More Boaler!

I am so glad I discovered Jo Boaler’s work.  She is inspiring me to rethink the way I teach math.  For my Inquiry class, we had to pick a peer-reviewed journal article to summarize.  I immediately searched for Boaler.  I found one of her most famous articles, and below I share my summary of her research.

Boaler, J. (2008). Promoting ‘relational equity’ and high mathematics achievement through an innovative mixed-ability approach. British Educational Research Journal, 34 (2), 167-194.

Summary of Research

This study followed students placed in mixed ability collaborative groups in math and compared them with students in traditional math classes over four years. Through lesson observations and student interviews and assessments, Boaler found that the students placed in the heterogeneous collaborative groups not only outperformed the students in traditional math classes in mathematics achievement but also showed increased relational equity, treating each other with more respect and offering each other more support.

Design of study:

Quantitative and qualitative data on 700 students were collected over four years at three California high schools. Two high schools used traditional teaching methods, which incorporated 21% lecture, 15% teacher questioning, 48% individual work, and 11% group work in ability-grouped math classes. The third school used reform methods, with 4% lecture, 9% teacher questioning, 72% group work, and 9% student presentations in heterogeneously grouped math classes.

Data Collection:

Over 600 hours of lessons at all three schools were observed and analyzed. Sixty students were interviewed every year, sampling from high and low achievers and different cultural and ethnic groups. All students were given questionnaires in the first three years of the study asking about their experiences and perceptions of math. All students were assessed for mathematics understanding, and selected focus groups were given applied mathematics assessments that were videotaped.

Emergent Themes:

  1. Students who work in heterogeneous groups learn to take responsibility for their own learning and the learning of others in the group resulting in higher mathematics achievement for all members.
  2. Students who work in heterogeneous groups learn to value other people’s perspectives and respect diverse cultures and backgrounds, which has positive implications for their future participation in a global economy.
  3. Students who work in heterogeneous groups show improved communication and collaboration, as well as enhanced critical thinking and problem solving.
  4. Teachers need to conscientiously support and model group work in order for it to be successful.


Despite starting out at lower levels, students at the reform school outperformed students at the traditional school within two years. More students (41% vs. 27%) at the reform school went on to advanced math classes by the fourth year of the study. Achievement differences between ethnic groups were reduced. The students at the reform school showed increased relational equity: helping others learn, taking responsibility (59% vs. 5%), and respecting other’s viewpoints.


Teachers require training to instruct students in the practices that make group work effective. Teachers need to help groups avoid unbalanced workloads, and make collaborative norms explicit. Teachers must also act as advocates for respect and responsibility, paying attention to how students are working together and not just to the mathematical outcomes.

 Implication of Study for Stakeholders:

The traditional teaching method in which teachers dispense information and individual students process it is not the most effective way to improve academic outcomes, and it does not address social justice, social and emotional learning, or citizenship goals at all. Grouping students heterogeneously by race, ethnic group, gender, and ability, and allowing them to work together to process a given topic improves outcomes in all of these areas, as well as engaging students in the development of twenty-first century skills such as collaboration, critical thinking, and communication. Teachers need support and training to enact these radical changes to their teaching methods so that they are implemented effectively in their classrooms.


Write Your Own Textbook

One week to go before school starts up again and I am beginning to get nervous.  I am also getting depressed because I feel like I am a poor teacher and I don’t know how to teach.  I know, POSITIVITY, please.  So, I am looking through back issues of Mathematics Teaching in the Middle School, which I was delighted to find my NCTM membership gives me access.

I picked Mathematics Teaching in the Middle School because as I stated before, it seems much more relevant to the level of mathematics that I teach.  And I’m hoping to find some ideas to use for the whole class discussion we have been studying in our Summer Institutes.  There is an online flipbook edition, and individual articles can be downloaded as pdfs.

Producing a How-To Book by Julie A. Mallia, Don Pawloski, and Peggy Daisey is from the February 2012 edition of  Mathematics Teaching In The Middle School. I tried to do this last year with my Geometry classes, having them write their own textbook for each unit.  But it was just a list of things to do, with very little original writing (put in your own words…) and most of it was just transcribed verbatim from the textbook, with very little evidence of understanding.  This article proposes a project that would be a summing up at the end of the year (or in my case of my Algebra double-block, end of semester).  Ideally, it would be done in concert with their English teachers, so they would make sure it was original writing.  Also, the students get to pick their topic (from a list I provide) so that they can select something they are confident that they understand. 

August Journal # 2

I received the second of my two journals.  This one is Mathematics Teaching in the Middle School (a name that the NCTM is looking to change), and as I predicted, this had much more that was relevant to my classes.  In fact, I don’t think there is one thing in this issue that I couldn’t use in my classroom.  So I’ll just hit the highlights.

I-THINK I Can Problem Solve, by Sararose D. Lynch, Jeremey M. Lynch, and Johnna Bolyard explains a method to scaffold student discussion and to encourage metacognition.  It compares this method to Think-Pair-Share (a technique that I realized, upon reading this article, I’ve been doing wrong all this time.  Facepalm.).   I-THINK stands for:

I   –   Individually think about the problem
T  –  Talk about the problem
H –  How can it be solved?
I   –  Identify a strategy to solve the problem
N –  Notice how your strategy helped you solve the problem (I love this step!  Thinking about what you just did rather than rushing on to the next problem)
K –  Keep thinking about the problem.  Does it make sense?  Is there another way to solve it?

The authors also discuss how to implement this method, which I really appreciate.  It suggests starting with a classroom discussion on “Thinking about Thinking” and then moving on to a problem with no one solution.  Using this to introduce I-THINK and using visual aids and graphic organizers to reinforce the framework is recommended.  They also gave a rubric for scoring student work.  Even the authors said they only do a I-THINK lesson once per week.

Develop Reasoning through Pictorial Representations by Wendy P. Ruchti and Cory A. Bennett is another way to, as the article put it, “illustrate students’ understandings or misunderstandings.”  This reminded me of Twice as Less in that it provides a window in to the through processes of the students.  And again, it gets the students thinking about thinking, so that they can analyze their own thought processes.  The authors point out that this method is invaluable in teaching fractions, as it lets teachers know the flaws in student reasoning, and it gives the students a method for checking their own work.

There’s another article about building vocabulary through graphic organizers, but since it goes over the same ground as the article in Mathematics Teacher I won’t go in to it here.  There are a couple of projects and a nice open-ended discussion activity, plus a “palette of problems”  that could be used as openers.  Lots to use!

August Journal

Earlier this year, I bit the bullet and joined the National Council of Teachers of Mathematics , or NCTM .  I had resisted joining due to the cost, frankly.  And I wasn’t wrong.  My membership cost $115 for one year.  In all fairness, the cost was so high because I decided to subscribe to two of their journals rather than one.  I received my first journal this week.

Mathematics Teacher is geared toward high school teachers like me.  Unlike me, the teachers in the target audience clearly teach higher-level classes.  The majority of the articles relate to pre-Calculus or Calculus classes.  Generally, I teach lower-level classes:  Algebra 1 and Geometry.  But I still found some stuff to be excited about.

Vocabulary beyond the Definitions by Nancy S. Roberts and Mary P. Truxaw has some solid recommendations on incorporating techniques such as word walls and graphic organizers into the math classroom, along with some hints about typical trouble spots for English learners.  My Algebra students tend to do fairly well with vocabulary, but my Geometry students struggle mightily with the volumes of new words thrown at them during the year.  In my classroom this isn’t a phenomena restricted to English Language Learners, but as the saying goes, SADIE teaching is just good teaching.

Improving Student Reasoning in Geometry by Bobson Wong and Larisa Bukalov details a method that allows for differentiated learning in the classroom that is student-directed and allows the teacher to address one concept at a time.  (This is similar to the A-B-C technique discussed at the Summer Institute)  The authors acknowledge the challenges involved with creating such a curriculum, namely the large amount of time it takes and how to properly assess, but there is so much potential with this method.

I also found even more opportunities to spend money, namely in some interesting books advertised or referenced in the articles.
I’m intrigued by Danica McKellar’s books. I wonder if my students would find them relevant and/or interesting.
I’m curious about the advice in Success from the Start: Your First Years Teaching Secondary Mathematics, by Rob Wieman and Fran Arbaugh.  Even though I’m mid-career, I got very little subject-specific training in my credentialing program.

I don’t have a hyperlink for the second title because I found the NCTM online store difficult to navigate.  There is not a search function (so I couldn’t just put in the title or the author), and categories can not be filtered (for example, “Alphabetical by Title” or “Alphabetical by Author’s Last Name”).  I can’t understand why NCTM would go to the trouble to advertise a book they clearly want to sell, but then make it nearly impossible to find on their site.