Blog Archives

Day 25: Music and Mathematics

This evening was the PSCTM Fall Dinner: Where Math Meets Music: The Harmonic Series.  Sean Richarz who teaches Band, Jazz Band, Music and Popular Culture and Guitar along with being a performer in the area as a trumpet player, conductor and the lead guitar player in his Pink Floyd Tribute band “Green Floyd”shared his thoughts about where math meets music.  We looked at scales, the harmonic series, sound waves (our beloved sine waves) and transformations in light of music composition.  It was really fascinating to experience how the mathematics sounds!

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Day 179: Summer Reading

This summer I want to get lost in reading!  I’ve been compiling a list over the year based on tweet recommendations, blog referrals and face-to-face endorsements.  I’ve whittled down the list to these 5, although I can already tell that I may add a couple others.

  1. A More Beautiful Question – Warren Berger
  2. Mindset – Carol S. Dweck
  3. The Falconer – Grant Lichtman
  4. 5 Practices for Orchestrating Productive Mathematics Discussions – NCTM
  5. Embedded Formative Assessment – Dylan Wiliam


What are you reading this summer and why?

Addendum: I’m adding “The Doodle Revolution” to my list.  I’ll be posting more about this over the summer.  I am so excited about it.  It merges my left and right sides of my brain and gives me an outlet for my artistic side.

Day 174: A More Beautiful Question

I’d say over the past 10 years, I have been reflecting about and trying to improve my questioning techniques with my students.  It seems to be a common thread in much of my lens as I plan each year, attend PD and read.  I have noticed over the years that my questioning is getting better (although it’s got a long way to go…will I every be a master questioner?!).  a more beautiful questionI bring this up because over the last year as I became a proficient Twitter user, I saw lots of tweets about teacher and student questions in the classroom and references to blogs about classrooms where math curriculum is explored through student questions.  This was a new twist on questioning that has sparked my interest.

One book (and author) that seems to be referenced often is A More Beautiful Question by Warren Berger.  I decided I wanted find out for myself what the big hoopla was all about!  I added it to my list of books for my summer reading.  I ordered it and some others through Amazon and it came yesterday!  Excitement!!  I began to read just the beginning and I ended up reading two chapters.  And I need you to know that I don’t have that kind of time right now to be reading that much, but I just couldn’t put it down.

Here are some quotes I’ve already jotted down:

p.8 by Warren Berger giving the definition:

A beautiful question is an ambitious yet actionable question that can begin to shift the way we perceive or think about something – and that might serve as a catalyst to bring about change.

p. 16 by Stuart Firestein on the potential of a good question:

One good question can give rise to several layers of answers, can inspire decades-long searches for solutions, can generate whole new fields of inquiry, and can prompt changes in entrenched thinking.  Answers, on the other hand, often end the process.

p. 23 by Tony Wagner, a Harvard education expert and Paul Bottino, a Harvard innovation expert on “expertise”:

Known answers are everywhere, and easily accessible.  The value of explicit information is dropping…the real value is in “what you do with that knowledge, in pursuit of a query.”

p. 18 by Stuart Firestein on questioning:

Questioning is a very sophisticated, high-level form of thinking.

And this particular quote got me wondering, if I value questioning and want my students to know that I value it, I must assess it.  Which led to me asking, How can I assess a “good” question?  What are the components of a “good” question?  How can I help my students grow their ability to ask those “good” questions?

What are some books you are planning to read this summer?  What is motivating your choices?  How do you anticipate using the gained knowledge in your practice?

Day 132: My Twitter Trembles

Guess what?!  An old dog CAN learn new tricks!  I’ve been teaching way north of 30 years, but I’ve always prided myself about keeping current, especially with technology.  Yet I have to say, these last 5 years, with social media and phone apps seemingly taking over our lives, I’ve begun to feel like I’m missing out on connecting with this new generation of teachers, and students.  I do not have a Facebook account (but lurk behind my hubby’s account – aren’t I pathetic?!), but I did start a Pinterest account about two years ago (at the urging of another teacher friend) and really love the anonymity and ease of access.


But, try as I might to understand and use Twitter as a PD source and to connect with educators around the globe for the last two years, I’ve felt my age.  While at the TI 2013 conference, I attended a PD that encouraged us to sign up for Twitter and gave us the basics (and I mean basics).  Since then, I just couldn’t seem to understand how to follow conversations and was deathly afraid to post.  I didn’t even know what some of the abbreviations (like DM and #stuff) meant.  I’d even attended the Twitter for New Users meeting groups at various conferences.  Once again, I was a lurker on Twitter, a wall flower watching everyone else seeming to have a great time.

So I decided late last  year that I really wanted (needed) to figure Twitter out.  I connected with our Technology TOSA, pleading with him to help me get into the Twitter world. He has really helped me get a sense of what is going on without making me feel my age.  I pinned a lot of sites about using Twitter in the Classroom/School in Pinterest to at least get more familiar with the lingo (and this helped me understand how not having the right words can be a barrier to access).  I practiced at a conference posting and such.  Still had trouble following conversations or finding posts I wanted to see again, but my Tech TOSA hung in there with me.Tweet

This year I posted photos of sessions I attended at our regional conference, learned how to hashtag (although I’m still learning to attach the #name to the conversation).  And I actually joined my first chat group called #Statschat.  Because of this risk-taking, I’m connecting with people who write impressive blogs (that I’ve followed for years).  When I was having trouble with one of the Geogebra pages I wanted to use with my kiddos, I contacted the original author @giohio via Twitter and he had it fixed within an afternoon.  This evening I’ve been collaborating with another AP Stats teacher about a great graphic he’s putting together to connect hypothesis tests with the sampling/experimental method.  It has been a kick.  And I’m glad I stayed tenacious about learning this new medium for collaboration.

What risks have you taken this year to improve or grow your professional life?

Day 126: Do We Have It Yet?

So yesterday’s Professional Development opportunity was AWESOME!  Today, back to reality.  My AP Stats kiddos have been left on their own a great deal this past 6 class day with me gone for 4 of them.  But they are troopers.  They were left with the task to learn about and understand confidence interval procedures for two means.  Today I wanted to summarize with them what they know, what they need to fine-tune and what eluded them.  So I wrote on the white board Step 1, Step 2, Step 3, and Step 4.  And they told me what needed to be filled in using the Additional Practice problem from their notes.

Helium Balloon Problem

They actually articulated well the key and crucial parts of the process.  We had a little calculator issue as you can see, which led to using the calculator more efficiently and effectively. We also revisited the distinction between the standard error vs. standard deviation along with finding the critical t-value.

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We’ll see how the Frappy’s go next week.


Day 125: Intellectual Need in the Math Classroom

I can’t believe I’m out of the classroom AGAIN, but I just couldn’t miss the opportunity to hear and interact with Dan Meyer in his workshop: Intellectual Need in the Math Classroom offered by our local educational service district.  So many good things today, but I did come away with one nugget that I’ll be ruminating on for the rest of the year (teaching life?!).  Dan started the workshop by posing the question: How do we engage students in difficult mathematics?  And he suggested the three biggest responses (by teachers and textbook companies) are:

  1. Make math real world
  2. Make math job related
  3. Make math relevant

We then took a humorous look at these “suggestions” in action: interesting covers (because we want the kids to like what we have to offer, so we’re desperate), have career interviews or real-world connections dropped into the middle of a unit (make the work seem related to job acquisition), try to have real-world problems (do something to a real-world picture to link it to the math we’re studying – but almost lying to the kids?) or try to take an uninteresting problem and re-work it to try to connect to what kids might relate to (but does an image of Starbuck’s coffee make the problem about exponential growth any more engaging?).

In Dan Meyer fashion, he offered a Dandy Candy Video and activity (which can be found at 101Questions) to look at strategic moves to engage students at the beginning of the class.  And through the debrief of our experiences, we revisited the question:  How do we engage students in difficult mathematics?  And Dan’s answer is:

  1. Start a fight
    • instigate an intellectual (or emotional) fight
    • get them arguing with each other and you
    • use student answers to get their opinions out
  2. Turn the math dial up slowly
    • start with their everyday experience
    • make the problem vague and bring in the math as needed
    • have students guess high and low answers, best and worst, etc. along the way
    • slowly add vocabulary and layers of “math” to the experience
    • you can always add to their experience.  You can’t subtract what has been done. So think before you give the “math” component
  3. Create a headache – to provide the “aspirin.”
    • ask questions to get the students to think more deeply
    • ask student to describe how to do something without the math tools – they’ll “beg” you for it eventually, if you don’ violate #2
      • for example, ask students to describe a precise location without a precise tool (grid system.
      • in the Dandy Candy example, ask to determine “the best.”
    • challenge their thinking to the next level: more precise, more efficient, best way

It was an invigorating day, with lots of conversation and pushing on our everyday practice.  I really appreciated Dan’s disclaimer that you don’t do this every day – you’ll burn out quicker than a flame in a hurricane (my analogy).  He suggests only one of these kinds of activities per unit.  Lure the students to the mathematical water through an engaging and meaningful activity, but then its okay to provide direct instruction as needed to develop skills and sense-making activities to develop concepts.  I have always believed balance between activity-based instruction and direct-instruction makes for the most productive and growing classroom.

I’m always thrilled to talk with other educators to learn new things and today provided that opportunity for me.  It was a delightful and fruitful day.



Day 121: Fortunately, Unfortunately

Today I gave a presentation called “Fostering a Math Practices Mindset” in which the participants explored the CCSS Standards of Mathematics Practices followed by examples of how I’ve tried to incorporate these practices with and without technology.

fostering a math practices mindset

I sort of felt like the children’s book: Fortunately, Unfortunately.fortunately unfortunatelyUnfortunately, the technology available to me during the session didn’t work so well, so I limped along with slow internet and missing applications on the loaner computer.  Fortunately, the participants were great with lots of enthusiasm and energy, even for a late afternoon session.  I know they got a lot out of it ‘cuz they said so!  Lesson learned…bring your own computer!

And tomorrow is the “Pi Day of the Century” so we are to have pie on 3:14.15 at 9:27.

Posted 3/14/15:  Here is just one of the many pie-spreads throughout the conference


Day 120: Candy Store for Math Teachers

I just love this conference.  There is so much to see, do and learn.

Opening speaker was Jo Boaler talking about Fixed and Growth Mindsets in Mathematics

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Then the rest of the day I attended a variety of sessions using and not using technology, using and not using the math practices, using and not using engagement techniques.  It is really interesting to see how certain strategies in presenting make such a difference

Once strategy I found intriguing is “My Favorite NO-Learning from Mistakes,” presented by Sharon Bruce.  Have to think about how to make this happen in the secondary upper math classroom.  Of course, with a good idea, there is always a way!


Day 119: Learning Progressions as a New (Old) Practice?

One of the PD’s we had today was called Focus on Learning: Pathways, Checkpoints, Adjustments.  A big part of the workshop was around writing learning progressions for a particular lesson.  Jennifer Wilson shared some of the work she and Jill Gough have done in creating posters around the Mathematical Practices learning progressions.  Here are some examples created by Jill and Jennifer:


This is the biggest idea that stuck with me during this day.  Lots to ruminate on!


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