Data Strategies – Determine a Root Cause

Here a few strategies that we used at the data workshop to determine a root cause for a problem:

Fishbone Analysis – In the ‘fish head’ we wrote a problem that we discovered after digging through our school’s data.  (Problem Statement: Our students’ scores on the ELA MCAS are improving but are not improving on the Math MCAS). From here we identified the major categories that feed into this problem and wrote these into the “ribs” of the fish. (Curriculum, Teachers, Students, Leadership). Next we brainstormed the possible causes for each category. For example, under the Curriculum category – open response is challenging for our students, curriculum drastically changing with CCSS, statistics is on MCAS but is not in curriculum, etc. After we worked through all of this, we discussed which of these categories may be the root cause. This tool results in a powerful visual representation of problem and causes. I loved using this!


20 Reasons – This tool would be particularly useful when we have a lot ideas about possible causes of a problem that we need to get off our chest. Sometimes it’s hard to focus a conversation when there are several people involved, each coming in with lot of different ideas that they are passionate about. The idea here is to get all these ideas on paper before talking them out. All you need to do is write down the problem, then start a list. Participants can call out their ideas and every idea is listed. Once all the ideas are listed, each member identifies which of these they think is the root cause and indicated by a check mark.  From there the conversation is facilitated until a root cause is agreed upon.

Why, why, why – This tool is very similar to a conversation I recently had with my 4 year old niece. She asked a question, and I responded. Then she followed up with, “Why?” This continued for WAY too long. And this is exactly how this tool works. The problem is presented to the group, then they are asked, “why?”. The first ‘because’ is recorded, then “why?” Continue this process until there are no more reasons to give. Then circle back to original problem and ask why again. It is recommended to repeat this process 3 – 5 times, then facilitate a conversation about which ‘because’ is the root cause of the problem.


PD – Data Workshop

I spent the last three days in a Teachers 21 workshop – Using Data to Improve Student Achievement. Four teachers from our school attended together, which moved the workshop from being a good experience to amazing! We learned how to navigate several different data sources, strategies to make an overwhelming amount of data manageable, templates to help us dig deeper and create next steps, and time to use these resources to make sense of our school’s data. By being there with a team, we worked through the data together, made observations, inferences, and discovered some glaring problems. We each could have identified these problems beforehand based on anecdotes and experiences, but seeing it represented in data made it so much more powerful. Instead of being based in a feeling, our conclusions were based on facts. It was another example of how vital it is to have the time to dig into data as a team. We left with some great products to bring back to our school that summarize our three days of work in an organized and presentable fashion. We also have some preliminary action steps to address what we found and to hit the ground running in September. Data can be overwhelming and scary to a lot of teachers. Over and over I see that time is most valuable resource for a teacher. We had three solid days to learn about data together and each one of us left excited about the work we did and asking about how and when we can do this again!

Thinking Routines – Introducing and Exploring Ideas (part 3)

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

5) 3-2-1 Bridge (Activating prior knowledge, questioning, distilling, and connection making through metaphors) – Works well when students have prior knowledge but instruction will move it in a new direction; can be done over extended time during the course of a unit


I love the added components to the traditional 3-2-1 prompt! I think the key to using this effectively at the beginning and end of instruction is to be sure the students have some prior knowledge for the initial 3-2-1 and that the lesson has enough new information for a different set of responses. One way that I’ll use this thinking routine is to give (and collect) the initial 3-2-1 prompt as an exit ticket that the day before a lesson. This could act as a quick pre-assessment that I could use then use to adjust the next day’s lesson. I love the bridge component! What a great way for students to make connections to the new content and identify what they’ve learned. 

6) Compass Points (Decision making and planning, uncovering personal reactions) – Solicits the group’s ideas and reactions to a proposal, plan, or possible decision

Visible6I teach 10th graders, the year when they take the state math test, so this activity could be an opportunity for the students to express their Needs and Worries about the test. I think this could help them realize that they are all anxious about the test and allow them to self-assess (and in turn inform me) of the skills they already know they need to improve. I wonder if they would have anything to add to Excitement…maybe this could be where they come up with the benefits of learning these skills and not referring to the test itself. In regards to the test, I’m not sure if I would have anything positive to add here, but maybe they could help me think of what’s exciting about the MCAS. I will delete ‘stance’ and just focus on the Steps or Suggestions phase as a tool that allows their responses to direct the strategies I implement as we prepare throughout the year. I’m planning on doing this in September to give me a sense of this group’s learning preferences. At first I thought this activity would be a stretch, but now that I’ve written it out, I kind of like it! I think there is the potential of building a culture of naming the reasons why something may worry us (a.k.a. math!), looking at this from other perspectives, then being proactive and planning how we are going to proceed.

Another suggested for the use of Compass Points is for the introduction of a potential or new program to faculty. At our school, each department collectively chooses a goal in the fall. As with any consensus, not everyone can “win”. I’m thinking of using Compass Points after choosing the goal to look at it from a number of perspectives. By detailing Excitement, it will remind us of the reasons for choosing this goal and it will be good to refer back when we are frustrated with it in February. =) The Needs and Worries will give insight into the professional development that teachers need to reach this goal. The Steps/Suggestions may help us to split up the work involved in the steps or to create groups of teachers with similar plans.

Although not math related, I also teach an advisory class every other week. I think this could be a great routine to use in that class. We are given general themes or topics to work with, and I bet this will work great with one of them.


7) The Explanation Game – (Observing details and building explanations) – Variation of STW that focuses on identifying parts and explaining them in order to build up an understanding of the whole from its parts and their purposes

Visible7If you think back to the See-Think-Wonder routine, the students were presented with a new or ambiguous figure/photo/diagram. Yet in the Explanation Game, the students are looking at a figure/photo/diagram that they may recognize, but are not sure how or why it is the way it is. “…students may examine the features of a mathematics diagram to determine what those features do or what purposes they serve.” I think this routine could be a great tool when working through real-world data. I anticipate using this as my Pre-Calculus students work with real-world piecewise graphs. The first component of the routine, Name It, is for students (in pairs) to name all the different parts they are observing (e.g. ‘increasing linear function’, ‘horizontal line, ‘decreasing linear function’). This could even be extended to finding the equation to accompany the part that they ‘named’. The second component, Explain It, is where the students would then work to apply the real-world situation to what they see on the graph (e.g. ‘the income increased at a consistent rate for 3 years’, ‘for 5 years the income remained the same’, ‘the income decreased at a consistent rate for 2 years’). Students often struggle with applying the situation to the graphic representation. I think the exercise of naming each distinct part first would make this transition less overwhelming for students. Plus, by working in pairs they would have the opportunity to talk it out with their partner, which always helps! I think the last components, Give reasons and Generate alternatives, often are passed over in math classes. I think this is the part that would make a “real-world situation” actually feel real to students. I often have students interpret a graph, but rarely ask them why these things may be happening. When I create my own graphic representation of data (in the real-world, if you will), I do want to see the trends, but I also want to figure out why those trends may be occurring. Just thinking about these last two components makes me feel like I’ve been robbing my students a fun part of data analysis. I’m definitely going to take the time to work this thinking routine into my practice this year!

**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 third 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 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

A Map of Thinking – Part 2

As I mentioned yesterday, I’m working my way through the book Making Thinking Visible by Ron Ritchhart, Mark Church and Karin Morrison.  Since the main objective of this strategy is to reveal what our students are thinking so that we can further their learning and dispel misconceptions, then it makes sense that we should first be very familiar with the types of thinking. The authors divided the types of thinking into two main lists based on the thinking goal. The first list is focused on thinking moves involved in understanding. The second list is focused on another group of goals, specifically solving problems, making decisions, and forming judgements. Because this is not a hierarchical list, I wanted to make a non-linear representation of these lists to help me wrap my mind around it and to refer to when I’m lesson planning. Below are a couple screen shots of the prezi that I created for the second list. The screen shots of the first prezi are on yesterday’s blog post.

As I was thinking about this list of thinking moves, I kept thinking about the Standards of Mathematical Practice. Has anyone thought about these connections or done any work to connect them?

prezi pictureprezi picture2

A Map of Thinking for Understanding

My thoughts on the book Making Thinking Visible by Ron Ritchhart, Mark Church and Karin Morrison. – continued

The questions – “What kinds of thinking do you value and want to promote in your classroom? What kinds of thinking does that lesson force the students to do?”

The challenge is posed to teachers – to step back and consider, what do I want the students doing mentally? What type(s) of thinking? When I ‘stepped back’, I wasn’t sure exactly what my choices were. I could flounder around a bit, throw out some ideas, but I wasn’t very happy about that. And neither were the authors. What they found is that teachers would often flounder (making me feel better), then ask if the answer was found within Bloom’s taxonomy. Well, it is and it isn’t. They worked to put together a couple of lists to make visible the forms, dimensions, and processes of thinking. They say time and again that these lists are not comprehensive and could be fleshed out further. But, these lists will help us determine the types of thinking that we are going to later work to make visible in our classrooms.

The lists are broken into two ‘Maps of Thinking’. The first is based on thinking moves to aid in understanding and the second to solve problems, make decisions, and form judgements. I was struck by this concept and wanted to wrap my head around these thinking moves. I also wanted to create some sort of document that I could have nearby as I’m lesson planning. Below is the document I created for the first list (link to the prezi below). The second document will come along soon.

Understanding Map



A Map of Thinking involved in Understanding Prezi



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


E-Z Calculus

My husband is learning about network analysis of psychopathology, which has calculus in its foundation. He asked if I could take some time to tutor him this summer, which I am very excited to do! I checked out a number of books from the library and found one that I really like – Barron’s E-Z Calculus by Dr. Douglas Downing. I was looking for something that would explain some of the big ideas of Calculus in a way that was digestible on Sunday mornings in the summer. I’m not sure if this is the case for all Barron’s books, but this one is set in a mythical kingdom. I would bet that many students have felt that they were in a strange land upon entering their Calculus classroom, so this seems appropriate. This mythical kingdom is the backdrop for Downing’s explanations of the in’s and out’s of Calculus. So far, we are both finding it easy to understand, meaningful, and a little silly. Of course, there are also practice exercises at the end of each chapter, which do the trick but are not the main draw. If I ever end up teaching Calculus I will definitely use this as a resource with my students. So far, it gives an accurate big picture all the while using silly examples that make sense, stick in one’s memory, and doesn’t sacrifice the content.




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