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.

Five Things I Learned at TMC16: Monday 7/18/16 & Tuesday 7/19/16

1. The Variable Analysis Game: Joel Bezaire presented The Variable Analysis Game: a simple game that encourages pattern seeking and algebraic thinking. The class is presented with three (or more) rows of numbers, with column headings labeled a, b, and c. Students must find the relationship between the numbers in the same row and develop an equation that uses all of the numbers in each column, and works for every row. I'll leave it to Joel to explain more and give some examples, found at variableanalysis.info.

I plan on using this game in my classroom because it allows students to work at different speeds, communicate their reasoning without "giving it away", and has a natural extension.

2. More Coloring Books: Edmund Harris (Current Prof at the University of Arkansas and my roommate for this TMC16) has released Visions of the Universe, a sequel to Patterns of the Universe. Both are adult coloring books steeped in mathematics. This iteration includes map projections, fractals, a representation of all the ways you can add to 11, and a number of lesson plans for bringing a bit of art into math class.

3. Nominations: Kathryn Belmonte encourages students to put more thought and effort into open ended assignments by nominating each other to share their work in front of the class. See the details here.

4. Birthday Functions: Hannah Mesick shares one of her favorite ways to celebrate her student's birthdays: by using them as an analogy to teach functions. Find more in her Prezi here.

5. The MTBoS is not a Plug and Play: Dylan Kane delivered a great keynote about moving beyond resources and into solid petagogy. His slides are here and the video is here (thanks to Glenn Waddell).

6. (Extra Credit) Using Photos: Sarah VanDerWerf uses images to remind herself of the little things she tends to forget while teaching. I'll try to find examples of these, but the internet is a big place.

Five Things I Learned at TMC16: Sunday 7/17/16

1. The difference between further and farther: Farther is used to refer to actual distance, but further refers to a metaphorical, or figurative, distance. Often, further can be used to discuss depth in a curriculum. For example, "we can take the similarity of congruence further by extending it to non-polygons, such as circles"

2. The Price is Right and Probability: The mathematics used in the game show The Price is Right varies greatly, and allows for a natural way to differentiate a probability or statistics course. Flip Flop gives a nice introduction to the main concepts of probability (and a great ending in this clip). Games like Bonkers let us think about random guessing versus strategy. Others, such as Punch a Bunch, have multiple levels of probability built in which can be great for a low-floor high-ceiling task. Better yet, they have the individual clips online. Thanks to Denis Sheeran for pointing this out. Even better, this Slate article has strategies for every game, along with how viable that strategy is. This would have been great to use at last year's TMC trip to The Price is Right... if only we had made it onto the stage.

2a. If you want students to play the games themselves, check out online games, such as this Deal or No Deal simulation. This provides a great discussion about expected value.

3. Instilling racial competency in teachers doesn't have to be painful: I attended a professional development session not too long ago that aimed to make teachers aware of their own privilege. Instead of meeting this objective, it inadvertently made examples of teachers with less socioeconomic privilege. Some participants left crying, others confused.

In contrast, Jose Vilson and Wendy Menard led a very comfortable, yet deep session on Racially Relevant Pedagogy. They started by placing the words Race, Ethnicity, Gender, Class, and Religion on different walls of the room. Then, Jose would pose a question and ask teachers to respond by walking towards one of the words on the wall. Questions started with, "Which one do you most identify with?" but quickly became deeper, "Which one do you talk about least with your parents?" Along the way, teachers were prompted to volunteer reasons for their answers, but rationals were never forced, and responses were always appreciated. The setting was so comfortable, I didn't realize how many "difficult" topics we had addressed until it was over. I hope I can take the spirit of this session back to my school in the fall.

4. The Dean's Feedback Meetings: Feedback meetings can be an especially strong way to create a culture of learning, both as a math teacher and as a grade level team leader (dean). Anna has graciously put her resources up here.

5. Primary and secondary teachers have a lot we can learn from each other: Tracy Zager began her keynote with the graph below from Math With Bad Drawings. She went on to discuss how both content and pedagogy are important for primary and secondary teachers, and how we should use both physical and electronic interactions to connect and ask for feedback from teachers outside of our own grade level. The keynote discussed a number of rich anecdotes that probably deserve a separate post. However, Tracy's two final calls to action were concise and clear:

- Look at who you're following on Twitter and diversify those voices to include different grade levels.

- What are the current obstacles to cross-grade level collaboration? What can I do to tear them down?

6. (Extra Credit) Explore Math: Sam Shah created a site called Explore Math. It serves as a launching point for students to browse topics not traditionally covered in a secondary mathematics curriculum. He asks students to create about four of these open-ended explorations a year, and finds that it works best as a low stakes, high reward activity. More information can be found here.

7. (Extra Credit) Voroni Maps: Dave Sobol gave an amazing presentation that focused around finding perpendicular bisectors between two points on a map. They're called Voronoi Maps, and you can use them to divide up a the USA into sports regions, find the closest airport to you, and more. The best part is, you can then compare the Voronoi Maps to actual maps (e.g. media markets) and discuss the differences.

8. (Extra Credit) Nice Ride: You can rent a Nice Ride bike in Minneapolis for three days for only $10. What a deal!

Five Things I Learned at TMC16: Saturday 7/16/16

1. Since the start of my career, I have believed that math lessons should be relevant and engaging to students. For some time now, I have struggled to deliver a level of relevance on a consistent basis. My ideas often seem either contrived, or a carbon copy of someone else's lesson I have graciously borrowed from online. However, Dennis Sheeran presented a framework for increased relevance. The acronym is below, but more information can be found at his site. For now, just an overview; these five thing posts are supposed to be short, after all.

Infusing your life

Natural flow

Sudden changes

Television and pop culture

Awareness of your surroundings

National events

Two or more disciplines 

2. GeoGuessr is an online game that uses Google StreetView images. Players are dropped in a random location and are challenged to estimate where they are in the world. Players receive points for each guess, and the more accurate the guess, the more points are received. Besides being moderately addictive, this game makes an excellent data collection tool and jumping off point for a number of statistical discussions. We explored the relationship between accuracy of guess and points awarded. It turns out the model is not linear (as might be expected), but follows an exponential decay as the estimation moves farther from the actual location. More interestingly, the coefficnets of the model will change depending on what version of the game you're playing (America only, whole world, just cities, etc), but the general exponential trend remains stadic.

3. Julie Wright gave a quick but rich presentation about feedback quizzes. About twice a quarter, students in her class take a quiz in which the left third of the paper is left blank. Upon completing the assessment, student work is alphabetized and scanned. Instead of placing a numerical grade on each tests, Julie uses an annotation software to type comments into the blank column in each student's paper. Besides shifting the focus from grades to feedback, this system also allows the teacher to copy and paste comments, allowing for more efficient grading. After printing and passing back the papers, students fix their mistakes.

Upon reflection, I'm not sure that I am comfortable with only giving feedback in this manner twice a quarter. Yet, I do see and understand the massive time commitment required to give specific thoughtful feedback on every question for every student. Perhaps a compromise is to provide feedback on a single exit slip question every day and provide students the opportunity to reflect during the next class. This would allow for feedback but decrease the grading load on the teacher. However, the logistics involving scanning the papers requires a bit more thought.

4. Sara VanDerWerf has a backwards bike, and no, it's not a bike that runs in reverse. A traditional bike is steered by moving the handle bars and pointing the front wheel in the direction you want to go, but the backwards bike has a set of gears that cause the front wheel to turn in the opposite direction of the handle bars. The bike was first created and ridden by Destin from the YouTube channel Smarter Every Day. In his video, Destin explains that it took eight months for him to un-learn how to ride a traditional bike and learn the backwards bike. Surprisingly, it only took his young son two weeks. He attributes this to a higher level of neuroplasticity in kids than in adults.

Back at TMC16, Sara explained that riding the bike can teach us lessons in persistence and struggle. Just as with the bike, our students may sometimes know how perform a task, but are still unable to complete it. As both Destin and Sara have said: knowledge is not understanding. Just because you may be able to verbalize the mechanics of the backwards bicycle, it doesn't mean you can ride it. This rings true for both teacher and students. First year teachers may know what good teaching looks like, but may find it difficult to meet their own expectations in the heat of the moment. Similarly, students may understand a set of mathematical procedures, but freeze when these procedures are applied in a multi step context. It is crucial that we understand this struggle and this frustration, and provide safe spaces for students to fall, dust themselves off, and get back on the bike.

5. Minneapolis is a great place to run.

6. (Extra Credit) I learned how to wobble... poorly.

Five Things I Learned at Descon16: Friday 7/15/16

1. Desmos is now accessible to students who are visually impaired. Pressing command + F5 activates a text to speech function that will not only read the functions as you type, but will also identify if your cursor is in the denominator, subscripts, superscripts. Pressing option + T will activate an audio tracer that will audibly read the coordinates of points on your graph as you press the left and right arrow keys. Pressing option + H will play an audible tone whose pitch will change as the function increases or decreases from left to right across the graph. I hear that Chris L was trying to get a graph to play "Mary had a Little Lamb". Unfortunately, I can't figure out how to make these work on a Windows OS. Hmm...

2. teacher.desmos.com now includes bundles of activities. Bundles are a collection of activities in a suggested order around a specific topic, such as functions or quadratics. Each bundle has key understandings for that topic, as well as commentary on how each activity fits into the larger sequence. Of course, it's up to the teacher to fill in the gaps, but I'm interested to see how well the sequencing, pacing, and scaffolding provided by these bundles holds up in the classroom.

3. Access to Desmos at home can be a big step towards equity in the classroom. Sara VanDerWerf, besides being one of the most passionate and kind people I have met, gave a great keynote that touched on technology and equity. She noted that a large barrier to students is often home access to technology, which is why she implored the group to have all students

4. Alice Hsiao introduced me to the Chicken McNugget Problem (Also called the coin problem), which is stated below:

McDonalds sells Chicken McNuggets in boxes of 6, 9, or 20. Obviously one could purchase exactly 15 McNuggets by buying a box of 6 and a box of 9.

Could you purchase exactly 17 McNuggets? How would you purchase exactly 53 McNuggets?

What is the largest number for which it is impossible to purchase exactly that number of McNuggets?

What if the McNuggets were available in different sized boxes?

I intentionally haven't added any links as not to spoil a solution. After all, I'm still working on one now...

5. Drinks in Minneapolis are annoyingly cheap during happy hour, especially when there is a minimum charge. Thanks to everyone at the Desmos team for their time and generosity!

6. (Extra Credit) Make a list in Desmos with L = [1, 2, ... 10]. Check back in with Sara's Desmos Dictionary. Make your own marble slides, card sort. Hide some folders. Talk about Desmos, Geometry, naming points, and back end vs. front end. And lastly, read The Art of Evangelicalism.