overnight, and certainly not when it comes to education. Please give us a chance.

Algebra Art Project

This year, my district is using a new curriculum program that has scheduled time for project-based learning at the end of each semester. During the fall semester, we didn’t have time for a project in Algebra 1 (we barely made it through the major standards on the curriculum calendar) and we didn’t really like the projects they had uploaded anyway. We wanted more creativity in our project, instead of having all our students do the same assignment and get the same answer. (That’s boring. Also one of the Algebra 1 teachers has 4 sections of Algebra 1 and would have to grade 100 of them. Awful.)

This semester, we found that we were going to have a couple of weeks at the end of the semester between the last standards and preparing for finals, so we decided we really did want to do a project. In the fall, we had seen an idea for a project in the NCTM Mathematics Teacher journal (two months in a row – apparently it’s the popular thing right now) that we thought would be perfect. The basic concept is that the students use graphs to draw a picture. This is a nice low-floor/high-ceiling activity, as students can use graphs whose shapes they are already familiar with or do additional research to learn about new types of graphs. They can also just type in a parent function and then play with the parameters until their graph looks like what they wanted, or they can use the algebra skills they’ve learned this year to write the equation they need. (Hint: no one chose the second option. I will say that I was ok with this, as I really wanted them to learn about transformations on graphs by playing with them, which is really the best way to make those connections.)

This semester in Algebra 1, our students have learned about linear and quadratic functions. When we taught quadratics, we did so with a very heavy emphasis on the relationship between the equation and the graph, so our students had a very solid foundation in graphing parabolas. I wish that we had emphasized graphing during the first semester when we were learning about linear functions as well. I know my own function fluency has come about largely because I’m so comfortable relating functions and their graphs, and I wish my students had that knowledge to fall back on. Things to consider for next year, I guess.

Introduction Activity

We created a worksheet for students to work through in the computer lab for about 2-3 days to remind them of how transformations on the different functions work. We also made sure to include circles so that students could have the shape to use in their artwork. I figured it would be really hard to make a lot of their ideas without circles, which turned out to be even more true than I expected.

Our introduction was very long, and included a “reference sheet” for students to fill out so they would have the information about all the transformations in one place. We wanted to make it shorter, because it really did seem repetitive and took longer than the time I had booked in the computer lab, but I felt like some of our students would need to go through and analyze each transformation separately. I did have a few students who skipped to the reference sheet and used sliders in Desmos to see what was happening under each transformation. These students tended to be the ones that I knew I could trust to do that and still make the connections between the numbers and the effect on the graph (or were working with a student who would help them see this).

Download our introduction sheet and reference page: Desmos Intro Exploration I’m not really very happy with this, as I said, but feel free to make changes to suit your needs.

Project Guidelines

We gave the students 3 requirements:

  • Your drawing must contain at least 12 equations.
  • Your drawing must use at least 3 different types of functions.
  • Your drawing must be creative, artistic, and original.

(That last one is another reason we taught them how to graph circles – lines and parabolas only give you two types of functions. Also, yes, I realize that a circle is not a function.)

We required the students to tell us what they planned on drawing before their first day in the computer lab to work. This was partly to prevent them from using a graph someone else had created (my rationale is that I made them commit to an idea before they had a chance to look online at what other people had created) and partly just because I wanted to make sure my students had a starting point for their first day. I was gone that day, so I didn’t want them telling the substitute that they didn’t know what they were doing.

We intentionally left our assignment very open-ended. Again, this allows us to make sure there was a low point of entry for the assignment, but also allows students to take the project as far as they wanted. We assumed we would have some students who would do the bare minimum and some who would just go crazy with it, and we were right.


I spent two days in the computer lab doing the introduction pages linked above. My students didn’t finish and could have used another day on this activity. I finally told most of them to skip to the reference sheet, as that was more important anyway. None of them got to the last page where they were supposed to make a face.

The next day the class was back in my classroom where we went through the reference sheet as a class so I could make sure they had all the information they needed. I also made them commit to an idea on this day.

The fourth day the students were in the computer lab ready to start working, and I wasn’t there. A couple of students finished the intro pages instead of working on their projects, but most seemed to get started ok.

The next week, I gave them 3 more days in the computer lab (Monday/Wednesday/Friday), so they had 4 total days that were intended to be dedicated to the project itself. I was also really glad I broke this time up instead of doing 3 straight days – they needed that time to process and troubleshoot before getting back into the lab. (The other 2 days we reviewed factoring to prepare for the final exam, so it really helped to break up the review days as well.)


Now I get to show you all the pretty graphs. Let’s be honest, that’s really the whole point of this blog post.

My students know me well – one student graphed a TARDIS.

I also ended up with 3 different versions of Captain America’s shield.

One of the other Algebra teachers shared her student’s submission with me because she knew I would appreciate it.

This student got to learn about sine curves to make her stem.

This student took the gold medal for number of equations – 45. Not to mention this teddy bear is adorable.

Hurdles to Overcome

We had a few issues related to getting Desmos to do what we wanted it to. Shading concentric circles different colors proved to be an issue that is far beyond the capabilities of students in Algebra 1 to solve, and with the number of students drawing Captain America shields, well, let’s just say it was a problem. I had trouble figuring out how to make it happen, and had to ask the Desmos staff for help (on Twitter). They responded with assistance, and stated that “For art purposes we would like to make that a bit easier.” I just think it’s awesome that they consider art to be such an important component of their platform. Who says you can’t be creative in math?!

The Best Part

I’ve spent 10 months working with these students. When they don’t get something right away. I know how quickly they shut down and talk to their friends instead of continuing to try. I fully expected to spend every day in the computer lab watching them chat instead of work because they “don’t get it”. I knew going into this assignment that I was going to refuse to be helpful, and they were going to hate it. For the day that I was gone, I even left my sub specific instructions NOT to attempt to help them, and not to listen to their complaining.

This didn’t happen. (I mean, sure, nearly every student needed help at some point, but it wasn’t constant, and they didn’t whine.) I think they helped each other a lot. When they did ask me questions, they were specific questions – “How do I get this line to [blah blah]?” or “I want this parabola to look like this [pointing]. How do I do that?” or “I need to put hair on this face. How can I do that?”

Seriously, I cannot even begin to tell you how much time I have spent trying to convince them that specific questions are the best thing ever. I’m also a huge fan of pointing at the coordinate plane to show me what you mean. If you can visualize it in your head and explain it in words, I can help you translate it into math notation.

They asked each other, they used Google, they pulled up other people’s graphs and used them as a model to figure out what was wrong with their own equations, they used their reference sheets (not as much as I would like, but they did)… When they did finally call me over, we would talk through what they wanted to accomplish, and how to notate it. I asked a lot of questions and tried to avoid giving answers. I would give them a parent function, say, “Change these numbers until you get it where you want it,” and walk away. And they did it. They figured it out. I am so impressed by and proud of the level of perseverance that I saw in the computer lab over that week.

Even better, the results of this project have been so much fun to grade. Math teachers don’t get to grade art projects very often.