How do teachers make sense of this stuff that gets designed by other people and use it for their own purposes?
I usually take a quick look at the resource and think about if and where it may fit. I may make a note to myself to consider including it when I reach the appropriate part of my school year, and put together a unit plan, then I’ll return to it and consider its appropriateness with my students to achieve a specific learning goal.
There is an abundance of resources available online of varying qualities. I usually do not jump into and investigate a new thing just because somebody pinned it or shared it on Twitter, but I am always afraid I may be missing out on something awesome. I’ve learned that if it is awesome, I’ll continue hearing about it over time and be able to benefit from learning about its implementation in other classes
I am most likely to use a resource either because I learned to know and trust the source, understand the approach and can adapt it to my style and the needs of my students (Desmos activitiesShell Centre, Illustrative mathematics) or, because I read blog posts from other teachers discussing how the resource fits in their lesson the strengths and weaknesses of the lesson, what they plan to do next.
After developing a list of learning targets for the school year and developing a high level pacing plan, I think about how students make sense of the content, and what content they already know that I can relate it to or build upon. I then sequence the learning targets for the unit. Next, I find, adapt or develop a synthesis project for the end of the unit that incorporates as much of the learning targets as possible from the unit and hopefully also draws in learning targets from previous units & grades too. Rolling cups or spiky door, for example.
After that, I look for hooks. I try to create conditions where students ask me to help them improve at the learning targets I have planned. I try to create a need to learn the thing. After that, I find, adapt, or develop application lessons where they can apply the learning targets as a culminating activity after developing basic understanding of a target – 3 acts and Mathalicious lessons fit well here.
I then lay it all out in a calendar incorporating days for direct instruction, notes, basic practice, and standards based assessments where they are appropriate. I usually build in a few unplanned days in order to allow for some flexibility throughout the unit to make space to dive into teachable moments that may arise.
I have been trying to use CPM’s new Precalculus textbook as a guide for my pre-calculus class. During my planing period, I opened to the next lesson to try to put together a plan for my upcoming class.
I only had about one hour to prepare for this class period.
This was the opening prompt:
Gerrit wants to simplify the complex fraction
but is overwhelmed by the fractions within a fraction. Work with your team to help Gerrit write an equivalent expression that is a rational expression instead of a complex fraction. Be ready to share your strategies with the class.
I stopped to think about how this would work with my class:
I thought back to when I addressed this topic last year – and how poorly it went. Students were frantically writing steps to memorize procedures. Although I am experienced enough to avoid saying “copy-dot-flip” or “invert & multiply“, I know this is how my students learned to divide fractions and it is how they approached these problems.
I decided I did not want to teach this the same way this time.
I considered just doing the lesson as described by CPM because I really didn’t have much time to plan and it was good enough. They are HS seniors. I can’t change their perspective on math and learning at this point – right? I tell myself this sometimes. To just keep it simple, but I never listen.
I read Dan Meyer’s blog post “If Simplifying Rational Expressions Is Aspirin Then How Do You Create The Headache?” I pictured myself asking students to evaluate a complex expression for specific values and whether or not they would be surprised (or even care) if I could evaluate the expression quickly. They have been beaten down by math for most of their lives. I thought this would just be another instance where a math teacher made them feel less competent, and that did not seem like a productive way to pique their interest.
I thought about when I recently re-read Lockhart’s Lament, especially this part:
“I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them … Mathematics should be taught as art for art’s sake. These mundane “useful” aspects would follow naturally as a trivial by-product.”
I thought about Tracy Zagar’s session that I recently attended at ATMNE’s Conference where she talked about how connections through multiple representations made explicit can help students to develop understanding. Tracy also asked participants to think about related topics, often taught separately resulting in learners thinking of math topics as separate concepts. We identified related math topics that are taught separately such as:
graphing lines: f(x)=mx+b and transformations of functions f(x)=a f(x-h)+k
“There are two kinds of simplicity: that of naivety; and that which, by penetrating beyond superficial differences, brings simplicity by unifying.”
At this point I was losing valuable planning time, but I decided it was more important for me to make sense of dividing fractions for myself, and help students to experience the joy in understanding, than it was to have a flawless detailed lesson plan, so I found and worked through Graham Fletcher’s Making Sense of Invert and Multiply. During my lunch, I created and thought through as many cases of dividing fractions as I could come up with. Then I selected a few to use with my students.
I decided against starting with the opening question provided in CPM’s textbook above, I thought I would close with that question.
I decided to open the lesson asking students to think about and share how they would represent the number of groups of 1/2 that are in 3/4. Instead of beginning my class with these learners feeling intimidated and overwhelmed, they were curious.
Students discussed and compared representations, made connections and got genuinely excited at the silliness of being in a college credit pre-calculus class and that we were making sense of fourth and fifth grade mathematics. Mid-discussion, one high school senior yelled “I am in 12th grade and I just now understand how dividing fractions works!” Shaking her head with a mix of frustration towards how math is taught and satisfaction that she understood division of fractions.
Once they were ready, students worked in pairs on whiteboards to think through CPM’s opening question. They took time, consulted and corrected each other and all ended with the same simplification in different ways. Then the craziest thing happened:
They asked me for more complex fractions to simplify!
In the spirit of Geoff Krall’s Problem Based Curriculum Maps, I attempted to organize my geometry curriculum and learning targets along with associated activities, tasks & lessons. In order to keep this as a useful document, I tried to only include the tasks that I have actually used in my geometry classes. I am interested in adding & deleting from this document regularly to keep if useful for me (and hopefully others). I plan to have the second semester completed this summer, as I am trying to develop this as I go this school year.
[update 7/25/16: It is finally completed!]
I have shamelessly stolen from all over the MTBoS, Math Vision Project, Engage NY & the Unit Blueprints Project and tried to give credit as much as possible.
Please share any criticisms, activities that I should add, activities that are misaligned, etc… in the comments.
Soapbox debate: provide class with a debatable prompt, and a minute to think. Then student must stand and state their claim and warrant to the prompt. It’s more fun if you randomly call on students.
Always, Sometimes, Never statements: students summarize previous idea, then state their argument.
Point-Counterpoint: use would you rather questions. Students must alternate arguments, so they have to disagree with previous person.
Table debate: assign student to teams to develop arguments, and then have the teams debate.
How to encourage debate:
add debatable terms to questions – best, worst, most efficient, should, biggest, smallest, most, weirdest, coolest, always, sometimes, never
Change boring math into a debate – Given an equation, ask,
What is the best way to graph this?
Which number would you change to change the graph the most?
This graph will never go below the x axis
Full Scale Debate:
Divide the class into 4 teams. Provide students with a carefully constructed scenario and 4 different stance’s to argue (example provided with musician recording contract).
Students read a variety of texts or resources on a topic, then consider questions in a large group discussion.
examples of questions to consider:
What are some strengths and weaknesses of each presentation?
When would each text be appropriate to use?
What difficulties may students have?
This sounds like a really interesting thing to do in math class. I need to learn more and see it in action so that I can implement this effectively.
Critical thoughts to creating a successful debate culture:
the accumulation of many intentional, small teacher moves over time sets the culture of student talk
When you want students to talk to each other, the teacher must SIT DOWN. make yourself small, and not the center of attention. Encourage students to talk to each other. If it is a whole class debate, have the student talking STAND UP. Slowly back out of the center, have students call on their peers.
Start early
Keep it simple. Use basic soapbox debate for the first month or two.
explicitly talk about what active listening looks like – be very specific (not writing, looking at the speaker, knees pointing toward the person speaking…)
ideally, dedicate about 5 minutes per class 1-2 times per week
provide structure (argument = claim + warrant ) and verbal cues
Occasionally, have students do a quick write providing an image and a word bank. This will help students to practice communicating mathematically.
Keynote: Jose Luis Vilson
We need to talk about race with our students and give them a safe space to grapple with their thoughts. In math instruction, the goal is to teach students to grapple with tough problems for which the solution is not already know and work towards a logical and reasonable resolution. This same principal can be applied to social justice issues.
Some questions/statements for students:
I just want to hear what you have to say
Why do you feel this way?
Where is your compassion/empathy?
We need to become comfortable getting uncomfortable and evangelizing for our truths. Avoiding confrontation and being polite can be destructive in the end.
She shared a very complete and organized collection of quality, basic stuff. Progressions, lessons, strategies. I can’t wait to use and adapt it for my first year of teaching pre-calc in a while.
Experience Connecting Representations – David Weiss
post more equations than corresponding visuals (task is to match the visual to the equation/expression)
provide individual think time – What do you notice?
Time to discuss with a partner (teacher circulates, listens & asks a pair if they would be willing to present their thoughts to the class)
display verbal cues:
Presenter
We saw ___ so we connected ____.
_____ matches ______ because ______.
Audience
They noticed ____ so they _____.
Their connection works because ______.
5. Get presenter’s to the front. One can only speak and the other can only point. They explain their thinking for one pair. Keep this light, safe & fun. If a student does not explain clearly enough or missing key elements, just let it go, they will most likely come out in later explanations.
6. Ask a student in the class to re-explain the presenter’s thinking
7. Teacher record thinking while a new students explains.
Repeat from step 5 with a new pair of students.
Once all problems have been paired and described by the class, have the pairs try to create a visual mode for the remaining equation that was not paired to a model
Close by having students complete a written reflection.
Explore Math – Sam Shah
Sam talked about a low stakes high reward assignment that he gives his students. They have to complete 4 or 5 mini explorations on any math topics of interest to them (with incremental due dates) and complete a brief written description or some evidence of what they did.
He turned the advanced math classes into a ‘club’ called varsity math and created t-shirts, stickers, party’s and a summer camp to go with it. He also made recruiting posters and placed them at the middle schools in order to motivate students and create a buzz around taking more advanced math classes.
This is a great idea! I recently convinced 10 students at my school to take a more advanced math class and I think I will have to figure out how to adapt this concept to fit my tiny group in an effort to get this group to grow in future years.
She opened my eyes to recognizing the different skill sets that elementary and secondary teachers have and the importance of valuing these skill sets and why we should try to break down our comfort barriers to get over ourselves and learn from each other.
I think I need to write a whole additional blog post on how individuals’ comfort seeking needs really limit our happiness, growth, empathy and success. (an ongoing theme this conference)
In this session we progressed through these steps to develop a model for ranking roller coasters, but the big idea here is more about how to facilitate this process. It would apply well to geometry tasks including 3 act’s such as best square or Mathalicious’ Face Value (my post on this task).
We need to be thoughtful and intentional, not just resource collectors.
This resonated with me as I am an avid idea collector, but I struggle with how to make a curriculum coherent. I want to work on criteria for coherence and re-evaluate the content of my current classes.
It is our Holiday break & I should be planning for the Intro to Computer Programming class that I’ll be teaching starting in 2 weeks. Problem is, I don’t really know how to code. I can write simple programs in a TI-84, and I wrote some C++ & Matlab programs back in college.
I think programming is important & I teach in a very small alternative high school (about 100 students). I have supportive administration. I noticed a significant proportion of students who expressed interest in programming, especially after completing the Hour of Code. Being an alternative school, there are a lot of students who need additional math credits in addition to students who need elective credits & I want them to choose math! Last semester I taught Financial Literacy, which supplies beneficial life skills, but I can’t teach it twice since students need a variety of math elective options. So, programming it is! I asked for it & I got it. IT filled to capacity within a week of scheduling next semester.
So here its what I’m currently thinking about the structure of the class:
The class meets for 90 minutes Tuesdays, Thursdays & every other Friday.
I’m going to be heavily dependent on CodeHS’scurriculum. At least this first time. I’m not going to drive myself crazy developing everything from scratch. This is effective when I know what I am doing, but I don’t. I’ll gradually develop this class as I learn more about Javascript & HTML.
Since this is for a math credit, I’m going to start every class using Fawn’s Math talks. If I plan to learn a lot by teaching this class, why not try something new for warm-ups too?
Since this class will be largely student paced, I need a way to hold them accountable and to stay motivated. I am going to create a student form for daily reflection in the last 5 minutes of each class. This will include: What they learned today, Where they struggled, if they helped or got help form any peers, what activities they completed, room for their comments, and room for a reply from me. I’ll post it here when I make it. hopefully that will be soon.
I wish I could think of a fair way to encourage collaboration, but I haven’t been able to figure that out. Maybe we will have discussion time on the every other Fridays where students present where they are stuck and we work together to fix it?
I plan to do a hybrid standards based grading (SBG):
50% of their grade being the standards: all of the programming skills students should demonstrate: for loops, indentation, commenting, while loops, if/else loops, debugging….
30% Successful Completion of the Challenge projects within each CodeHS unit – I am considering a grading scale where students can choose their grade based on how many of the projects they complete. I don’t know. I may just assign all of the challenge projects because this is their chance to use the 8 Standard of Mathematical Practice and what is cooler than that?!
20% successful completion of the learning activities.
This all may change as I work through it with students & I’d really like to make more of it my own. I can;t remember the last time I was so dependent on a pre-designed curriculum. I get a little nauseous thinking about it!
[update 1/7 I made a progress log. I’m not sure what I’m missing, so Im only going to make copies for 1 or two weeks, and then I’ll see how they are used & what I should change ]
Waayyy too much time and thought went into this! I have to remind myself that teaching & covering are not the same and that I have to let some things go in order to tech other content well. I’m just not going to do Law of Sines & Law of Cosines in geometry. It isn’t happening. I’m trying to be OK with this. I’m also not going to teach simplifying radicals. There. I said it.
So, here’s the plan:
We are going to start the semester with TV Space to motivate the Pythagorean theorem. I’ve asked. My students have not heard of it. I know, they should have, but they don’t know it. They are going to dominate it in a few weeks. We’re going to follow that up with Mathalicious’ Viewmongus, which, at this point, students should be able to work through pretty comfortably. I know this is a lot of Pythagorean Theorem, but I’m trying to build persistence and a culture of problem solving as the semester gets going. Then I am going to support my students writing a program in their TI-84’s to use the Pythagorean theorem given 2 coordinates (AKA distance formula). I plan to use the sheet shared by Jasmine Walker as a guide.
Right after that, I’m hitting ’em with Taco Cart. Dan & Fawn have done wonderful work. I’m so so so excited about finally getting to do Taco Cart. Yes, its the holiday break & I can’t wait for school to start to get in some serious problem solving with me being less helpful!
Then we’ll spend a day reviewing midpoint & programming in that baby to the TI’s. Next we’ll spend a class period on Pam Wilson’s distance & midpoint activity. I may make it into a scavenger hunt, as mentioned in the comments, we’ll see how ambitious I am when we get to that. Depending on how the pretest for this unit goes, I may do a 1 day activity reviewing slopes of parallel & perpendicular lines, then we are off & running with coordinate quadrilateral proofs. I’m sure I’ll throw in Illustrated Mathematics’Is this a Rectangle? The big project for this unit will be a choice between Jasmine Walker’s quadrilateral programming project or Mathy McMatherson’s Facebook project described here & incorporating his reflective follow up here.
Then we’re hitting up some right triangle trig. I think I am going to go old school with this unit. I’ll introduce the concept using Tina Cardone’s geogebra exploration, but then I think were just going notes & practice: roundtables, problems around the room, row games. Once we’ve got it down we’ll build clinometers to measure tall stuff outside (flagpole, maybe?). That’s probably all we’ll get in in the next 6 weeks, but I intend to follow this up with this awesome trig task as soon as I can, maybe the start of the next 6 weeks. I’m like a kid in a freggin’ candy store with all of the awesomeness spewing out of the MBToS! Y’all fire me up!
Here’s the learning target tracking sheet for the next 6 weeks:
Crazy Math Teacher Lady 