Rethinking Lesson Planning

Every lesson plan template I've ever seen looks similar to the following:

Taken from: www.lessonplans4teachers.com

While teachers use these templates less and less as they become more experienced (unless, of course,  they are faced with district mandates/evaluations/"we don't think you work hard enough so here's something else to do").  That said, I find that the paradigm of developing lessons in a linear fashion remains. I'm going to reach my goals by doing X, then Y, then Z. Issues I have with tempates aside, I have been recently thinking about alternative ways to conceptualize lesson planning (like any scaffold, a template can be helpful when starting but can also be limiting. I still rue the fact that in high school I was taught how to write 5 paragraph essays REALLY well, but was never given the freedom to break from this structure).

Here's my initial stab at a way to think about lesson planning that is less linear and, hopefully, more supportive of best practices around student exploration and constructivist learning. For now, I'm calling it a lesson web.

Created with Mindjet MindManager

At the center of this web is a central skill, concept, or habit. Some of your goals for the lesson (which you can write down separately) will be directly connected to this central topic. From this central topic, you can brainstorm three things:

• problems 
• potential methods for solving problems within the realm of the central topic
• potential misconceptions related to the central topic

Relationship links (the red arrows) connect specific problems to specific methods and misconceptions. Building these relationships is crucial, as it will serve as a check to make sure you are giving students problems that address potential misconceptions and desired method (while I personally don't think teaching specific algorithms is necessary, I understand that being familiar with a standard algorithm can sometimes make communication more efficient).

At the next level, problems branch into possible variations, extensions, and generalizations.  In my classroom these are developed by both me and my students.  These problems can also be connected to a new central topic, making this not only a template for lesson planning, but really a curriculum map.

Misconceptions can also link to problems that will help expose or eliminate those particular misconceptions.

Things I like

• When giving students the flexibility to create their own problems and develop their own methods, the teacher needs to anticipate what students might do. This structure supports this. Teachers can even use this to outline the order in which students will share out (for example, starting with students who used method 1, then method 2, etc).
• This structure is easy to adapt. After the lesson, it's easy to add new methods students came up with that you didn't think about or misconceptions that students had. This also gives the teacher a tool to use the problems students create as a springboard for future topics (making new connections between problems and topics).
• While I have the topic playing the central role, you could just as easily start with a problem or a misconception and build the web from there.

Things I don't like

• There's no assessment (formative or summative) built into this. This is something that a teacher will have to think about in parallel to building their web.
• Sometimes my primary goal for a class might lead to a web where one or more of these fields don't naturally fit in.  For example, if my goal is to promote communication between partners, I can imagine problems that might help meet this goal, but I'm not sure what a "misconception" would look like or what different "methods" students might develop. 
• This is a lot more work than just going with my gut and adapting on my feet. :)

Next step? Developing a lesson web for a specific topic.

   

The Common Core Standards

The final draft (for now) of the Common Core Standards for English and Math were released today. I flipped a coin, so I guess I'll talk about the math section. I'd read parts of earlier drafts, but spent much of today reading through the final version. Here's a summary and my first take:

We start with a two-page introduction, with lots of quotes from people who helped write these standards, and then a paragraph on mathematical understanding.

Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from.

Me like...the hard part now will be writing valid and reliable assessment questions.

A page describing how to read the standards follows, and explains that grade level standards will be broken up into three hierarchies: standards, clusters, and domains.

"Standards define what students should understand and be able to do. Clusters are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject. Domains are larger groups of related standards. Standards from different domains may sometimes be closely related."

"grade placements for specific topics have been made on the basis of state and international comparisons and the collective experience and collective professional judgment of educators, researchers and mathematicians"

This concerns me a tad. We're making decisions based on what's been done in the past (isn't the whole point that what we've done in the past hasn't worked particularly well...I really hope this is more than just trying to get all 50 states to do the same mediocre thing) and what's being done in places like Singapore (where reportedly everyone is good at math). Now while Singapore Math and I are not friends on facebook, there are aspects of this curriculum that I think are fantastic. Leinwand and Ginsburg wrote a good article about what they believe make it successful. I wonder if the core standards goes far enough to link problem solving with concepts and skills, especially since the authorsexplicitly claim that these standards "do not dictate curriculum or teaching methods." I feel that this was a political decision, and question whether this was a really bad idea. If we are trusting "math experts" to determine the content that is taught in schools to students who will be entering the workforce in 2030, why not entrust experts to dictate best teaching practices? I've heard there's lots of research on the subject.

Three pages are then devoted to defining and describing "standards for mathematical practice."
They are:

problem solving, reasoning and proof, communication, representation, connections... adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

These practices are organized into eight categories that the authors elaborate on.
1 Make sense of problems and persevere in solving them (very Polya-esque).
2 Reason abstractly and quantitatively (ability to decontextualize and contextualize).
3 Construct viable arguments and critique the reasoning of others (conjecture, proof, and critique).
4 Model with mathematics (connect math to "the real world"...not sure how this is different from #2 and I definitely worry about how this will be done in lame, inauthentic ways).
5 Use appropriate tools strategically (can you in good faith make this 1 of the 8 principals of doing mathematics and not allow students to use these tools on high stakes tests?).
6 Attend to precision (accuracy, on the other hand, is overrated...I jest, precision is a practice that will be beneficial to students in every realm of their life).
7 Look for and make use of structure (understanding symbols/structures and characteristics of these symbols/structures).
8 Look for and express regularity in repeated reasoning (looking for shortcuts and generalizations).

It's unclear whether the authors believe that these practices will lead to a better understanding of the content (which will be assessed) or whether they believe these practices to be an integral part of doing mathematics (which would then imply that these practices should be assessed independently of content). Boy, I could probably be convinced to do some unsavory things to see the latter. The authors leave this decision largely to textbook writers: "Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction." which, in my experience, are beholden to test writers. Unfortunately, my guess is that these will be treated much like the current "mathematical reasoning" standards that are"inherently embedded in each of the other strands", in my humble opinion a specious claim confirmed by the low cognitive levels involved in some of the standards and most of the test questions.

Also...where's pattern sniffing? Did I miss it? That was a surprise.
What about problem posing? Ok...I'm less surprised that this wasn't in there. At least I still have a purpose in life.

And I still can't help think that this isn't as good as Cuoco's Mathematical Habits of Mind, but I digress.

Anyway, that ends the first section. The remainder of the document is devoted to grade level standards. This post is already way too long, so I'm going to stop here and do a separate write-up on the grade specific standards.