Thinking Routines – Synthesizing and Organizing Ideas (part 2)

Routines for Synthesizing and Organizing Ideas (Chapter 5) continued… (part 1 here)

4) Headlines (Summarizing, capturing the heart) – Quick summaries of the big ideas or what stands out


I like this routine because it can be short and sweet or extended to a longer, collaborative activity. It originally was used as a way to wrap up PD meetings, allowing groups a succinct way to share without further discussion (good idea!). Individually or in pairs, give students time to create a headline that captures their new learning. Facilitating this work is important so the students create headlines that capture the learning and not just a title for an activity. For example, “Exponential Patterns: Predictable or Not” speaks to what was learned while “Investigation of Exponential Growth” describes the activity.

Then they share their headline and its meaning (their reasoning) with a small group. After the small groups share, post the headlines together and prompt students to find themes. It’s really important to emphasize that the goal is not a clever headline, but a forum to gain different perspectives on what was learned. One headline can’t capture everything, but collectively the big ideas will surface. In a whole class discussion the teacher can either ask students for the story behind their headline, or ask the class what another student’s story might be, then let the student that wrote the headline add anything that was missed.

A few examples/ideas that will work in a math class:

  • Students write the “words behind the headline” on the back of the paper to give the teacher clearer insight into the student’s thinking. This should be limited to a couple sentences to keep the activity relatively short.
  • Triads create a few headlines, then choose one to fine tune and add to the class headlines. While facilitating the teacher asks students to explain why they chose that one and not the other to reveal why they see that as most important.
  • Use after hands-on activities or labs to ensure they discovered the big idea.
  • Create a headline for a state test question to help reveal how they might solve the problem.


5) The 4C’s (Connection making, identifying key concept, raising questions, and considering implication) – A text-based routine that helps identify key points of complex text for discussion; demands a rich text or book. [Simplified vocabulary for the 4C’s – draw the connection they made, what they didn’t agree with, what was most important to them, when they had learned something new or important]

Visible12 Math doesn’t easily lend itself to complex texts, but I can think of a few times to use this. I’m going to show a data video and this routine could help us debrief. Also, in How Not to be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg I came across his response to the common question, “when will I ever use this?” I found it pretty inspiring. I’m going to have the students read it this year and I’ll use the 4C’s as a way to debrief this article.

 PD – Another suggestion is to use this as a means to lead discussions of professional readings.

6) The Micro Lab Protocol (Focusing attention, analyzing and reflecting) – Can be combined with other routines and used to prompt reflection and discussion


I’m really looking forward to using this routine!! This is a useful structure in directing discussions to ensure equal participation and that everyone will contribute. Students learn how to be better listeners and how to build on another’s ideas. Love it!

They provided several math examples for this routine =)

  • After an experiment or lab, this can be used as a summary. Give the students 5 minutes to review the lab, look at notes, and questions, then run the Micro Lab
  • Give groups of students a set of problems that are different but related (e.g., functions or transformations). They distribute the problems among themselves and they have 10 minutes to work on it. Then during the Micro Lab they explain what they did, why they did it that way and where they got stuck or confused. The silence can be used for note taking in this case.
  • Reflection prompts for Micro Lab: How are you becoming more accomplished as a mathematician? Where do you want to improve?

The teacher that used Micro Lab said that it led to collective problem solving and better math talk. The discussions that followed showed insight and students made connections between the problems in the set. This makes me even more excited to try it out!

 A couple of suggestions – The more time that is given to first reflect on paper to the questions, the better the discussions will be. Be strict with the time limits and silence. Debriefing about the silence is important. “The purpose of silence is to take in what was just said and to recent, getting ready to hear the next speaker a pair of fresh ears.” p.150

 PD idea – use Micro Lab to facilitate group reflection on a strategy. For example, “how is your classroom changing as a result of your work with these ideas?” 

7) I Used to Think…, Now I Think… (Reflecting and metacognition) – Used to help learners reflect on how their thinking has shifted and changed over time20140729-220448-79488708.jpg

This is a reflective routine that focuses attention on the thinking more than the activity. This routine can be used when they’ve had a chance to confront misconceptions or shift their thinking in fundamental ways. I bet this could be great with student portfolios to have the students refer to their past work and recognize growth. I’ve been focusing a lot on open response questions, trying to shift how my students think and feel about these problems. This could be a good way to reflect on how they used to approach these problems after I’ve taught them to use our open response protocol.

The authors suggest that when introducing this routine the students should share out as a whole group to model how to explain their thinking with guiding questions from the teacher. Once they get used to explaining their reasoning then this could be used as a think-pair-share.

 “…development of understanding is not just an accumulation of new information but often results in changes of thinking.” 

**I would love to hear if you have used these routines OR if you have any ideas of what math topics for which they could work well.**

(Note: This is the second of two posts that detail one-third of the ‘Thinking Routines’ outlined in Making Thinking Visible by Ron Ritchhart, Mark Church and Karin Morrison. After each number is the title of the Thinking Routine followed by the ‘Key Thinking Moves’, then ‘Quick Notes and Descriptions’. All of this information is directly quoted from the book Table 3.1 Thinking Routine Matrix. Then are the details of the routine and questions to accompany it. Finally, in italics, is a brief brainstorm of how I thought I might use this in a math classroom and/or in math department meetings.)

Thinking Routines – Introducing and Exploring Ideas (part 2)

Routines for Introducing and Exploring Ideas (chapter 4) continued…

3) Think-Puzzle-Explore (Activating prior knowledge, wondering, planning) Good at the beginning of a unit to direct personal or group inquiry and uncover current understandings as well as misconceptions


Similar to a KWL in structure, the key difference in Think-Puzzle-Explore is in how the questions are asked, which shifts the focus first to discovering students’ prior and partial knowledge (and misconceptions), then encouraging curiosity and planning. I’ve always found it hard to use a KWL in math class. Even when I was confident my students had prior knowledge about a topic, I found that they were reluctant to write things down in the ‘Know’ column, stating that they didn’t know if they were correct. The responses for what they ‘want’ to know lacked in depth and curiosity. Overwhelming the response was “I want to know how to solve it” or “how to get it right on a test.” I’m not great at going back to these types of activities at the end of a unit, so I never did follow through with the “Learned” column. I’m happy that the issues I have with a KWL chart are addressed with the Think-Puzzle-Explore routine. I think the shift to the phrase ‘Think’ will illuminate partial knowledge and misconceptions, which is really what I need to know when we start a new topic and is an essential tool in driving instruction. “… (‘Think’) gives permission to have a go, raise possible responses to the question, safe in the knowledge that you are not guaranteeing that you have the absolute facts but rather some thoughts about it.” I think the Puzzle section could be powerful as a whole class discussion providing a chance for students to build upon each other’s ideas. By recording a class list of ‘Puzzles’ on chart paper we could refer back to this list to check items off as they are discovered and add more ‘puzzles’ that arise as instruction continues throughout the unit.

4) Chalk Talk (Uncovering prior knowledge and ideas, questioning) – Open-ended discussion on paper; ensures all voices are heard, gives thinking time. A conversation conducted silently on paper. “It provides flexibility to move from one idea to another in a nonlinear way, to formulate questions as they arise, and to take the time needed to think through the collective information produced.”


With no names written on the posters, this routine gives students freedom to take risks and ask questions that they may not feel comfortable voicing in a verbal discussion. This routine gives every student equal contribution time and the chance for me to hear every ‘voice’, both of which are difficult to accomplish in other formats. The prompts can be single words and phrases, yet posing questions can take the conversation up a level. I used this activity once and didn’t love the result. I think the problem was that I stuck to phrases (e.g. Exponential Functions, Exponent Rules, etc), which seemed to stifle the conversation and they thought I was looking for a particular answer. I’m going to give this routine another try by posing questions instead. I find open-ended math questions hard to create, but I think it would result in a deeper conversation.

The authors suggest using Chalk Talk as a means to reflect on topics or learning moments – I love this idea! I envision using this activity before a cumulative test as a means to discover what the class remembers collectively about a variety of topics and to give them a chance to ask questions too. I anticipate misconceptions or gaps appearing in their work, that I could then use to structure the review activities that would follow. I also love the idea is to use this at the end of a term to reflect on what they learned. The authors provide sample questions to use for this purpose: “What have you been most surprised by in this unit of student? What is hard for you to master in this topic? Where would you most like to see improvement in yourself? What skills do you have around this topic that you could share with others? How do you know when you really understand something?” Nice!

I think this activity can be quite useful in a professional context as well. In a PD I attended last year, the instructor used this activity to pose questions about bullying and homophobic language. I found it to be very effective in giving every participant the opportunity to voice ideas in a safe environment. As a learner, I liked moving around the classroom, reading others responses and having the chance to respond or build off of these. I found it to be reflective and a chance to ‘hear’ many more voices than in a whole group discussion. An added benefit was that the follow-up discussion stayed on track, which I attribute to our collective focus on the chosen questions and realizing that a lot of people had meaningful ideas to share. Also, participants referred to ideas written the poster that struck them, which directed the conversation and fostered the idea of collective knowledge and experiences. I think this could be used in math focused PD, possibly to start the school year with questions focused on our department goals, the focus of the year, changes we are going to implement this year, etc.

**I would love to hear if you have used these routines OR if you have any ideas of what math topics for which they could work well.**

(Note: This is the second of three posts that detail a third of the ‘Thinking Routines’ outlined in Making Thinking Visible by Ron Ritchhart, Mark Church and Karin Morrison. After each number is the title of the Thinking Routine followed by the ‘Key Thinking Moves’, then ‘Quick Notes and Descriptions’. All of this information is directly quoted from the book Table 3.1 Thinking Routine Matrix. Then are the details of the routine and questions to accompany it. Finally, in italics, is a brief brainstorm of how I thought I might use this in a math classroom and/or in math department meetings.)

Thinking Routines – Introducing and Exploring Ideas (part 1)

Routines for Introducing and Exploring Ideas (chapter 4) 

This is one of three posts that detail a third of the ‘Thinking Routines’ outlined in Making Thinking Visible by Ron Ritchhart, Mark Church and Karin Morrison. After each number is the title of the Thinking Routine followed by the ‘Key Thinking Moves’, then ‘Quick Notes and Descriptions’. All of this information is directly quoted from the book Table 3.1 Thinking Routine Matrix. Then are the details of the routine and questions to accompany it. Finally, in italics, is a brief brainstorm of how I thought I might use this in a math classroom and/or in math department meetings.

1) See-Think-Wonder (Describing, interpreting, and wondering) – Good with ambiguous or complex visual stimuli. This routine emphasizes the importance of observation as the basis for the thinking and interpretation step that follows the close looking. This routine was designed to draw on students’ close looking and intent observation as the foundation for greater insights, grounded interpretations, evidenced-based theory building, and broad-reaching curiosity.


I came across a very similar idea (that I LOVE) last October at the Baltimore NCTM conference. The protocol is called ‘I Notice, I Wonder’ out of Drexel University in Philadelphia. It’s based on math problems and built out of this same concept – teaching students to look closely, make observations and then allow their curiosity to drive the rest of the conversation. The ‘I Notice, I Wonder’ protocol became a regular part of my practice this year, and I saw a big difference in how my students approached and discussed problems (I will go into more detail about this in a future post).

There is a slight difference between these routines, as ‘See-Think-Wonder’ adds the ‘think’ component, which is described as an interpretation phase. Questions to lead this discussion include, “Based on what we are seeing and noticing, what does it makes us think? What kinds of interpretations can we form based on our observations? What else is going on here? What do you see that makes you say that?”

Based on what I learned from implementing this last year, I have a few suggestions for implementation: When introducing the problem, provide silent time for the students to look closely. I did have students record their observations, but first I insisted upon one minute with pencils-down, so they could just look. A big part of this process is the discussion that follows because the students build off of each other’s ideas. The quality and depth of their responses grows over time, so don’t be discouraged if it doesn’t go great the first time out. I always recorded their responses on chart paper to validate all the contributions and have a record of the conversation to refer to as we worked on the problem. During the discussion, try to be non-judgmental and record everything! There is not a right or wrong response with this activity, which allows all students the chance to contribute without worry. I used this routine in many different ways throughout the year. The key was to present an open-ended visual without given questions (sometimes I would use the visual representation from a multi-step problem and remove all the questions). Ideas: Graphs (anything real-world works great); different data representations; Dan Meyer’s Three Acts; Diagrams; Sequences represented by drawings; Tables, etc.

2) Zoom In (Describing, inferring, and interpreting) – Variation of STW using only portions of an image


I like that this routine lends itself to practicing elements of a growth mindset. As new portions of the visual are revealed, the students learn to be flexible with their hypotheses and re-work their ideas to encompass the new information as well.

This could be used as a means to introduce to piecewise functions. It could be implemented again when the students learn to write equations for piecewise functions. It could be used in a statistics unit with a carefully chosen data representation to look at what is occurring at different sections/areas of the data. With some creativity, it could possibly be applied to teaching proofs, complex geometric figures, composite figures, tessellations, and/or transformations.

The ‘Picture of Practice’ example in the book details Zoom In used in a math classroom. The teacher wanted students to realize that math is all around them and to see the big picture of math. She used M. C. Escher’s Day and Night image, which has a variety of rich elements to discuss. Great idea!

Day and Night

Thinking Routines – Digging Deeper

Currently I am reading and re-reading Making Thinking Visible by Ron Ritchhart, Mark Church and Karin Morrison. I picked up this book because there was a lot of talk this past year about making thinking visible and I wanted to learn more about it. This is going to be one of many posts concerning what I’ve learned from this book. I’m finding that the ideas in the book are drawing together a lot of things I’ve been thinking about and trying to accomplish with my students.

Routines for Digging Deeper Into Ideas (Chapter 6)

The post details one-third of the ‘Thinking Routines’ outlined in the book. After each number is the title of the Thinking Routine followed by the Key Thinking Moves then Quick Notes and Descriptions. All of this information is directly quoted from the book Table 3.1 Thinking Routine Matrix. Then are the details of the routine and questions to accompany it. Finally, in italics, is a brief brainstorm of how I thought I might use this in a math classroom and/or in math department meetings.

1. What Makes You Say That? (Reasoning with evidence): A question that teachers can weave into discussions to push students to give evidence for their assertions. ‘Fosters a disposition toward evidential reasoning – correctness of an answer doesn’t lie in a lone outside authority, but in evidence that supports it; empowers the learning community to examine the reasons and evidence’

photo 1

This strategy could easily work its way into almost every math discussion and make a big shift in seeing/understanding what/how students are thinking!

2. Step Inside (Perspective Taking): Stepping into a position and talking or writing from that perspective to gain a deeper understanding of it

photo 3

This could be a way to explore functions and/or function transformations (Pre-Calculus, Algebra 2). Give a group of students a particular function and each member steps inside the function from a different representation (table, graph, verbal, algebraic). The group could share and ‘get to know’ their function from each perspective, then each group shares with the class.

Geometry – A way for students to explore parallel and perpendicular lines (or maybe a slope activity) through the graphs, tables and equations. They could possibly narrate their perspective in first person (Each time I climb up 3 units, I step right one unit. When I look up I see another line above me. Each time I climb up 3, so does that line. I’m never going to catch it!)

3. Claim-Support-Question (Identifying generalizations and theories, reasoning with evidence, making counterarguments): Can be used with text or as a basic structure for mathematical and scientific thinking. Frames the enterprise of mathematics as being about speculation, generalization, analysis, and proof

Claim, Support, Question

Math Idea from text – introduce a math problem with a variety of entry points and strategies; give individual work time; discussion of their findings, ideas and generalizations; documents claims; more work time with a focus on evidence to support or refute initial claims

*a specific math classroom example on p. 195-197

4) Circle of Viewpoints (Perspective Taking): Identification of perspectives around an issue or problem

photo 2

Math applications are a stretch…the concept is based on seeing something from different perspectives, so it could work as an introduction in Geometry viewing 3-D figures from different sides.

5) Sentence-Phrase-Word: (Summarizing and distilling): Text-based protocol aimed at eliciting what a reader found important or worthwhile; used with discussion to look at themes and implications.

photo 1-1

Math applications N/A;   Professional Development Idea – a protocol for discussing readings with other adults; share in rounds with a chance to comment after each person shares; document to help find themes

6) Red light, Yellow light (Monitoring, identifying of bias, raising questions) – Used to identify possible errors in reasoning, over-reaching by authors, or areas that need to be questioned. This is used to spot occasions to be skeptical, ask questions and for application. (Note – highlight text or identify with Y or R., then document responses)

photo 4

Math applications N/A

Potential Professional Development Uses from text

a. Facilitating PD ‘when teachers share their classroom efforts, student work, or reflections around professional reading…to move conversations beyond merely agreeing or disagreeing…creates a sense of safety to navigate difficult ideas.’

b. Discussing of action plans and proposals – conveys that there are natural points of dispute that need working through

7) Tug-of-War (Perspective taking, reasoning, identifying complexities) – Identifying and building both sides of an argument or tension/dilemma; to suspend taking a side and think carefully about the multiple pulls or reasons in support of both sides

photo 2-1

Math applications N/A; Possibly use for an advisory lesson


The beginning of a Pre-Calc Notebook – Symmetry

This was my first time teaching Pre-Calculus and I had very few resources to draw from, so I relied mostly on the textbook. Throughout this year I struggled with figuring out exactly how much content should be on each page and how many examples should follow. I was worried about running out of room in the notebook at the end of the year, so sometimes I was stingy on examples or we would do other examples on handouts. I’m definitely going to adjust that this next time around by not cramming as much into each page and providing more examples on following pages.

This is the beginning of Unit 1 – Functions, in my Pre-Calculus notebook.

#1: Point Symmetry – On the inside of the blue paper is a Frayer Model graphic organizer for the definition of point symmetry. (I’ll update this with a photo of the right side of this page soon, which are just some written examples.)

PreC_#1.1_Point Symmetry_R










#2: Line Symmetry – This page includes another Frayer Model and a four-section foldable. The graphs that were on the inside are pictured below and we wrote the definitions in the middle sections. The left side was an example from the textbook with a graph and color-coding.

PreC_#1.2_LR_Line Symmetry


PreC_#1.2_LR_Line Symmetry_INSIDE
















#3: Even and Odd Symmetry – Again, a couple Frayer models and some examples of graphs. The examples on the left side all started with the same function in the first quadrant. The students had to complete each table and graph to create a function that was even, odd, or neither.


FrayerModel_2 FrayerModel_3


ISN – Introduction Pages

This school year was my first full attempt on Interactive Student Notebooks (the year before I started it second semester in Algebra 1 and Geometry). This year I taught Pre-Calculus and Geometry. Many of the resources I’ve used were found on blogs. You’ll be able to tell because these are the pretty ones. =)

Inside Front Cover Pre-Calculus: On the inside the front cover of the ISN I decided to place a handout that I would have the students refer to regularly.  The first was for the Pre-Calculus class. A wonderful colleague, Ms. Ford, shared this resource with me. In Pre-Calc there is a lot of sketching and graphing, and this was a great guide informing them of my expectations. Anytime they needed to graph many would open back up to this page and do a great job. And those that didn’t, well…


Inside Front Cover Geometry: Below is the handout for the inside front cover for the Geometry class. In this class we do a lot of work with answering open response questions in preparation for the state test. Our math department worked really hard this past year to develop an open response protocol. This is the student friendly version that we came up with and they would open up to this page when we worked on any word problem.


Page 1: Numbers About Me: The first page was an icebreaker activity from Sarah Hagan’s “Math = Love” blog called Numbers about Me. The students wrote five different numbers and the significance of that number to them. The example I used was the number 5, because I come from a family of 5 kids. I’m thinking about making this into a “speed dating” activity next year, where they will introduce themselves to another student and share one of their numbers and the significance, then rotate.

INTRO_Numbers About Me

Page 2 –  Course Guide:  I found a version of this course guide on the amazing blog “Everybody is a Genius”.  With a few edits, it became a great course guide for my classes. It’s easy to locate all the important information and much more student friendly than the traditional syllabus. (I do have a more in-depth syllabus that I share with the students, but it’s not in their ISN.)

INTRO_Course Guide2