Formulaic Teaching

UntitledAll trig, all day.  Trig in Algebra 2, trig in Geometry.  I’m a pretty happy camper when I’m talking about the unit circle and all its marvels, the interwoven patterns that emerge with every bit of new information.  I’m not sure all of my students feel the same (actually I’m fairly certain many of them don’t,sadly), but Saraphina gave herself a private Double Thumbs Up when the derivation of the sine of the sum of two angles became clear to her. Gotta love that kid.

In my slower track Geometry classes this week, we’ve been working on the Law of Sines.  We derived the formula together, and have been using it first to find side lengths, and then, using the inverse, to find angle measures.  I began with the entire class working together, but the addition of the use of the inverse function definitely split the class into several levels of comprehension.  So today I developed a tiered practice set (link below) and got them working – relatively independently. (Truth be told, I could use a clone or two Untitledin those classes.)   This will take another full day of practice, and we haven’t even touched problem solving.  I’m beginning to wonder whether introducing the Law of Cosines after this topic will have as much value as I had hoped.  We are moving into Volume and Surface Area next, for which I have more hands-on and accessible problem solving activities.

In my Algebra 2/Trig ‘gifted’ track classes (intentional air quotes added), we began the exciting unit of trig formulas.   I absolutely hate giving students formulas without context, and without deriving them.  But here’s the thing – or things: (a) in the unstoppable march towards the Regents exam, there is no time to do this and still complete the curriculum; it doubles the lesson time, and (b) many of my ‘gifted’ students are not quite ready for the algebraic fortitude that deriving the formulas requires.  I know they will revisit this next year in Pre-Calculus, and will definitely cover this ground in the future.  And I have learned – the hard way – how easy it is to lose half of the class doing this.

And yet –

I won’t lower my expectations.  I respect all of my students too much to say they can’t do it.  So I have been reflecting on this week of practice with formulas and the value therein; it’s not terribly inquiry-based, or discovery-oriented.  But after listening to a lot of conversations, and answering their questions with more questions, I’ve developed a more nuanced view of what might appear to be mechanical work.

EverymanIn my Geometry classes, for instance, my students really, really want to solve these problems correctly, and consistently.  This practice is pushing them to extend their understanding of triangles beyond the all hallowed Right Triangle to EveryTriangle.  Many of them still struggle with orienting the angles and their ‘opposite’ sides, regardless of the tips and strategies I’ve suggested.  They are calling upon the basic triangle content from the fall term to aid their understanding, and the connections, for many, are visible (and rewarding to see).  There are students who need more intervention than I realized – this activity surfaced some problems very quickly and clearly, like not being able to distinguish between an angle measure and a side length.  [How a student has gotten to junior year in high school without this being addressed is for another post, or rather, another rant.]  When the bell rang today, the kids asked for another day to complete the practice.   Growth mindset in action.   I love their perseverance, and I love to watch them help each other.

In my Algebra 2/Trig classes, we began with the Cosine Sum/Difference formulas yesterday, and moved on to the Sine formulas today.  I gave them a full day of just using the Cosine formulas, introducing the idea of using them to prove relationships (such as cos (-x) = cos (x)  at the end of the class.  I began the sine formula lesson in my first class of the day with deriving the difference formula; the students worked with me, but we had little time for practice.  In my other classes, I let them practice first, leaving an open Screen Shot 2016-03-31 at 7.43.45 PMquestion on the board.  After a while, students – who were comfortable enough with the work to think about this – began to ask for assistance in working this through.  I found myself talking to pockets of students, only to look up and see others joining the group.  (Link to practice worksheet below.)

And these days of practice have had value beyond their primary objective of mastering the use of the formulas.  The connections in trigonometry are prismatic, and the more one explores and works with it, the deeper they become.  Working on these problems reinforce the concepts of reference and quadrantal angles, Pythagorean triples, and the big idea of proof.   The students support one another well (thanks to Visibly Random Groupings almost every day!), and are less hesitant to ask for help from me.*

Side note: In my Geometry classes, as I mentioned, I need a clone.  EVERYONE wants help, and NO ONE wants to ask a fellow student first.  The students become angry when I won’t tell them whether an answer is correct, but rather suggest a strategy for checking it.  In diametrical opposition are my Algebra 2 students, who are LOATHE to appear stupid, and hesitant (complete understatement for most of them) to ask questions in front of the class.

tumblr_o27lw8t5mi1qlxdvro1_1280When I put my plans together for this week, I was concerned about these double duty practice classes.   I never imagined, actually, that I would be blogging about it.  And yet watching my students work on what I was afraid was too mechanical has actually been a lesson for me, one which – naturally – poses more questions.  How do I insure that my students get the procedural fluency they need all the time, while keeping the classroom full of inquiry and math talk?  And the eternal question, how do I get them to do some of this learning independently?


I guess the questions are what keeps me going.

It’s personal.

Every now and then, someone from the #MTBoS writes a powerful and deeply moving piece.  Sometimes I know these people, sometimes I don’t, but reading their stories can be deeply moving for me, and opens up a connection.  Someone takes a brave step, and shares something important, not knowing how they will be viewed, or judged, but willingly taking that personal risk.  I’ve admired these people, and wondered whether I would ever share my story – what purpose it would serve, would it help anyone, would it be for anyone but me?  I guess there’s a little bit of that in every blog post I write.

But now I think I get it.  There are certain things that you may want understood about yourself,  which may or may not make some people uncomfortable, about which you don’t want to be repeatedly explicit.  And while there is always the possibility of negative reaction, the discomfort of the masquerade outweighs the fear of consequence.

Seven and a half years ago, I was given the diagnosis of invasive lobular carcinoma – breast cancer – and was treated with surgery, reconstruction, chemotherapy, and follow-up hormonal therapy.  Given family history and the fact that over 12% of American women will get breast cancer at some point in their lives, this isn’t really a shocking event; if you are female, someone in your circle (1 in 8) is going to experience this.  It was during my third year of teaching, and I was fortunate to have wonderful doctors, good insurance, and amazing personal support at home and at work.  The kindness and generosity that was shown to me during my illness was humbling, and I will always be grateful for every bit of it.

I had a recurrence scare – which turned out to be not – two years later, which launched me into the next level of treatment – monthly visits with my oncologist (who, thankfully, I adore) accompanied by injections and  blood tests.  My visits were medically boring (again thankfully), and, as he examined me, we discussed everything from our children to our

travels (he gave me a list of Park City restaurant recommendations on a prescription) to the state of public education.  I was scanned several times a year – CT scans and bone scans, and then PET scans – because the cancer markers in my blood were continually on the rise. There were no changes in my pictures, so my monthly sojourns to Long Island (I followed the doctor when he relocated) fell into nice predictable pattern.

Until February 8.

For the first time in 6 years, there was a change in my scans.  Dr. Raptis ‘was suspicious’ that  it was a recurrence, and I spent the balance of the month visiting the radiology department at Mount Sinai Hospital, to finally receive a biopsy-confirmed diagnosis this week – metastatic breast cancer, two small lesions in one of my vertebrae.  The dark place looming in my imagination since 2009 has manifested itself, finally – something I almost feel like I was waiting for, except I wasn’t.  Not really.  I was hoping to outsmart it.

The good news – it’s treatable, the next level of treatment involves pills (although I would have endured the massive injections forever if they kept illness at bay), and I have been told that the biology of my disease is that it grows very slowly, and is even now barely perceptible.  I have no symptoms, am getting the best treatment available, and I’m stilling seeing one of the top breast oncologists in New York.

The bad news – well, who wants a diagnosis which includes the word ‘metastatic’?

For a long time (like until this week), I didn’t talk about my medical situation with people unless (a) they were with me eight years ago or (b) they were really close friends.  It wasn’t a deliberate act, or a secret, but it was in the past, and [hopefully] irrelevant.  But things are a different now.  I’m different now.  I want to move forward with my life – enjoy my children, be devoted to my work and various forms of professional development, find time to quilt, read, exercise, be with friends.   The days of believing time is infinite fade for everyone as they get older, and that awareness is ever more keen for me.    I don’t, however, want it to change my outlook, except for the better.

So why write this?  Why post it, and create an indelible virtual record?  I guess I don’t want this to be a secret. and I don’t want to have to explain myself.   I want to say ‘no’ to things I don’t really want to do (but think I should) and ‘yes’ more frequently to me.  Sometimes I need to go silent online, and sometimes I just need to tell the whole truth.  And I don’t want to have to explain myself when I do.

Whenever I have read a post in which someone has shared something of themselves, I feel touched by them – entrusted with their confidence, even if that confidence was shared with the virtual world – and grateful for the opportunity to be part of their community.  I’m going to quote Megan Schmidt, someone who is absolutely fearless in her honesty, and in this, a role model for me.

“Let’s lift each other up in a way that helps us grow from the inside and helps us appreciate our own abnormalities as perfection.”

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Me and Ruby the Empath Cat



Not going to jinx it but…

CafKo6RUkAAGETRThe fall term finished amidst a lot of anxiety – students anxious about their grades, me anxious about their reactions to their grades. Regents week was surprisingly restorative, and I was able to give a lot of thought and attention to what I wanted to change/try/implement in the spring term.  I am teaching, in addition to Algebra 2/Trig, two sections of the ‘non-Regents’ Geometry course.  This was a course I developed last year, but the pacing and units have been changed to be better aligned with the Common Core standards.  Thankfully, the composition of the class has changed as well.  Last year, the roster included a combination of sophomores who were perceived to be unlikely to succeed in the Regents class, off-track juniors and seniors who needed a math credit, as well as an over-compliance-limit number of students with IEPs and English Language Learners.    This mixture made for some very interesting times in class, but you can look in my blog archive if you want to read more about the past.  This year, the students programmed for the non-Regents Geometry class have come through a four term Algebra 1 class (don’t ask), and, for the most part, are on track for graduation.  Because of my Problem-Solving elective, I didn’t teach the first term of Geometry, but the reports on the fall classes from the two teachers who did were good.  I want to keep this momentum going, so I spent a lot of time visualizing lessons that were accessible and rigorous, with students doing the heavy lifting in class rather than me.  The first unit includes right triangles and trigonometry, and begins with a rather very dry topic – radical expressions.

Not only can radicals be a difficult topic to motivate, but the range of student experience and understanding can be very wide; this is a topic this is included in the Algebra I curriculum.  My lessons needed to be intriguing enough to keep those students who remember the content engaged, while opening the door for those who do not remember, or perhaps never understood it in the first place.  Thanks to the master of creating interesting activities out of even the driest of topics, Don Steward, my first two lessons were well-structured, well-scaffolded, and engaging.  And thanks to starting the term with the 100 number activity shared by Sara Vanderwerf and Megan Schmidt, and classroom seating using  Visibly Random Groupings every day, my students are already talking to each other mathematically, as well as beginning to work cooperatively and supportively.

UntitledBy opening with a warm-up which related square roots to area, I was able to give the topic a visual context, as well as introduce the idea of a cube root.  We reviewed perfect squares through 144, and I was impressed to see how seriously the students were taking this content that I am certain they have seen before – without being prodded or asking “do I need to copy this?”, they all created tables of the perfect squares in their notebooks.  [Side Note: I am going to do a modified INB in this class (I’ve been given notebooks and supplies thanks to  a Donorschoose grant which was funded Untitled2in record time!), easing up on page numbers and exactly copied content.] We then looked at determining between what integers some irrational square roots fell, which helped the students understand that not all numbers have perfect squares, but still have square roots.  The final activity on the Untitled3first day of the lesson was an exploration of the representations of square roots.   It was great to hear the students working together on each of these small activities.  I had some challenge worksheets in hand for a few students who knew the content fairly well, but needed both review and to be more deeply engaged so as not to hijack all class participation.  This intervention, too, was successful. (Dare I say ‘score!’?  I don’t want to tempt fate.)

Picture8Today we continued with simplifying non-perfect square roots.  Again, student understanding at the outset of the lesson ran from a truthful “I know how to do this already” to evident misconceptions.  Despite the fact that a lot of the content on this warm-up previews next week’s work, it got most of the students working – even if they didn’t remember exactly how to simplify radicals, they began messing around with the first example.  The increasing difficulty of the problems kept the students who were ready interested.  I wrote the perfect squares through 144 on the board for reference in looking for perfect square factors, although [and I’ve seen this before] most students resist using that support and instead use guess and check with a calculator.  When they worked on p204some practice examples, I gave them the worksheet in a page protector with a dry erase marker, and distributed only enough so that they needed to share.  Friday afternoon before a long weekend, after a very snowy morning – they worked all period.  I’ll take it.

An interesting note regarding the puzzle worksheet (I’m sure you’ve seen these or some like them; a successful solution to all problems rewards the student with a silly pun.  However, for my English Language Learners (13 of which were put in the same class without any kind of heads up to me), the puns are meaningless.  I think there’s a literacy lesson in here for me to develop.

A huge amount of thought went into these lessons on my part; I’ve taught radical expressions almost every year of my teaching career (now numbering 10!) but these lessons were more engaging, I think, for the students AND for me.  I like that I was able to improve on something I’ve done so many times before, even with borrowing all of those wonderful materials, and I’m pleased that I saw an appropriate result.  I’m looking forward to revamping more lessons for this group of students – their involvement and effort is a wonderful reward.


#MTBoS 2016 Blogging Initiative Week 2 – My Favorite


I’ve been watching the posts on this topic go up all week, and, as usual, have been taking notes on the wealth of resources my online community has been sharing.  It’s been a rough week, despite the Monday holiday; it’s been an end-of-term-grade-anxiety, getting-over-stomach-virus, finally-winter-has-hit kind of week, and to be honest, nothing was feeling too favorite to me except the weekend ahead (and delicious it is, I must admit, cozily watching the snow outside).  But I realize now that I have had a chance to breathe, that there are MANY things that are favorites of mine, so I’m going to highlight just a few of them.

My Favorite Strategy/Routine I love my weekly math maintenance routines.  Regardless of the group of students working on these, the engagement level is high.  I didn’t invent this202_answerstrategy – it was gleaned from at least one of my colleagues in the Blogosphere -most notably, Jessica over at Algebrainiac.  And my favorite part of this weekly routine?  My students inevitably ask me if I am making the solution videos over
at  These oh-so-cynical children became quite passionate this year over the legality of stirring down the marshmallows so that they melt a bit in order to fit more into this cup of hot chocolate.  Fun times.


‘Fata Morgana’ by Teresita Fernández at Madison Square Park

My Favorite Site for Last Minute Resources Hands down, this has to be Don Steward. Whenever I need something for a challenge,  a little differentiation, a different entry point, or another way to look at/practice a familiar topic, this is the place to go.  Where does he come up with these things?  Where do all those great banner images come from?  I don’t know the answer to either question, but I do know that this blog never disappoints.

IMG_7067Favorite Classroom Moment(s) Even though at times it has been the bane of the semester, my 6th period problem-solving class has also provided some of the term’s best times.  On several occasions, the solution paths to problems posed in this class have sparked huge debate.  The kids love to complain about the work, but when I distribute the problems for insertion into their notebooks (there is nothing like glue and scotch tape to unite a fragmented table of kids, even at age 17), they immediately begin cutting, pasting and arguing.   The conversation over City of Truth versus the City of Lies from Math is Fun (another great resource for challenges AND content) can only be described as passionate and delightfully cacaphonous. Screen Shot 2016-01-23 at 1.42.29 PM

Non-Math Class related Favorite Video of the Week I haven’t been watching Stephen Colbert since he made his move to late night network TV, but this video may have changed all of that.  Makes me laugh out loud every time I watch it, and I’m up to my fifth go round so far.  Cheers!

A Day in the Life

It’s 6:40 am and I’m waiting for the next bus, having missed the last one by seconds.  It’s been that kind of a week: a stomach bug combined with January blues/end of term apathy.  I’m hoping this day gets some zing to it – after all, it’s my blogging initiative img_7038kick-off! – but on a cold Friday when you’ve been sick, well, some teacher days are just like that. Best part of early morning commute:  quiet.


Today is staff college pride day; I don’t actually have garb from my own alma maters so I am sporting a Rutgers t-shirt, with a large red R emblazoned across the chest (ubiquitous on campus) for Marilyn,  layered with a modest grey sweatshirt with the artfully quiet MICA logo, for Geo.  In case you’ve never read my blog before, I’m pretty proud of both of those kids and their educational accomplishments.

IMG_7040Two relatively brief bus rides and I’m at school.  The sun is up (almost), and I’m looking forward to my first Algebra 2 class; they’ve been a great group to start my day with this term – hard-working and always willing to try.  Plus they get my jokes, such as they are. You can’t ask for much more than that at 8 am.  I’m going to try my best to spice up higher degree polynomials for them.  A puzzle? A game? Card sort?

Once I’m in my office, with 35 minutes to the first bell, I decide re-inventing today’s lesson is not the best use of my time; I’d rather take the opportunity to open the floor to some college-bound conversation.  I cue up a range of animations that Geo has producedIMG_6858 copyat MICA; I know on a Friday morning, the students will find watching them a treat (as will I).  I head down to the classroom with my trusty cart-o-supplies (bottom left of photo), looking forward to the cheery greeting I always get from the early crew.    I treat the early-comers to a brief animation – Geo’s work with scientists from NASA working on the Fermi Telescope.

As the kids pile into the room, they begin pulling out the tasks that are due today, and comparing work.   I’m excited to see what they’ve done with this assignment – I’ve named it “Jeopardy”; I supplied four answers with accompanying topics – they need to write the questions.  I’m looking forward to blogging about this after I’ve gotten a look at their work.

They settle down, and the lesson begins.  It’s not a favorite of mine – reviewing a lot of procedures for finding the roots of quadratic functions and applying them to functions of higher degree – but the kids need the review, and to be honest, I’m not up to a super creative discovery lesson after this week.  I’m pleased to see the students working together, going through their notes, and synthesizing the many bits and pieces relating to functions as we work through examples together.  They are, as I have said, a good-natured crew, and I make a mental note to do something nice for this class to thank them for starting my days well this term.

With a few minutes to spare in the period, I briefly share my educational resume – the school I most recently attended (Pace University) and those not so recently, SUNY Albany and Columbia.  I spend more time telling them about MICA and the Mason Gross School of the Arts at Rutgers; I think it’s as important for them to hear about creative fields in education as it is for them to think along specific career paths.  And I give them one more video.

The next class is pretty much a repeat of the first, although my third period students are a little bit more serious than the previous bunch.  I don’t know if it’s the hour, and they’re more ready to work, or the children themselves.  This class does benefit from my running through the lesson once already – even though I’ve taught these lessons before – we may, on occasion, dig into the content in greater depth because of this.  Predictably, perhaps, however, my relationship with the first class is warmer.

After the first 2 classes, my teaching day is 40% over.  And it’s only 9:40!  (Isn’t math grand?)  I have a double period free, which allows me to really dig into grading, work with students, read, plan – whatever the order of the day is.  Today,  some logarithm exams beckoned during the first half of my break .  I was pleasantly surprised at how well many of the students did on the extended response portions of the exam, and how efficiently the grading went.  Such will not be the case with the aforementioned task;  a creative assignment SHOULD MEAN widely different submissions, requiring a lot of individualized grading.  My rubric will hopefully stand me in good stead.  During the second portion of the break, my two monitors, Rachel and Tiffany joined me.  These girls are juniors in one of my Algebra 2 classes.  They are both transfers from another high school, and lovingly do whatever task needs done –  updating iPads, taking things out and putting other things in to page protectors, picking up photocopies, and cutting worksheets with the paper cutter (this is, interestingly, their favorite…)download.  They never take advantage of the fact that they are in my office and have access to – well, everything, and they are as reliable as the day is long.  I miss them when they have lab on Wednesdays.  Today, after the requisite errands, they asked me to look at their exams.  We went over them together, and while they were not 100% happy with their own performances, they characteristically did not try to exert undue influence.

The other 60% of my teaching load occurs in a long block after this break.  I teach two classes of Problem Solving, and finish the day with one more section of Algebra 2.  I’ve written mostly about the Problem Solving classes this term; my relationship with them is, sadly, frequently unpleasant.  With six days in the term to go, the students were working on a problem set utilizing the strategy “Identifying Sub Problems”.  In an effort to elicit original work from everyone, there were four versions of the worksheet (similar problems, different numbers).

Attendance has dwindled severely in these classes, but those students who are coming to class are working – many in an effort to keep their grades above 65.  Most of the students who are working require a lot of support, which I am happy to give.  I won’t reiterate my disappointment with the classroom culture that has evolved in both sections, although there is a perceptible difference between them.  In the earlier class, despite my frustration with the work ethic I have not been able to elicit from the students, I have a fairly open and honest rapport with many of the kids, and there is, ironically, mutual appreciation and affection.  The students know that my behavior in the classroom comes from a belief in their native intelligence and a refusal to let them off the hook regarding what they might claim is a lack of ability.  And even though they might wish I would give them a passing grade and leave them alone, they respect my appraisal of their intelligence; many, many teachers do not give them that benefit of the doubt.  However, this is not the case with the students in the later section of the class.   I have not been successful in getting many of these students to shut off their phones, do their own work, or accept my guidance.  I know many of them think I am unfair, and plainly, a pain in the ass.  I’m not going to belabor it any more.

My final class of the day – another section of Algebra 2 – has a different flavor than the morning sections, and affirms my belief that high school math classes can be most successful before 11 am.  They are an engaged group of students, and work well together, but the energy level in the room after lunch time is not always well suited to deep mathematical thinking, especially on a Friday.  We talked at the start of class about college options – this class actually questioned me about the sweatshirt I was wearing before I introduced the topic.  The kids were very interested in the creative paths my own children took – after all, they (my progeny) had a math teacher as a parent, went to well-respected public high schools in New York City, and were making their way with artistic rather than professionally oriented majors.  I loved the thoughtful looks on my students’ faces as they asked questions and digested the answers.  Afterwards, they worked their way steadily through the lesson described earlier in this post (oh too many words ago) in order to watch Geo’s JuniorThesis animation, definitely posted here before.

I finished my day at school bubbling in my attendance sheets, grading a few more exams, and checking in with my new colleague who is struggling with a tough program – a lot of slower track Algebra 1 (who thought four terms of Algebra 1 would be a GOOD idea for ANYONE??) and some off track seniors.  She came from a very rough high school, so this placement may be less stressful, but she’s definitely not been given any slack in her course load. Politics sucks.  I made sure my supplies were on hand for Tuesday’s classes, and closed up shop in the Bat Cave, as my office is affectionately called.

I actually made it out of the building at a reasonable hour, savoring the start of the three day weekend, and contemplating the long view of the last week of the fall term.  It’s been an intense one, personally and professionally, and to be honest, I can’t say I’m unhappy it’s almost over.  The good thing about teaching – we get do-overs every term, every week, every day.

If you’ve read this far, well, thanks for sticking through it with me – a day in the life of this teacher.  It’s a ridiculously long post, I agree – but it only reflects the nature of the work.  How does that saying go – you don’t have to be crazy to teach, but it helps?

And here’s a animation for YOU!


Proud Participant of the 2016 #MTBoS Blogging Initiative

I, Wendy Menard, resolve to blog in 2016 in order to open my classroom up and share my thoughts with other teachers. I hope to accomplish this goal by participating in the January Blogging Initiation hosted by Explore MTBoS.

You, too, could join in on this exciting adventure. All you have to do is dust off your blog and get ready for the first prompt to arrive January 10th!


Why I love logs

I have had two wonderful days of discovery and student-centered learning while introducing logarithms in my Algebra 2 classes, thanks to Julie Reulbach and Kate Nowak.

Last month, Julie asked me for my log word problems (which include zombie attacks and other infectious issues). I usually use these after I have introduced logs, but ever-wise Julie used the problems to create the NEED for logs. Brilliant. Using Julie’s idea, I modified my original lesson similarly.

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The students intuitively realized that they were using the inverse of an exponential function – several wrote the equation quickly (and then proceeded to use guess and check) or created a table. Those students who made tables saw that the answer was between 7 and 8 hours, but when I told them I needed a more specific answer, they, too, moved to guessing and checking. (This process led to a nice mini-lesson on the use of the TABLE and TBLSET functions on the graphing calculator.)

After reviewing the answer to the problem, I moved immediately to Kate Nowak’s Super Fun Log Puzzle.  Screen Shot 2015-12-19 at 9.05.36 AMThe kids worked through the ‘puzzle’ fairly quickly, and were able to articulate what they were doing. When I revealed that they could replace the word ‘power’ with ‘log’ – well, I love that moment – the moment that they realize that something they thought would be difficult wasn’t really that bad, or that their friends were wrong when they said logarithms were log10_tables_4_figureterrible and mysterious. [There is a moment in the lesson when I show them a log table from the back of a textbook – the kind I used back in the day. They are suitably horrified by life before epidemic calculator use. Another moment I love.] And the room was filled with math talk all day long – students helping students, students arguing with each other, students making sense of something new. A pretty easy day for me as well.

Today, I followed up with another of Kate’s activities (I owe you SO much, @k8nowak!) – discovering the properties of logs. The students began the work the moment they entered the room, and they were engaged in the process all period long. I steadfastly bounced all questions back to them, referring them to their tablemates as resources. I nipped calculator usage in the bud, except as a checking device.

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And, as I tweeted earlier, all I did was walk around the room listening for 30 minutes, occasionally redirecting if I saw a table going seriously off track, or asking some pointed questions to spark some recognition. Towards the end of class, we recapped their discovery as a whole class, and the relationship of the ‘laws of logs’ to those governing the use of exponents were universally clear. And the concept of inverse functions was reinforced once more, when viewing the relationship between logarithms and exponential functions.

After the difficulties I’ve been having in my problem solving classes, hitting a home run three times in a row in the Algebra 2 classes felt as good as you might imagine it would.

At the end of the day, I found myself in a conversation with an officemate who teachers Honors Pre-Calculus; he is finding that his students are weak in the skills that they should have mastered in Algebra 2, surprisingly so for that level of class. These are the students who will be going into AP Calculus next year, one third of them into BC Calc in fact, and this teacher is trying to strike a balance between making sure that their requisite skills are strong enough to handle the difficult conceptual work to come. He cited logarithms as a particular area of mystery and antipathy for his classes. I shared these lessons with him (the lessons are quite unusual in my school in their level of discovery), and I could see that, like the zombies, the idea might be infectious.