6th graders, under the facilitation of their Math teacher, Laurel Janewicz, have learned to take data, analyze the data and tell a story with it. They are demonstrating their understanding of Math concepts, data graphs, misleading graphs and communication skills.
Laurel chose to give authentic, relevant and meaningful data (not invented data) to her students to analyze from the results of a Challenge Success survey taken the previous school year at the school. The survey compiled data about the school’s extra curricular activities, homework habits, parent involvement, student engagement, sleep patterns etc.
Laurel’s plan was to have students analyze the data and then create different types of graphs to be able to communicate their findings in a presentation. Students were to tell a story of the data. The rubric below showed students Laurel’s expectations in terms of content, communication/presentation and a blog post.
Laurel also made connections to standards clear:
The bottom of my rubric has the content standards for statistics and data, but Common Core also has 8 Mathematical Process standards and this project hits on a lot of them:3. Construct viable arguments and critique the reasoning of others.Make conjectures, justify conclusions, communicate them to others4. Model with mathematicsIdentify important quantities in a practical situation and map their relationships using diagrams, graphs,etc.Analyze those relationships mathematically to draw conclusions5. Use appropriate tools strategicallyBe sufficiently familiar with tools appropriate to make sound decisions about whether these tools might be helpful, recognizing both the insight to be gained and their limitations.Identify relevant external mathematical resources, such as digital content on a website, and use them to pose or solve problems.Use technological tools to explore and deepen understanding of concepts.
Laurel, in her own words, lists some of the observations and comparison from teaching the same unit in previous years.
What is different this year?I used real data that is relevant to them because I created a survey which they responded to and shared the results with the students and assigned each student a question/results to analyze.I pulled all the parts of this unit into one project. Instead of making and analyzing graphs for one set of data (real or fake), finding and analyzing measures of central tendency for another (real or fake), creating and analyzing misleading graphs for another (real or fake), they do all of it for one real, relevant set of data.I added the element of making the data tell a story- using it to communicate or persuade. Data and a narrative go best together.I incorporated use of technology so they could share this on their blog not just with their classmates and the Graded community, but with a global community.I dedicated a lot of class time for working on this and shared student work along the way so students could see exemplars and offer and receive feedback.I designed specific questions for students to offer feedback on the projects on the blog posts.
From the perspective of modern skills and literacies upgrades:
Good teaching is good teaching. Adding technology to bad teaching still will not increase student learning. Adding technology to good teaching can add new layers and open up new dimensions of connections and learning. Laurel’s lesson on data analysis and graphing (including misleading graphs) was well planned, developed and executed to begin with. The lesson could have stood on its own and would have addressed the Math standards.
By tweaking the lesson, as Laurel described above, so many more instructional methods, skills, literacies and standards were addressed:
- making thinking visible
- being able to visually tell a story with data
- communicating that story via an electronic media for a larger audience (potential global connections)
- communicating math concepts
- going through creation cycle: data analysis, creation, sharing, publishing, feedback, revision
- student choice
- media literacy: choose appropriate media, possibly “media/app smashing”, by mixing several tools/media to create one project
- network literacy: writing for an audience, receiving feedback, responding to feedback
- information literacy: analyzing data, recognizing misleading data, visualizing data, interpreting data from multiple perspectives
- digital citizenship: be aware of copyright of digital images (Creative Commons, proper citation)
Natasha, one of the sixth grade students summed up her experience in her blog post:
In math, we have been working on a project with data from the responses we got from the Challenge Success Survey. I thought that this project was extremely interesting because we got to incorporate our knowledge of most of the things we had learned about in that math unit. I really liked taking on my project from a different perspective. I also got to experiment with different websites that were really cool. I got to learn all about misleading graphs, graphs and so many other things that I hope you find as cool as I did.
Is it Fake or just Misleading? By Yael
Let’s Get into This by Rens
The Challenge is Complete by Felipe
Interested how this story continued to unfold? Watch for an upcoming blog post of Blogging in Math class, with student samples and model lesson video of Laurel introducing her expectations for quality blog commenting in Math.
This is the second part of the blog post : Visible Thinking in Math
She started out with laying a foundation from the start of the school year.
Listen to her students explain the why, how and what next of metacognition in Math class.
How could she give her students practice in articulating their mathematical thinking? We chose to use iPads and Explain Everything app.
- Students took an image of the Math problem
- Students recorded themselves solving the Math problem. Emphasis was placed on articulating their thought process, including when they thought “I really don’t know where to start”. Helping making their “fluency” of following thinking like that with strategies to continue audible.
- Once the video of them writing and talking themselves through solving the problem (correctly or incorrectly solved), the project file was saved as a video clip and exported to the camera role.
- Another student was then charged in starting a new Explain Everything project on the same iPad and importing the previously saved video clip from the Photo Gallery.
- It was the new student’s job to watch and listen to the thought process and annotate mathematical thinking and strategies observed.
- The new video (original video clip plus annotations, written and oral) was saved as a new video clip and uploaded to Google Drive to be able to be embedded into a blog post
Examples of one of the final video clips (make sure you listen to oral annotations by student #2… about 3:13 minutes into the recording).
Laurel presented at the AASSA (Association of American Schools in South America) conference this past month with an elementary school colleague, Kelli Meeker, about her findings and experience of Redefining Reflection
Laurel also developed a few questions as follow up to help her students reflect on their blogfolio on the metacognition “project”
What does metacognition, thinking about your thinking, mean to you and how has it helped you in math?
Metacognition, thinking about my thinking, ……
What does your “inner voice” say to you or what questions does it ask you as you solve a problem?
I have an inner voice that …..
How has reflecting on your thinking while solving a problem helped your mathematical thinking?
Reflecting on my thinking/listening to my inner voice while doing math ….
What have you learned about yourself as a mathematician from this project and from this whole year?
This project/This year I ….
Below are a few excerpts of student responses. Click on the students’ name to see their entire blog post and embedded video.
Thinking about my thinking is reflecting in my own words. It is thinking about how your thinking can improve and what your thinking has mastered. When I am thinking about my math thinking like when I am screen casting a video on Explain Everything, my inner voice tells me to break up the problem and then read the specific part and work on that part. Afterwards, I think about if this is a good strategy or not. I think that this Explain Everything project has helped me a lot because I solved a problem and then I listened to my thinking while solving the problem
In math, Ms. J taught us to kind of talk to our “inner voice.” I only talk to my inner voice in difficult problems, I sort of ask for help. When I’m with my inner voice, I try to think differently, and usually can get a way for my answer, but I need to concentrate a lot. While I reflect on my thinking I always think in a better way. This helps because I always question myself and see if I’m really correct. I get to a more profound way of thinking.
We have been focusing on metacognition while doing math. This means thinking about our thinking, and asking our selves, “What am I doing, and why?”Using metacognition has really helped me analyze my results in math and it has also made my work a lot more error-free. Whenever I do questions now, and I am not sure how I got my answer, or if it is right, than I always think back to what I did to find out the answer, and if I could do anything better. This is also a habit of mathematical thinking that I find that I am very good at, and I use a alot.
Metacognition, thinking about thinking. When Ms.J first introduced this to us I was like, What The heck! What does she expect us to do? But now I see that it’s a useful skill that has improved not only my math skills but my other classes as well. Very early on i realized that I loved to talk. Ever since i was little i knew this. So it’s one of the reasons why sometimes I think I get bad grades in math. I hate being alone, and in fact am afraid of being alone, so not talking is a symptom. I usually struggle in silence because I like to work through my thinking aloud. Which was why I benefited from this project so much.
I think that I can apply metacognition to lots of different things, like sports that I play, like basketball. During a game, I can ask myself: “Why isn’t this working? What can I do to improve?” The next quarter, I can work on improving in those aspects to help the team win the game.
I realized while doing the project that in my head I am thinking about more than one aspect of the problem at a time, as we call it in math class, my inner voice. It was constantly checking if what I was doing made sense and figuring out other efficient and coherent ways to solve it, so if I had any difficulties or needed to revise my work I could use them. By, also, hearing my second voice I was able to understand the problem on another level, meaning I could draw the right visuals, analyze it, and do it with a different method.
When I first came here from 5th grade, I soon realized that I was not really listening to my thinking, actually not at all. I still did not know what metacognition actually meant and could not define it in first quarter. Now I can define it, and know what it is. So then, I started to think more deeply what I am doing and why I am doing this while doing these problems in my head. This has really helped me because it can not only help you to see the reasonableness of the answer but also to read more carefully.
Metacognition helped me, because, when I make a mistake in the problem, I don’t really notice it, unless someone else shows me what the mistake was, or where it was. After hearing myself in the problem, I can tell if I made a mistake. For example, if I misread the problem and didn’t notice, then heard what my thinking was, I would’ve noticed the mistake I had made. Metacognition, to me, means understanding what works, and what doesn’t work in your head.
When I would reflect my thinking on the iPad, it helped me by looking over my homework’s, my tests and etc. It would help me now and then. My inner voice would ask me “Does this answer make sense?” “How did you get this answer?” When my father would ask me “How did you do this problem?” I would say “I don’t know?” That when I realized that I need to ask myself these things. Now metacognition helps me a lot, like when I am asking my dad for some help and when I am doing a problem by myself
I have an inner voice. I think that the whole purpose of the iPad projects, was to find my math inner voice, and use it. I think I found that inner voice. I’m pretty proud of myself for that because it was with my first projects, it was pretty hard, though now, for sure I found it. It helps me wonder, and think: Should I use this chart or this chart? Which method works best?
While doing these problems, I have sort of an “inner voice.” Not in the crazy, psychopathic way, but the thinking way. I tell myself to do this or do that, or check my work. I say hundreds of things to myself in my head. And I always ask myself how I did this. I explain to myself, and try to find mistakes. Mistakes teach you that to become great at math, you need to make mistakes. Albert Einstein once said,”A person that never made a mistake never tried anything.” I know I’ve made mistakes that that inner voice saved me from.
We are having conversations, looking at student samples, tweaking how reflection and thinking about their thinking impacts student understanding and learning as well as create peer-created resources for future students (think Alan November’s thoughts about leaving a legacy).
A million thanks go to Laurel and Adam for sharing their thoughts, questions, trials, failures and success in the process and most importantly their willingness to make it transparent for others to learn with and from their process.
Do you have student samples of making mathematical thinking visible? Please share the link for all of us to learn from and have quality examples to model after.
More examples of students “writing” in Math:
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?
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.
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…