Revamping Mathematical Thinking Through the Cloud

Along with the new mandated grade level content standards, the authors of the Common Core included transferable math principles coined Standards for Mathematical Practice. The overarching assertion is that these fundamental skills, at the core of the Common Core math standards, will launch American students to be college bound and meet the professional expectations of the 21st Century. The National Research Council claims that “the past 30 years indicate that U.S. students can adequately perform straightforward computational procedures, but their comprehension of underlying mathematical ideas is limited.” Clearly, the deficit in mathematical thinking is an educational problem that educators need to alleviate. I hoped to learn more about how to better support my particular group of learners with these practices through my Mills Teacher Scholars inquiry. -Tommy Gonzalez

“When I grow up I want to be a math teacher,” Ronelle typed in his digital math journal in Google drive. Within this same mathematical journal entry, written towards the end of the year, he Googled an image of math formulas scribed onto a window overlooking a collegiate gothic style university reminiscent of a scene from A Beautiful Mind. My year-long focus on mathematical thinking through daily journal reflections seemed to influence my students’ understanding and appreciation of math, but this had not happened without carefully calculated changes to my math pedagogy, student expectations, and instructional tools.

A Murky Start

The task of teaching mathematics to third graders consistent with the new Common Core regime was, needless to say, initially agonizing. Nevertheless, my focus was simple: Evaluate my students’ mathematical reasoning to ensure that they understood the daily math lesson and concept.

When I began my Mills Teacher Scholars inquiry, my methodology was quite broad. I would ask my students to explain their thinking on a particular problem that was embedded on an assessment. For instance, on a performance task question asking students to “Explain how you figured this (completing an array) out,” Jennifer’s response was the following: “I figured it out because you need to full (sic) the box.” This explanation lacked clear reasoning that would allow Jennifer to reveal the process that she executed in order to arrive at her solution.

Other times, I would ask my students to orally explain the strategies and skills they used. This turned out to be an inadequate means to compile data for my inquiry because their testimony was mostly inconsistent with their work. I needed a source of data wherein I could regularly check for my students’ mastery, or lack thereof, of math skills and concepts.

Breakthrough

Journaling seemed like the most feasible way to capture my students’ math reasoning because it created a documented portfolio of student learning and growth, and would serve as the consistent source of data that I had been seeking. Managing twenty-four daily written reflections is easier than coordinating twenty-four oral presentations. In addition, a survey of my students revealed that most of them thought that daily reflections on worksheets and assessments helped them learn math.

Worksheet Reflection

In the early stages, my students’ reflections were unorganized, paper was wasted, and students did not understand the purpose of having to write the entries. I gave my students direction on how to properly write entries with titles and dates that corresponded with their entries.

During my Mills Teacher Scholars inquiry sessions, I devised some criteria that served as a checklist to analyze my students’ mathematical thinking and shared these with my class by modeling exemplars of each indicator listed below.

Indicators of success:

1) Uses precise academic language to show math thinking in journal entries and assessments.

2) Documents strategies in journal entries and on assessments.

3) Models math with numbers, formulas, pictures, and/or words in journal entries and on assessments.

I would publicly praise my students whenever they would met a specific prong of the criteria and provided my struggling learners with direct guidance to enhance their reflections. For purposes of differentiation, I asked my students to simply use pictures, words, and numbers to articulate either what they learned, what confused them, or what they still wanted to learn. The results of directly teaching journaling techniques and upping the rigor were immediately apparent in their improved journal entries. Furthermore, students perpetually asked if they could share their journal entries with the class and, thus, we fused author’s chair with math, which beget fruitful math discussions.

Before Teaching Journaling Techniques

After Teaching Journaling Techniques

My inquiry would lead me to discover that the math journals allowed students to reveal their thought process, advance their thinking and their depth of knowledge. More critically, the journal reflections were a shift from the California State Standards, which stress a simple computation paradigm, to the problem solving and real life application that the Common Core demands.

Journaling in the Cloud

As a proponent of EdTech, I understood the need to have my students type their journal entries on computers, both for real world applications and to prepare my students for the Smarter Balanced Assessment Consortium (SBAC). The SBAC requires students to explain their math reasoning in constructed responses and in some problems they are asked to create a digital drawing to answer a portion of a question. It was time to toss the pencil and paper and replace it with a keyboard and digital canvas. That is, our math journals became digitalized.

Digital reflection with digitally created image.

My approach was in line with that of Keith Devlin, a well-known technology advocate and mathematician who advises:, “The only way to acquire mathematical thinking ability is by a process of exploration – lots of trial-and-error and reflection.” Soon my students were Googling images and pasting them onto their journals to represent their math reasoning without being asked. This type of exploration appeared to deepen their understanding of concepts and enable them to visually apply their knowledge within their reflections.

Digital entry with Googled images.

I used Hapara, an online teacher dashboard for Google Applications, to organize my students’ Google Docs math journals, which allowed for instantaneous feedback and provided me with evidence of their learning. I utilized the universal sharing function on Hapara to share exemplar entries with the entire class and we followed up with class discussions around this high quality work. Moreover, our transition to a digital platform increased students’ engagement in thinking about mathematics, raised student creativity, and allowed students to identify and synthesize online sources with their learning experiences.

Moving Forward

A few of my students had an aversion to writing reflections or needed extra support. One particular student consistently scored the highest in formative and summative assessments. Nevertheless, her reflections were barren, perhaps because she expresses her thinking more succinctly with numbers. The others who struggled with math reflections tended to be English Language Learners and I noticed that the quality of my ELLs’ reflections improved when they first collaborated with their math partners or conferred with me regarding a particular reflection technique or strategy.

Looking closely at the progression and expression of student math reasoning through digital math journals, I am convinced that this is a viable tool to influence student learning and creativity in mathematics Not only are online math journals timesavers, but they are entirely appropriate in our digital age. Further, our digital learners reach a wider audience when they create on cloud-based platforms like Google Docs or collaborate on a class math blog. We owe it to our mathematicians to provide them with instruments to think rather than simply teaching them computational rules and procedures. When it comes to mathematical thinking, digital mathematical journals are one of many means to that end.

Tommy Gonzalez is currently a fifth grade teacher at Fred T. Korematsu Discovery Academy, a Title 1 school in Oakland, Ca. This is his second year as a Mills Teacher Scholar and the first as a teacher leader at his site. He strongly believes that EdTech, learning online with computers, can create a democratic platform that allows for an exciting personalized and equitable education. He takes this passion into his classroom every day.

*Header image source ddoss