Results of My Research

Hi reader,

It’s been a while since I’ve updated this blog. I’ve been busy with my Ed.D. program, but I’m now finally done with all the work! I only need to present my findings next weekend to fully earn that diploma. Below is a summary of my findings for those who are interested.

In Medford, Massachusetts' middle school students…

  • Positive attitudes toward math are higher in 6th graders than in 7th graders and higher in 7th graders than in 8th graders, with a significant difference between the latter two.

  • Attitudes toward math tend to be similar between males and females, with the exception of math anxiety which is higher in females.

  • Students in the accelerated/advanced math classes report more positive attitudes toward math, with the exception of value (how important they perceive math to be for their futures).

  • Asian students reported the highest levels of positive attitudes toward math and Hispanic/Latinx students reported the lowest. Black and White students were somewhere in the middle. This trend, however, was not true for math anxiety where no significant differences were found among the races.

  • Of the factors studied, self-efficacy/self-concept correlated strongly and positively with achievement in math. Other factors such as value of math and enjoyment of math correlated weakly with achievement.

  • Income did not correlate with any of the attitude factors studied. This might be because reliable income data was not available.

  • There was a strong positive correlation between enjoyment of math and self-efficacy/self-concept beliefs in math. That is, students who enjoy math are likely to also believe they can be successful in math and vice-versa.

My presentation slides, which include much more detail on the bullets above, can be found here. I’m also happy to share my dissertation through email.

Carolina Vila

Introducing Gerrymandering in Class

I attended a gerrymandering conference last summer. It was thought-provoking, eye-opening, and overall the best conference I've ever been to. Created by Tufts professors, the Geometry of Redistricting workshop invited hundreds of educators, computer scientists, lawyers, and other professionals from around the country to learn about and brainstorm solutions to the complicated issue of gerrymandering. According to Merriam-Webster, to gerrymander is "to divide (a territorial unit) into election districts to give one political party an electoral majority in a large number of districts while concentrating the voting strength of the opposition in as few districts as possible." In short, the way you divide land can determine the winner of an election. Because the gerrymander-er is working with dividing pieces of maps, they are working with math, and as an educator at the gerrymandering conference I was encouraged to bring what I had learned back to my classroom.

At some point during the last two days of the conference, facilitators asked me and other participants to gerrymander a ten-by-ten grid as a sort of warm-up exercise. Using something similar with my students seemed like a no-brainer - it was simple, fun, and enlightening. Below is the activity I eventually used with one of my classes (on the last period of a Friday!)

1. Before diving into a ten-by-ten grid, I started with a simpler one. I gave each student a sheet with three ten-by-five grids with the first two columns filled with x's (as pictured below). I asked students to divide the first grid into five equal sections so that 3/5 of the sections are dominated by blank blocks while the other 2/5 of the sections are dominated by x blocks. If these were voting districts, it would seem that dividing the blocks up in this way would be "fair" (the majority of districts are dominated by the majority group with perfect proportionality).


2. For the second grid, I asked my students to divide the fifty blocks into five equal sections in which all of the sections have more blank blocks than x blocks. This sounds a bit confusing at first (to mostly anyone, I think) and I had to repeat the direction a few different times in slightly different ways. After a minute or two, though, most students were able to figure out that simply dividing the sections in their horizontal rows two at a time does the job. This can be seen in the second grid of the picture above: Each section is now dominated by blank blocks.  If these were voting districts, the blanks would have control of all the districts.  As you might expect, students believed this set up to be very unfair.

3. For the third grid, I asked students to divide the fifty blocks into five equal sections so that a majority of sections have a majority x blocks.  In the third grid of the picture above, you can see that 3/5 of the sections are dominated by x blocks.  Now, the minority controlled the majority of the districts. Students thought this was unfair but also "cool." There's something fun about helping the underdog win.

4. I expanded on the third grid at this point in class. When, if ever, should the minority control most of the districts? Who actually does this with maps? (Answer: both major political parties in the United States have gerrymandered.) Which real districts are gerrymandered? What do they look like? How do people feel about it? What, if anything, is being done about it?

5. After a lively discussion and looking at real gerrymandered districts with the help of the internet, I gave students another sheet. This time the sheet had two ten-by-ten grids with x's in 40 of the 100 blocks. This exact arrangement of x's was taken from the activity facilitated at the Tufts conference. I asked my students to divide the 100 blocks into ten equal sections so that (1) on one grid the majority of sections were dominated by blank blocks and (2) on the other grid the majority of sections were dominated by the x blocks. Pictured below is one work in progress, mistakes included.


6. Once mistakes were cleared up, students were very excited to show off their gerrymandered maps. I shared different arrangements from student volunteers using my document camera and we talked about strategy in performing the act of gerrymandering. For example, students shared how they were able to concentrate a large number of the dominant group in fewer sections by packing them together. This gives the minority group a chance to spread themselves over more districts, an actual gerrymandering technique that's called "cracking and packing."

7. I closed the lesson with a reminder that because real districts don't exist in perfect square blocks, the math and computer science behind redistricting is immensely more complicated. I also reminded my students that this is an ongoing issue/problem, and one to which we will return as more and more articles come out about gerrymandering. 

Update: the issue of gerrymandering came up in the recent Alabama senate race between Doug Jones and Roy Moore. Although Doug Jones won the majority of votes, had the race been decided by district votes Moore would have won 6 to 1!

If you teach, I hope you'll considering facilitating a similar lesson soon. Students loved it.

Carolina Vila

My Experience with Bilingual Education

Lawmakers in Massachusetts, where I live and work as a middle school teacher, recently voted to bring back bilingual education. This post is not about the pros and cons of bilingual education. Instead, this post is about my personal experience with bilingual education when I was in first and second grade during the early 1990s.

Before I go on, I should say that I do not remember many specific details since I was only about six or seven years old during my time in a bilingual classroom. Rather, I remember feeling certain emotions and receiving certain impressions. For instance, I remember knowing - don't remember how I knew, but I knew - that my mother was in constant communication with my teacher. This immediately dropped off once I was placed in an English-only classroom and my Spanish-speaking mother could not easily communicate with my teacher. I believe the connection between home and school in those critical years was part of why I succeeded. 

Perhaps more importantly, however, is the overall impression I received that my home language mattered. I learned through bilingual education that my Spanish culture was something worthy of study. It was a confirmation of a major part of who I was, and who I still am. I believe this has incalculable value as educators strive to help raise children who understand their personal histories and feel whole. Humans who feel whole learn better and treat others better. 

Both my first and second grade bilingual teachers were also such positive role models with whom I connected closely because they each could have been an aunt or family friend. This speaks to the benefits of hiring more minority teachers as well.

When I think about first and second grade in a bilingual classroom, I feel warm and lucky to have experienced the high-quality bilingual education I received.  I'm thrilled Massachusetts is one step closer to bringing it back.

Carolina Vila

Beautiful Questions

I recently read a KQED News piece by Katrina Schwartz entitled "How to Bring 'More Beautiful' Questions Back to School" -- connected to the book "A More Beautiful Question" by Warren Berger.  I was motivated to write this post due to one particular past experience and my belief in the power of good questions.

First, let me describe my past experience.  I was lucky to visit Uganda for 30 days during the summer of 2009.  I taught at a school there but also visited a few others to get a more complete picture of Uganda's school system.  On one visit to another school, I sat in on a geography class at a local high school.  The class had dozens of students crammed into a small room and the teacher stood in front of the class with his chalk in his hand.  I, along with another tourist-volunteer, took a seat with the students and anxiously waited for class to begin.  The class was about an hour long (I say about because staying on any kind of schedule is not a priority to many Ugandans) and the teacher lectured for about half that time.  The lecturing seemed normal to me and I assumed it would continue until the end of class. 

About a half hour in, however, the teacher concluded his lecture and asked students for their questions.  A few students had questions and he happily answered them.  The teacher then asked for more questions.  And then more.  And then more and more.  I remember thinking, "Geez, what if there aren't any more questions?!" and "Why is he just assuming they have more to ask?"  There were pauses here and there but eventually someone would ask yet another question.  That went on for over thirty minutes. 

Wow.  Three things struck me during and after that class.  1. Who told the teacher to reserve half his class for questions? Was this normal in Uganda or was this part of his personal teaching style?  2. How wonderful is it that some of the best questions came ten or twenty minutes into the questioning portion of his class?  What does that mean for my class, where I sometimes only allow a few minutes for questioning?  3.  What kind of message is this sending to the Ugandan children?  How does this affect how they view questioning and the role of questioning in school?

Although it occurred years ago I'm reminded of this event often, especially when I run out of time in class or read an article on good questioning like the one mentioned above. While I highly recommend reading the linked piece (here it is again), I will give you the gist of it.  The articles states:

-Little kids love to ask questions but questioning "drops off a cliff" when they go to school (so sad!)
-Time constraints make it difficult for teachers to ask good, deep questions in their classrooms
-People feel vulnerable when asking questions which makes this process even more difficult
-Teachers and parents should value great questioning because we want to create an informed citizenry who isn't afraid to ask questions
-Asking good questions takes practice
-There are five things teachers can do to help bring about good questioning in their classrooms.  They are:

1. Make it safe -- use smaller groups so students feel more comfortable asking questions or make questioning the point of the activity (versus getting the "right" answer).
2. Make it cool -- tell students that people who ask new questions are cool and are the ones who can change the world.
3. Make it fun -- turn it into a game or frame questioning as playing the role of a detective, for example.
4. Make it rewarding -- give a "best question of the week" award or add a bonus question that asks students for a question on a quiz or test.
5. Make it stick -- make the above part of your normal classroom routine so it becomes a habit for your students.

Good questioning skills are something that can be developed across all ages and subject matters.  It would build curiosity and confidence.  I think this would be a fantastic school culture goal, don't you?  If you teach, how do you use questioning in your class?  Do you feel, as I do, that you can improve in this area?

Ms. Vila


My students took the Partnership for Assessment of Readiness for College and Careers (PARCC) test last spring.  Items from that test were recently released and shared within my school.  Although my students will take the MCAS 2.0 this year (an updated version of the old Massachusetts state test) we are being told that the new test will be very much like last spring's PARCC test.  Naturally, I was curious to see the released items.

With all the talk of preparing students for the "real world" and building critical thinking skills etc. etc., I'm surprised to continue to see questions like the one below on these big tests.  Here is an actual released PARCC item:


OK, I get that we want students to simplify and manipulate expressions, but wouldn't putting something like this on a state test encourage teachers to "drill and kill"?  Here's another released item:

Very similar, isn't it?  Another one:

Absolutely soulless.  Where is the critical thinking, the context or application?  Isn't that what these new tests were to both encourage and assess?  How can you look at these questions and not understand why many students lose interest in math?  Can't we do better?

I'll admit that other released questions were a little better (here they all are for those interested).  About 25% of the questions I looked at were like those shown above.  That may not sound like a high percentage to some, but it does to me because I think that percentage should be zero!

Am I wrong? Am I missing something?
Ms. Vila

People Keep Asking Me About This

Teachers on Instagram have asked me one particular question more than any other: where did you get those whiteboards?!

I love getting the question because I love giving the answer.  A few years ago I was extremely close to buying expensive whiteboard material.  On a whim I went to the teacher store instead (Lakeshore Learning in Saugus, MA) and found those wipe-off posters meant to be used as a sort of homework or announcement board.  I bought six of them at about $3 or $4 each and stapled them to my back corkboard.  They've been FANTASTIC for a few years now.  If you were to get a close look you'd notice that they're starting to wear and tear a little bit so I'll probably buy new ones for next year.  Looks like they're costing me about $20 every few years - not expensive at all!

Here are a few pictures I found online of the type of wipe-off poster I'm discussing.  Not exactly the same but very similar:

I highly recommend them!
Ms. Vila