Making Mathematical Thinking Visible with The Explanation Game

In Minnesota, third graders need to be able to represent multiplication facts by using a variety of approaches, such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line and skip counting (3.1.2.3).

For students aged 8- and 9-years-old, this mathematics benchmark is quite dense. Even the "I Can" statement, written in kid-friendly language, is still fairly complex and abstract.

To help her students understand the concept and be able to successfully demonstrate the skill described in this benchmark, one third grade teacher used the Thinking Routine The Explanation Game (from Making Thinking Visible by Ritchhart, Morrison, and Church.)

This teacher wanted her students to look closely at features and details of the different ways to show multiplication facts, so The Explanation Game seemed like the perfect thinking routine to elicit that type of thinking from her students.

During the Explanation Game, students take a close look at something they're trying to understand and:

Name it. Name a feature that you notice.
Explain it. What could it be? What role or function might it serve? Why might it be there?
Give reasons. What makes you say that? Or why do you think it happened that way?
Generate alternatives. What else could it be? And what makes you say that?

With her students, the teacher made six posters, each with a different way to represent multiplication facts. The teacher purposefully left off the name of the particular multiplication strategy shown on the poster.

The students then had to answer these questions:

• What is the multiplication equation shown? 
• What strategy are you using to show that equation? Explain the strategy to a partner. 
• Write another equation and solve it using that strategy. 

Finally, the students labeled each poster with the name of the particular multiplication strategy. By allowing the students to independently construct their own understanding of the multiplication strategy first and then naming each strategy, the teacher was guiding her students' learning with inquiry, a powerful vehicle for learning in the IB Primary Years Program.

Below are the posters after the learning engagement.

Teachers in other grade levels could use this same Thinking Routine and lesson structure to address mathematics benchmarks at their particular grade level, particularly when the benchmark requires students to represent a number or a mathematical operation in a variety of ways. The list of benchmarks below is not an exhaustive list, but is only provided to show the variety of settings in which this Thinking Routine and lesson structure could be used.

Kindergarten (K.1.1.2): Read, write, and represent whole numbers from 0 to at least 31. Representations may include numerals, pictures, real objects and picture graphs, spoken words, and manipulatives such as connecting cubes. 

1st grade (1.1.2.1): Use words, pictures, objects, length-based models (connecting cubes), numerals and number lines to model and solve addition and subtraction problems in part-part-total, adding to, taking away from and comparing situations.

2nd grade (2.1.2.1):Use strategies to generate addition and subtraction facts including making tens, fact families, doubles plus or minus one, counting on, counting back, and the 
commutative and associative properties.

4th grade (4.1.2.1): Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. 

5th grade (5.1.1.1):  Recognize that quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction or mixed number, or a decimal. 

6th grade (6.1.1.4): Determine equivalences among fractions, decimals and percents; select among these representations to solve problems. 

Step In & Back

Since the beginning of the 2013 school year, teachers at our IB PYP school have been investigating about how students think and what routines we can use to help our students think more deeply about big ideas (called concepts in the PYP). One way we've been exploring how to do this is by reading and discussing the book Making Thinking Visible by Ritchhart, Morrison, and Church.

Recently, as a 6th grade teacher began the transdisciplinary unit Who We Are (for more info about this unit, see our school's POI), she used the Understanding Map to identify what kind of thinking she wanted her students to do. The Understanding Map was developed by the Visible Thinking and Cultures of Thinking Project at Harvard's Project Zero.

At the beginning of the unit, she wanted to give her students the opportunity and time to wonder what they were curious about regarding health, decision-making, and interactions with others. So, she decided to use the thinking routine Think-Puzzle-Explore. This routine allowed her to identify what students already knew, what they were wondering, and where students thought they could go to find answers to those questions. For more information about this thinking routine, read my post from May 2014: Think-Puzzle-Explore.

As students thought through what they knew and wanted to know, the teacher quickly began to see that many students were curious about Ebola. Wanting to leverage this curiosity, the teacher planned to teach the metacognitive and comprehension strategies from her reading curriculum unit with texts related to the current Ebola epidemic, which she found at the site Newsela. 

As students began to read, identify main ideas, and find details that not only supported the articles' main ideas but also answered their initial questions about Ebola, they seemed to come up with even more questions. It is important to point out that it was when the teacher gave content that was meaningful and worth knowing about, students suddenly were engaged in reading, understanding, questioning and most importantly thinking!

Now that the students had some background knowledge on Ebola, the teacher and I began to collaborate and we returned to the Understanding Map to decide that our next step was to ask the students to consider different viewpoints regarding this global issue. To help students do this, we decided to use the global thinking routine Step In & Back, developed by researchers working in Project Zero's Interdisciplinary and Global Studies Project (Veronica Boix-Mansilla, Flossie Chua and Melissa Rivard).

Since we expected students to consider a viewpoint different from their own, we selected a radio interview of Patrice Juah, a Liberian woman who recently traveled to the United States. As students listened to the interview for the first time, they took notes on what Ms. Juah saw, thought, cared about, and wondered. We also listened and took notes, modeling for the students how to listen carefully and think deeply.

After, students interacted with each other, noticeably showing respect for and valuing others' thinking as they listened carefully to each others' ideas. Once students had initially shared their thinking, we listened to the interview again. After the second time, students again shared their thinking. Below is some documentation of some students' thinking from the Step In part of the global thinking routine.

Next, as a whole group, we had a debriefing discussion on the process of using Step In as a framework for considering another perspective. Student responses included:

• "It helped me know what I should be listening for. Without it, it would have been hard to listen to."
• "Listening to the interview twice helped, because if I was writing down one idea the first time we listened, I probably missed another important idea. When I listened to it the second time, I could hear other important ideas I had missed."

Then, students were asked to Step Back and examine their own perspectives. Students were given two prompts:

• What questions do you have for/about this person? Who could you ask?
• What doubts did this raise about your ability to take this person’s perspective?

Students were given a couple of moments to write down their thinking and then shared with the group. The students' thinking regarding the 2nd prompt was:

What doubts did this raise about your ability to take this person’s perspective?

• The interviewer and the interviewee were speaking quickly.
• The accent was hard to hear and understand.
• I haven’t lived through the things she’s lived through.
• She didn’t talk about all the little details - we still have more questions!

With regards to the first prompt, we wrote down all the students' questions, synthesized them, found Ms. Juah on Twitter and posed the questions to her. Our interaction is below:

Click here to read Ms. Juah's poem, The Ebola Ride.

After reading about how one teacher had her students consider different viewpoints by using the global thinking routine Step In & Back, how could you use this routine to get your students to think about other angles of a particular topic?

NOTE: The words in red above are some of the cultural forces that play a part in creating a Culture of Thinking in our classrooms. To learn more about these Cultural Forces (there are 8 in all), read Intellectual Character, Making Thinking Visible, Creating Cultures of Thinking (all by Ron Ritchhart) or visit the page on his website that talks about these cultural forces.

Making Visualizations Visible

The ultimate goal of literacy instruction is to arm students with the literacy processes, strategies, and skills that they will be able to intuitively and independently use to extract meaning from text. "Text" can include the written word, videos, interviews, speakers, field trips, conversations, discussions, and more.

The teaching and learning of these literacy strategies is best done in authentic contexts. On page 70 of Making the PYP Happen, the authors state that "it is the school's responsibility to provide authentic context for language teaching and learning in all areas of the curriculum," [emphasis added].

In a Primary Years Program (PYP), the program of inquiry provides the authentic, engaging, challenging, and significant context for learners to develop and use language. In other words, teachers should situate literacy instruction right inside the units of inquiry. However, it should be clarified that although comprehension and metacognitive strategies are to be taught within the unit of inquiry, it cannot be assumed that learners will implicitly pick up on these strategies while studying content concepts. PYP teachers still must provide specific time and opportunities for their learners to explore, discover, practice, and construct understanding of these literacy strategies. When we teach literacy in this way, students are learning literacy processes and content concepts simultaneously.

Recently, I took the opportunity to teach a first grade reading lesson from our school's literacy curriculum. The objectives of the lesson were that students would retell story events and visualize & analyze characters during reading. Rather than use the text provided by the series, which would not have been authentic, challenging, or significant for the first graders, I choose a text that would help the learners understand their current Unit of Inquiry's Central Idea: Responsible choices promote balanced living.

A List from Frog and Toad Together by Arnold Lobel.

In the story A List, Toad creates a list of all the things he'll do during the day. I selected this story because the list he makes contains many responsible choices that would represent a balanced lifestyle. (SPOILER ALERT: Toad never completes all the items on the list because it blows away in the wind and he subsequently goes melodramatically ballistic, which should not come as a shock for those of you who know Toad well.)

Before reading the story, I discussed with the students the importance of being able to really understand a story while we read. "After all," I said, "that's the point of reading! To understand what you read." In their book Strategies that Work, Harvey and Goudvis say that "different readers rely on different strategies to help them gain better understanding." One such strategy is visualizing. "Visualizing strengthens our inferential thinking. When we visualize, we are in fact inferring but with mental images rather than words and thoughts." The authors continue, "When we visualize, we create pictures in our minds that belong to us and no one else," (p. 130-2).

However, if we let students keep their visualizations locked up in their minds, students may become mentally overwhelmed. Rather, my goal of the lesson was to get students to make their visualizations visible so that they could more fully engage in thinking. "...it is difficult [for students] to do their best thinking because of cognitive overload, that is, because the thinking demands exceed their capacity. When our thinking is distributed, when we do not have to rely solely on our internal mental resources, we free ourselves up to engage in more challenging thinking," Ron Ritchhart in Intellectual Character: What It Is, Why It Matters, and How to Get It, p. 100-1).

I prompted students to visualize what was going on in the story as I read, but then to draw those pictures from their mind on a blank sheet of paper. After reading, we talked about what their pictures should include, if they really, truly understood the story. Together, we decided that students would need to have the characters of the story, the setting (time and place), and what went on in the story (the events).

To show students how to self-assess their understanding of the text, one student showed his visualization to the group while I modeled the kinds of questions the students should ask themselves.

Did I understand the story?
Did I include the characters?
Did I include the setting?
Did I include the important events of the story?

By reviewing all the students' visible visualizations (I included other examples below), the classroom teacher and I could clearly and quickly see who was able to adequately extract meaning from the text and who was not. Using this powerful information, the classroom teacher could form guided reading groups to support students' unique and specific literacy needs.

After reading about making students' thinking and understanding visible by having them draw out their mental visualizations, how could you make your students' visualizations visible with the "texts" they read, listen to, and view?

Teaching Literacy in Authentic Contexts

Recently, the third grade team was reviewing language standards (from the MN K-12 Academic Standards in English Language Arts, 2010) as they collaboratively planned how they would teach their students about nouns. The third grade language benchmark 3.10.1.1 states that a third grader must "Demonstrate command of the conventions of standard English grammar and usage when writing or speaking" including: 

• Explain the function of nouns ... in general and their functions in particular sentences.
• Form and use regular and irregular plural nouns.
• Use abstract nouns (e.g., childhood)

Before students are able to tackle these complex grammatical ideas, teachers knew they needed to first teach some prerequisite skills to their students. The teachers decided to focus their upcoming grammar instruction on being able to identify and create common and proper nouns.

In our Primary Years Program (PYP), "learners' needs are best served when they have opportunities to engage in learning within meaningful contexts," (p. 68 of Making the PYP Happen). The authors go on to explain that it is the school's responsibility to provide authentic contexts for language teaching and learning in all areas of the curriculum. Not only is this a requirement of the PYP, but teaching literacy concepts, processes, strategies, and skills in meaningful, authentic, relevant, engaging, and challenging contexts is considered best practice. After all, it is in authentic contexts that we adults actually use those literacy processes that we learned in primary school, so why shouldn't our students learn them in this way?

In their current unit of inquiry, Who We Are, students are constructing an understanding of the central idea that community forms when people realize they have things in common. During the unit, students study those who have left their community to come to a new one.

One way students learn about this is by reading the book The Long Way to a New Land by Joan Sandin. Since students were already reading the book, the teacher had the students use the text to learn more about common and proper nouns. Students were instructed to read through the book, identifying common and proper nouns by sorting them onto a t-chart that the students created from a blank sheet of paper. Common nouns on one side and proper nouns on the other.

When the students were given the chance to develop this literacy skill in a meaningful context they were motivated and engaged. And because the task was relevant, significant, and tied to the other learning that they were doing, students learned! When the teacher tested the students ability to identify and create common and proper nouns, all of her students (100%) demonstrated proficiency.

After reading about how one teacher successfully taught literacy, specifically grammar, in authentic context, how could you or have you explicitly taught literacy concepts, processes, strategies, or skills in authentic contexts?

The initial stage of an inquiry

In 2012, representatives from the International Baccalaureate evaluated our Primary Years Program. Among the 52 recommendations that came from that report was the following:

The staff review their planning process to ensure pre-assessment, with a variety of assessment strategies and tools, is included in order to consider students' prior knowledge so that learning can build on what students know and can do.

Before new learning can begin, students' current knowledge, skills, and conceptual understandings must be considered. However, this is not the only thing that happens in the initial stage of an inquiry. Provocations should also invite students to wonder and be curious.

At our public school, we do not often guide students through open inquiries, where they can investigate that which is intrinsically of interest to them. Therefore, it is extremely important that we work hard at the initial stage of an inquiry to get kids interested in the concepts/topics taken from the academic standards on which our units of inquiry are based.

In order to pre-assess her students knowledge, skills, and understanding and to get them interested in an inquiry under the Transdisciplinary Theme Who We Are, one third grade teacher lead her students through a Visible Thinking Routine Chalk Talk (from Making Thinking Visible by Ritchhart, Church, and Morrison).

Central idea: Community forms when people realize they have things in common.

An inquiry into:

• Maps
• Why people settle
• Family histories and the histories of others
• A student's place in the world

This teacher posed questions rooted in the Key Concepts of the PYP that fit with the lines of inquiry for this unit. Her purpose for leading her students through this silent conversation was to make their thinking visible so she could assess what her students already knew, were able to do, and understood. She also was inviting her students to think, to wonder, and to be curious about the topics they would be investigating.

As this teacher demonstrates, Chalk Talk is certainly a great assessment tool that gets students interested in the topics about to be investigated.

Below are some additional strategies, tools, and ideas that teachers can use to enter into the inquiry cycle with their students, from different researchers and educators who have created different models of the complex inquiry process.

Creators: Teachers at my school (created during a staff development day)
Initial stage called: Invitation
Teacher:

• Invites students to wonder about a topic
• Visual thinking routine: I notice, I think, I wonder
• Creates a safe environment to take risks/ask questions
• Allows a space for kids to think outside the box
• Fully engages students
• Ignites authentic interest in topics
• Creates a desire to know/understand something
• Captures the hearts, brains, and spirits of kids
• Builds background
• Starts with aspects that are interesting to students
• Identifies topics to study (if based on standards)
• Considers learners’ prior experience and current understanding
• Pre-assess students to learn what they know, understand, and can do prior to the study

Student:

• Wonder about a topic
• What is it like that?
• How does it work?
• Identify what they already know
• Are fully engaged
• Get excited
• Become curious
• Are motivated

Creators: Stephanie Harvey and Harvey Daniels
Initial stage called: Immerse: invite curiosity, build background, find topics, and wonder
Teacher:

• Plans instruction and teaches with central curriculum concepts and focus questions in mind
• Gathers and organizes curriculum materials and resources
• Immerse kids in multiple sources to build background knowledge
• Invites curiosity, questioning, engagement
• Models own curricular inquiry
• Conducts think-alouds with text and materials related to the curricular topic
• Demonstrates how to ask questions about curricular topics
• Facilitates small-group formation to ensure heterogeneous groups with compatible interests
• Confers with small groups and individuals

Student:

• Express their own curiosity
• Explore, experience, and learn about topics using texts, visuals, Internet, artifacts, etc.
• Read, listen, and view to build background knowledge about the curricular topic.
• Talk, write, and draw in response to instruction
• Wonder and ask questions
• Meet with teams to set schedules, ground rules, and goals.

Creators: Kath Murdoch
Initial stage called: Tuning In
Teacher:

• Establishing the 'known'
• Connecting to students' lives
• Create a sense of purpose for inquiry
• Invite first thinking
• First invitation for questions
• Ask:
• What theories do we have?
• How do you already understand this?
• What connections can you already make?
• How could we find out more about this?

Creators: The 5E Instructional Cycle
Initial stage called:Engage
Teacher:

• poses problems
• asks questions
• reveals discrepancies
• causes disequilibrium or doubt
• assess prior knowledge

Student:

• calls up prior knowledge
• has an interest
• experiences doubt or disequilibrium
• has a question(s)
• identifies problems to solve, decisions to be made, conflicts to be resolved
• writes questions, problems, etc.
• develops a need to know
• self reflects and evaluates

Creating rubrics with students

In 2012, representatives from the International Baccalaureate evaluated our Primary Years Program. Among the 52 recommendations that came from that report was the following:

"The school provides further opportunities for students to participate in the assessment of their work, help develop assessment tools, and reflect on the assessment of their work. The school continues to support students in documenting peer/self-assessment activities to enable them to reflect on these over a period of time."

When our school’s staff had a chance to reflect on the recommendations contained in the report, one staff member commented: “I know we can create a rubric with the students, but I struggle with the ‘how’.”

Assessment in the classroom will include, among other things, developing clear rubrics, (Making the PYP Happen, p. 45). In that document, a rubric is defined as, "an established set of criteria for rating students in all areas. The descriptors tell the assessor what characteristics or signs to look for in students' work and then how to rate that work on a predetermined scale. Rubrics can be developed by students as well as by teachers," (p. 49). 

Developing a rubric with students is important because they deserve to know the criteria on which their behavior, creations, performance, and thinking will be judged. When students have a hand in creating rubrics together with their teacher, they have a better understanding of what is expected of them and they also feel included.

One third grade teacher recently created a rubric with her students, outlining the expectations that they have for each other during "work time".

To begin, the teacher asked students to give her examples of "work time" that they had already experienced in third grade. Responses given included:

• Morning work
• Personal histories (unit of inquiry)
• Grit reading (independent work time)
• Portfolios

Next, the teacher set the purpose and explained what a rubric was with the following slides: 

To help guide their rubric creation, the teacher gave the students the criteria (focus & commitment, independent work/reading, small/guided reading group) and the predetermined scale (zones 1, 3, & 5). 

To create the descriptors for each zone, the teacher first asked students to identify exemplar behaviors for each category (Zone 5). Students were instructed to turn and talk with a neighbor to discuss ideas before sharing with the whole class.

Once students had collectively identified the behaviors that were favorable, students worked together to fill in descriptors for Zone 1, the behaviors to avoid. Again, students were invited to talk with a neighbor before a whole-class discussion ensued.

Finally students collaboratively thought about the kinds of behavior that one would exhibit to be in Zone 3.

Next, the teacher showed pictures of the students that she had previously taken during work time and asked the kids to place the students in the pictures in a particular zone, according to the criteria that they had just discussed. This practice allowed the teacher to formatively assess the students' understanding of the descriptors they had just created and would eventually be judged against.

Rubrics are important assessment tools and involving students in the creation of rubrics helps them better understand what is expected of them. After reading about how one third grade teacher created a rubric with her students, how could you or have you created a rubric with your students in your classroom?

Zoom In

A fifth grade teacher recently used the visible thinking routine Zoom In (from Making Thinking Visible by Ritchhart, Church, and Morrison) with her students to help them describe, infer and interpret multiplication arrays.

Read about the experience in the teacher's own words:

Beforehand, I created a simple picture array of identical plants, 3 rows by 6 columns, 18 total plants.

To start off, I told the students that we would start digging into our first math unit and we would be using the visible thinking routine Zoom In to help us do that. I explained that we’d start by looking at a small part of a picture and we’d reveal a little bit more each time. Their job was to observe and jot down what they noticed on their whiteboards.

First, I showed them just one plant.

The students focused on observations of what was in the picture - describing the plant.

Between each round, students shared out their observations and I recorded them on the SMART board. By having students share their thinking, it allowed their classmates to hear what others were thinking and noticing. This helped in making their own observations as they added on to what other students had shared. It also allowed me to ask students to tell more and explain what their thinking meant or how they came up with their predictions.

The next zoom moved across the row to show one more plant, doubling the plants but not yet revealing a full row or column. I asked them to again observe: "What did they see? What new things did they notice? How did their thinking change based on the new parts of the picture they were seeing?"

There were still many plant-based observations and a few mathematical words started to pop out (doubling, multiplying).

Again, I recorded their thinking on the board, as students shared their thinking.

Round 3 revealed two full columns. I reminded students at the beginning of this round that we were looking at the picture as mathematicians and asked them to think about their observations with a math mindset.

Again they wrote down observations, noting how their thinking had changed and how they might think about the picture mathematically. During this round, students independently began predicting what they thought might come next in the picture without any prompting from me.

A synthesis of the class's thinking:

Round 4 revealed a full row, in addition to the 2 full columns they had seen in the previous round. 

During this round of observations, students had discovered the pattern of rows and columns, and most observations focused on predictions of how many plants were going to be in the full picture. Some students were observing using row-and-column strategies, while others were using multiples of 3 or 6.

Everyone's thinking together:

In the last round, the full array was displayed.

By this point, all students could name it as an array and write the corresponding number sentence.

Everyone's thinking after the last Zoom In round:

To wrap up, I asked students to draw another array for 18. All students were able to create another array, which showed me they all understood the concept of a rectangular array and equal rows and columns.

After reading about how this 5th grade teacher used the visible thinking routine Zoom In to help her students describe, infer, and interpret in math, how could you use a visible thinking routine to help your students think deeply about mathematical concepts?