We Really Love Mr. Miller

Posted: March 30, 2016 in Uncategorized

Mr. Miller passed away today. We all process grief a little differently. I think it will help me to write about how much he meant to me and the entire St. Mary’s community. Orion Miller had taught mathematics for a total of 45 years, that is amazing to me. If his health hadn’t failed him, I’m not sure he would have ever stopped teaching. He often mentioned the idea of retiring, but it was always a few years in the future. A few years would go by and retirement was still just a few years away. Teaching for him was much more than a job, he fed off the energy of the students and was passionate about students seeing the beauty of mathematics.

At the end of the last school year he took me aside and asked if I was interested in being the head of the math department at St. Mary’s. I was, but I wanted to know why he was thinking about stepping down. He said that it was his idea to step down, that he thought it would make for a better transition. For the last 15 years, Orion has had a very slow growing type of cancer. Maybe he knew that the cancer was starting to take a hold of him? He never complained about it. I didn’t know it until later that he was going through Chemo during the time that he was teaching this past fall. One thing I did know for sure, if I was going to be head of the math department…I would have big shoes to fill. Orion was a large hulk of a man, but his legacy is what looms even larger.

One of his mantras has stuck with me. For me, it encapsulates the kind of person that he is (was). It was my first year at St. Mary’s and I was talking to him about how great it was to teach at a place where there is time built into the schedule to help the girls who are struggling with mathematics. He said this to me, “I offer all the St. Mary’s girls free tutoring for life.” It didn’t matter if they were in college and needed help, he was always there for them. At Thanksgiving, he officially retired from St. Mary’s. It is a testament to the kind of person he is that some his AP Calculus girls came by his house each week for tutoring. That was his version of what retirement was supposed to look like. He never stopped helping students while he was alive. That was Heaven on earth for him.

I thought it might be appropriate to share a video that our high school math department made a few years ago. We sang a tribute to Mr. Miller to the tune of ‘Rudolph the Red-nosed Reindeer.” Appropriately, we changed the words of the song to, “We  really love Mr. Miller!

That sentiment is still true. We all really love Mr. Miller and his sweet wife, Nancy. Orion had a way of being ornery while letting you know that he really cared. In the video, you may have noticed that I wore the same clothes he did, that was on purpose. It was a shock to him when I went off script and jumped on his lap. Today, I was a little shocked to get the news that he had passed away. I knew that his health was deteriorating, but I guess I always thought it would happen sometime in the future. I may not show it, but I already miss him. Still, his legacy lives on.


I spoke at the MCTM conference in Oxford, Mississippi this fall. One of the great things about speaking at conferences is the opportunity to learn from other presenters. I attended Jennifer Wilson’s session on the slow-math movement. In the session, she talked about the importance of using NCTM’s Principles to Action. I had read about these and thought they had good ideas that could guide instruction. For some reason, I began to internalize a few of them and see if I could implement them more in my classroom practice. Don’t you love it when that happens? Setting a goal and being willing to change the way you have taught things in the past almost always leads to improved teaching. (I am trying to think of a counter-example and I can’t think of one. Even when I try something that fails, I usually learn something  that will shape future lessons.)

There were two principles that I began to think about in particular.

  • Facilitate meaningful mathematical discourse. I want to get students talking about math more in my class. This year, I re-arranged the student desks in pairs so that it would be more natural for students to talk. Sometimes the physical environment in our class can be an obstacle to what we want to accomplish.
  • Elicit and use evidence of student thinking. I have been making a concerted effort to ask students how they solved problems. Then, following up by asking if anyone solved the problem a different way. I think this is starting to pay big dividends in my class. Students are more willing to share and I think they feel more invested in understanding the mathematics.

Could I combine both principles in one activity? I used my AP Statistics class to give it the good old college try. We were learning about the Geometric model. Which, is basically calculating the probability of the ‘first’ of something happening when there are only two possible outcomes.

I started with an easier example to get the ball rolling. I asked the students to discuss the following scenario:

  • Golden Grahams cereal has one of 5 different Star Wars figurines in each box of cereal. If you want to get the Luke Skywalker figurine, about how many boxes would you have to open?

I didn’t realize that the students would need clarification. One group asked if all 5 figurines were equally likely to be in a box. Another group suggested that they thought they might get lucky and get one on the first try. Instead of clarifying the question, I asked, “Do you think that you answered the question that I asked?” They self-corrected the question to mean, what is the average number of boxes that it would take to find the Luke Skywalker figurine. Pretty soon, most groups came to the conclusion that it would take 5 boxes (on average) to find the desired prize.

Since the AP Statistics formula sheet doesn’t provide the formula for the Geometric mean, I wanted to make it a priority for the students to really understand how to find the Geometric mean (with or without a formula).

geometric 2

I gave each group Goldfish crackers. We did a short simulation and determined that 40% of the Goldfish were smiling. Did you know that not all of the Goldfish crackers even have a mouth? Incidentally, after many simulations, I am pretty confident that the factory produces smiling Goldfish at about a 40% rate.


I want to share some student thinking to this question I posed:

  • What is the probability of picking my ‘first’ smiling Goldfish on the third one that I pick from the box? (given a 40% rate.)

After having time for discussion in their groups, here is some of their thinking that they shared:


What an invigorating class! I wish every class was that involved in making the math their own.

Earlier this week I had a young teacher observe me. He asked me, “How do you improve at teaching?” My answer was probably a little scattered. Other teachers who have taken the time to mentor me and answer my never-ending string of questions have certainly had a profound effect on me. Reading books (that Jill Gough recommends) helps me to frame my learning. Attending conferences has equipped and inspired me to try new things in my teaching. And, lastly, blogging has helped me reflect on my teaching practices in a way that I wouldn’t have done otherwise.

STEM doesn’t happen by accident

Posted: November 15, 2015 in Uncategorized

I just spoke at the Salt Lake City Leadership Summit. A huge side-benefit of speaking at a conference is that I get to learn from the other presenters. A few of them spoke about the importance and challenge of incorporating STEM into the curriculum. As the new head of the math department at my school, I had already met with the science dept. head to talk about ways that we might be able to work together. Talking about the idea of STEM doesn’t accomplish much.

Jeff Lukens, a fellow T^3 Instructor, shared some thought provoking ideas. He made me realize that math and science teachers sometimes teach the same thing, but we use completely different language. Math teachers say x and y, science teachers call them the independent variable and the dependent variable. In math class we talk about slope, in science class it is the rate of change (later on, we use that term in math as well). Language is a barrier that we must overcome if we are to bridge the gap between math and science. Isn’t that what STEM is all about? Science and math coming together. It sounds so easy.

At my school, we are fortunate, science classes are right across the hall. In many schools, they are in separate wings of the building. Despite their close proximity, I am ashamed to admit that I have done very little collaboration with the science department. Separated by just a few feet, but they might as well have been in another building.

This week, one of my AP Stat students came and asked me if I knew how to do standard deviation in Excel. She knew how to find standard deviation in her TI-Nspire calculator from my class. I asked her why she needed to know. Evidently, they were using standard deviation in AP Biology. Interesting. It was a sign. This was my chance to collaborate with the science department!

I made a point to go visit our awesome AP Biology teacher, Ms. Wright. She was excited to hear how I taught and use standard deviation in math class. In fact, she said that she was thinking of visiting me to ask about standard deviation. All I had to do was walk a few steps over to her room. I had to be intentional about it.

Look at the different graphical representations of standard deviation:

st dev

Walking over and starting the conversation is the first step. I am learning the language that science uses and vice versa. Ms. Wright and I are planning our next steps. I’ve taken the first step, it will be interesting to see where this leads…

Is Mathematics the most important subject in school? In a 2013 Gallop poll, Americans rated math as the most valuable subject they took in school. But does anyone believe that Math is twice as valuable as reading? Or that math is twice as valuable as writing? The College Board certainly does! Let me explain…

You may be aware that the PSAT/SAT exams have been redesigned. These tests are used to predict which students will succeed in college. Part of the rationale of the redesign is that they wanted to strengthen the assessment’s predictive validity. As the Test Prep Coordinator at my school, I have been trying to read up on the changes and how it will affect our students. None of the articles that I have read have highlighted how the new scoring system of the new PSAT/SAT exams places far more importance on Mathematics.

Under the old scoring system, Math, Reading & Writing were all equally weighted (each subject was 1/3 or about 33% of the total score). Under the new scoring system, 1600 is a perfect score, with 800 possible points coming from the math tests. The other 800 possible points come from equal parts of Reading and Writing. Here is a summary of the redesigned SAT scoring system when compared to the ACT scoring system.

Old SAT Scoring:                 Math (33.3%), Reading (33.3%), Writing (33.3%)                                      

Redesigned SAT Scoring: Math (50%),     Reading (25%),    Writing (25%)                  

ACT Scoring:                          Math (25%),     Reading (25%),    Writing (25%),       Science Reasoning (25%)

sat vs act

As you can see, math takes up a much bigger slice of the pie (1/2 of it)! What does this massive scoring change mean for students? The obvious conclusion is that students need a deeper understanding of mathematics. The redesigned SAT makes mathematical fluency, writing equations, and knowing the meaning of variables more than just simply solving equations. It makes sense for students to spend more time prepping for the math section for two reasons:

  • Math is now 50% of the SAT score (twice as important as the Reading or Writing sections)
  • Math section has changed more dramatically than each of the other two sections (it was a shock to my students how different the questions were in the math section)

Having taught at an all-girls school for the last six years, I think there is a bigger issue at play here. I wonder how the scoring of the redesigned SAT tests will affect the gender gap? What is the gender gap?

sat math gender gap

As this graph demonstrates, the gender gap is particularly evident in the Math section of the SAT (Girls score on average 32 points lower than boys). https://www.aei.org/publication/2013-sat-test-results-show-that-a-huge-math-gender-gap-persists-with-a-32-point-advantage-for-high-school-boys/ The gender gap is problematic because the SAT is supposed to predict college success. Can any test really do a good job of that? I’m not so sure that is possible. The problem is that girls do better in high school by almost every statistical measure (top 10%, GPA, take more AP classes, more take 4 years of both math and science). They actually do better in college as well. The SAT exam? Not so much. Is that fair?

FairTest.org points out the financial implications of the gender gap. Millions of dollars are at stake. In the 1990’s, more males received National Merit Scholarships. As a result, the College Board added the writing section to the SAT exam. http://www.fairtest.org/test-makers-revise-nat-merit-exam-address-gender-bias-fairtest-complaint-will-lead-millions-more-gir. I didn’t realize that Title IX education law was applicable in this setting. By making changes to the SAT (adding the writing section), the College Board was indirectly admitting that their exam was gender biased.

Why is the College Board redesigning the SAT? Again, it could be coming down to money. The SAT is losing market share to the ACT. Of course, there could be other reasons, but it would be hard to ignore that this may have contributed to the change.

sat vs act stat


How will the new SAT changes affect the gender gap? I think it is a case of one step forward and two steps back. First the good news. The redesigned SAT has eliminated the guessing penalty. The ACT exam does not have a guessing penalty and they have done a better job of minimizing the gender gap. One theory is that males tend to be bigger risk takers and are more willing to take a guess if they are not sure of the answer. On the other hand, girls tend to be more cautious and can be unwilling to answer a question unless they know they are correct. In my opinion, eliminating the guessing penalty should help minimize the gender gap. Why? Because with no penalty, everyone guesses on every question. The risk-taker advantage is removed.

However, changing the scoring system should have a negative impact on the gender gap. The gender gap is most evident in the math portion of the SAT exam. By making the math section count as 50% of the SAT score (instead of 33.3%), the scoring change only exacerbates the gender gap problem. I think there are a lot of good things that are part of the redesigned SAT, but looking at the big picture, I think that the gender gap may grow larger. Why isn’t anyone talking about the scoring change for the redesigned SAT? I hope this blog post gets the conversation going. Please share your thoughts in the comments section below.

If you teach mathematics at any level, then do yourself a favor and acquaint yourself with the work of Jo Boaler. I train teachers around the country and have been surprised how many folks have never heard of her transformative work in the mathematics classroom. My friend Jill Gough tested this in her session at the Martin Institute Conference. She had 40 participants and not one of them had heard of Jo Boaler. Seriously? How is this possible?

Who is Jo Boaler? Let me start by sharing one of her most significant ideas. This year, I have a goal of putting up posters in my room that represent ideas I want my students to embrace, to create an environment that promotes learning. Here is the first poster that I have created.

boaler poster

How did I make this supercool poster you ask? Well, I had just heard Dave Burgess (the Teach Like a Pirate guy) speak and noticed that his posters were awesome. I leveraged technology and sent him a direct message on twitter to ask him how he created them. He told me his wife, Shelley, used Postermywall.com to create them. I tried it and I am pleased with the result! A nice example of how social media connects learners. I didn’t think to ask Dave until after the conference was over.

Jo Boaler teaches a free online class for Stanford University called, How to learn math: For Students. Many of my students have taken this class and it is a game-changer. In the class, she refutes some long-standing myths like:

  • You are either a math person or you’re not a math person.
  • If your not fast at math, you’re not good at math.
  • Boys are just better at math than girls.

As well as address the causes of math anxiety. If you want your students to adopt a growth mindset like Carol Dweck talks about, I challenge you to let your students take this course. There are six lessons in the online course that take 20-30 minutes each. Here are some comments from some of my students who took the course:

students quotes

By the way, this is not a paid advertisement for www.youcubed.org. I just believe in sharing good ideas and resources. My students have benefited from taking her class. Maybe your students can to? In the interest of full disclosure, Jo Boaler is one of my math heroes. I actually got a chance to meet her at the T^3 International Conference in Ft. Worth. Here is a pic.

jo boaler

If your students have taken Jo’s course, or you have something to add, please comment below!

Competitive vs. Cooperative

Posted: August 29, 2014 in Uncategorized

I am a pretty competitive person. This may horrify some parents, but I don’t ever try to lose to my kids on purpose to build their self-esteem. It was a bit painful last Labor day when I ran a 5K race with my then 12-year old son and he beat me. It was the first time he ever beat me in athletic competition. Afterwards, he said, “Dad, I didn’t even try my hardest.” Ouch. Tough for me, but significant for him because he knew he had earned it. I still remember the day I beat my dad in tennis for the first time (& my grandpa in shuffleboard). Not easy tasks!

At my son’s school, they have prizes for meeting AR reading goals. Here are the three prize levels:

  • Doubling the goal
  • Meeting the goal
  • Progressing towards the goal

What?! They get a prize no matter how little they have read! It reminds me of one of my favorite lines from the movie, The Incredibles:

In the classroom, our personalities have a big impact on teaching. Old habits die hard. I think there is a place for competitiveness in the classroom. But which is more important? Competitiveness or cooperation?

I think I just learned this lesson (again) yesterday. At the end of my lesson on product rules, I asked a tough challenge question to get them thinking. Here is the question:

Which of these product rules could be used to quickly expand (x+y+3)(x+y-3)? Now, try expanding the expression.

Product Rules

In the first class I taught this lesson, ONE student was able to apply the correct formula and get the answer. There was no talking or sharing of information by the students. There was a competitive vibe in the room, students really wanted to be the first one to find the answer. That ONE student really benefited from the challenge problem! But, what about the rest of the students who weren’t the first to finish?

Maybe you see where I am going with this? I realized that another approach might be needed. When I taught this same lesson to another class, I used a think/pair/share thinking routine like I read about in Making Thinking Visible by Ron Ritchhart. All I did was ask the students to think quietly about the problem presented for 30 seconds or so. Then, I asked them to pair up and talk to their neighbor about which product rule they thought they could use. I am not exaggerating when I say that pretty quickly EVERY group came to the conclusion that difference of squares could probably work. Is it possible that some groups overheard other groups talking? Sure. But the students were not afraid to express ideas to each other and it was a very collaborative environment that had been created in my classroom.

Then, each group tried to execute the difference of squares rule on the challenge problem. We didn’t have 100% success (closer to half). But, when someone made a mistake, their partner often helped them learn from it and move on. If you are curious, here is the work:

product rule 2

Which is better? A competitive or a cooperative classroom? I hope the answer is obvious to you. I, like a lot of teachers, continue to be a work in progress. I have learned this lesson before, but sometimes it is easy to fall back into my old habits. Or, choosing the easy instead of the best way.

Writing a blog about this has helped me see this issue from a different perspective. We all need to reflect more on our teaching. Thanks for reading, please comment below if you have something to share.

Failure is an option

Posted: August 13, 2014 in Uncategorized

The new Domino’s Pizza ad caught my attention. Here is an excerpt from the video below:

At Domino’s, failure is an option. We know that not everything is going to work. In order to get better, in order to move ahead, you are going to make mistakes. You ever hear of the cookie pizza? “I don’t want to talk about it.” If we gave up after every mistake, we wouldn’t come up with something new…

What an unusual approach to an ad! Did someone at Domino’s read Mindset by Carol Dweck? In my classroom, mistakes are just opportunities for learning. If I want my students to view mistakes this way, shouldn’t I model that as well? Still, like the girl in the commercial who didn’t want to talk about the cookie pizza fiasco, I have been reluctant to share my failed experiment.

Last spring, I had a student teacher and we put our heads together and came up with a great idea! I have always been more likely to try something new if I can bounce ideas off of other folks.

What was our great idea? TOTALLY flipping the classroom. Have the students teach the material! Sounds like an incredible idea, right? It certainly had potential. Is it a bad sign that on the end of the year survey, most of the girls in the class put making the teaching videos as their least favorite part of the year?

Students were split into groups and assigned a lesson in the Conics chapter and told to make a screencast or video using their iPhones an so that the other students could view their lesson. They were given strict deadlines as well as specific targets that they had to cover in their video lesson. Here is an example of one of their screencasts:

Here is a short list of drawbacks:

  • Most groups didn’t meet their deadline because of technical difficulties with uploading a video
  • Since the videos weren’t posted on time, the students had difficulty viewing them before coming to class as they were expected to
  • There were some mistakes in the math on some of the videos
  • The students complained of being confused after watching a lesson

Here is a short list of positives:

  • The students collective sigh of relief when they found out the next chapter would not be taught by students

I had hoped that this approach would mean the students would be doing lion’s share of the work. That didn’t really pan out. They did ask questions with greater urgency than they had ever had before. Was it the fear of possibly failing the test?

We had to spend an extra day or two on review, but in the end their test scores were not much different than their previous tests. This reminds me that learners are resilient. This may have been a failed teaching experiment, but I don’t think it caused any permanent harm.

What can we learn from all of this? Maybe the idea has some merit, but could be tweaked? Another math teacher at my school used a similar idea and was successful. She split the students into groups and assigned them review topics at the end of the semester. They did not use a screencast, but used presentations to review for her final exam. It may have worked better because the students didn’t have to try to learn the lesson from scratch on their own and then try to turn around and teach their classmates?

If we gave up after every mistake, we wouldn’t come up with something new…

Is it OK to fail? If you haven’t had failures in your class, does that mean that you aren’t taking many risks? What serves our students best? What else can we learn?  Share in the comments section below.

I just got back from the T^3 International Math Conference in Las Vegas. I had a great time reconnecting with friends and learning in the conference sessions. If you haven’t been to Vegas, it is pretty crazy. Here is a pic of the shower in the Rio Hotel:


Why is there a window in the shower that opens to the living area? #Crazy.

In the opening session of the conference, I was challenged by the thoughts of Alan November. He used PollEverywhere.com to solicit feedback from the educators in attendance at the conference. First, we used our cell phones to answer this question: Who works harder in the classroom?


As you can see, in most classrooms, teachers are working much harder than students are. Alan said we tend to underestimate our student’s contribution to the ecology of learning in the classroom. When you think about it, learning is not usually a passive activity. We need to get students more engaged so that they don’t ask questions like, “When I am I ever going to use this?” Alan November then asked the audience a second question: Who SHOULD be working harder in the classroom?


The results were not surprising to me. Most educators think that the students should be working at least as hard as teachers are. Seeing the data, it was clear to me that classrooms are not working the way we would like them to. Students need to do more of the heavy-lifting of learning than we are letting them.

Alan November’s provocative idea was that we should harness the energy of our students to help other students. You learn more when you teach. Kids can make great teachers. He mentioned Steven Pinker’s Curse of Knowledge idea: the more you know about your subject, the least prepared you are to meet the needs of first-time learners. I’m not sure that I fully agree with that idea, but I do think that it is challenging for teachers who do know their subject to place themselves in the seat of a new-learner. It takes effort (and empathy) to try and do that.

One of Alan’s solutions (I like that he didn’t just state the problem) is to allow our students to teach the class (virtually). The example he gave was a video of a sixth grader, Bob, who made a video (MathTrain.TV) to teach his classmates how to find the prime factorization of a number. He talked to the students who made these videos and they said things like, “When you have to teach something, you really have to learn it.” Any teacher will tell you that is a true statement. Are we under-utilizing students? Would this work in a high school classroom?

I titled this post, “Are teachers working too hard?” I’m not sure that is the right question. Maybe a better question is, “How can we get students to work (at least) as hard as teachers are?” If you have ideas or thoughts about how this could be done, please comment below.

What super-power would be the most useful to have in the classroom? Part of me thinks that I would like to be able to read students thoughts (although this idea is also a little scary). I am reading a book called, Making Thinking Visible, inspired by Harvard’s Project Zero. In it, the authors chronicle various ‘thinking routines’ that could actually develop students’ thinking. I wonder how much of each lesson my students are actually engaged and thinking about the mathematics? If I had this super-power, I am positive that I would be hyper-motivated to developing and training students to think deeply about the lesson. Can you imagine seeing the thoughts of whole groups of students as their mind drifts off? How depressing! Hmmm. Maybe I don’t need to know everyone’s thoughts after all, I just need to be more committed to keeping students engaged in the lesson.

On second thought, maybe it would be a good idea to be able to turn back time? You remember when Lois Lane died in Superman’s arms because he didn’t arrive in time to save her? So, in his anger, he proceeds to circle the earth hundreds of times until time actually reverses. See the video (with awesome special effects) here:

How many times have you graded a quiz or test only to realize that the students did poorly because your didn’t do a good job of preparing them for the assessment? How great would it be to be able to go back in time and add/fix/adjust so that the students would be properly prepared for the assessment? That would be supercool, but I am not sure it is necessary. I mean, if you are doing a proper job of infusing formative assessment into a lesson, you should know (not just be guessing) how students are doing and gauge your instruction accordingly. And, given the off chance that students do poorly anyway, we should reflect on that and adjust accordingly the next year. If you teach multiple sections of the same class, you probably do make adjustments from one class to the next. I am more and more convinced that the art of teaching is found in making (usually) small adjustments on the fly.

Ok. Let’s see then, what should be my choice for super-power? I actually had to google to come up with this ideaself-duplication. I don’t think there would be a down-side to this one. Can you imagine being able to help all of the students individually? Finally, differentiated instruction comes to fruition! That would be the ultimate! I could craft different lessons for each student. Time is such a valuable commodity in the classroom. Imagine being able to make the most of every moment for each student in the class?

But, we live and teach in reality where super-powers are hard to come by. In my writings, I would like to explore leveraging technology to enhance our own teaching abilities. Share tips that I am learning from the books I am reading (this has been a key to my growth in teaching). And, I would like to share experiences that cause me to think. I want to thank Jill Gough (Experiments in Learning by Doing) and Jennifer Wilson (Easing the Hurry Syndrome) for inspiring me to start a blog. My hope is that we all can aspire to become Super-teachers.

If you have a cool idea for a super-power, please share in the comments section…