Category Archives: Uncategorized

Paper is the enemy.


Actually, I love paper.  Writing, drawing, folding… love it.  I’m even running an after school origami club this semester.  And as a math teacher, paper is actually the preferred medium (second to whiteboards of course).  While technology gives us many tools that speed up and deepen math learning, much math isn’t suited to the computer screen or the computer keyboard. So paper is necessary, and as a result, organizing papers is totally necessary.

While it does seem like organization is something that many teachers bring as one of their natural skills, this is not something that came naturally or easily for me. It has taken me years of not being together to develop enough layers of systems that my classroom is organized, and as a result my students are organized – and this year has been a huge improvement for my students.  We are set up so most things just work, and we don’t spend our precious class time passing back work or figuring out where the handouts are. Here are some of the systems that have had the most positive effects on my students and our classroom this year in case they might be useful for someone else.

Inboxes and outboxes

I definitely needed systems for the day-to-day passing back and forth of written math thinking and feedback. I have a separate inbox for each class, and each student has a mailbox (…or a mail folder). My grade 8 class is small enough that they each have a mailbox, and my high school students each have a folder in a pass-back crate. I have fewer than 70 students, so this is manageable for me, but I think it would be worth it, even with greater numbers.

The mailboxes have also had an excellent unforeseen side effect: when a student is absent, I just put any handouts into their mailbox. I print exactly enough copies for the whole class, and don’t need to worry about keeping track of extra copies of work from past days or notebook pages or homework sheets that students did not get. Students know to check their mailboxes when they return to school after an absence.

Extensions, extra practice, remediation

This has been a way for me to put some responsibility on students to figure out what they need. Because we rarely use the textbook, students rely heavily on me to give them appropriate problem and practice sets. Before this year, I was often frantically printing extra sets, or creating and copying sets as I needed them. This year, in addition to the regular classwork and homework, I am preparing leveled sets of work. I include foundational practice for students who need to brush up on their pre-algebra, extra practice sets for the current work we’re doing, and extension and challenge work for those who are ready. Students know that if they finish with their in-class work, that they can self-select from the appropriate boxes or get extra practice before an assessment.  I followed three students into my classroom yesterday, and asked if I could do anything for them.  They said, “No thanks, we’re fine.”  They went and took a variety of extra practice and extensions, wished me a good afternoon, and left. It was a pretty good feeling to seem unnecessary.

Daily Work

Like most of us, I often have my classes back to back to back without a break in-between. That means that I have to be super prepared before my first group of students arrives in the morning. Even when I made copies ahead of time, I found myself misplacing the work for the current class, or not able to lay my hands on it at the moment when I needed it, and wasting the first few minutes of class finding the right papers. No longer. I put each class’s work in a slot so it’s ready when they get there.

 

Homework

Same deal. I transfer these from the class slot to the HW slot at the start of class, and students know to take a copy on their way out of class.

 

 

 

 

 

File Cabinet

Now I’m just showing off. Before this year, a file cabinet was just a place to stuff papers until another time when I would come back to throw them away. Maybe this isn’t such a big deal to you, but this is the first time in my life I’ve been able to find anything that I’ve filed before.  Totally worth the time in labeling!

Although none of this is at all original or revolutionary, maybe something will be helpful for you.  :) Please share more organizational systems in the comments or find me on twitter if you have more ideas to share.

Students Know

Students know.

This post was written as part of the 2017 MTBoS Blogging Initiative, in response to the prompt “We all fall down.”

When I was finishing my graduate degree, I was hired to teach my first college art class.  I was really proud of being the “professor.”  I would come to class and say all of the things that I thought the professor should say.  I had had some really wise teachers who showed up to class and gave really wise commentary about art, and I thought that what I was supposed to do was to emulate their behavior, and say wise things.  I remember that there were a lot of students so we had divided up the first formal critique into two days.  On the first day, I acted the part that I thought I had been hired to act.  I said things that sounded like what an art professor should say.

The students saw right through it.  They could tell that I was just playing a part.  It was a miserable day both for them and for me.  I really reflected about this, and was determined for day 2 to be different.  But I was scared too.  If I didn’t say the things that were my idea of what someone in that role should say, what would I have to contribute?  I went to class ready for it to be even worse than the first day.

I don’t know if I said anything wise or deep or even if I gave any meaningful feedback that day, but I was honest.  And students were totally different.  They were actively listening to me and to each other.  They could immediately sense that there was a difference between the two days of critique.  What they wanted was not my idea of an art teacher.  What they wanted was honesty.  As soon as I just acted like myself, they were willing to be on the journey together – even if they had an inexperienced captain.

As a math teacher, I have been so grateful to have made these mistakes in the art classroom.  I had so many art students who were much more gifted artists than I, that I didn’t feel threatened when my math students would catch on to something new before I did, or would think of a more elegant solution than the one I had in mind.  I learned to be comfortable with not knowing as an artist, and have translated these important lessons to my math classroom.  When I catch myself trying to be wise or trying to sound like something I’m not, I remember that first critique day.  Students know.

Read and Share: Some Important Voices in My Classroom

This post was written as part of the 2017 MTBoS Blogging Initiative, in response to the prompt Read and Share.

My math teaching jobs have been at small schools, where I have been the only one teaching my courses.  While I have had great colleagues, I have never had a group of math teachers at my own school to collaborate with.  As a result, the MTBoS has been hugely important to my development as a teacher, and while there are far too many amazing and generous educators who have changed my teaching to list here, I thought that I would share a few of the voices who have had the greatest day-to-day impact in my own classroom, and have linked their name to a recent post that taught me something or caused me to reflect on my practice.  Most of these will not be a surprise to anyone who knows what the letters in MTBoS stand for.

Here’s a student sketching one of Dan Meyer’s Graphing Stories, which has been projected on the whiteboard

Dan Meyer: For me, he is among the most important voices in contemporary math education. I incorporate his ideas about how to make math education meaningful and relevant for my students into my classroom everyday.  But in addition, I follow closely his open minded approach, and his attitudes toward having a productive conversation even with someone who starts from the premise of disagreement, or even a critic who begins by hating on him.

Fawn Nguyen – What can you say about Fawn? Her brash and direct writing give her the room to discuss what it means to really care about your students, and to share he joy of mathematics and the empowerment that comes from learning how to be a problem solver. I will read every word that Fawn writes for us.

A pattern from Fawn’s excellent visual patterns.org

Nora Oswald – No one can gamify math like Nora. Her activities (or at least the structure of her games) make appearances in my classes at least a couple of times a year. Her structures seem to provoke the healthy kind of competition – where students want to push themselves without keeping the other team down.

The Desmos Team – The desmos team models how to be learners.  They are continually responsive to the community and to improving the calculator and the experience for users.

Kalid Azad – I just recently discovered Kalid’s work when I was looking for a better way to explain the graph of the sine function with radian scale. He has a knack for sharing straightforward examples and ways of thinking about math that focus on conceptual understanding.  The linked post definitely had an immediate impact on my class the next day, and I am making my way through his older work to see where else it might lend new insights for me and for my students.

Ben Orlin – Math with Bad Drawings is insightful, entertaining, and true.  Harder to say exactly what I bring into my classroom, but I find myself thinking of his posts often when I’m with students.

David Wees – In addition to the contributions of the New Visions work to my own Algebra curriculum, reading this blog regularly adds a tweak to one of my instructional routines, or adds depth to my formative assessments.

Jo Boaler – her work with youcubed is a really important voice in promoting equity in math education.  As a feminist, my goal to promote equity, uncover unconscious bias, and create opportunities for ALL students is at the core of why I became a teacher, and in fact at the core of my personal values outside of being a teacher.

Whose voices are most important to your teaching, and how do they show up in your classroom?

MTBoS 2017 Soft Skills: My Grade 8 Exit Trip

Here is an excerpt from something I read to students before we get on the bus to leave for our end of middle school overnight trip.  It comes from Joe Ehrmann, a former NFL football star and volunteer coach for the Gilman high school football team .  This comes from the book Season of Life, which I originally read about at Delancey Place.

” ‘What is our job?’ Biff asked on behalf of himself, Joe, and the eight other assistant coaches.

” ‘To love us,’ most of the boys yelled back. The older boys had already been through this routine more than enough times to know the proper answer. The younger boys, new to Gilman football, would soon catch on.

” ‘And what is your job?’ Biff shot back.

‘To love each other,’ the boys responded. “

This post was written in response to Sam Shah’s week 2 mission from the 2017 MTBoS Blogging Initiative: #MTBoSBlogsplosion.  Sam suggested that we focus this week on “soft skills” – the things that we do to help kids grow that aren’t necessarily directly teaching math.  I’ve written here about an overnight trip that I organize at the end of the grade 8 year, including some specifics of the activities that we do to help bring us together.

I believe that as a teacher, helping kids to be confident, and to care about and respect each other is equal in importance to any math skills.  I try to design many lessons to provide opportunities for both of these things – a balance between problem solving, individual reflection, and social behaviors.  These are typically not at odds with each other – teaching problem solving and math skill building does help to boost confidence, and opens more doors for a whole student, however I try to be deliberate about also choreographing social opportunities.  I also include lots of small things in our classroom routines that are there just to build relationships – things which are not specifically math, but which are directed at helping kids to understand how to treat each other, and how to be self-fulfilled, and how to reach their “personal best.”  The Grade 8 exit trip is designed with this in mind.

I am deliberate about using the words field “work” rather than field “trips” when I am off campus with students.  This language makes it clear that we are not passive observers, but instead that we are purposeful in our activities.  I tried to design this trip with this in mind, and had to articulate what I want students to get out of this trip:

  • I want to mark the end of middle school, and entrance to high school as an important benchmark.  This is a big deal for students, and I want to honor this.
  • We should make sure to look back.  They should have some opportunities to reflect on their time as middle schoolers, and to celebrate their personal successes, and the successes of their peers.
  • We should make sure to look forward.  What are they looking forward to?  What fears do they have?
  • We should make opportunities for them to showcase their content skills – writing, performing, problem solving.
  • We should showcase that we value students as whole people – young adults who have idiosyncratic strengths.
  • We should have fun so we have a positive framework to look back on.

We start the day by gathering around a metal can.  Students are given a slip of paper, and asked to write down something that they want to leave behind from middle school.  I tell kids that the move to High School is an opportunity to re-invent yourself, and that we all have things that we want to leave behind and change about ourselves or about our feelings toward some others.  We then toss our slips into the can, and set them on fire – metaphorically “burning our baggage.”  While we are in our circle around the baggage can, I read the rest of the anecdote quoted above to the group.  The message is that whatever ideas we have about each other, we can also find positives, and for this trip, we set aside any issues, and focus on loving each other, and showing that love to each other.

Next, we all draw names from a hat, which contains everyone’s name from the class.  During the bus ride, students are asked to think of something that they appreciate about the person whose name they have drawn.  When we get off of the bus, we begin with an “appreciation circle” where students are asked to share these acknowledgements of their peers.  Our next task is a hike.  I like to find a place where we have to climb up a trail to get to the top of a mountain.   We choose a hike that is as difficult as we feel we can manage for our least in-shape student (or teacher chaperone).  This gives us a metaphor to discuss when we get to the summit.  How climbing the mountain felt like a big deal, and there was complaining, but that we felt accomplished after we got to the top, and about how we helped each other to make it all the way there.  We make some explicit parallels to their middle school experience here, and then we have our second appreciation circle, where students are asked to reciprocate the appreciation that they received at the base of the hike.

After a snack, students are asked to write a short story in their journals, using the gathered group as the characters in their story.  The story includes a favorite memory from middle school,  a fear about High School, and something that they’re looking forward to in High School.  These always manage to include some of the inside jokes of the group, and highlight some of the idiosyncrasies that we love about each other.  We share back some of these, and then have a late lunch.  I always make sure to prepare some of the food for lunch.  Even if I just make a fruit salad or something, this is how we show love in my family – by feeding each other.  I let students know that this is one way that I am letting them know that I love them.

We give the kids some free time after we get to our hotel before dinner, and finish the evening with some problem solving tasks and reflection.  On day 2, we choose a site and organize some drama and analytical Social Studies activities around where we are.  After lunch, and before heading back to school, we include a closing circle where we reflect on the trip, on our time together, and on what lies ahead.

Preparing to do a Shakespeare reading in an ancient amphitheater!

In considering my students, I am most proud of those who leave my class having gained respect for themselves and for each other, and demonstrate this respect.  This does often go hand in hand with gaining confidence as problem solvers or as math thinkers, but I admit that I have had students who have totally blown it as math students (one or two who have even failed my class), but who still come to have lunch with me sometimes.  I still find some success in having built relationships, and I hope that maybe they’ll learn some math later when they’re ready.

MTBoS 2017: My favorite… tool for teaching transformations

My favorite tools for teaching transformations from parent functions are the Desmos Marbleslides. This is the first year that I have been able to use these activities to cement our learning across function families in our Algebra 2 classes. While these aren’t exactly skill and drill practice, they do seem to give students similar opportunities to do the repetitive work that is needed to build procedural fluency.

Just a few of the reasons I love these marbleslides…

  • They are consistently motivating, fun, and engaging
  • There are opportunities for creative solutions
  • They present open problems with multiple solutions, battling the idea that all math problems have exactly one answer that is in the back of the textbook!
  • As a teacher, I am always interested in and surprised by student solutions – very different from much of my grading
  • Students demonstrate perseverance through these challenges – they really want to come to solutions, and will keep working until they succeed

I made my first custom Marbleslide for students to practice transforming absolute value functions. My activity is basically an exact copy of the Desmos team’s work, but with Absolute value equations. The custom activity was very easy to build, and I am turning over some more creative ideas to explore now that I have done this.

I am pretty sure that part of our success with our understanding of transformations has come from the course map this year. We are basing this year’s sequence of topics around families of functions. We began with an informal study, just looking at shapes and appearances of graphs, and what kids of situations might be modeled by different function types, and have been adding formal analysis of each family with each new unit.  Starting with this big picture has given students a framework to fit each family into – they are connecting what is similar and what is different as they dive into each new kind of function.

It has been amazing to see – we have just gotten into trigonometry, and by the time we got to the sine function, kids were so comfortable with shifting graphs around the plane that I didn’t need to do any explicit instruction – they knew to play with the constants to get their graphs to shift in different ways, and with very little prompting from me, they argued out the differences between period and amplitude shifts.

I am excited to see how these understandings will transfer to the Desmos Drawing project this year. Last year’s students set a pretty high bar, but this year’s 10th graders are already demonstrating a deeper understanding – and 3 months earlier.  Stay tuned!

MTBoS 2017 Blogging Initiative

Happy New Year! Time to share back and collaborate again. Looking forward to seeing ideas from new bloggers and to checking in with old friends.  Note: this post originally appeared on the ExploreMTBoS site.

 

Welcome to the Explore the MTBoS 2017 Blogging Initiative! With the start of a new year, there is no better time to start a new blog! For those of you who have blogs, it is also the perfect time to get inspired to write again! Please join us to participate in this years blogging initiative! […]

via New Year, New Blog! — Exploring the MathTwitterBlogosphere

2016-17 Algebra Notebooks: Scaffolding Organization

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Confession: Since I began teaching math, I haven’t really managed to use a textbook. I do generally claim on my syllabus that we are using UCMCP or Saxon or Kendall Hunt as our “anchor” text, but year after year, I find that it is just too hard to integrate house-created or MTBoS-sourced materials with a textbook’s sequence. I do hand out textbooks at the start of the year (my current school uses UCSMP). I make sure to give the occasional homework assignment from the book so they remember that they have one as a resource. I direct them to the related lessons in the book as we work together in class, and I reference the parts of the book that students can use to help them prepare for semester exams when I prepare the exam review materials.

I also use some problem sets from the book, but truth be told, when I have tried, I just have not found printed textbooks to be effective learning tools for students – although I admit that it’s possible that I just haven’t found the right book yet. But I think that there is more to it – the pre-printed book format has to include all of the information, all at once. It takes away the possibility to choreograph and reveal information in a controlled way – a way that builds suspense, piques interest, and doesn’t spoon-feed.  This pedagogical idea closely correlates with the guiding principles for creating math activities as articulated by the Desmos team. Digital media allows for this type of sequencing of information, but we can also do this in person by doling out questions, information, and formal notes at the right moment during our lessons. We still share and give explanations, but whenever possible, not until AFTER a student has had the opportunity to make some sense for himself or herself.  Building our books piece by piece allows for this unfolding process.  I have also found that most students arrive in my class with some variation of the idea that math only exists in the textbook, and is not related to their lives outside of class at all.  Creating our own books has been another tool to help combat these beliefs.

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To be real, this approach does translate to a huge amount of work. At this moment, there is no resource that is set up for us to use in this way, and putting together a coherent and cohesive curriculum for ourselves is a full time job in and of itself – even before delivering said curriculum. I totally understand why a teacher might choose to just use the book. Spending so much time doing this means that we are not spending time on other important parts of our job – like giving meaningful feedback, communicating with families, or collaborating on interdisciplinary work – all of which are arguably just as important. But I just have not found a book that works by itself. I think that I can do better for students by curating materials from multiple sources.

To get to the point of this post, what this has meant is that I create a ton of printed materials, which students have to keep organized. This has worked just fine for half of my students – the ones who have already built good organizational and study skills. The other half end up with binders full of papers – much of it meaningful, but often in no particular order, and they don’t know what to do to go back to review or to prepare for assessments.

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This year, one of my professional goals is to help my students to organize all of this material. I required all students to bring a math notebook (at least 100 pages), and a math binder. The notebooks will only include material, which is correct, polished, and can be used to study from and the binders will be where we store all of our working and thinking – we are basically building our own personal textbooks.  I let students know that the notebooks will serve as an ongoing assessment of understanding, and are therefore treated as a graded assignment.  They know that they will be expected to periodically present their notebook to be checked.

I was inspired by @mathequalslove’s notebooks, and used her basic design for the unit dividers. The learning goals for the unit are listed on each divider, along with space for us to fill in the big picture generalizations (at the end of each unit). My school has made the decision to track students from grade 8 (honors and non-honors sections), but I do my best to leave the door open for students to be upwardly mobile by making the honors-level work available to all students.  In the notebooks, this translates to a second page for each unit, which details the honors-specific learning goals.

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Included in each unit:

  • Unit divider with learning goals
  • Honors-level extensions with learning goals
  • Essential Questions
  • Unit Vocabulary
  • Various graphic organizers/ note-takers for content (Although I appreciate many teachers who get crafty with their “interactive” notebooks, I don’t tend to use foldy things. It takes a huge amount of time to just glue in the flatty things)
  • Worked examples

A side result of this work is that these notebooks have made me a better teacher as well. Once again, I had to take my unit plans, and really make careful decisions about what needed to be included in the notebooks. Although I don’t stick to the order in which the skills are listed, my organization has to be in place at the start of each unit.

Here are the materials for Unit 0 and Unit 1 for Algebra 1 and 2. I’ll publish these as we complete each unit.  Please do let me know if you use these, or if you have suggestions for improvement.  I have also included a Unit X in our notebooks, which includes materials for general problem solving and reference materials (times tables, trig tables, unit conversions, etc.) I’ll likely share this at the end of the year, as we are continually adding to it.  Lots of questions still linger, and I will be grateful for your input.  In particular, here are my current quandaries…

  • I know that my binder sections are not right yet, (I have divided their sections into homework, classwork, assessments, reflection, and “other.”) and could definitely use some advice there.
  • After two units with each class, I already see some changes I’ll make for next year.  Do you see some things you would do differently?
  • Does dividing up the standard vs. honors-level targets in this way make sense?

Thanks in advance for your thoughts!

alg-1-unit-1 alg-2-unit-1-families notebooks-unit-0

Shifting Populations

I’ve been wanting to write and think about this for a while now, and with the end of the school year, I finally have a little time to reflect on and share some more work from this year’s Algebra classes.  Population shifts make for potentially compelling and authentic math modeling tasks.  Last year, I had great success  with grade 8 students, who compared linear and exponential models of the population growth of developing countries, and made predictions for the population of their chosen country in the year 2050.  This work was adapted from Kyle Moyer and Zack Miller(@zmill415)‘s Booming Populations project.  I wrote up some reflections on that project last year.  This year, I collaborated with some colleagues to adapt this work for my current grade 8 students, and to extend this work for my grade 10s.  The rubrics and guidelines are at the bottom if you can adapt them for your use. :)

Computer Work

My grade 8s studied the populations of two groups of snails, one in a tank with no predators, and one in a tank with some fish (…who apparently find snail eggs to be tasty).  My colleague from the science department, Heather Charalambous was kind enough to host this study, and to use science class time to support some of the conceptual thinking around how and why the snail populations changed (…and to count the snails!).  Kids used spreadsheets to create linear and exponential models, compared their two models, and made predictions about what would happen to the snail populations.  We checked their predictions against the actual number of snails at benchmark dates, and examined discrepancies between their predictions and the actual outcomes.  Materials are below.

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For grade 10, I collaborated with two excellent colleagues, Julie Jonsson, and Rachel Iannacone to create another permutation of this project, looking at the question: What events have the most significant impacts on the populations of cities?  In grade 10, students study U.S. History, so we tweaked this to fit into their coursework.  Students were asked to choose one American City and  to examine their city’s population from 1850-1940.  As with the 8th graders, they created linear and exponential models to help them to analyze and make “predictions,” about what they thought would happen between 1940 and 1960.  They then compared their predictions to historical data, and made arguments about the reasons for any differences.  Students who were ready created some polynomial models as well – although these models potentially fit the data better, they are complex, and challenging to defend the contextual choices.  The culmination of the work asked students to look at their city’s population changes through 2014, and to make a future prediction for what they think will happen to the population over the next 35 years.

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Improvements from last year…

  • Collaborating with teachers from science and social studies really helped to make this work deeper for students.  Especially with the grade 10 project, students were forced to look beyond the math to examine why populations shifted.
  • The math felt like it was in service of the compelling questions, and I think that students really felt like their math skills helped them to quantify and analyze an interesting problem.
  • The grade 10 project guidelines and rubric were carefully honed down for clarity and depth, and were designed as a precursor to and preparation for the I.B. Extended Essay, which most of our students will complete in grades 11-12.  This improvement was largely due to the collaborative efforts of my two partners, Julie and Rachel, who were just awesome to work with.  We were able to really hash out our different opinions and priorities, without anyone feeling threatened or marginalized, and to keep working until the project met all of our standards — this was one of the best professional collaborations I have experienced.

Better last year…

  • We did not have a culminating event for either of these projects this year.  Last year, we organized a “population summit,” where students presented their findings to a panel of “experts.”  Having to present their work publicly in this way really made students up their game.  This year, we did put up their work on the math wall, but somehow it wasn’t quite the same as public presentation.  Although presenting takes time, I really want to build this into the project if we can in the future.
  • Although the students did get some choice in their cities, there were a few who did not get cities that they were that interested in.  This made for less engagement, and I want to figure out how to really make them feel like they have some control next year (…even if it is just the illusion of choice).

The Materials…

SHIFTING POPULATIONS FINAL ASSIGNMENT + RUBRIC

SHIFTING Essential Questions

SNAIL Project Guidelines

Let me know if you use or adapt this work for your classes, or if you have ideas for how to improve or deepen this work, and please send me a note or find me on twitter if you’d like to see some student exemplars.  Happy to share.

Algebra 2 Concept-Based Map (Draft)

I’m in beautiful Pomos, Cyprus, having finished my second year at an international school in Nicosia.  Pomos is an inspiring place to work and to plan for next school year, and I am anxious to share the work I am doing with you for your thoughts and feedback.

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This post reflects a current draft of next year’s work for my Grade 10 Algebra 2 class (Algebra 1 to come soon). I want to begin by gratefully acknowledging some of the most important sources for materials and inspiration for me.  My online MTBoS community is wonderfully generous, I have some top-notch local colleagues, and it is a truly great time to be a collaborative math teacher.

  • Although I veer in several key places, my starting point for this map was the work of David Wees (@davidwees) and the New Visions for Public Schools’ Algebra 2 curriculum.
  • The kernel of inspiration for the work was inspired by Glen Wadell’s (@gwaddellnvhs) big picture thinking, in particular THIS POST, which has been churning around in my head since last June when I first read it.  The way that he begins the year communicates a clarity, which connects his whole course together in a way that I wanted to emulate.
  • Thanks to Pam Wilson (@pamjwilson) as well for sharing her linear equations unit, which was a big help for me.
  • Henri Picciotto’s post on “Forward Thinking”, which helped me to focus on always keeping the big questions and concepts of the course at the front of my mind when planning.
  • My colleague from the English department, Laramie Shockey, and her help in understanding Lynn Erickson’s Concept-Based Curriculum Model (which is a required strategy at my school, but which I had not found useful until Laramie’s mentoring).  This process was really clarifying and useful for me. I was doing many, if not most of these things already, but this is a concise way of framing it, which helped me to pull the pieces together.

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By its nature, this map remains a work in process and is a living document. To keep this as relevant and lasting as I could, I worked to pare this down to the most important concepts for the course – but any curriculum map has very limited meaning until a group of students actually arrives. The process of creating this map really helped me to gain clarity about each part of the course, and what I want students to learn.  It helped me to know what to remove from the course, and what to prioritize.  This has to work differently for each school, and yours will naturally have to be different from mine, but here is some of what guided my choices:

  • Sequencing – whenever possible, begin with informal before formalizing both in the small (day by day) and the big picture. 
  • Include multiple exposures to ideas – for example, identify linear functions visually in Unit 1, formalize and practice skills with linear functions during Unit 2.  Compare linear functions to exponentials in Unit 3, and model with linear functions in unit 7.
  • My map is based on Virginia State Standards (My school’s standards of choice) with the addition of the CCSS Math Practices, but in addition, my curriculum for grade 8 and grade 10 is geared to prepare students for the IB math program in grades 11 and 12.  In addition, I teach some very specific skills to support the grade 10 science curriculum.
  • I teach equations first, and then functions. I find that students can work with functions more fluently once they are comfortable with the algebra. Although this is different from the New Visions work, I have had success with this sequence, and it seems to work with the populations I teach.
  • We include right angle trig and a study of vectors during A2 to support grade 10 students who take physics during the second half of the year.
  • Generally I pared down the language in this document.  While I like specific academic language, this version was developed with kid-language in mind. I want kids to actually be able to say the things that are written as generalizations when a visitor comes in to ask them what they are learning and why.
  • In addition to the sequence of topics, I included a “Unit X,” which emphasizes the importance of problem-solving and cultivating the habits of mind of a mathematician in grade 10 math.

I found that the concept-based model helped me to focus on what I wanted students to know and to do, but I haven’t yet made the whole of the model useful for me.  It’s quite possible that I don’t fully understand the concept-based system, but I don’t understand the importance of the one word conceptual lens or one word for Macro/Micro Concepts, so I’ve left these out of my maps.  My school asks that teachers use a specific model (Atlas Rubicon – Yuk, $%#&@, and Blech!!) for our internal sharing, so I do have to include them in that version. Feel free to send an email if you’d like to see these as well, but for me they weren’t that useful. Please let me know if you understand these better than I do and can lend some insight.

My process:

This process has to be personal and specific to your situation, but here were my steps.

  1. Name the concept (unit)
  2. Sketch/draft generalization for the unit.  Brainstorm Essential or Guiding Questions
  3. List important topics, facts, procedures
  4. Write the related generalization for each topic, fact or procedure
  5. Revisit unit generalization based on what happened during steps 3-4
  6. Translate topics/facts into “Critical Content” (What students should know) and “Key Skills” (What students can do)
  7. Design formative and summative assessments
  8. Cycle through 1-7 until they (mostly) match each other and I am (mostly) satisfied with them
  9. Correlate with my standards to see if I’ve missed anything
  10. Cycle back through 2-9
  11. Add important unit vocabulary
  12. Organize the “Possible Learning Experiences” – this is the most fun for me – I love to source, modify and/or create and choreograph the experience for my students.  This document does not yet include this part, but I will publish it here soon.

Steps 1-7 are cyclical for me, and I think you could start anywhere as long as you cycle through these until they all match – this was one of the real moments of clarity for me. I would write a unit generalization, and then realize that it didn’t match the facts/topics. I had to decide which one I had to change, which forced me to make a clear decision about what I wanted to prioritize. I wanted to connect my guiding questions with my essential understandings. If one was in there without a clear reference to the other, I tried to visit them until there was a match, or I felt that there was a reason to include one without the other.

I would love your feedback on this map.  Does the sequence make sense? Am I missing anything critical?  Is my language kid-friendly enough? Academic enough?  Do you do things in another order that works better for you?  Thanks in advance for your thoughts!  Here are the maps in Keynote and .pdf format.

Algebra 2 Conceptual Course Map DRAFT 3.key

Algebra 2 Conceptual Course Map DRAFT 3

Math + Art + Desmos… Connections.

“I love math and art, and I’m glad that I was introduced to Desmos, a way to use both subjects at the same time.” – Marianna, Grade 10

Slide01

Callisto, Grade 10

Drawing with graphs has been a powerful way to motivate students’ interest in understanding how equations relate to functions, and how manipulations of equations lead to transformations from a parent function.  I jumped on to Fawn Nguyen’s Des-Man project as soon as I saw the idea, and have done some incarnation of this work each year.  Each time I’ve guided students through this process, it’s gotten better and deeper, both through the development of my own approach, and from improved tools like the Desmos Des-Man interface (…which I’ve heard is currently “in the shop” undergoing some improvements) and more recently tweaks to this idea like the “Winking Boy” challenge, created by Chris Shore (@MathProjects), and posted on the Desmos Activity Builder by Andrew Stadel.

Slide02

Amit, Grade 10

This year’s work was definitely the strongest yet, and I owe the major improvements to my reading of  Nat Banting’s post, which extended this project to another level for my students. In the past, I have asked students to create a graph, which had features of a face or a building or a plant.  This year, I asked my grade 10 students to choose a graphic, photo, or work of art, which they had to replicate using only equations.  I asked that they choose an image that was meaningful to them for some reason, and then helped to guide them to something that was challenging, but that they could accomplish – a natural moment for differentiation, built in to the process.  In the earlier versions of this project, students had been motivated by trying to make their face look angry or happy or sad, but they didn’t have a specific place where their equations had to end up. Asking kids to commit to re-creating something forced them to be purposeful and deliberate in every choice.

Slide04

Ilyas, Grade 10

They took the responsibility of recreating their chosen image seriously, and honestly, their work exceeded my expectations. There were regular exclamations of satisfaction echoing around the room as we worked on this. They persevered. They definitely attended to precision. They argued with each other about the best equations to use. They reflected about how to make the best use of Desmos. They practiced the habits of mind of successful mathematicians.

Slide09

Anastasia, Grade 10

When we shared the in-progress work for some peer feedback, kids were actually applauding each other when their work came up on the screen.  Not because I reminded them to be a supportive audience, but spontaneously.  Seriously.  And when they saw the staff creative picks at Desmos, they asked me whether they might be able to submit their work.  The whole class was taking pride in creative math work.

Slide03

Karim, Grade 10

I asked that students reflect in writing on their learning during and at the end of the project.  I haven’t asked for students to do enough writing in math so far this year, so when they seemed to be really struggling with this, I made a fill-in-the-blank “reflection assistant” to scaffold their thinking and writing, and to give them some ideas about what to include in their written analysis.

A few highlights from their reflections:

  • “I was quite surprised that I could replicate a drawing by using graphing.  If somebody asked me to do it last year, I would say that it is a “mission impossible.”   However I was able to do it.”
  • “As my piece of art, I chose the logo of the football club Barcelona because I am a big football fan and FC Barcelona is a club worthy to be recreated through the use of quadratic equations in vertex form. In addition, the logo was an appropriate challenge for me, containing easy and smooth curves but also difficult shapes, like letters or circles. When the project was assigned, I was skeptical that it was possible to recreate an artwork, just by using equations. But now that I am done and a proud owner of a recreated art piece, I strongly believe that it is possible (obviously).”
  • “I found out that desmos is a really good tool to practice and sharpen your understanding on any equation and in my case it was the vertex form of a quadratic. Desmos allows you to experiment and find new ways to fix the problems or even work more efficient in order to surpass the problems in the first place. I am proud of the detail and sharpness of my work in general. I tried really hard to make the whole piece smooth and detailed. In order to do so, I zoomed in a lot and by doing so, I identified minor mistakes and was able to fix them.”
  • I chose “Pumpkin Pepe” as the subject of my project because it provided the right level of challenge for me and it was really fun to do. Overall, I really liked this project because it solidified my knowledge of graphing equations and has made me more comfortable using parabolas. I found that my understanding of quadratic equations really improved while I worked on this project because before, I wasn’t sure which variable shifted the parabola which way, but now I understand.
  • “I found that my understanding of parabolas and linear equations really helped me improve, and made me more confident during my work on this project. At first parabolas seemed to not make any sense to me, but now I feel like I really understand the way they work. Now I have the capability make connections with all these equations in the real world.”
Slide08

Marianna, Grade 10

Here are the project guidelines, the rubric, some peer editing forms, the “reflection assistant,” and a .pdf, which has a range of student work.  My rubric borrows from the I.B. Math Internal Assessment Guidelines, as one of my tasks as a grade 10 teacher at my school is to do some specific preparation for the I.B. program in grade 11.  Thanks in advance for any feedback on this project, and on the guidelines and rubric.

Desmos Drawing Project Guidelines and Rubric

Desmos Peer Feedback

Desmos Drawing Project Reflection Assistant

Desmos-art-project-student-work-2015-16-updated