As luck would have it, I caught Dan Meyer's "Full Stack Lessons" talk at CMC North just shortly before I was planning to use Barbie Bungee with my 7th graders.^{[3]}His talk spoke at length about this task as an example of how the same task can be engaging and valuable in one classroom but be drab and worthless in another. The presentation was great and pushed me to think about the types of actions and thinking my students do when they complete tasks in my classroom.

I've thought about this before, and I worry. I worry a lot about two things I deny my kids. I deny them with my too-carefully scaffolded and overly-planned lessons. I deny them opportunities to ask their own questions and to develop the method for which they will solve questions. I've fought this before (see: Fermi Estimation Problems) but a new school, a new text, new families, new students, a new coast, and more and more new stuff has made me feel like I've lost a bit of my mojo for building this into my classroom and my practice.

Dan's presentation was a wonderful reminder of the opportunity I had. The opportunity to "be less helpful" and to, therefore, help my students. So, I spent the drive home from CMC North percolating over how I would structure Barbie Bungee to de-scaffold it for my students' benefits. Here's the structure I landed on. I hope it is helpful to you as you consider implementing this or similar tasks:

- I arranged the students into groups of 4. Partners are my preference, but I worried about handling twice as many groups
- I purchased a unique doll for each group. This eliminated cross-copying of values and ideas since they had different heights and weights
- I broke the task into 3 distinct "drops". Where kids did the following for each one:
- Agreed on a number of rubber bands needed for their drop (2 x 45-minute classes)
- Dropped their doll and recorded the results in slow-mo (30 minutes or 2/3 of a class)
- Completed a reflection about what they did, why, if it worked, and what they wanted to do next time (15 minutes and homework)

**I provided MORE scaffolding at each drop**^{[4]}Yes, I know this sounds wrong- On the first drop, I said "figure out how many rubber bands you need."
- On the second drop, we brainstormed mathematical tools that might be helpful. Then I said, "figure out how many rubber bands you need."
- On the third drop, I had them complete a T-table, plot the points in Desmos, graph their own line of best fit, and then have Desmos calculate a linear regression, so they could, "figure out how many rubber bands" they needed"

This is not the normal progression I follow in my classroom. But, I think it is right for this task. There were so many good conversations; I can't possible recreate them all here. But, I do have one example that I think is illustrative of what I was going for.

We had a serious lack of meter sticks in the classroom, so students got inventive and began taping rulers together. I noticed that they were taping the rulers end-to-end which, frankly, makes lots of sense. Except....

Rulers leave a length on either end (presumably for manufacturing purposes) that isn't part of the 12" distance. But, my students ignored this gap. Did I point this out to them? Yes, but only after they were done with all of their measurements.

Why wait so long? Well, I wanted it to be too late to redo the measurements. So, instead of starting from scratch because of what their teacher told them, they had to adjust their model to deal with an error they discovered in their methodology. They had to figure out if their error made their measurements too long or too short. So much good thinking and conversation came out of this. Maybe more than any other part of the activity.

Too often, I get lulled into thinking that learning is this really linear and well-defined trajectory. My kids really learn the most when I remember that class can (and sometimes *should*) be messy. Their own ideas lead to natural teachable moments and errors where they are curious and engaged and passionate.

At a talk, I once quoted Lord Petyr Baelish who said, "Chaos isn’t a pit. Chaos is a ladder." Like all Westerosi men of lessor houses, I certainly need reminders of this wisdom. Perhaps this could even become my house words. I need to remember to welcome chaos into my classroom when it is to the benefit of my students.

**Propaganda Time: **Androids (yes, I guess iPhones too) can shoot HD slo-mo video. So, of course, I had to put a video together for my school:

**"Next Time"s**

Next time, I want to more explicitly connect the work in Barbie Bungee to other investigations. Specifically, the stacking cups investigation we had done previously. I mentioned this in a reflection question, but I want to spend more time on this. Perhaps even make it the a key part of my assessment.

Next time, I think I want to mix up the groups between each "drop." The thing I am most proud of in my classroom is that students regularly explain ideas to one another (SMP3). Mixing up groups will allow kids to say, "Oh, my group tried this and it ___________."

Next time, I need to build in a lesson on linear regression prior to this activity. The kids thought it was cool, but didn't really "get it"

This list is heavy on games and puzzles which you might describe as "mathy". But, since there is no such thing as a math person, these are great gifts for all!

If you don't know Randall Munroe's blog by the same name, you're welcome! In this book he shares answers to his favorite questions from readers.

Currently on Kickstarter (you won't get it until the summer), this is a detective style puzzle set in a old-timey cartoon meets film noir universe. My excitement for this has no bounds.

An absolute gem of a game. Players work together to play cards from their hand by color and in numerical order. The catch: only the other players can see your cards! This game is challenging and puzzle-y and fun.

If you like escape rooms or similar experiences, check out the Exit series. The Abandoned Cabin is the first one to try. Its got some pretty clever puzzles to solve. Lasts about 1 hour, so it's easy to find time for it.

A 2 player board game in which you are making a quilt out of polyominoes. You and your opponent draft pieces from a common pool and there is lots of strategy to be had in how you restrict your opponents options.

Don't let the fact that this is a video game turn you off; this game is incredible. The designer put together an incredibly diverse set of puzzles from a simple grid maze. You, your family, and friends will be working on this for days.

My all time favorite video games. The premise is so simple. You have a portal maker. It makes a blue portal and a red portal. When you go through the red portal, you come out the blue portal and vice versa. You use that tool to navigate 3D maps. The puzzles are awesome and the often hilarious. **J.K. Simmons** even voice acts in the second.

Are these board and video games too fancy schmancy for you? Pick up this thick book of puzzles from the godfather of recreation mathematics. Then track down my favorite question about sectioning any obtuse triangle into all acute triangles.

I got a demo of this at CMC North. It's awesome. You get a small square and have to fold it to match a pattern. They are tough, but not overwhelmingly so!

A brand-spanking-new game about arranging polyominoes into levels. The more pieces on higher levels, the more points. Takes about 15 minutes, but you will play lots of times!

Before we get to the questions, it is important to think about restrictions. My experience has been that interviews are short, really short. Often, they are scheduled for just 30 minutes, but by the time everyone arrives, finishes the small talk, and actually get to interviewing, there are only 15 to 20 minutes. So, let's limit ourselves to 5 questions. And, let's order the questions with the most crucial ones first. That way if time runs low, you have collected the most important information.^{[6]}It is also important to note that, as the interviewer, there are certain questions you are not allowed to ask. Some of them are straight forward: don't ask how much money a candidate has. Others are less intuitive. Even if it is just polite small talk, you cannot ask a candidate if they have children. The assumption is that any question you ask will be used to determine whether an applicant is hired, so you cannot ask about protected traits, beliefs, et cetera.

Let's get to the questions:

Most teaching interviews involve a sample lesson, but this only gives a small window into a teacher's classroom. I want to know the routines and practices that teachers use and build on throughout the year. Do you use number talks once a week? Do you build students' understanding of proof through structured discussions and writing? This questions gets at what a teacher believes is most important for teaching mathematics, but, by asking what you will notice in the classroom, you force the applicant to describe actions and practices not just beliefs.

Curricula are very powerful in transforming teachers' actions and lessons. I want to know what type of curriculum an applicant prefers because that suggest a great deal about their beliefs. Most, if not all, teachers draw from resources outside of their curriculum. So, I also want to hear that a teacher knows where to find strong tasks.

Regardless of the course or grade level, I want every math teacher working to strengthen our students' mindsets about mathematics. This does not happen by magic. It takes thought and reflection. I want to hear a candidate connect mindset to different aspects of their teaching, especially feedback.

Feedback and assessment are incredibly powerful aspects of teachers' practices that impact student learning and confidence. I want any potential candidate to have a well-articulated approach to assessing and providing feedback to students. I also want to hear examples of formative assessment and how those assessments are used to guide future lessons and work.

The details of this question depend on the position the teacher is applying for, but it should give the teacher the opportunity to demonstrate their pedagogical content knowledge (see below) for a big conceptual idea that students in their courses will be wrestling with. I want to see that this candidate knows more than just how to "do the math" covered in a course. I want to hear that they know how to help students make sense of big ideas they will be teaching.

From Ball et al. - (2008) - "*Content Knowledge for Teaching : What Makes It Special?*"

You might, for example, ask a fifth or sixth grade teacher the following: "5th and 6th grade students spend a significant amount of time studying operations involving fractions and decimals. Still, many find story problems that require these ideas challenging. How do help students become better solvers of story problems involving operations with fractions and decimals?"

Those are my go-to interview questions. But, like many, I adjust the questions based on how a candidate responds. When I am the one being interviewed, I am thinking about these questions too. I want to be sure to communicate my beliefs about assessment, curriculum, mindset and feedback even if I am not asked directly.

What questions do you like to ask or be asked? I asked this question on twitter and was reminded of some wonderful questions including:

@MathFireworks Can you give an example of a student who struggled during class and how you engaged with them mathematically?

— Anna Blinstein (@Borschtwithanna) April 20, 2017

The format of the conference feels fairly different from other conferences I have been to. For example, my presentation is just 30 minutes long, which is a perfect length for me to share out ideas about how teachers can revamp their homework practices and assignments.

As always, I like to share two things when I present: my slides and a document summarizing my talk. If you can't attend my talk, **I suggest that you begin** with this document which summarizes everything I am presenting about. You can also check out my slideshow:

I find that my students understand each step independently, but need lots of experience discussing, practicing, and figuring out which step to do when and why. To promote these discussions, I had my students construct flowcharts.

Think, for a second, about how different the processes are for these computations:

\(4\tfrac{1}{6}-3\tfrac{3}{4}\)

\(4\tfrac{2}{3}-\tfrac{1}{3}\)

In the first expression, you need to make common denominators, regroup the mixed numbers in some way, subtract the whole numbers, subtract the fractions, possibly regroup again, and simplify. In the second expression you only need to subtract the fractions.

I have found that my students struggle with adding and subtracting mixed numbers because they can't understand when certain steps are needed and when others are not. Trying to teach this through direct-instruction fails for most students. There are just to many if-this-then-this points. Students need to process those ideas on their own, you can't just tell them how and expect them to memorize the procedure.

So, I took this challenge as an opportunity to introduce my students to flowcharts. Flowcharts are a way of displaying algorithms and constructing them requires computational thinking. You must consider what possibilities exist and how your algorithm will process the inputs (problems) you give them. This was hard for kids, but they loved it.^{[8]}I find teachers write, "They loved it!" or similar a lot in their posts. But, really, there is no way to audit this. It's like a cooking show on TV. The hosts LOVE every dish they make, but they could really be terrible. #DeepThoughts

Each box in the flowchart is a command (ex: Make common denominators.) or a question (ex: Are the denominators the same?). Commands should only have one output while questions should split based on the answer. In the example below you can see that this group began by asking if the inputs were mixed numbers.

I had the students work in pairs on individual whiteboards. They used mini-post-it-notes for the boxes. This was CRUCIAL to their success. They are reworking, rethinking, and revising so often that they need flexible materials.

Groups also developed algorithms of varying degrees of complexity, detail, and sophistication. My goal in designing this lesson was to get the students to think through the different conditions encountered when adding and subtracting mixed numbers. Most groups were still revising and reworking their flowcharts when class ended and that was okay! They did not need a perfect flowchart to achieve the goal of the lesson.

The above group's flowchart lead to my favorite conversation. If you look at the chart, you'll see that they have MANY redundant boxes. Of course, that still works, but part of computation thinking is eliminating redundancy and promoting efficiency. After talking this over, this group scrapped their nearly finished flowchart with the goal of using as few post-it notes as possible!

Other groups, like the one below, had a harder time recognizing unneeded redundancy. They made separate charts for adding and subtracting fractions. Still, this allowed us to have conversations about whether one flowchart could have handled BOTH operations.

In class, we have been discussing different strategies for when the fraction you are subtracting by is larger than the fraction you are subtracting from (ex: \(4\tfrac{1}{6}-3\tfrac{5}{6}\)). Some groups tried to build ALL of those strategies into their flowchart. But other groups realized that the always-convert-to-improper-fractions strategy is easiest to describe in a flowchart. That led to dead-simple flowcharts (see below) and also allowed us to discuss how optimal algorithms for humans and optimal algorithms for computers can vary.

I'd love to use flowcharts again in the future, but you need a certain type of algorithm for flowchart creation to be valuable to students. They are best for describing algorithms with many if-statements or conditional steps. I can imagine factoring quadratics or solving equations might be procedures that would benefit from computational thinking. Maybe you can think of other processes that would benefit from a flowchart. I know geometry courses use something like flowcharts for proofs.

Does anyone out there is flowcharts for other mathematical procedures?

A big part of my problem is that I like to over-explain anything I post up here. But, really, the people reading this blog are more than capable of filling in gaps. So consider this post the start of my own personal blogging initiative; I'm going to work to get more short posts up here.

First up is a worksheet on division and divisibility rules. Sometimes students just need to practice, but when they do, I try to make sure that it is not mindless. So I'll often have them only complete problems that meet a certain condition. I made this when my 5th grade students were studying whole-number division and learning about divisibility rules.

Print or download the document.

This worksheet shows a number of my common homework practices.^{[10]}I'm talking about homework at NCTM's Innov8 conference in November. I typically provide students with answers and give them some sort of meta-task when the assignment would otherwise be rote.

My colleague is doing some probability investigations with her 4th grade students. She's got a number of games from Marilyn Burns' probability book, and she is having students students investigate the experimental probability of the games.

One of those games is called Shake and Spill. Grab six 2-sided chips (or coins) and flip/spill/spin them so that they land on a random side. Record the result and repeat. My colleague is having her students pool their results to better estimate the relative probabilities. Cool!

But, I wondered if they would generate enough data (and generate it fast enough) to see the Law of Large Numbers in action. So, I built a website that would simulate this game over and over and over again with a live graph. You can check out the website here.

I'm planning on building more of these. They are an awesome way for me to practice my Javascript and I think that they may be helpful to others. What other probability simulations would you want?

Learning multiplication facts is often a stressful process for elementary students. It is often the time when many kids first become anxious about math. As students become stressed, it can lead to arguments between children and their parents, and no one wants that!

It does not have to be this way. Learning multiplication facts is a wonderful opportunity for parents to have mathematical conversations with their kids! This post will help you learn how to help your child learn their facts faster and how to be happier doing it.

Many different researchers have developed different models for how children's brain learn multiplication facts. While some of them are complicated, it is worth giving you a quick overview here. In general, there are three levels of understanding:

**Level 0 -** The student cannot generate an answer to a fact. This implies that they do not yet understand how multiplication works.

**Level 1 - **The student can generate an answer to a fact using a laborious procedure, usually skip-counting. They know that multiplication can be thought of as repeated addition, so they use this strategy to get the solution. For example, if asked 5 x 4, they may count by fives four times (5, 10, 15, 20). This allows them to solve facts, but is slow and challenging, especially for larger facts.

**Level 2 -** The student can generate an answer to a fact using more efficient strategies. At this level, facts still take more than 2 seconds to solve because students are using a strategy to figure out their answer. Their strategies will often use already memorized facts. For example, they may solve 6 x 7 by using 5 x 7 -- "5 x 7 = 35 and then I need to add one more seven, so 42".

**Level 3 - **The student has the answer memorized. They can recall the fact in less than two seconds and are not doing any math in their heads. They are simply retrieving a fact that they have memorized.

It is easy to see that level 1 will be much too slow to be a successful level of knowledge, but many parents and children wonder why level 2 is not sufficient. Research shows that students who reach level 3 have more success in future math classes.^{[13]}To be clear, this is only a small part of what determines success in math class. But, we know that it matters. So, we want all math students to reach level 3 for the 0 to 10 facts.

So, why is memorization better? In other words, why aren't the strategies in level 1 sufficient? It comes down to the brain. When students at level 2 use multiplication facts while studying other concepts, they need to dedicate more of their brain power to answering math facts.

Take for example, learning how to multiply multi-digit numbers (ex: 123 x 45). If you multiply these two numbers, you are recalling six facts.^{[14]}5x3, 5x2, 5x1, 4x3, 4x2, and 4x1. Students who have their math facts memorized (level 3) can dedicate all of their thinking to understanding and learning the algorithm. Students at level 2 have to learn and understand the algorithm while ALSO solving the multiplication facts using their strategies. Having level 3 knowledge of math facts likely makes learning future ideas easier.

Many people hear this and come to the conclusion that they should aim right for level 3. They pull out the flashcards and the Mad Minutes and start drilling away. But, this is not the way to help!

Level 2 is a crucial stage of development for students. It is in this stage where students develop their understanding of how multiplication works. Even though level 3 is a goal, trying to skip level 2 will weaken your child's understanding of mathematics. Because, it is at this level where students learn strategies that are useful beyond just memorizing multiplication facts.

As parents, the best thing you can do to help your child learn their multiplication facts is to have conversations about the strategies you can use to solve a multiplication fact. Through these conversations you will bolster your child's understanding of mathematics and help them move up the levels of understanding introduced above. So, here are the strategies you and your child can employ while solving multiplication facts:

**Zeroes, Ones, Twos, Fives, and Tens - **By the time your child starts studying multiplication, they will (hopefully) have a solid understanding of addition and have reached level 3 for most addition facts. This knowledge can serve as the foundation for the first learning multiplication. Multiplying by 0, 1, 2, 5, and 10 are the easiest facts to learn. For example, multiplying by 2 is the same as adding a number to itself. Students should already know this from their addition facts. Starting with these facts gives your child easy reference points to help with other facts. This is not technically a strategy, but these facts often serve as anchors for learning other facts.

**Turn-around Facts - **Multiplication is commutative. This means that you can rearrange the order of the factors and get the same result. For example, 5 x 4 = 20 and 4 x 5 = 20. This knowledge nearly halves the number of facts children need to know.

**Doubling** - Facts can be found by doubling known facts. For example, 4 x 7 can be solved by knowing 2 x 7 = 14 and then doubling 14. This strategy can be used anytime one of the factors is divisible by 2. That's a lot!

**Skip-Counting Up -** You can often skip-count up from a known fact to find an unknown one. This strategy is often used for 6s and 7s. For example, I know 5 x 8 is 40, and I can use this to figure out 6 x 8 by counting up one more eight from 40. This is an extremely important strategy because it introduces students to the Distributive Property of Multiplication Over Addition. You and your child don't need to be thinking of the Distributive Property when you use this strategy, but know that you are giving them experience with an idea that is crucial in Algebra. Here's what those steps look like more formally:

6 x 8

(5 + 1) x 8

5 x 8 + 1 x 8

40 + 8

48

**Skip-Counting Down - **If you can skip-count up, you can also skip-count down. This is a great strategy for 4s and 9s since you can count down from 5s and 10s respectively. For example, I can figure out 9 x 7 by calculating 10 x 7 - 7. This also gives students experience using distribution.

**Halve - **If you or your child is a good doubler, then you may also like using a halving strategy. This is great for 5s if you know your 10s, 3s if you know your 6s, etc. Example: I know that 6 x 8 = 48, so halving 48 helps me figure out that 3 x 6 is 24.

**Combining Other Strategies - **As you get experience with these strategies, you may find yourself using multiple strategies together. For example, if I wanted to figure out 23 x 7 I might use doubling to figure out 20 x 7 (7 x 10 x 2 = 2 x 70 = 140) and then skip-count up three more sevens (140 + 3 x 7 = 140 + 21 = 161).

If that feels overwhelming it is likely because you have not had much experience thinking through multiplication strategies. It is not caused by a difference in math ability. After teaching for nine years, I have no reason to believe that this level of thinking is out of reach for any child or adult.

Notice that ALL of these strategies work for facts beyond ten too. **That's the power of level 2: these strategies extend beyond the zero to ten facts we expect our students to memorize.** Spending a lot of time developing the level 2 strategies helps children become better mathematicians while also learning their facts.

Alright, now that we have the background knowledge we can figure out exactly how to help. The key idea here is that you want to move your child through level 2 and into level 3 not skip level 3. To accomplish this, you will want to have conversations with your child about facts. Here is an outline for how you might discuss 6 x 8:

**"Do you remember the product of 6 x 8?" **- Begin by asking them to try to recall a fact. This pushes them towards level 3. But, if they can't recall it that's okay!

*"I don't remember"*

**"Can you figure it out using a strategy?" **- Since they don't have the fact memorized, it is time for them to try to use a strategy to figure it out. You may have to wait a LONG time. That's okay. We already know they do not have the fact memorized, so give them as much time as they need to think through their strategy.

*"46"*

**"How did you figure that out?"** - Notice that I didn't say "wrong." They will figure out their mistake when they discuss their strategy.

*"I know that 5 x 8 is 40 so I added one more 8. And, that is 48" - *They may not even notice that their answer changed. That's okay here. I am focused on the logic of their strategy.

**"That's an awesome strategy! So 6 x 8 is 48."** - Notice that I praised their strategy since that is what I want them to focus on.

**"Are there any other strategies you could use figure out 6 x 8?"** - By asking for more strategies, I will reinforce the fact and also help them develop a wider variety of strategies. Again, I'll need to wait very patiently here.

*"Well, I know that 6 x 10 is 60. So I could count backwards two more 6s. [long pause] 54 [long pause] 48. 6 x 8 = 48"*

**"That's another cool strategy! Are there anymore that work?"**

"*[Long pause] I can't think of another one"*

**"I thought of one! I know that 3 x 8 is 24. Since 6 is double 3, I can double 24 to figure out 6 x 8. That's how I figured out 6 x 8 = 48"** - If there is an obvious strategy they missed, you should introduce your child to it. You might ask them to explain the strategy back to you if they like it. Hopefully, your child is being exposed to multiple strategies in their classroom too.

*"That's a cool strategy. You are the awesomest parent ever." *- You probably won't get this response but you deserve it!

**"Now that we have talked about strategies, can you remember 6 x 8 without using a strategy?" **- Now that I've helped solidify level 2 understanding, I want to give them practice recalling the fact to practice for level 3.

*"Yeah! 6 x 8 = 48."*

It takes TONS of these conversations to move children through level 2 to level 3 knowledge of their multiplication facts. Trust your instincts on where to begin. If your child is really struggling to learn their facts, start with facts you know they can do well and repeat those facts often.

Make sure your first few discussions are successful. And, **never** **get frustrated.** If they cannot do what you are asking them, then they cannot do it. If they can't skip-count up when multiplying by six, then talk about skip-counting up for their three facts.

Most importantly, enjoy these conversations with your child. It is amazing to watch children develop and then apply new strategies as they learn. You will be astonished at how sophisticated their thinking becomes after just a few conversations!

"Coherence, with respect to mathematics curriculum, generally means that connections are clear and receive emphasis from one year to the next, from one concept to another, and from one representation to another. High-quality materials are coherent pedagogically, logically, and conceptually."

It is hard to argue with this opening statement. We all know that students benefit from consistency as well as well-planned and organized learning experiences. When each lesson builds on prior knowledge and experiences, students are more likely to learn and understand mathematics. Understanding is building networks of connections between concepts in the brain. Coherence definitely supports that.

"The increasing availability of online instructional materials—some of which are of high quality and some of which are not, and many of which can be downloaded at no cost—has added a new dimension to the curricular landscape for mathematics teachers and school districts."

"Online instructional materials" is such a general term with lots of different materials under its umbrella: teachers-pay-teachers, desmos, auto-generating worksheets, blog posts, Khan Academy, Illustrative Mathematics, Three-act-math, WODB, Estimation 180, etc. There is a huge variety of quality and style here. I worry that lumping them together is going to muddle the point for many teachers. If you use any of these resources and feel like you are being criticized, it is so easy to get defensive and shut out new ideas.

"Some of the most engaging conversations about mathematics teaching today are taking place within online communities where teachers share instructional resources and ideas that they have either created themselves or found on their own online."

#MTBoS shout-out! Definitely agree with this statement that "engaging conversation" is happening online. However, #MTBoS benefits from being a self-selected community. The teachers that participate in it have opted-in. This greatly increases the likelihood that new members subscribe to beliefs that are similar to the overall community's. It's easy to have productive discussion when most participants are on the same page to begin with.

In contrast, a colleague that has been assigned to teach the same course will not necessarily hold the same beliefs as you. This is a great opportunity for growth but is probably less likely to produce productive discussion unless protocols are put in place like professional learning communities.

"A recent survey by the RAND Corporation found that the vast majority of math teachers, at both the elementary and secondary levels, reported they used materials that they developed or selected themselves to implement the Common Core State Standards for mathematics. There is no question that this practice is widespread."

From my anecdotal experience, it seems as if a very small group of teachers "use" a curricula. By "use" I mean follow the day-to-day lessons and activities set forth by the curriculum (with few exceptions).^{[17]}To be clear, I am NOT saying that when you "use" a curriculum you never integrate additional lessons or activities. Just that it is NOT the norm and takes up a small percentage of your class time.

Some of this is caused by lack of access to good curricula, but much of it is not. I have encountered stigma against using curricula. It seems that there is a belief that only beginner or lazy teachers use curricula, and that the best teachers always create their own materials. It is almost a hipster-type mentality, once a curriculum is published and goes mainstream, it cannot be good. That seems insane to me. Every curriculum has its strengths and weaknesses, but some of them are REALLY good. It does not feel right to dismiss them all off-hand.

"The dilemma is that while districts, schools, and teachers have greater access than ever to tools and resources for selecting and developing instructional materials, the skill required to develop a high-quality curriculum is both complex and often underappreciated. The widespread availability of online tasks therefore makes having and working with a coherent curriculum at the school and district level even more important because it is the curriculum that establishes the learning goals in a coherent progression and helps teachers see and understand the multiple pathways that students might take through the progression."

This is the most interesting paragraph in the whole letter. I completely agree with his statement that "the skill required to develop a high-quality curriculum is both complex and often underappreciated."

Think about how you spend your time as a teacher. Mine looks roughly like this:

- Actively teaching or working with kids: 50%
- Email, grading, copying: 30%
- Lesson design and preparation: 20%

Curriculum developers are spending 80% or more of their time developing lessons and have years to develop a curriculum.^{[18]}That 80% figure is a guess, but I feel confident that curriculum developers' percentage is significantly higher than the percentage of time I spend developing lessons. Also, do not forget that they can spend years developing a curriculum whereas a teacher has to be ready for day 1 of school. If only 20% of my work time is dedicated to developing lessons, how can I possibly produce work of comparable quality consistently. I can make a few great lessons here and there, but a day-to-day program is going to be near impossible.

The answer is that I can work collaboratively with peers (in my school or over #MTBoS). That's what's happened with things like Geoff's Curriculum Maps; but, even with how well those are thought out, coherence is still a real concern. There are so many gaps and inconsistencies in those maps. That's not a criticism of Geoff, it's just an inevitable result of compiling lessons from so many different sources that are not designed to build on one another.

But, I think Larson stops short of what he actually wants to say here. He loves the active discussion happening over #MTBoS (with good reason) but does not like the trend of going away from published curricula towards collections of lessons. Even when those lessons are individually strong. His thesis is summed up in the quoted paragraphs below: given that teachers are moving away from published curricula, it is crucial that they are extremely thoughtful in how they build their own curriculum.

Personally, I think this is impossible. We already struggle to find the 3+ million qualified teachers we need to teach our students. It is not possible to ensure that all of those teachers are trained well enough to individually (or in small groups) select tasks to build their own curricula.

"Ideally, teachers who select online instructional resources and engage in online community discussions would not be working in isolation but within well-developed professional learning communities in their schools. This sustained colleague-to-colleague communication would increase the likelihood of the selection of high-quality tasks that fit within mathematical learning trajectories and support the school and district’s curricular goals for students."

"Perhaps the greatest danger is the potential for vast inconsistencies in instruction and highly variable learning experiences for students that in turn can lead to differences in student learning outcomes."

This is the politest way of saying that when teachers use different self-generated and self-organized materials there is going to be significant differences in the quality of their teaching. With so many teachers developing their own curricula or combining materials to form something like a curricula, some students are going to get lower quality instruction.

"Without question, curricular coherence is highlighted and enhanced when teachers work collaboratively and regularly with colleagues at the school level to plan instruction, implement the task, anticipate student work, respond to student learning needs, and provide consistency in curricular aims and instruction for students—no matter what teacher students might be assigned."

Collaboration is great, but I worry that it is not enough to overcome the issues with self-curated curricula. Developing cohesive curricula takes too much time and care to be done while also teaching and grading and meeting with parents and supervising recess and doing all of the other millions of things that teachers need to do.

But, even if you agree with me, there is still the issue that many many teachers are building their own curricula. One reason that is happening (I think) is that teachers are not getting what they want from the curricula they have to work with at their school. So, how can curricula be more appealing to teachers so they get used more. I have some ideas:

**Digital**- I have yet to see a curriculum that really embraces the digital world. Many now offer digital portals, practice problems, and pdf copies of the texts, but those resources are clearly built AFTER the curriculum is designed. They are not integrated into the actual designs. A modern curricula would be designed from the ground up to be a digital medium. Videos would be integrated into the lessons. Students would be using resources like Desmos, Google Docs, and GeoGebra as part of the actual lessons. Teachers are driven to use their own materials because they see the benefits of these digital tools and do not want them to be slapped on to the textbook lessons.**Better Hooks/Launches/Act 1s**- The hook is the question or situation that sets up a lesson. It creates the need to investigate and learn some mathematics. Much of the appeal of the teacher designed online resources is that they have great hooks. Most traditional curricula have unengaging set-ups. They say things like "Two students in Mr. Schwartz's class are arguing about whether..." or "Sometimes business owners need to calculate tax..." These are hardly inspiring.**Include More Open Problems -**Many curricula lack open questions. Every problem or prompt has an obvious answer and strategy for solving. They especially lack open-beginning and open-ended questions. Teachers know how valuable these types of questions are -- especially for engaging with the standards of mathematical practice -- so they have to work them in on their own.**Access**- In an ideal world, a great curriculum would free to download. Is there no model for which this can happen? Couldn't a group of designers self-publish? Could we give grants to pay for the development so that the end result could be free for every teacher in every school? I do not know enough about curriculum design to answer these questions. But I suspect it is possible and that the publishing companies would REALLY dislike it.**Concise**- Many curricula I have used suffer from being too full. They have writing prompts, side-projects, investigations, practice problems, more homework than you could ever assign, check-in quizzes, pre-assessments, grading rubrics, additional practice, lesson plans, quick lesson overviews, and on and on. It would take most classes 3+ school years to do everything provided with a curriculum. So teachers start skipping things and that undermines the coherence of the curriculum. I realize that a lot of those materials are included because they help to sell the curriculum, but I think many teachers would be happier with something that had the fat trimmed off of it.**Simple and Clear Teacher Materials**- Trying to go through the teacher resources for a curriculum is a nightmare. There are often thousands of pages of materials to read. And, while the details you want are often there, you do not have the time to find them. A better curriculum would have much simpler lesson plans, chapter overviews, and explanations. That way, teachers can more quickly and easily see what the goal of each lesson is and why the lesson was constructed in the way it was.

Math education needs to get better, but I do not think the answer is for millions of teachers to develop their own curriculum. There are obvious benefits to working with established curricula, but curricula also has to step into the modern age and better meet the needs of teachers and students. We can get there, but there is still a lot of work to do.

So, I did my favorite thing. I broke down my last presentation and built it back up anew. This time, instead of focusing on one problem type (Fermi problems), I aimed to develop criteria for identifying good problem-solving tasks. Then, I chose good problems that exemplified those criteria.

Here are the slides for my workshop. Additionally, when I give presentations I like to type up a document that summarizes my talk and includes all of the links and summarizes what I shared. You can see that reference document by clicking on the words you are reading right now.