The first thing I look for in a teaching resource is it’s connection to the other activities in my class. What hole is this filling, and does it tie naturally into the narrative? In the same way, there’s nothing wrong with ‘answer getting’ if it isn’t the end of the process. Let’s face it – math isn’t very satisfying for most people if they aren’t getting answers (and most math teachers are satisfied if their students can get answers.) Is there a cultural expectation that our students are tying those answers into a similar narrative? Expecting kids to create a consistent ‘story of math’ without getting answers first is unrealistic at best. The obvious follow-up question is: how do we create a culture where students see getting answers as an intermediate step to understanding math?

I think that this is a really, really interesting observation. It raises up all sorts of questions for me, each leading down a different path of inquiry:

1. Twitter only lets you type 140 characters. Would there be more theory-talk on a different platform?

2. Would more theory-talk about task design lead to less resource requesting? Don’t know if I believe it, but here’s the hypothesis: if teachers believe that task-design is complex, they’ll be more reluctant to ask the world for resources.

3. Are we talking about criteria for task-design or criteria for task-selection? Or are they the same?

4. You’re saying that certain twitter conversations are unproductive. Can you point to, or describe, some twitter discussions that are closer to the ideal? (Extension: Are there better and worse ways to use twitter?)

5. What can teachers do to develop their theories about teaching? Do teachers need to be nudged into theory?

6. Would it be pleasant for teachers to talk theory? Would theory-talk lead to uncomfortable disagreements?

Thanks for the thought-provoking post!

Let me clarify that comment about #MTBoS. We are all trying to ramp up our classrooms, to get our students learning and sharing, but usually, we are in the minority. We are salmon fighting the current! I was showing the whole math department Desmos, teacher.Desmos and Function Carnival. I had some of the teachers gasping with delight! The math dept head cut it short and then apologized later, saying that even the best programs won’t be used by “some” teachers! I just needed 5 more minutes!!

The search for effective “answers” can be used as a tool to build effective understanding. So, first, I notice that I get a lot of wrong answers. Then, in place of genuine understanding I find ways to at least get right answers. Then, I get tired of having to look so much, so I try to create my own right answers that match the answers I’ve been finding, and in so doing develop a sense of understanding about what makes those answers “right”.

Same process is certainly true for me, the math teacher. I got tired of giving weak lessons and activities. So, I go looking so that I can steal effective lessons and try them in my class. Then, I start to notice the patterns, and get tired of spending so much time looking, so I try to make my own effective lessons. As I try and notice the shortfalls, I make updates, which is a manifestation of a growing understanding of the essential elements of an effective lesson.

And the understanding of a decent lesson doesn’t necessarily remove the need to resource-find. On the contrary, it likely would make the process more fruitful. So perhaps a teacher who goes out looking for resources isn’t evidence of a teacher lacking understanding, but could be a sign of a teacher whose understanding is growing.

Dan, I have been following since maybe 2010 and I could take one of your 3act lessons or ideas from your high school Algebra class and it would just be a side trip into something interesting before going back to the boring book. Once I began to use more coherent materials that made me think about math in a deeper way and had experience with methods that failed many of my kids, I find that looking at the mathematics under the lesson and making sure that that is solid allows me creativity and the courage to go outside the text to use others’ ideas with more confidence. The reason why your graphing exercise is so potent is because even though the context is very fun, the deeper need for a dual measurement system and the fine tuning of the graph tool itself as the lesson progresses is an outcome of understanding the mathematics underneath the graphing.
It is easy enough to memorize the elements of graphing and teach the mechanics and different forms of linear equations and have no idea of the value and meaning of those tools.

I would have to agree with Andrew Shauver. As a student, I find greater understanding while searching for answers. This post reminded me a lot of my high school geometry class. In this class, we were taught several rules and the teacher demonstrated why the rules were true. Then we were set loose to figure out some problems. We had to use our understanding of algebra to figure out how to find the answer using the rules that we had just learned to be true. I am not yet a teacher, but it seems to me that what makes a good resource good (or a good lesson good) is that it works. When the resource is used, do the students have a greater understanding of the concepts presented? Unfortunately this can not be judged only by whether they found the correct answer. Judging understanding requires class discussion and asking questions. Why did you use that formula? How did you find that answer? How do you know the answer is correct? Etc.

My opinions on this have changed from when I first started to teach. Early in my teaching I kept a very narrow focus: what does today look like, tomorrow? Oh man I have no idea how to introduce this topic, I’ll go talk to a colleague. Rarely did I talk to them about more than a lesson’s worth of detail at a time, and sometimes it was far shorter: “I need a worksheet on ____, you got any?”

My biggest change in opinion is about scope: you can string a ton of great lessons into a horrible tapestry, especially if those lessons were constructed independently. What comes right before this, what comes next, what came three months ago, what came last year, what comes three months from now, what comes a year from now? If the focus is on higher-order mathematical thinking, these questions have huge impact on a lesson’s effectiveness.

These questions also make it almost impossible to judge an effective-looking lesson in a vacuum, or to recommend a specific lesson on a topic above another. It’s hard for me to recommend a specific lesson from Illustrative, or a three-act, because the whole is rarely being taken into account. For example, one could probably take lessons from Illustrative, and build a curriculum using those precise lessons. In my opinion, this curriculum would be great on the microscopic level and horrible as a cohesive whole. Every single one of those lessons could be effective, but they’re not telling a consistent story.

I also see this with the ways schools use textbooks, and this is really the textbooks’ fault. “We’re going to start with Chapter 9, then do Chapters 3 and 4, then 12, then back to 5-8.” What the hell is that, nobody would read a novel that way, but we do it all the time when designing math courses. This is possible only if a textbook is agnostic to the order of its units, and the consequences are disjoint pieces that cannot build on one another by design. The textbook won’t expect students to remember what they did a month earlier, because it’s impossible to know! The teacher must then add this cohesiveness to the curriculum on their own, and that is truly difficult to accomplish.

In my opinion, teachers do far too much editing or adapting of materials. It takes a huge amount of time to do so effectively, and that is time that could be spent in many other positive ways. Unfortunately this seems to happen by necessity at times, due to nonexistent or poor materials. I don’t feel teachers should be required or expected to write their own curriculum.

It’s hard to say how or if this problem can be solved, but one thought is to look for chapters and units as resources rather than looking for lessons. The same cohesion problems can occur, but at least the entire unit will be internally consistent.

The trap I fall into is finding a ‘fun’ resource and forcing to work into a lesson. This rarely goes well and I often wish I had spent that much more time starting from scratch.

Really, we need to literally go back to questions such as ‘Why am I teaching this?’ ‘Where does this fit into the students learning journey?’ and ‘How am I going to structure the learning so that the student wants to learn this?’ before we even think about where resources fit into our lesson. This takes a lot of time to think about and process. Time and space many teachers just don’t have.

I had a lovely discussion about this very issue with you, Dan, et al at the NCTM conference in Louisville last year. So glad to continue it here.

As discussed at length here, part of the issue is that we often take up lessons without reflecting on the purpose/intent of the tool. Worse, we enact lessons without a deep understanding of the mathematics at play, which cripples our ability to provide feedback in class that centers on the core ideas. We may ask a great first questions, but the second and third, arising from student thinking, fall short. Looking for resources on a unit or larger scale may help make sense of the progression, but I’m not sure it helps us uncover the deeper structures at play.

The underlying and unsaid trait of humanity here is that we take any resource and bend them to our own practice. I’ve seen a 3 ACT over scaffolded, a card sort turned worksheet, a “sometimes, always, never” busted up with the hammer of over-simplification. I’ve also seen many a well intending teacher get into a great lesson, only to find they lack the classroom strategies (usually around feedback questions and discussion) that empower them to reveal and give space to the student thinking.

So –
Learning takes place in layers. A lesson lives within a classroom experience. To be effective depends on my mindset, my methods, and the tools I have in the moment.

Great lessons in the context of learning around mindset and methods are the instruments we use to “do” our work. But the reflection and coaching conversations where we “learn” about our work are critical as well. Without them, we use scalpels like hammers.

But this work is much harder, much more personal, much more in the moment of the classroom. Can we harness the power of tech to share this work as well as we have to share the tools?

RE: harnessing tech to share learning work

The good news is YES. We definitely can.

Video clips of lessons, capturing teacher reflections where they discuss the decisions not the actions, virtual coaching, virtual professional learning communities. We have great access now. The work I think is on our conversations. Critical conversations where we name and articulate core ideas that drive our strategies and resources. Peer reviews where we challenge each other’s practice and provide feedback that goes beyond the superficial “this is what I do.” Comparing the enactments of common lessons?

I’m excited by the potential of where tech can take us, but the work will be around us learning how to learn through the new medium.