The conversation about visible thinking in Math started with one of our teachers at Graded, The American School of São Paulo, Adam Hancock, wanting to know how he could incorporate having students’ use their blogfolios in Math class. It seemed natural to have students write for Humanities (Language Arts and Social Studies), but writing did not seem part of what Middle School Math was about.

How could “blogging” go beyond taking a digital image of a Math problem on paper or a quiz and writing about “how the student felt about solving the problem or passing the test?”or ask themselves what they could have done better?

One of the first steps was to bring more “language” into the Math classroom. In a Skype call with Heidi Hayes Jacobs, she said that Math should be taught more like a foreign language.

Students need to know vocabulary words and become fluent in “speaking Math”, in order to be able to communicate their thoughts and ideas.

Videos and screencasts are great tools to articulate, visualize and then share ones’ thinking when working to solve a Math problem. Below is a video of Adam, modeling solving a mathematical equation.

Google Glass- Math Equation from langwitches on Vimeo.

Making Mathematical Thinking visible had the following purpose for Adam in his classes:

1. give students a truly differentiated math experience and expose them to a wide variety of math concepts.

2. encourage self directed learning and allow them to demonstrate their understanding in a way of their choosing.

3. make their learning process visible and allow students to reflect on their growth and learning in the process of solving the problem, by using the KWHL routine (What do I know? What do I want to know? How will I find out? What have I learned?)

KWHL by Mary

*Prezi by Isabella*

More student blog posts:

- Nico’s KWHL Chart and Problem (chart, video, text)
- David’s Math KWHL (Chart & video)
- Andre’s KWHL Chart ( video, text)
- Lucas’ KWHL Problem (image, video, text)
- Alexandre’s KWHL Problem: Quadratic Equation (graph, audio)

The process of making mathematical thinking visible, as well as the artifacts’ quality, was hopeful, awkward, “messy” and challenging…

Adam and my observations:

- Students were working in different areas of math, and most of them had to learn something new, and tie it to what they already know in order to explain their problem.
- It is not a natural skill for students to be able to “speak” Math.
*There is a need to expose and encourage students to use mathematical language to communicate.* - The ability of being able to articulate and make a thinking process visible is a skill we need to support our students in becoming fluent in.
*It was challenging for students to think about and articulate their learning value instead the production value of their artifact.* - Some students focused in their reflection on documenting the steps of what they did as they were solving the problem, not on the necessary thinking that was involved.
*Some students don’t/didn’t see the reason why they should be reflecting on their learning in Math.* - It seemed unnatural to ask students to write a reflective blog post tagged on the end. It seems artificial and one more thing to do as an add-on, versus reflection as part of the learning process.
*Option of breaking the reflection process into different blog posts along the way, which later on can be linked to each other to demonstrate the process path.* - When students are given a lot of freedom to demonstrate their understanding, a lot of them need structure and some clear guidelines or else the product does not turn out very well.
*This may improve with practice and more opportunities for them to work independently.* - Many students didn’t fully follow the KWHL routine, and only posted an explanation to their problem. In some cases the explanations were wrong. In many cases, they didn’t actually post the KWHL page, and so they lost sight of “the point”. Maybe because this was a new process, a lot of students produced “the bare minimum “.
*Generally speaking, students who are conscientious and engaged did well and produced meaningful blog posts. If they did the KWHL process correctly, they documented what they didn’t know before they began researching their problem, and then demonstrated what they learned in the process.* - There is a sense among many students that this is actually ‘more work’ than just taking a test, and therefore it is harder.

These observations are helping us continue to strive for meaningful activities and strategies that support student learning. I am often reminded of Vicki Davis’ blog post, Fail Foward, Move Foward. The word “fail” has a connotation in education, that has to change, along the paradigm shift of how we learn best and differently. In the spirit of Failure is Mandatory in the Culture of Innovation, we are celebrating these “failures” and seeing them as challenges to continue to talk, think, rethink, repeat, throw out, tweak and re-imagine…

I am excited to see how we will continue to make thinking visible in Math and help students write /blog about their thinking strategies in order to become fluent in the language of Math. A big thank you goes out to Adam for learning along side!

Stay tuned for Part 2 in Visible Thinking in Math…

I’m excited to read that Adam is starting to blog in his maths class. Thanks for sharing the KWHL chart, I haven’t seen that before and that is a neat way to help my students structure their writing. You mentioned that we should teach math like a foreign language and create opportunities to speak math and I totally agree. But we shouldn’t limit this to increased use of math vocab words and a forum to writing about what was learned. Blogging also creates a great avenue for mathematical thinking and reasoning which is vital in MYP, DP and the Common Core. It has been a struggle for me to get my students to move away from blog posts that outline the process and what they’ve learned into blog post that explain and justifies the process as well as makes their learning visible.

How do you know?is one of the hardest questions for my students to answer and blogging is a great way to tackle this question.@Ange Thank you for weighing in on the discussion of HOW can we support/encourage/model for our students to think about their thinking and how can one make that thinking visible for others. Completely agree that we can’t limit speaking math as a foreign language to math vocab words and writing. We know enough people trying to “learn” Spanish, French and other language with that method and end up not being able to actually speak it. As a former World language teacher, it is about the exposure to other fluent speakers, first hearing the language, then getting lots and lots of opportunities to hear, see, feel, smell, etc. the language, then opportunities to practice the words learned in order to internalize them. The best method is variety, not one strategy alone. F2f conversations, videos, text, images, illustrations, etc. will all contribute in helping our students become fluent in Math.