Doing it the hard way

As the first full week of classes came to an end yesterday, I was working with my Honors Geometry class on Friday afternoon.  We had all worked this problem for homework:


As I redid the problem myself prior to class, I found myself redrawing the diagram on a larger scale (which I find my students don’t often do), and I also knew it would be a question that not everyone could solve.  As the class went over homework together in small groups, I asked four different sets of people to put their solution on the board.   One group imposed a scale of 0 to 100 on the problem, placing point R at 0 and point S at 100.  They then found the numerical placement of the other midpoints in order to find the result that WZ=3x.  When they presented their ideas, they said they were thinking of 0-100%, which is what led to that scale.


A second group did something similar, but chose to place point R at 0 and point S at 8.  They also reached the same solution, that WZ=3x, but all of the points they placed on their “ruler” were at integer values.  pic6

A third group did an identical scale, although one boy shared as the group presented that he had actually used 0 to 24 on his paper, not 0 to 8.  pic3

The fourth solution on the board was drawn by one person [let’s call him Steve] who had gone to the board early in the class so he could explain his work to his two partners, who hadn’t figure out the problem.  Steve opted for a more algebraic path to a solution, but reached the same end result.  He had included some statements which included defining some lengths as being 5/8 of other segments in the problem.pic5

As Steve explained his solution to the class, I could see the confusion in other’s eyes as to how he had approached the problem.  We worked through this solution together to make sure everyone understood Steve’s idea, and then paused.

I asked the class to consider the three different solutions we had up on the board (since two were the same).  Was any path a better one than another?  Are all of the solutions reasonable?  What should we take away from having done this problem? One girl raised her hand right away and said, “I think we should all be using Steve’s method because it’s the hardest.”

As soon as the words were out of her mouth, I realized how much of the typical student experience with high school math could be summarized by that one line.  To many students (and perhaps to many teachers too), the hardest path is the one we should be taking.  I felt pained as I heard this girl say this out loud, but at the same time, I felt grateful that she was willing to put that thought out there.  I immediately shared my feelings with the class, and our conversation continued, but I find myself still thinking about her comment now.  I shared the thought right away with a few of my colleagues, almost in a joking way.  However, it feels like there is so much more to draw out of the remark.

If most students think that they have a path to a solution, but they doubt themselves because it isn’t “hard enough”, then I have work to do as their teacher.  If we all share solutions in class as described above, debrief, and students still leave the room thinking that the hardest path is the one they “should have” taken, then I have work to do too.  Is there any way to combat this feeling in students of math?  Why doesn’t an easier path seem plausible?  What has happened in students’ prior math experiences to make them think that harder is better?

As I experiment with a variety of different classroom structures this year, I want to be sure to remember to find the moments to point out the multitude of solutions, and try to emphasize that the path that makes sense to each person has validity and does not have to be hard.  I also want to try to work to change my own mindset.  When I wrote my solution to this particular problem, I wrote this:


It didn’t even occur to me to place a ruler-type scale on top of this image to simplify the work.  I think that is the math teacher in me, feeling like I should be able to express the situation algebraically, even there is a simpler path.

Overall, I’m grateful to my students for showing me an easier path to a solution, and eager for the year ahead to try and help all of my students understand that math does not need to involve choosing the hardest path.



After a few days in New York with some wonderful math Twitter folks, I have had just the push I wanted to get my mind thinking about the start of the year.  I have my collection of tangram-like materials, so I can do the Master Designer activity I just learned about.  In fact, I want to have my department do the activity together (or maybe Broken Circle) so they can help their students think about group norms.  I’m already excited just thinking about that!

I also want to bring an idea I had last year back into my “start-of-the-year” mind.  My Honors Precalculus class was in the midst of doing a long-term project, where they were working pairs.  As I looked at each group, I could see the way that each person brought something to make that group even better.  For example, there were two girls were exploring the odds of winning a tennis match if you win the first point.  Three things stand out to me as I think back about this particular partnership.  First of all, these two girls are (and were) powerhouses.  They are going to change the world and do amazing things.  I can see it in how they go after an idea, asking great questions, helping others, and not being afraid to be the “skeptic” in class, not letting some niggling concern go and going after a concept completely.  Wow!  Just thinking about what they may do someday gets me excited.

The second reason that this pairing stuck in my mind is because of how their project idea came to be.  They had followed my initial brainstorming instructions and landed on this tennis idea (which they eventually pursued).  I chatted with them briefly and tried to sway their work in another direction.  In retrospect, I don’t know why I did that.  Instead of letting them draw me in, and helping them find the path to the math they could explore, I tried to get them to pick another idea where the math was more immediately apparent to me.  I don’t want to do that again, so that moment was a great turning point for me.

Lastly, and the reason I’m writing this blog post, this group was a team in the best sense of the word.  One of them had concurrently taken Honors Statistics, so she could engage in a statistical analysis of the question they wanted to pursue. The other student was enrolled in Introductory Programming and was making great progress in that class.  Simultaneously, these two students were able to do an incredibly thoughtful analysis of the tennis question.  One wrote a simulation to run repeated random trials and investigate, while the other student wrote the statistical analysis (using R) to assess the question.  

As I described this group working together, I found myself noting how they made such a great team.  It let me think of each of the other partners in a similar fashion and see this everywhere.  Another set of two boys together were extremely interested in data analysis, and both had strong skills using a spreadsheet.  Their work together was improved by being a joint project.  A third group benefitted from a Python coder in the midst.

The reason I write all of this is to remember that feeling.  I want my class to feel like we are all on the same team.  I’m glad one person is there because they know more about coding than I do; you are glad I’m here because I may be able to explain hard ideas better to you than the teacher can.  As I think more about group norms and the start of the year, I know I want to make sure to be explicit about the team goal for all of us.  As an entire class, we will be an impressive group.  When we work in smaller teams, we still all have skills to be sharing out.  In fact, this may be the most important thing to have students share about themselves early in the year.  What are you good at?  What can you do, but don’t really like to do?  What do you think of as an area in which you’d like to improve?


Day 1 of the mini-TMC in NYC is over.  Already, it has been a great time and well worth the trip.  Here are some thoughts percolating in my mind right now, as I make final tweaks to my presentations for tomorrow:

  • I’m still intrigued by the basketball problem @carlolitwitter had us consider.  I want to do more of the math.  My pattern-seeking self wants to find more, and think about ways to display what I have found.  I am again reminded that Pascal’s Triangle is everywhere….  It also made me think and reflect with @bkdidact about doing math with my department, with tasks that are accessible to all.  Lastly, working on this problem brought my colleague Jeanne to my mind, and her love of her circle cevian problem as a way to warn students about thinking they see a pattern that doesn’t hold true. (Short version: draw a circle, adding points to the circumference. For each point added, connect that point to every other point and count how many distinct shapes/areas are created and look for a pattern in the results.)
  • I am perhaps not surprised but a bit saddened to hear how many people aren’t happy at their current schools, regardless of their tenure at that institution.  I continue to feel like that I like what I do and love the school at which I teach.
  • I heard much frustration directed at the DOE.
  • Talking with @alittlestats made me think again about the role that Statistics teachers, or at least the good ones, get put in.  If you do something well, you will be asked to keep doing it.  Teaching the same thing again and again can be fun but even the best teachers need to pause to reflect and rejuvenate.  That said, my pause from teaching Calculus is ending and I cannot wait to dive back into that class this year.
  • I enjoyed talking with @_b_p about joining a new school, the unexpected benefits of changing jobs and interviewing, parenting and math.  It’s always great to meet more teachers!
  • I think I’m most excited about @NicoraPlaca‘s Master Designer task to start off the school year and help develop group norms.  Her task description is here.  This idea has been in my mind for a long time, based on numerous science museums I’ve visited where two people sit across from each other and try to build the same structure based on verbal descriptions only (sight unseen).  I cannot wait to do this to start the year in each of my classes!

Ending on a high note

In a recent post, I shared that I opted to end the year with my Honors Precalculus students making some short videos to share out their research on a few personal finance topics.  We spent a total of 3 class days on this work – 2 days to work with partners and create spreadsheets and/or Google Slide presentations, and 1 day to record the videos.  Tonight, for the last night’s homework assignment, I asked everyone to watch each other’s videos and add to a discussion thread about what they saw.  A few excerpts are below:

“I had heard a lot of the terms you discussed in your video before, but I wasn’t sure exactly what they meant.  It was helpful to see all the different loan types organized in one place because it made them easier to compare.  I also really liked the case study because that is something that is really applicable to a lot of students.  I was very surprised by the table in the Perkins loan tab that showed how much total money you pay based on the length of the loan; I did not realize the total amount of money increases that much as you extend the length of your loan.”

“I thought your video was really cool.  Student loans are a very applicable topic to our age group, and it was nice to get to learn more about them.  With this presidential election, I know there’s been a decent bit of talk about student loans among the various candidates, but I don’t know a ton of details.  It would be interesting to consider the various presidential candidates and their positions on student loans, and then take another look at the numbers like you guys did.”

“I really like how you not only explained what a Roth IRA and a 401k are, but you also explained the math behind part of the Roth IRA. I liked the slide where you talk about the math behind doubling your money every 7 years and found the spreadsheet very informative. I also really liked how you created a hypothetical situation, because it allowed me to understand the whole process even better. It was really helpful when you explained how you created the spreadsheet for the hypothetical person (i.e. talking through each cell)”

“I never thought that there could be a such thing as good credit, so that was interesting.  In my project, there was a little talk about credit, but I didn’t know what that really meant until now so this was very helpful.”

“I learned a lot about how retirement planning works for different people in different financial positions. I learned what Roth IRA and 401k were and how they are used to saving money for later use. It’s cool that your Roth IRA money doubles every 7 years, and it seems to be easier to withdraw money from this account setup. The change in time of when people start saving is crazy to how much money they can lose from starting just a couple years later. The huge spreadsheet you guys made really helped me understand how retirement savings work per year and per individual. If can get an extra 240k by saving 10 years earlier, it makes me want to start saving now!”

This was such a great way to end the year – hearing student voices share what they learned.  I had hesitated to assign homework for the last day of school, but I did like ending with this project.   I think the students learned a lot from their own research, and learned from each other.  It makes me feel much better about sending these students off to future math studies, having taken the time to do this important work.

One of the last points that struck me were the variety of learning opportunities in this mini-project, which I added not expecting to use many of the math skills we worked on this year.  I decided I felt okay about that. Nevertheless, there were a variety of useful ways that students were using and improving upon their skills in the math classroom (and beyond), including:

  • practice working a project with a partner
  • first time use of Screencastify to make short videos
  • another opportunity for students to share their work with a larger audience than just me
  • virtual discussions happening outside of class time, which will make tomorrow’s class even better
  • homework assignment which will take a finite amount of time (the total length of all videos plus about 15 minutes total for discussion posts)
  • more savvy use of well-labeled spreadsheets with absolute cell references and formulae, to allow the user to consider a variety of scenarios
  • looking at patterns in data and trying to explain what might be happening and why (in real life) – one group was looking at consecutive differences to understand growth rates
  • explaining math catch phrases, such as the claim that “Roth IRAs double in 7 years.” Seeing the (basic) math behind that makes it crystal clear.

I’m sure there is more to add to this list that I’ve forgotten.  Overall, I feel like I made a step in the right direction, for the first time in a very long time, where the students were considering questions to which they wanted to know the answer.

#MTBoS30 – Day 30

I didn’t quite get all 30 days in, but I’m amazed at how many posts I did actually make this month.  Thanks to all who responded!  If nothing else, I’m stepping back from this experience with a reaffirmation of the #MTBoS community.  A post asking for ideas, feedback or help of any kind will be retweeted and strangers will reach out with a reply.  Committing to regular posts, and the use of the #MTBoS30 hashtag led to more people reading my posts, and I read more than I had in the past, too.  Overall, I do want to think about ways I can use a blog effectively, and perhaps part of that is just taking the time to read and respond to others.

I did not post over the weekend, as I took a very rare opportunity to not open my computer for two days.  I did still check in on my email by phone, and reply to the students who had a question for me.  Otherwise, I did not go online until tonight.  I hardly ever do that, even when I’m on vacation, but I really enjoyed it.  I suppose that is my parting reflection as the month of May draws to an end.  Being connected on Twitter and in other ways has millions of benefits.  However, it remains important to unplug sometimes.   The most rejuvenating step could be the step backwards.  I hope that it will allow me to finish the final stretch of the school year in a refreshed mindset, anticipating the joy of summer and different pace of life to come.  Thanks to all who read and replied, and I do promise to post on a regular basis to keep this habit going.

In the real world

#MTBoS 30 – Day 27

Tonight, I found myself thinking about some of the courses taught at my school.  I want to find ways to make them more engaging for students, more of the time.  My mind has continued to return to more student choice in the course they take (so not everyone is taking “Precalculus”, but instead choosing to study a particular flavor of the course).  As much as I like this idea, as a former scheduler, I’m already overwhelmed by the logistics of adding some more restrictions into a junior’s schedule.  I also recognize that the course still remains very fixed, with a constant curriculum that does not adjust to the students in the room.

So, my new thought for today is that the title “Precalculus” can be fine; in fact, we can likely take advantage of an opportunity to avoid changing the name of the course.  Instead, we could work on changing the lived experience.  Can we incorporate some more “real world” aspects? This was on my mind as I hopping on Twitter tonight and happened upon an old Dan Meyer post about Real Work v. Real World.  His ideas were intriguing, as usual, but it was actually a comment from someone else (DG Reid) that resonated with me even more:  “The question isn’t just: Is the problem real? What engages students is: Does the answer provide useful information?”

At the same time, I have another colleague who is having the amazing experience, for the first time in 33 years of teaching, of seeing how much joy comes out of a project where the students were able to choose what they wanted to study.  It is the combination of all of these ideas that pulls me in.  How can I work with my amazing department to help put some of these ideas into action?  What opportunities are we missing?  What could we reframe?

The last idea in my mind comes from my youngest colleague, who is in her first year as a fellow, having just graduated from college.  She asserted recently that because we create many of our materials for class, the students are expecting all information to come to them in a neatly packaged form, ready for use.  They don’t know what to do when confronted with novel, open-ended situations because we have unintentionally led them to expect clarity and brevity.  For our students to engage fully (which they really want to do – we are so fortunate!), we need to let them start asking the questions, and guide them forward when they need it.

#MTBoS30 Day 26

As I look through my seemingly random but lengthy list of things that need to get done, I find myself prioritizing the things that make me happy.  That could be something I want to do (today, that was wrapping the math prizes in fun white gift bags with orange polka dots on them), or it could be doing something that will make me happier once I have crossed that task off my list.  To tackle that, I began writing solutions to a take-home test that is due soon.

One item on my to-do list is to choose furniture for a mock math classroom that will be set up over the summer.  In anticipation of the perhaps distant future construction of a math building on campus, we are adding another classroom to our inventory.  We won’t have as much square footage as our ideal, but we can play with almost everything else.  I found myself smiling as I told our project manager that I really want it to look like an elementary school classroom:  colorful, fun, and inviting.  Even if only that remains at the forefront of my mind throughout the design period, I am sure we will end up with a classroom we like.  I visited a few other areas on campus to get inspiration today, including our K-2 building to look at fun carpet design, and some health classrooms to see what comfortable furniture they recently purchased.  At the top right are some desks that are intriguing to me with their unusual shape.  Now, it’s on to find area schools using some of this furniture so we can see it in person and make some decisions!  Here’s to making the final stretch possible by emphasizing what makes you happy.

Day 24 – #MTBoS30

I’m definitely slowing down!  I missed a night of blogging as I powered through until way too late to finish grading my students’ projects.  I’m tired but glad that’s done!

One of the fascinating parts of grading this project was to read each student’s reflection on the process and their work.  One of the aspects of the project structure that they liked the most was the project log, which was one of my colleague’s ideas for another project.  We asked each group to create a shared Google doc, shared with me and with each of them, which was a series of empty cells in a table which looked like this:

Date Detail
5/2 Accomplished in class:
To do for homework:
Plan for work in class tomorrow:
Questions for your teacher:
5/3 Accomplished in class:
To do for homework:
Plan for work in class tomorrow:
Questions for your teacher:

The number of kids who praised this structure was amazing!  I heard it most often from kids who I think of as highly organized already.  They liked the opportunity to keep themselves focused and on track with the help of this scaffold.  The project period lasted for two weeks, and students reported (after the fact) that they had some busy nights with tests or papers due in other classes. This log allowed them to stay on track even when they had to not work as much on a given evening.  It made me realize that even a relatively small additional structure, which remains highly flexible, can still make an open-ended project run more smoothly.

How to approach the week

This is not going to be a math post.  I ended up taking a weekend off from everything school-related.  Friday night was outdoor movie night at my sons’ elementary school, which was great.  Saturday morning began with errands, followed by a baseball game, and then we hopped in the car to head south to celebrate a family member’s graduation.  Pulling back in the driveway around 9 pm did not make me more inclined to think about my week ahead.

Instead, I’m going to focus on trying to shift to a positive mindset for the next few days regardless of how much needs to be done in a very short period of time.  It will all get done, it did not get done this past weekend, but it definitely get done before next weekend.  The motivation of having a long weekend with no work to do is a strong one.  Now, if I can just avoid feeling guilty for not having done any work this weekend.

Falling behind…

I already am feeling guilty about not posting yesterday.  I guess that’s a good thing?  Thank you, #MTBos30.  Now, I’m not at home, and still feeling the compunction to post something.  I read someone’s post about their student feedback, and was compelled to read feedback from one of my two classes (which has wrapped up because it’s full of seniors).  I read many positive things but one strongly negative one, and I can’t seem to stop thinking about it.

The comment mostly speaks to how little this particular person learned in my class.  The “stay positive” part of me is inclined to push it to the back of my mind and ignore it, and move on.  The realistic side of me is trying to own this feedback, understand what this student has said to me, and try to learn from it.  What do I need to do differently so that next year in May, I don’t get feedback like this?  I don’t know if I can make adjustments right now or even give myself to the time to fully consider the impact of the input.  My instinct right now says to set it aside, and mull it over in a few weeks, when my mind is more free, less stressed and ready to try to learn from the experience.  The main thing is to not set aside forever.  I think I can do that.  We’ll see!