Curriculum Mapping BOOT CAMPS - Heidi and Marie in NYC. Click here for more info.

Telling a Story with Data

Cross posted to Langwitches Blog by Silvia Rosenthal Tolisano


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 others
4. Model with mathematics
Identify important quantities in a practical situation and map their relationships using diagrams, graphs,etc.
Analyze those relationships mathematically to draw conclusions
5. Use appropriate tools strategically
Be 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.

graphing graphing2 graphing3 graphing4 graphing5

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
  • differentiated
  • personalized
  • 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.

Student examples (created in Wideo, Google Presentation, PowToon, Piktochart, Prezi) of presentations:

How Much Time are Graded 6th Graders Spending on Homework? by Maya W.

Come to Graded by Jack

Is it Fake or just Misleading? By Yael

Let’s Get into This by Rens

You Can Never Get Too Deep When it Comes to Data! by Tashi

Homework? Time? What’s Going on? by Laura

Do you do as much Homework as I do? by Alyssa

The Challenge is Complete by Felipe

math-felipe-infographic math-felipe-infographic2 math-felipe-infographic3


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.

Upgrading Blogs Through Lens of SAMR

Cross posted to the Langwitches Blog by Silvia Rosenthal Tolisano

The step from using a static website or emails as a mean to share announcements or calendar items to sharing the same type of items on a blogging platform is not far nor a steep step. My ultimate goal for using a classroom blog or student blogfolios though, is that of creating transformative teaching and learning opportunities, not to have a platform that substitutes a composition book or paper journal. To make the difference visible and clearer, I am looking through the lens of the SAMR model.


First, a Classroom Blog seen through the lens of SAMRblogging_through-SAMR_lens1_111913_105226_AM blogging_through-SAMR_lens2_111913_105258_AM

What about Student Blogs?

Substitution– technology acts as a direct substitute for the task

  • A student uses the blog as a tool to substitute a handwritten/typed and printed assignment. The student copies and pastes a Google Doc or other file from a word processing program into a blog post. The comments on the blog are closed or not being utilized. Students might upload a scanned image of an analog test score, worksheet or other analog artifact. Students might answer a prompt or question posed by teacher to the entire class. There is no added value to the learning process versus the analog task.

Augmentation– technology acts as a direct tool substitution with functional improvement

  • Students use their blog as a platform to publish assignments (research papers, essays, responses) for their teacher to see. The blog is a place to push out information, possibly for Student Led Conferences or a showcase/process portfolio. They insert or embed images, videos, presentations or audio to support their written text. They possibly insert hyperlinks to additional resources. Students tag and categorize their posts with searchable labels. The blog platform becomes a digital organization of students’ online learning records, which is centralized, archived and searchable. Teachers use the comment section to give feedback to their students about their performance. Classmates read each other’s papers and leave comments.

Modification– technology allows for significant task redesign

  • Students use hyperlinked writing as part of digital writing process to show and connect their thinking to topics, influences, relationships and process between previously published content and external resources. Students communicate beyond the written word, in multimedia and transmedia ways. It is evident in their work that they are writing with a global audience in mind and their work encourages conversation, invites multiple perspectives to add and influence their work. Students receive constructive feedback from peers as part of the writing process. Comments inform students’ writing and original task of “paper” extends and “spills over” into the comment section, altering form of writing piece as well notion of “completion” of paper.Students are demonstrating writing skills for digital spaces, by observing digital citizenship, hyperlinked, networked, peer- connected and non- linear writing.

Redefinition– technology allows for the creation of new tasks previously inconceivable

  • The student blog becomes an embedded part of the process and a natural extension of communication and learning cycle by documenting evidence of learning, reflecting, sharing and receiving feedback in order to consider revision. Teachers and students actively and strategically disseminate and connect the blogs to a learning network for feedback and resources. The blog archives artifacts, reflections and connects learning over time. The blog becomes a natural extension beyond assigned academic work and is being used as a hub to document students’ learning, demonstrating self-directed and self-motivated lifelong learning habits as they are organizing, building and maintaining their own online learning records, a growing academic digital footprint and develop their personal brand as well as personal learning networks.

Student Blogs -SAMR2Student Blogs -SAMR


The Heart of the Close Reading Standard

Collaborative blog post by Mike Fisher and Janet Hale.


Despite the amount of publishing and vendor products that employ a contrary interpretation, close reading is really about HOW we engage reading skills. It is not WHAT we engage. Developing “close reading” as a skill is not an essential part of this standard. Instead, it is a methodology, a strategy that is a way in which to reach the heart of the reading standards and the heart of improving comprehension.


The Hows and The Whats


Let’s look at the Common Core anchor standard number one for Reading-Literacy:

Read closely to determine what the text says explicitly and make logical inferences from it; cite specific textual evidence when writing or speaking to support conclusions drawn from the text.


The WHATs are clear:

  • Determine what the text(s) says explicitly
  • Make logical inferences from the text(s)
  • Cite specific textual evidence when writing or speaking
  • What you write or say must incorporate evidence drawn from text(s) to support conclusions


The HOWs are muddier:

  • While “read closely” is explicitly stated, what literally should be read is left to interpretation as well as HOW one records his or her notations based on defined task, purpose, and audience. For example, one interpretation could be creating an opportunity for students to deeply analyze a speech transcript for its rhetoric and annotate (or annotext) to capture evidence. Another interpretation could be re-reading a section or sections of a narrative focused on characterization and have students using a semantic mind-map to make evidence-based notations.
  • HOW students will be assessed–both the actual assessments and evaluation tools (Who will be the assessor? Teacher…Peers…Authentic audience? What will be the judgement criteria? Rubric…Oral Feedback…Jury Panel?) are not included explicitly in the anchor standard; therefore, open to interpretation.
  • “When writing or speaking” is explicitly stated, which means that students must be able to not only meet the criteria of this anchor standard (R.CCR.1), they must also demonstrate their abilities in conjunction with relational anchor standards, such as SL.CCR.4: Present information, findings, and supporting evidence such that listeners can follow the line of reasoning and the organization, development, and style are appropriate to task, purpose, and audience.


By no means are the above the only considerations regarding interpretation of the HOWs involved in the close reading of texts. As a matter of fact, it is important to note that while this anchor standard used the term “text”, when reading grade-level specific standards associated with a related anchor standard for both Reading Literature and Reading Informational, R.CCR.7:


Compare and contrast a written story, drama, or poem to its audio, filmed, staged, or multimedia version, analyzing the effects of techniques unique to each medium (e.g., lighting, sound, color, or camera focus and angles in a film).


students must provide evidence by closely viewing media. This opens up Pandora’s box (RL.4.4) in that many teachers have not personally experienced this form of rigor regarding finding evidence in a media format, which involves its own set of terminology and understanding (e.g., how a specific type of shot–extreme long shot, long shot, full shot, mid-shot, close-up, extreme close-up–affects mood and tone).


Therefore, it is up to a teacher, or a teacher team, to interpret this (and other related) anchor standards. Students could determine what a text says explicitly through a digital-product assessment. Perhaps they could visually represent, through an infographic, logical inferences from two related texts. Another option could be to have students collaboratively (SL.CCR.1)  prepare a multi-media presentation that engages multiple HOWs to support the close-reading task.

The Hierarchy


As Mike blogged before, the words READ CLOSELY do not appear in ANY of the grade specific standards for R.CCR.1, further evidence that it is not the intended focal point. This anchor standard has more to do with building an increasing sophistication of how students deal with details in text (as well as media).


Let’s take a peek at the hierarchy through the use of Janet’s CCSS ELA Progressive Continuum App, which helps visualize new learning from one grade level to another.


(Note that bold print indicates new learning for a particular grade level.)


RL.K.1 With prompting and support, ask and answer questions about key details in a text.


RL.3.1Ask and answer questions to demonstrate understanding of a text, referring explicitly to the text as the basis for the answers.


RL.5.1 Quote accurately froma text when explaining what the text says explicitly and when drawing inferences from the text.


RL.6.1 Cite textual evidence to support analysisof what the text says explicitly as well as inferences drawn from the text.


RL.11-12.1 Cite strong and thorough textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text,including determining where the text leaves matters uncertain.


In the lower grade levels students are expected, with prompting and support, to ask and answer questions about details in text. In subsequent grade levels (grade three), students have to begin referring explicitly to specific details within text to answer posed questions. By the time students have reached grade five, they must be able to quote details accurately from the text in their speaking, writing, or multimedia products or presentations. In grade six the verb shifts from “quote” to “cite”, which alone creates interesting conversation with teachers on the interpretation of what this term truly means, and therefore, demands of students regarding evidence. Through grade 12 students are expected to continue to cite evidence using specific details from the text, but sophistication increases including the need to examine multiple pieces of “strong and thorough” evidence. In grades 11 and 12 , students must start discerning textual details, collecting and curating evidence to aid in determining which pieces of evidence (both explicitly and inferred) that most strongly support the analysis of the text, including “determining where the text leaves matters uncertain.”

The HEART of the Close Reading Standard


When close reading the previous paragraph, what would the key idea be? If you had to boil that paragraph down to a single-word emphasis, what would the word be? How about details? The heart of R.CCR.1 is that–it’s all about the details–questions about the details, referring to the details, quoting the details, citing the details, determining if the details leave matters uncertain.


The heart’s “pulse” is the rhythm students create that starts with answering and asking questions to ultimately discover how to best analyze texts. Students need a strong foundation (including quality modeling) in asking and answering questions in order to ready themselves to independently refer to texts to support their reasoning, including the abilities to quote accurately and cite evidence properly.

The HEART of the Close Reading Standard

Final Thoughts


It is extremely important that teachers collaboratively (both across grade levels as well as within a grade level) understand the heart of each Anchor Standard in Reading, Writing, Listening & Speaking, and Language rather than accept interpretations by someone else. Teachers, administrators, and curriculum specialists should be discussing their personal interpretations with one another and coming to agreement on what the anchor standards require and designing curriculum and instruction based on the mutual interpretations.


The implications are that locally-designed units of study or lesson plans, vendor products or state-adopted curricula may not be a perfect fit, which means there will be a need to closely read the resource’s details to determine where the text leaves matters uncertain. Based on your agreed-upon WHATs and HOWs regarding each anchor standard, what do these resources provide that meet your established criteria? Where are the products lacking or appear to be incorrect? Can those involved in the product or resource close-reading experience support their reading using evidence-based conclusions?


The heart of the close reading standard matters. It has a place and purpose, not only in Grades K-12, but for college and careers. Scaffolded skills that live in the “close reading” standard are necessary to ensure students are able to identify details and ultimately lead to greater comprehension of text in a sophisticated manner. But an array of close-reading skills are not meant to be THE only skill sets that matter. Close reading should take place occasionally, when appropriate for task, purpose, and audience. Any methodology used with too much regularity is doomed. Skill sets and their supportive strategies are meant to be strategic…targeted…focused. If teachers read closely with students every single day, it’s not a strategy, it’s a roadblock.




This blog post focused on the analysis (or close read) of only ONE anchor standard. There are 10 reading anchor standards, and collectively there are 32 English/Language Arts anchor standards. What opportunities for empowering educators regarding curriculum design and instructional practice can be manifested by asking them to participate in collegial discussions and deep understandings concerning all of the anchor standards?


Follow Mike on Twitter

Follow Janet on Twitter


Mike and Janet are the co-authors of Upgrade Your Curriculum: Practical Ways to Transform Units and Engage Students


Building Content Knowledge: Collaborate and Curate

cross posted to Langwitches Blog by Silvia Rosenthal Tolisano

Mark Engstrom. 8th grade Geography teacher and Assistant Principal at Graded- The American School of São Paulo, has redesigned his entire course.

Students move through the modules of this blended learning course on Geography at their own pace.  They build out content knowledge using a Personalized Map (through google maps) and the content delivered through this Digital Learning Farm method will be curated so that they can build out multiple pins on their map.  This content is then used as content knowledge to increase their understanding of the region.
He wanted to experiment with a different type of note taking to add to students’ documentation of gaining subject specific content knowledge. 


The class was divided into 3 groups. Each group contained one  person responsible to contribute by :

  • taking  notes on one google doc- each has a column
  • adding raw data (statistics, facts, charts, graphs, etc.)
  • adding images that visualized what was being talked about
  • writing on the backchannel
  • asking questions
  • linking to the course’s Essential Questions


collaborate-curate 3






Take look at the following video summarizing the class.

It is incredibly insightful to be going through and analyzing the backchannel chat after the class is over. It gives you a better understanding of:

  • what students heard
  • what students felt was important to capture
  • the discussion that evolved in the backchannel alone
  • the connections students made and shared








It was now back into each individual student’s court to CURATE their own notes. Students had access to all  documents from each group as well as the backchannel. It was up to them to go trough the information and take the pieces that they deemed important to add to their content knowledge.

Digital curation

 is the selection, preservation, maintenance, collection and archiving of digital assets.Digital curation establishes, maintains and adds value to repositories of digital data for present and future use.This is often accomplished by archivists, librarians, scientists, historians, and scholars. Enterprises are starting to utilize digital curation to improve the quality of information and data within their operational and strategic processes

Curating information has become a critical skills as part of information literacy. The ability of finding, evaluating, analyzing, remixing,  organizing and archiving information is more important than ever in the information overload era. The amount of information we are confronted with and that is being thrown at us is exponentially growing with no sign of stopping nor slowing down. We need to find ways to support students in becoming  curators of information.

One of the students, Ben, observed the following as he was going through the notes from the Backchannel group:

I found these very interesting because Florens and Tibet really try to link what is happening in India to our life in São Paulo which for me is a smarter way to learn things; by comparing them with your everyday life.



Socratic Seminar and The Backchannel

Cross posted to Langwitches Blog by Silvia Rosenthal Tolisano


Humanities teacher, Shannon Hancock, at Graded, the American School of São Paulo, read and worked through The Alchemist by Paulo Coehlo with her 8th grade students.

Not only did they read the text, learn about literary elements, but also learned to articulate and discuss in a professional manner the text with their peers. Shannon chose to use the Socratic Method, specifically a Socratic Seminar (Inner/Outer Circle Fishbowl) to hand the learning over to her students. She stressed to them: ” Educators don’t need to have all the answers, it is about asking the right questions.” Wikipedia explains the Socratic Seminar as follows:

This approach is based on the belief that participants seek and gain deeper understanding of concepts in the text through thoughtful dialogue rather than memorizing information that has been provided for them. While Socratic Circles can differ in structure, and even in name, they typically involve the following components: a passage of text that students must read beforehand and two concentric circles of students: an outer circle and an inner circle. The inner circle focuses on exploring and analysing the text through the act of questioning and answering. During this phase, the outer circle remains silent. Students in the outer circle are much like scientific observers watching and listening to the conversation of the inner circle. When the text has been fully discussed and the inner circle is finished talking, the outer circle provides feedback on the dialogue that took place. This process alternates with the inner circle students going to the outer circle for the next meeting and vice versa. The length of this process varies depending on the text used for the discussion. The teacher may decide to alternate groups within one meeting, or they may alternate at each separate meeting.

Shannon prepared her classroom by physically arranging the desks in an inner and outer “circle”…


… and prepared her students with the Socratic Seminar Norms for the discussion.


We tweaked the traditional format of the Socratic Seminar to include a backchannel. A backchannel is a parallel discussion, a collectively shaped comment on some ongoing conversations, not that different than the outer circle described in the Socratic Seminar. The backchannel in this case was the secondary digital discussion of the literary text. One student was the backchannel moderator in charge of making sure that Today’s Meet was projected and refreshed properly on the screen.


Watch the video below to catch a glimpse into Shannon’s classroom and their use of a backchannel for the first time.






Reflection of the Backchannel as part of the whole class text discussion:

  • All students had opportunity to contribute to the conversation (even the “silent” outside circle)
  • (Shy) Students who had a harder time articulating orally their opinions in the “inner” circle were able to contribute in written form
  • The skills to listen, observe, document, contribute, read, write, add value, ask questions and respond to others in the backchannel, all at the same time, is not a skill we are born with. It requires exposure and practice.
  • The backchannel log, gives an opportunity to review and assess individual students beyond the “in-the-moment”. It also gives students an opportunity to review and reflect on the experience.
  • The backchannel exposes students to a collaborative writing environment.
  • Possible extensions: Assign a student (or a group of students) to be the “Backchannel Cleanup“, responsible for saving, copying and pasting the log into a shared document. They then edit and format the log by deleting duplicate, unrelated or non-comprehensible comments. They can also organize the comments according to topics.

Analysis of the Backchannel Log:

There were many different layers going on in the Backchannel.

  1. Observation and comments about the Socratic Seminar behaviors
  2. Observations of literary discussion elements
  3. Documentation of inner circle discussion
  4. Added commentary of own opinions.
  5. Parallel conversation going in backchannel and inner circle.

Please note that the screenshots below are not in chronological order. They are shown to illustrate some of the points of the reflection and thoughts about the use of the backchannel.













I must admit, that I was in complete awe of the students and their teacher of how well prepared they were to come together and have a serious literary discussion round. The Socratic Seminar lesson could have stood on its own without adding any further layer facilitated by technology. It was the quality of the teaching and learning already present that allowed the backchannel to add another quality layer.

I can’t help myself, but I am already dreaming of further amplification.

What if ..

  • What if the class connects with another class who is reading the same book.
  • What if the one of the class can potentially contribute yet another perspective (possibly due to culture or geographical location) to the understanding and comprehension of the text. (Ex. Could our Brazilian class not contribute the perspective of the controversy of the Alchemist book here in Brazil to a class located in Sweden, for example, reading the same book?)
  • What if half of the inner circle (the fish) is in one class and half of the inner circle is participating via Skype or Google Hangout from a different class? (Synchronous)
  • What if the backchannel is comprised of students from BOTH classes (synchronous (Today’s Meet) and asynchronous (Google Document)?

Interested? Let’s dream up another layer of collaborative reading, writing and discussing literary text.

Visible Thinking in Math- Part 2

Cross posted to Langwitches blog by Silvia Rosenthal Tolisano

This is the second part of the blog post : Visible Thinking in Math

Another Math teacher (sixth grade) at Graded, The American School of São Paulo , Laurel Janewicz, has been passionately piloting metacognitive thinking and reflection in her own Math classes.

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.


What Now?

How could she give her students practice in articulating their mathematical thinking? We chose to use iPads and Explain Everything app.

photo 2

photo 3


photo 5



  1. Students took an image of the Math problem
  2. 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.
  3. 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.
  4. 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.
  5. It was the new student’s job to watch and listen to the thought process and annotate mathematical thinking and strategies observed.
  6. 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: