Takeaway #1 from #NCTM2019: Equitable teaching practices.

This is my first post in a LOONG time.  It’s been quite a year. Without getting too detailed:

  1. I left my last school (after 18 years) to help build a mathematics program at a new school. I love this new community, and I love the support I’ve received to build a math program guided by equity, inquiry, safety, and good teaching & learning.  It will be a long journey, but one I am glad I began.
  2. I know I needed to revisit lots of the professional development I’ve enjoyed in the past 10-15 years through this new lens. In many ways, I’m “Back at One.” In particular, I need to more deeply understand my responsibility to teach in a way that convinces my students that they will be okay taking risks, being themselves, and making mistakes.  Too often, the  fear of judgment, being dismissed, being deemed stupid / lazy / disruptive.

Last week I spent a killer week in San Diego attending The National Council of Supervisors of Mathematics and National Council of Teachers of Mathematics  research sessions, events, and presentations.

To summarize all of it?  I felt welcomed.  I felt appreciated. I felt like my colleagues and friends were ready to hold me accountable. I felt challenged. I felt like part of a team.

It reminded me of when I saw this on the Oprah show in 2000, with Toni Morrison:  “Does your Face Light Up when they enter the room?”  

The session that had the most impact, among a slew of very good presentations:

HANDS DOWN:   Dr. Imani Goffney, University of Maryland:  “From Oakland to Wakanda:”  Transforming Mathematics Classrooms to Become Empowering Spaces.  I cannot overstate the impact this presentation had on me.  I was most impacted by that fact that  I grew and blossomed as a teacher because of the compassion, kindness, and expertise of colleagues who took active steps to believe in my potential and see brilliance in me.  I was emotional upon reflecting how crucial these assumptions and extensions of trust were to my own growth as a student and as a teacher.

She expertly revealed common assumptions and practices that shut down these possibilities for students of color, and ways to move our classrooms from de-humanizing experiences to re-humanizing experiences. What I especially loved: It is grounded in deeper mathematical understanding, and deeper understanding of the students I teach. Two ways to re-leverage common practices:

  1. Work from the assumption of the brilliance of Black and Brown students (like we unconsciously do more often with White and Asian students).
  2. Collecting feedback on the impact of my own decisions in class.




Me, Dr. Imani Masters-Goffney, Dr. Nicole Bannister

This is what I will be working on.










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Twitter Math Camp 2017: “I want to cultivate a relationship with you.”

I’m back home in West Hollywood, enjoying coffee and recovering from a wonderful week at  Twitter Math Camp 2017. What the hell is Twitter Math Camp (TMC)?

Well, it’s a get-together of kindred spirits who have connected over the years  who have ambitious goals to help folks learn and love mathematics,  The loosely connected group gave themselves the name the “MathTwitterBlogosphere (#MTBoS).”  It’s one of the many communities of teachers that I identify with.

I have had the privilege of belonging to many different communities of teachers since I started teaching in 1995. Let me list some (with hashtags – some real some not):

  • The Gay Lesbian Straight Education Network  (GLSEN)
  • The Loomis Chaffee School  (LC)
  • The Loomis Chaffee Mathematics Department  (LCMATH)
  • The Harvard-Westlake School  (H-DUB)
  • The Harvard Westlake Mathematics Department (H-DUB Math)
  • AP Statistics Teachers all over
  • AP Statistics Exam Readers
  • The Park City Mathematics Institute Teacher Leadership Program  (PCMI)
  • the Los Angeles Professional Development Outreach Group (LAPDOG)
  • The Park City Mathematics Institute TLP Staff  (PCMIStaff)
  • Math for America Master Teaching Fellows Program  (MFA-LA)
  • AP Computer Science Principles Teachers Network  (APCS-P)
  • MTBoS: The Math Twitter Blogosphere (MTBoS)
  • Twitter Math Camp Attendees and Presenters  (TMC15, TMC17)
  • The Real HouseTeachers of WestHollywood (Yes! This is a group I identify with during the summers when we are not crazed with appointments and events).

When I look at all of these communities, I think a lot about my feelings of belonging and my feelings of alienation. Within each community, above I have experienced feelings of both.  Why? What made me feel like I belong? What made me feel like I don’t belong?  

I do not know whether these are causes or effects  of  my feelings; maybe a bit of both.

  1. Things I associate with feeling like I belong: 
  • People’s eyes light up and their faces brighten when they see me, and I do the same.
  • People say hello when I walk into the room.
  • I don’t need to prove or defend myself when I share something.
  • I may disagree with someone, but I can learn from them by listening and taking the time to understand them.
  • I feel safe honestly sharing feelings – even when there is conflict or discomfort.
  • I talk to members of my community, not about them.
  • I stay longer than I need to. Time flies.
  • We connect with each other “just because.”
  • We are all working hard, together.
  • I can be sad, or angry, or silly, or tired in front of them, and talk through it, and be okay.
  • I feel light, happy, and joyous.
  •  I care about you, and you appreciate and respect me. So I want to give you my very best. 
  • I feel, in my heart: I am enough, and I am worthy of being here with these folks. 

    2. Things I associate with feeling like I do not belong: 

  • I walk into the room and it’s silent: no hello, no looking up.
  • I feel like “I don’t want to bother them.”
  • I feel like this person will present an obstacle to my goals.
  • I am more concerned with “getting this done” than connecting.
  • I  assume there is a focus on “maintaining professionalism” and keeping conversations “focused and brief.”
  • I/others  want to talk  about people, not with people.
  • Ignoring, “cutting off,”  “distance,” and “icing” are viable behaviors to manage feeling hurt or disappointed.
  • Lots of whispering.
  • I worry that you are going to judge me, because you don’t respect me. So I will work hard to prove to you I’m deserving of your respect. Or maybe not, if that hasn’t worked in the past. Maybe I’ll blow it off. 
  • “When are we done?”
  • I feel like I want to get out of this room.
  • I feel sad. It’s hard for it to go away.
  • I feel the lies my brain is fixated on: that I am unworthy, not enough.  

   3. What does this have to do with Twitter Math Camp? I’ll start with what brought me here this year:  This was the first time I had participated “fully” in Twitter Math Camp: I stayed at the host hotel (instead of commuting, and I did back in 2015) and prepared a three-day workshop with Peg Cagle and Cal Armstrong, two deeply cherished colleagues and friends. Many of you have some really strong bonds within MTBoS. We have cultivated ours over 10-15 years.  IMG_0609

We forged these relationships together during some intense, emotional, and deeply thoughtful years writing the “Reflecting on Practice” curriculum as staff at the Park City Mathematics Institute from 2009-2012. Our  work at PCMI is one of the things I am most proud of as an educator.

We applied to co-facilitate a three-day workshop at TMC2017 about implementing rich tasks. But for me, it was  mostly to re-connect, work together, and re-experience the close connections we made over the years. They are two people I respect, admire, and love very much.  We’ve each succeeded and failed each other in our own ways through the years, but we have steadfastly supported, appreciated, and cared for each other in ways that are tough to replicate. When the shit goes down, I want these people in my corner.  I was able to feel safe implementing the workshop and to open up more to others at the conference.  My sense of trust and belonging with them helped me take more risks and develop a stronger sense of trust and belonging with the  MTBoS community. 

4. OK – enough about you, dude.  So what about TMC? I connected with a lot of folks who were also open, brilliant, willing, joyful, and interested in connecting with me. I saw some friends and colleagues I’ve known for a while (I’m thinking of Chris Luzniak, Sam Shah, Lisa Henry, Tina Cardone) emerge as true pioneers and leaders in the MTBoS community and the larger teaching community. I  deeply admire their leadership, brilliance and bravery.  And I got to meet others folks I only knew through tweets, or a single presentation I saw at a previous TMC, APStats,  or NCTM event. But I was thrilled to connect, in some way with folks like Jed Butler, David Butler, Julie Reulbach, Hedge, Glenn Waddell,  Brette Garner, Ben Walker and too many others I forget right now. All of us made connections, and said, at some level:

 I want to cultivate a relationship with you. 

“Whoa, dude… hold on there. I thought we were just playing Pandemic over IPA’s…”  

This phrase is scary to write:  it’s fraught with commitment and expectations that I am not sure I can meet. It’s fraught with uncertainty about the level of depth and connectedness that may develop.  It’s fraught with being the stalker-y needy colleague.

Ultimately, however, I think this is what we are asking for as we forge our “tribe” of colleagues, friends, mentors, and comrades at TMC.  When we open up to be vulnerable to people that (we hope)  are nurturing,  joyful, able and willing.

The details of how that relationship is formed, what it means, and whether it creates mutual joy is a function of  developing our skills in relationship-building.  

This wonderful synopsis of  “How to Love” by Which That Nanh  by Maria Popova from Brain Pickings  does a great job describing some of the complex work that we all must practice in order to cultivate the kind of relationships that bring us sustained joy, value, and feelings of belonging.  This one quote from the book is my favorite:

To love without knowing how to love wounds the person we love. To know how to love someone, we have to understand them. To understand, we need to listen.

… and listening takes time.  So how have I seem this played out among the folks at TMC?   It comes down to spending time with each other and making time and space to understand and appreciate each other. Meals. Games. Songs. Conversations. Conflicts (managed maturely and bravely).  Shared work tasks. Shutting up and Listening. Resisting reactions and pausing until I can genuinely listen and understand.

5.  Beyond TMC -: For many of you, these discoveries are not new. Recognizing their importance is new for me. I am leaving TMC with a stronger understanding of when I feel I “belong” to a community:

  1. How much time am I willing to spend being open, honest, myself, and vulnerable?
  2. How much time am I willing to spend to understand those I’m with? 

When I take the time to understand, ask, share, and take risks, my community improves.

It’s easy for me to  point to examples where others have, in my judgment failed to be loving, thoughtful, honest, warm, or kind.  But that does not give me license to do the same to them. I have the ability and the responsibility to do something else.

I can be braver about being more assertively welcoming and take a stronger interest in who is in front of me.
I can remind myself that those I don’t particularly enjoy are on the same team with me, and are, like me, doing their very best they can today.
I can do more to not dismiss or cut off someone who did something I did not like.
I can do more to listen and understand.
I can do more to share when I have an issue or uneasy feeling about a friend or colleague within my community.

This is my “one thing” I will be thinking about this year as I begin new work with my colleagues and my students.





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Possible Solutions: 2017 AP Statistics Free Response Questions

Hi Colleagues,

After the AP Statistics Exam questions have been released, I put out my own solutions and invite feedback, other approaches, and questions.

Nothing official in these solutions: I have taught AP Statistics for 21 years, and I have graded multiple times. Based in this experience, I put forth the solutions you see here.

Possible Solutions 2017 AP FRQ (First Draft)

I got a bit delayed with putting these out this year, but I did enjoy myself this weekend, so that’s good.  Workouts, friends, and rest are truly essential components of my days anymore. If you are a teacher, don’t underestimate their impact on your own health and well being.  Our profession sometimes glamorizes the false benefits of being the  “overworked martyr.”  I certainly perpetuated this in the past.  Trying to turn over a new leaf in recent years.

Thanks to Amy Crum for her solutions – I checked my own answers with hers before posting them.

Thoughts about the problems:

1: The Wolves problem.  (Basics of linear regression and scatterplots)  Haha!  I LOVE the idea of simply asking what the heck “linear” and “strong” and “positive” mean in this context. I predict that many students will not respond substantively to these questions (You know-  linear means it’s linear!)

2. Water and Soft Drinks. (one sample z-interval for a proportion)  Cool context:  there may be some issues with students correctly describing  the population of interest and the sample of interest in correct context. A common error for my students is to mis-understand the context, and then say something totally incorrect at the end of the problem.

3.  Melons.  (normal models, conditional probability) Great probability problem.  I like the subtle twist on conditional probability in part c).

4.  Pottery.  (making conclusions from boxplots, complex context) Another good,  challenging problem with exploratory data analysis. Students will need to articulate which numbers from which parts of which box-plot(s) are providing evidence for their conclusions.

5.   The Schizophrenia problem. Straight up simple chi-squared test.  I wonder if students will be expected to describe the association after completing the test, or if completing the test is enough.

6.  (Sampling with/ without replacement,  tree diagram probability).  This seemed like a shorter investigative task than in the past.  I wonder what will suffice as “complete” reasoning/ work for each part.  I especially wonder whether student swill be able to “transfer the lesson learned”  from parts (a) and (b) to part (c) without additional work to confirm that this difference  carries through to more complex situations.




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AP Statistics Exam, Free Response Questions: Any patterns?

The free response questions for the AP Statistics exam are sometimes hard to predict.  What trends do exist?

Here is A graph relating AP Free response question Number vs.  Chapter in the  5th Edition of The Practice of Statistics.

1:           Exploring Data (single variable, quantitative)
2:           Modeling Distributions of Data  (z-scores, percentiles, normal model)
3:           Describing Relationships
4:           Collecting Data
5:           Probability, the basics
6:           Random Variables, the binomial and geometric distribution.
7:           Sampling Distributions
8:           Inference: Confidence Intervals, 1 variable.
9:           Significance Tests, 1 variable.
10:         Inference for two groups
11:         Inference for categorical data
12:         Inference for regression

Here’s a graph of the relationship. Here are a few trends I observe:

  1. Question 1 tends to be exploratory data analysis OR an easier test about inference.
  2. Question 2 seems to feature probability OR sampling/ experimental design
  3. Question 3 seems to feature probability OR sampling/ experimental design
  4. Question 4 seems to feature an inference procedure often, but not always
  5. Question 5 seems to feature an inference procedure often, but not always.
  6. Question 6 always is a mixed bag, bot often involves some reasoning regarding inference.

Other observations:

  • “Probability, served three ways:”  Probability questions are often, but not always, three-parters that take a single scenario and then “serve up” questions involving three different techniques in probability.
  • Two sample t  vs Matched Pairs t questions are common.
  • Inference for regression, when addressed, is done pretty lightly, and often in question 6.  There are often many other parts to question 6 that are not about inference for regression.

chapter vs FR_question_number

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Smelling Like Data Science

This gallery contains 12 photos.

Originally posted on A Best-Case Scenario:
(Adapted from a panel after-dinner talk for the in the opening session to DSET 2017) Nobody knows what data science is, but it permeates our lives, and it’s increasingly clear that understanding data science, and…

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The Lesson of Grace in Teaching

Yes, I’ve started a new blog.  The platform here at WordPress seems to be nicer and more versatile.  I’ll be blogging here from now on. However, my first blog post was at Blogger, becau…

Source: The Lesson of Grace in Teaching

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My Possible Solutions: 2016 AP Statistics Free Response Questions

Possible responses to the 2016 AP Statistics Free Resposne questions, draft #1.

You can access the questions here.

Note: I construct these as a service for both students and teachers to start discussions. There is nothing “official” about these solutions. I certainly can’t even guarantee that they are correct. They probably have typos and errors. If you catch some, let me know! But if they generate discussion and help others, then I’ve succeeded.

My first draft: possible solutions, APStatistics FR 2016

Please read, critique, and suggest fixes!


I think that these very accessible questions are attempting to give students a chance to explain their reasoning and thinking with appropriate specificity.  I suspect that students can easily falter in the following ways:

#1:  I wonder if we’ll see students failing to be appropriately specific in using measures of center/ spread. I can see kids giving incorrect values for IQR, and not using range as something much more accessible. I can also see the rubric penalize for not quantifying the amount of increase of the mean.  It possible, so students should probably quantify the increase.

#2.  I wonder if we’ll see students not being appropriately nuanced in explaining the effect of the ads on preference.

#3.   I wonder if we’ll see students not identifying the variables correctly – they will probably identify summary statistics instead.

#4. I wonder if we’ll see students not showing mathematical pathways, and giving a surface-level explanation of part c)

#5. I wonder if we’ll see students not explaining thoroughly enough WHY np and n(1-p) must be greater than 10.

#6. I wonder if we’ll see students not being focused enough in answering the specific question posed in each part.


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