An Autopsy of a Sophomore Statistics Unit

This year, we flipped our sophomore geometry class around. Typically, we end the year with a statistics and probability unit, but this year we started with it. In fact, we extended the units from the end of August all the way until Thanksgiving break. Our beefed up units included:

• Creating data displays
• Probability using Venn Diagrams and two way tables to find unions, intersections, and conditionals
• One variable measures of center, spread, and shape
• Measures of location (z-scores and percentiles)
• Experimental design and survey methods

There were many reasons we made the change, but here are a few:

• Connecticut is using the SAT as its high school accountability test, and it has a fair amount of statistics on it (and very little geometry)
• Statistics has been the last unit in our districts 6-8 curriculum, and we all know what happens to units at the end of the year...
• We used to have 1-var stats and linear regression in algebra I, probability in geometry, and polynomial regression in algebra II, but teaching everything separately felt very disjointed.
• Our department generally agreed with Arthur Benjamin's ideas about teaching statistics.

Now for the autopsy (complete with emojis)...

😒 It was a struggle for our geometry teachers to change gears at the start of the year. We found that many of the routines that work in geometry don't work as well with statistics. I have typically used a Which One Doesn't Belong to start every geometry class, but I found it very difficult to find or make ones that worked in statistics. Over time, I found that other routines such as Would you Rather, Clothesline Math, and some 3-act tasks were better suited to starting a lesson.

😩 The shift in content was also tough for teachers. At our school, statistics is typically offered as a senior year course, and our 9th and 10th grade teachers hadn't seen some of the content in years. Remembering how to calculate z-score took a minute; remembering how to teach the meaning behind it took longer. Shout out to the teachers who provided resources that made it so much easier:

• Luke Wilcox's The Stats Medic
• Bob Lochel's Math Coach Blog
• Hedge's Approximately Normal

😁 On the other hand, students felt much more at ease as compared to previous years. We have previously started the year with coordinate geometry, which relies heavily on skills from algebra I.  I've started the year with Jo Boaler's Week of Inspirational Math for the past three years, but students that didn't fare well last year see the writing on the wall and begin to withdraw early. Starting with statistics circumvented that issue and promoted more of a growth mindset.

😃 Starting with statistics also felt natural, because at the start of the school year you're trying to learn more about your students. Those student surveys and "getting to know you" activities suddenly had more of a purpose. It felt natural to ask students what town they lived in (we're a magnet), then ask how long they rode the bus to get to school, then make a histogram of those times, and finally look at the center and spread of the data.

😏 Staying with statistics for an extended period of time let us talk about social justice issues. For example, after looking at the bus ride data we were able to have an informed discussion about the impact of magnet school busing on our students. Unfortunately, these conversations tended to be limited to one lesson. I would love to have students explore social justice issues further, but I'm not quite sure how I would go about it, if my sophomores are ready for that, or if there is enough time.

😓 Speaking of not enough time, we're now in a situation where our normal geometry curriculum is compressed by six weeks. Since the SATs don't focus on geometry, most higher-ups doesn't see an issue. However, I know that our team of geometry teachers will need to start picking and choosing what topics in geometry we can spend less time on. Wish me luck!

The conclusion: the stats unit will live to see another year!

Lessons Learned after Teaching without a Textbook

I was asked to write a guest post for Student Achievement Partner's blog on the pros and cons of using a non-text based curriculum. It was a great experience, and I had a fun time working through it with my colleagues. Check our the original post at the link below:

http://achievethecore.org/aligned/lessons-learned-after-teaching-without-a-textbook/

A Text-Based Curriculum Just Wouldn’t Work

I work as a high school math teacher and building-level team leader in an urban magnet high school in Hartford, Connecticut. Our district includes twelve semi-autonomous and rapidly growing secondary public magnet schools that are mandated to serve a culturally and economically diverse set of students. We draw students from 36 sending districts which include urban, suburban, and rural towns. Each school has a custom theme (e.g., environmental science, performing arts) and school structures vary. At one point, an attempt was made to standardize the curriculum throughout the schools using the Springboard texts from College Board, but due to the drastically different needs of each school, it was difficult to implement with fidelity.

As a result, we currently use a curriculum that was created almost completely by teachers and curriculum facilitators within our district. Each summer, a group of teachers comes together to write and revise the curriculum, which consists of a series of Google Docs connected by hyperlinks for easy editing and sharing. While the structure of our units and some of the materials are homegrown, many of the materials were recommended by teachers from different sources. Below are a few examples:

• Desmos Classroom Activities
• Illustrative Mathematics
• The Mathematics Assessment Project
• Engage NY
• Georgia Standards of Excellence Unit Documents

Each school is asked to preserve the structure of the units, the anchor activities, and some of the assessments we create. Yet due to the variability between schools, teachers are free to take or leave any additional resources included in the curriculum to suit the needs of their students. Teachers are also encouraged to leave comments on the curriculum documents throughout the year that will be used during the next summer’s revisions.

Initial Benefits

Fortunately, teaching without a traditional text has had unintended benefits. It has forced teachers to unpack standards and think deliberately about what strategies can be used to teach both content and practice standards. A sophomore teacher who once taught ratios and proportions “by the book” was pushed to think about the progression of the standards and even used the SAP Coherence Map to research how they are first introduced in sixth grade. During a recent meeting, a teacher remarked, “Writing my own questions has made me understand what the kids really need to know. Seeing structure in expressions is so much bigger than I thought.”

Moreover, we may think all of the chapters of a textbook are Common Core-aligned, but there are often topics that don’t attend to the Major Work of each grade. By ditching the textbook, we have effectively let go of non-aligned topics and opened up more time to focus in-depth on the standards.

The lack of a textbook has also made my colleagues and me quicker to adapt to changes in the education landscape. Connecticut has recently adopted the revised SAT as its high school accountability measure, and by having a stronger knowledge of our curricula, we were easily able to highlight which standards were linked to the SAT and find tasks that were both SAT- and CCSS-aligned.

Lastly, we’ve found that students need resources that are accessible anytime, anywhere. While you may be able to take a math book home with you, that doesn’t necessarily mean it’s accessible. By dropping the textbook, we have been able to focus on finding or creating our own tasks, applets, and online videos that are both student-centered and usable anywhere. Below are a few of our favorites.

• Schoology
• Google Classroom
• Desmos Graphing Calculator
• NCTM Illuminations
• EDpuzzle
• Edulastic

Lessons Learned

On the other hand, the overhead required to teach without a text can be high, and failing to commit fully can leave students in a worse position than if they had a textbook. While our current curriculum does a great job of attending to the Shifts of Focus, Coherence, and Rigor, the inherent flexibility makes it possible to bend the curricula within its limits and lose all three Shifts. My colleagues and I have created a list of tips for maintaining the Shifts without a textbook or rigid curricula to hold onto:

1. Unit plan with a buddy: Many of our smaller schools only include one teacher per course. As a result, there can be serious misalignment between years that leads to a lack of coherence. We make it a policy to plan each unit with another teacher of the previous or subsequent course.

2. Find both a lesson structure and a unit plan template that explicitly includes time for conceptual, procedural, and application tasks: Our curriculum includes ample resources that cover all aspects of Rigor, but it can be tempting to overemphasize procedural fluency by resorting to worksheet generators or online practice. Using both unit and lesson plan templates that include explicit sections for conceptual, procedural, and application tasks reminds us to check the curricular resources or find our own, if needed.Lesson Plan.

3. Comment on the curriculum: As previously noted, we house our curriculum on a set of Google Docs that are open to comments from teachers. Without a textbook in the way, our curriculum coordinator is able to use our suggestions to make changes in the curriculum by the following year and sometimes sooner.

4. Get good at pretesting: Our students come from a wide range of backgrounds, so it is difficult not to revert back to standards from previous grades, especially when your students seem not to remember prerequisite concepts. To maintain a strong Shift in Focus without a text, short formative assessments are crucial. One of my colleagues once reminded me, “How many of your students actually need help with below-grade-level topics? All? Half? A tenth? We should know! Because, for every prerequisite standard you spend more time on, you deprive your students of another, possibly more important topic.” Here are some of our favorite methods:

• Edulastic
• Plickers
• Kahoot
• Pear Deck
• Any of the quiz features on Schoology, Edmodo, or Google Classroom

5. Focus less on common texts, and more on common strategies: Often, math textbooks prescribe how a topic should be taught at the start of each section. Without the guidance of a book, our discussions have shifted to designing and piloting new instructional strategies and routines to improve achievement. We are only able to try one at a time, but we are slowly making progress.

6. Help out your newbies: The learning curve for teaching without a textbook is steep; your new teachers will need you. Introduce them to the templates, lesson plan with them, share resources, and offer to let them observe your lessons.

One of my colleagues recently told me that textbooks are like crutches: although they might save you some pain, you’ll never be able to truly run. I’ll take a scraped knee every once in awhile if it means I get to sprint ahead with my students.

Five Things I Learned about Coherence and Standards at the Core Advocates Catalyst Conference

Full Disclosure: This conference was hosted by Student Achievement Partners, who graciously funded two days of meals and accommodations for the duration of the convening.

1. The Instructional Coaching Guide: Connecticut's teacher evaluation system is based largely on the Danielson Frameworks. While these rubrics are often helpful, it is entirely possible to craft a lesson that scores very highly within Danielson, but does not actually teach any mathematics. Achieve the Core's Instructional Coaching Guide emphases the idea that the three shifts (focus, coherence, and rigor) are non negotiable, but that not every instructional or mathematical practice may be observed in every lesson. To that end, the rubric may be used in parts, and there are spaces to note when a specific practice was not observed, rather than giving unnecessary negative feedback.

The only comment I might make is to take out the numeric scoring guide, and only use the descriptors in each category. Connecticut has been a bit inundated with numerical accountability measures, and removing the numbers might facilitate better conversations while taking away the stigma of a rating system. See the rubric here: http://achievethecore.org/page/1119/coaching-tool

2. The Verdict on Time Fluency Tests (For Now): Timed fluency tests (e.g. Mad Minutes) were often used pre Common Core to determine a student's level of fluency. However, many critics noted that these timed tests can be damaging to a student's mathematical efficacy, and that they only reinforce the misconception that mathematics is about speed, not depth.

After some debate, the conclusion I came to is best described by thinking about fluency in a foreign language. A person who is fluent in a language is able to easily access basic facts and linguistic procedures within a context. While it may sometimes be useful to test how quickly someone can recite vocabulary terms in a foreign language, the learning should be done in context. Similarly, it may be appropriate to give a student a bi-annual benchmark, but learning basic math facts should always be done in within the context of "why" and "how". Moreover, if a student is having trouble with fluency in the upper grades, the solution is to work with these facts in context as much as possible without stopping curricula to focus solely on basic facts.

3. The Coherence Map: Student Achievement Partner's Coherence map is a great way to see examples of standards aligned tasks, learn about the prerequisite knowledge for mastery of a standard, and see how each standard will be used in later grades. Unfortunately, the map does not allow a user to start at a high school standard and trace the connections backwards. The best a 9-12 teacher can do is choose a 6th, 7th or 8th grade standard and trace it upwards to see what is is connected to in high school.

I had the chance to talk with Joanie Funderburk about her rational for leaving the high school standards out. Due in part to the many interconnections between the high schools standards, and in part to the lack of priority standards, they were unable to write an algorithm that accurately captured the connections. While the explanation made sense, I left still wanting an easier way to explore the connections between the standards. She suggested I create my own map, and while I wish these resources were already available, nevertheless, I think it might be a worthwhile exercise.

Another teacher suggested I check out Battelle's Vertical Progression Guide. Perhaps this is something to look into in the future.

4. Accessibility Strategies for Mathematics: We've seen many of these strategies before, but I like the succinct layout and the ease of use that comes with this document. I couldn't seem to find it on their website, but I have copied them into my own Google Drive here.

5. The Spectrum of Higher Education: There is a wide range of possibilities for me to take as I dabble with working both in and out of the classroom. I enjoy working with and coaching my colleagues on grade level teams and within the math department, but I'm not sure how these skills would translate to a whole school instructional coaching position, or to a district wide curriculum facilitator. And, if I wanted to move in this direction, I'm not sure what educational route to use. A sixth year? Administrator certification? PhD? Ed.D? The participants at this conference gave me a great deal of information, and more to think about than I could ever hope to write here.

6. (Extra Credit) Start, Stop, Keep, and Tweak: Adam Krupa shared one of his favorite frameworks for thinking about change or any kind (curricular, instructional, life changes). It's quite simple, and all it takes it thinking about implementing change in four specific ways:

Start - What should we begin doing at the end of this change?
Stop - What should we stop doing as a result of this change?
Keep - What should we continue to do at the end of this change?
Tweak - What needs to be reworked or modified during this change?

Five Things I Learned About International Geometry at ICME-13

Throughout the Thirteenth International Congress on Mathematical Education (ICME-13), I had the opportunity to listen and reflect on a number of competing geometry education paradigms. Despite the wide range of philosophies, curricula, and pedagogical strategies, I found that almost all perspectives can be placed on a continuum between a traditional “axiomatic-style” and a more reformed “discovery-style”, with some room for outliers. During a survey of current geometry curricula, Nathalie Sinclair (Canada) noted that, “Proof, and more generally geometry curricula, are bound to socio-cultural norms.” Interestingly, the style of geometry that each country favored tended to be a reflection of their culture. In this overview, I will explore where various countries fall on the continuum, and how their views are ultimately intertwined with their culture.

1. Eastern Europe: Eastern European countries (e.g. Russia, Hungary, and Romania) typically have a very axiomatic view of geometry. In these countries, most geometry curricula start with an emphasis on Euclidian axioms and proceeds to develop other notions from this standpoint. This style may have its roots in university education. During the ICME survey of geometry curricula, multiple panelists reported that secondary teachers in Russia often view themselves as content experts first and teachers second. A few also commented that teachers often use the pedagogical techniques of a more traditional university professor, techniques such as lecture and practice. Since Euclidian geometry as written in The Elements, it is reasonable to assume that there is a relationship between

2. Western Europe: Other European countries also employ a more axiomatic style, but for different reasons. Halfway through the conference, I had a conversation with Bernie O’Donoguhe (Ireland) about the role of Geometry in Ireland. She reported that secondary education in Ireland is largely motivated by two exams: the Junior Certificate and the Leaving Certificate. Both exams have mandatory sections in mathematics with an emphasis on axiomatic geometry. While many other western European countries have changed or done away with the geometry content in their end of course exams, Ireland’s history of moderate isolationism from mainland Europe may help explain why they have not followed this trend. It has only been very recently that the Irish government has discussed deemphasizing the role of axiomatic geometry on the Junior Certificate.

Most Western European countries (e.g. France, Spain, Italy) have recently reformed their curricula to either move away from the axiomatic method or decrease the role of geometry altogether. In a panel on the teaching and learning of geometry, Maria Bartolini (Italy) explained that the concept of proof in Italy is no longer tied exclusively to axioms. Rather, students often believe that accurate measurement and repeated trials can lead to a rigorous proof, and not just a mere conjecture. Nathalie Sinclair quickly added that Dynamic Geometry Environments (DGEs) such as Geogebra can be used to bridge the gap between empirical conjectures and more theoretical approaches to proof. By using check boxes, sliders, and drag features, students can both collect data to make conjectures, and get a visual-spatial understanding of how a particular theorem works. Bartolini agreed and noted that many Western European Countries have decreased the breath of geometry content on their exams, allowing for more classroom time to explore geometric theorems using DGEs.

3. Northern Europe: During the thematic afternoon, I learned that while many of the reforms in Western and Central Europe are new, the Netherlands has used Realistic Math Education (RME) for some time. Although the didactics of RME are not uniform across the country, the basic tenets of RME are the same. According to Marja Van den Heuvel-Panhuizen (Netherlands), RME should both start with and end with a context that is imaginable and real to the student. Problems must be novel, but can include an intra-mathematical or extra-mathematical context.

In a similar session, Guenter Krauthausen (Germany) explained that RME lessons in the Netherlands are often task-based with room for multiple approaches and natural differentiation. In natural differentiation, the teacher presents all students are presented with the same complex task. Then, students informally self-select the solution method, manipulatives, notation, and group members that they will interact with. This provides a very open and social learning environment, where geometric axioms are only taught if apply to the task, and only if the student requires it. As a result, students in the Netherlands are exposed to less content, but more depth.

4. East Asia: While there are stark cultural and pedagogical differences between the Scandinavian countries and the East Asian countries, their current approach to geometry is remarkably similar. Both regions teach through tasks and only discuss content that is relevant to the current task. Interestingly, most of the tasks in Japan’s national geometry curriculum contain some aspect of geometry. The Japan Society of Mathematical Education (JSME) led a session that featured six tasks from their curriculum. Although each task had multiple facets, every task required a non-trivial level of geometry content to complete. For example, one task involved folding a paper pentagon to make new shapes. Another task asked students to chop a cylindrical log into rectangular prisms of lumber. Unfortunately, I was not able to determine the prevalence of geometry in The Netherlands curricula, and thus I can only assume that Japan may emphasize it slightly more.

Another point of similarity between the Netherlands and Japan is the amount of mathematics content each country reportedly focuses on. According to Jinfa Cai, both countries cover less than 70% of the content on the Programme for International Student Assessment (PISA) and Trends in International Mathematics and Science Study (TIMSS), and yet they have some of the highest scores on these tests. In fact, Japan claims to explicitly cover less than 60% of the content on the most recent TIMMS. Towards the end of the conference, the JSME facetiously demonstrated how small and skinny their textbooks were compared to their larger and bulkier American counterparts. The display was a bit tongue-and-cheek, but the point is well taken. In Japan, less explicit curriculum can translate to higher test scores.

5. Afterthoughts: As the conference concluded, I was left with three lingering questions:

• How can two very different cultures produce such a similar geometry curricula? 
• Why do these countries perform so well on international assessments? 
• Can this success be replicated in other cultures, such as ours?

I suppose the answer to these lie in additional research about these countries, their curricula, and their cultures. In an effort to reach out to other teachers, I plan on disseminating the information I have gathered here through regional conferences and publications in New England. Hopefully, this will bring myself, and the math education community as a whole, one step closer to answering these questions.

Dylan Williams at PCMI

Dylan Williams spoke to the PCMI Teacher Leadership Program today via webinar. He answered questions, one of which was mine. Here are some major take aways and quotes:

- There are only two recommendations that educational psychologists can agree on. First, mass practice is less effective than distributed practice. That is to say, we should integrate our topics such that students have many opportunities to practice a particular skill. This notion is closely aligned with Ebbinghaus’s Forgetting Curve. Distributed practice also helps with truant students because no topic is addressed in only one day, making it easier for the absent student to catch back up.

- Second, frequent testing without giving grades helps students because it helps them practice retrieval. Students often practice memory storage when they hear a lecture, experience an activity, or study for a test, but they do not regularly practice memory retrieval. It seems that testing, when used the right way, can significantly benefit learning.

- Having students predict an answer before learning how to complete a problem, even when they have no obvious entry point, has been shown to increase student learning. The thought is that by predicting an answer, they cognitively struggling. This, in turn, leads to greater learning (more on that later). This is connected to test corrections. Even if students don’t perform well on an assessment, the act of trying and correcting their work aids learning.

- Higher levels of cognitive struggle lead to higher levels of learning. This can manifest itself simply: a student trying to solve a novel problem unaided will learn more than a student completing a procedural problem whose solution is already known. However, other types of cognitive struggle also impact learning. In a psychological study, students who were given a smudged print out of a story were better able to recall the story than students who were given a clean copy because they struggled to read the smudged copy.

- There is no psychological evidence that catering to individual learning styles helps students. In fact, it may lead to lower learning since it would lower the cognitive demand and struggle. Yet, this is not permission to ignore learning styles altogether. Working in a learning style that is not your own is tiring, and so teachers should frequently vary the style they use to ensure equitable learning for all.

- In both Japan and the United States, lessons often start with the teacher demonstrating how to solve a certain problem. In the United States, this is often followed by procedural practice. However in Japan students are challenged to find other solution methods. This leads to natural differentiation because higher achieving students can be prompted to find more complex solutions while the teacher scaffolds more basic solutions for students who are confused.

- The advantages of formative over summative, or comments over grades, is not absolute. The only thing that matters about feedback is what students do with it. If a student in AP Calculus would learn the most from seeing a number grade out of five on their practice test, then they should receive a numerical grade. If a student would benefit the most from teacher comments, then that is what should be done. Moreover, any assessment can be used as a formative or a summative depending on how you use it. Groups of four students could individually take a standardized test, then all pool their ideas as a group to create a fifth “best version” of the test. After finishing, the teacher would find any common mistakes and have groups present their solutions to each other. In this way, a traditionally summative assessment has become formative.

- The goal of feedback is either to give students information about where they are at currently, or how they can improve. However, it is disadvantageous to give both at the same time because students tend to focus on one type and ignore the other.

This talk gave me a great deal to think about, but I’ve tried to come up with a few succinct action steps to improve my teaching. These probably sound familiar:

Integrate the topics in my curriculum more, and don’t hesitate to expect students to remember essential topics from previous units.

Let students predict answers to novel problems before they start them, and allow them to struggle within a well-thought scaffold. 

Use more quick formative assessments. Even if I don’t grade them myself, the act of taking the assessment is important, and it can give students the opportunity to reflect on their peers and on themselves.