Welcome back Math Geeks! I’ve been thinking a lot about the objectives of lesson objectives, and I’m committing to writing a series posts to spark a conversation. I’m curious about how you frame lesson objectives to maximize student joy, and I invite you to tell me. If you missed my first post about ways to make the learning objective an invitation and not a mandate, I encourage you to check it out here. You can find part two here that’s about creating intellectual need for the objective before expressing the objective.
An Objection about Objectives
One of the reasons why we teach is because we want our students to experience the JOY of mathematics. Mathematics should be about questioning, wondering, and the joy of discovery…and math classes should leave students wanting to know more math and do more math thinking. We cannot build an appreciation of math through content standards alone. Math classes should be filled with opportunities for students to have voice and a choice. At the very least, they need a voice in making meaning of problems and a choice in how they go about seeking a pathway to a solution.
But sometimes we (or our textbooks) squash all the joy out of a math lesson. We rob them of their right to notice math things, wonder about math ideas, or do messy math stuff. And lessons that focus on “measurable outcomes” with “explicitly defined objectives” often euthanize mathematical curiosity.
The Textbook Approach: Killing Wonder and Joy
Let me offer an example. Suppose you’re teaching students ways to find the product of 2-digit numbers (4.NBT.5).
Your textbook probably breaks this skill down into measurable, discrete, tidy chunks. For example, GoMath has “essential questions” that frame the lesson objective for each lesson in their chapter on this standard. Here they are in order by lesson:
- What strategies can you use to multiply by 10?
- What strategies can you use to estimate products?
- How can you use area models and partial products to multiply 2-digit numbers?
- How can you use place value and partial products to multiply 2-digit numbers?
- How can you use regrouping to multiply 2-digit numbers?
- How can you find and record products of two 2-digit numbers?
- How can you use the strategy to draw a diagram to solve multistep multiplication problems?
By asking those questions, the textbook is taking all joy out of the learning of math. There is nothing to discover; there’s only filling in blanks.
Lessons with these essential questions and the accompanying “Students will be able to….” objectives will not foster joy and appreciation nor create the wonder to learn more math.
How Joy is Killed
Let’s analyze that list above and see what they’re trying to do. They start with a low floor number sense strategy about place value (1). They want students to make some estimates and demonstrate some number sense (1 and 2). They want students to learn different ways to reason about multiplication starting with a visual area model first to show partial products (3). They then try to leverage these partial products into more algorithmic procedural thinking about place value, regrouping, and recording your work efficiently (4, 5, and 6). They then conclude with some sort of application problem where they apparently want students to go back to visual representations (7). I presume this last step is to make sure that the curriculum meets the “application and problem solving” component of rigor (perhaps at the expense of creating joy).
All of those things are important and have value, and they’re logically sequenced in a tidy little pathway. It makes sense to me why they did it this way.
I have a problem with the fact that those lessons will kill a child’s love of mathematics. This is what we get when lessons and pedagogy become hyper-focused on specific, measurable outcomes defined by clear “SWBAT…” objectives.
Here’s the thing: Your 4th grade students have some knowledge about multiplying numbers. They drew arrays in 2nd grade and they went deep into the different ways to show multiplication visually in 3rd grade. (Watch Graham’s amazing video on the progression of multiplication here.) They also developed some useful skills around partial products. Of course, not ALL your students know or remember ALL of this, but they all know some (and some may know all).
How will you know what they already know until you give them a chance to show what they know?
The 3-Act Approach: Creating Wonder and Joy
What if we structured this chapter around the same general outcomes but made the objectives about reasoning, communication, estimating, and finding efficient methods? Couldn’t we get at the same skills highlighted in numbered list above, but do so in a way that gives students a choice and a voice in the learning? Can’t we make it an invitation rather than a mandate?
If you must put an objective on the board, I suggest starting the unit by writing this objective (or something like it):
“Today you will wonder about quantities, do some math, and share your reasoning with each other.”
I would focus on the wondering and the communicating up front. The “do some math” can be unpacked later in the lesson(s) simply by asking students “what math did you do today?” I’m pretty sure they will say that they did some multiplying of some (2-digit) numbers.
Imagine starting a unit with this image from Graham Fletcher’s website.
What does that moment look like? What kind of lesson would unfold? Here’s what I see:
Students will wonder about this image and talk to each other. They would eventually ask (among other things): “How many donuts is that?”
The teacher could ask them to make an estimate and watch how they count and reason about the problem while the students show what they already know without having to be told what to think or how to think. This visual invites students into making an estimate rather than mandating that an estimate be made. Making an estimate also makes students go slow and thoughtful as they ponder other things about the image. Do the donuts go under the white edges? If so, how far? Is there only one layer? If not, how many?
My point is, we can get students estimating without having to telling them that the objective is to use estimation to reason about products. We can make our objectives explicit through guided reflection.
From here, the teacher could share this image and say that there are three layers to the box. (Graham structures it differently in his 3-Act. For the sake of this post, I’m cutting some corners.)
Now if you’re starting your exploration of 2-digit multiplication, you will get to see what your students can show you. They most likely don’t have the formal skill yet. But watch them anyway. How do they make estimates? How do they reason about these quantities? What strategies do they bring with them from 3rd grade? And what might need to be areas of focus moving forward?
This problem doesn’t need to be solved at this time. It can hang there and be used to talk about the multiplication strategies throughout the next few days. After students explore these strategies, they can return to this image and the big reveal (Act 3) could be shared. Weave in practice with more images like this one.
If they’re struggling, perhaps dial it back for the next lesson when you explore by using these thoughtful images by Kyle Pearce. This tweet is a good starting point.
Or plow ahead with this amazing 3-Act task called Array-Bow. You can find my write-up of my experiences teaching Array-Bow here. It also makes for a good assessment because it really forces the need for multiplication strategies.
Conclusion
We have a moral obligation as math teachers to make learning joyful for our students. How we use objectives to frame lessons (or do we use lessons to frame objectives?) will have a tremendous impact on the level of joy in the classroom. Textbooks can’t do this well. It’s not just GoMath. I’ve published curriculum that I would modify to make it more joyful and engaging for my students. I’m offering not criticism so much as I’m offering a pathway for thinking intentionally about how we share learning objectives with our students and making sure we’re choosing practices that align to our purpose: to make math joyful.
How do you frame objectives in your lessons to create joy and foster of love of doing mathematics? What resonates with you (or irks you) the most about this post? Help us further the conversation.
I’m trying to better understand your claim that “lessons that focus on “measurable outcomes” with “explicitly defined objectives” often euthanize mathematical curiosity.” Are you saying that an objective that specifies the outcome explicitly is akin to “giving away the math” (which I agree with), or are you saying that defining measurable outcomes at all leads to teachers missing out on what students know that may not fit into our nice, neat SWBAT statement?
I grapple a lot with this idea of objectives. I think about the formative assessment cycle, and the need for clear learning goals and success criteria, as defined by Margaret Heritage. Research points to the idea that objectives will support our students in their learning, but I agree that not all objectives are created equally. Thanks for helping me think more about this.
Hi Chrissy! Thanks for the thoughtful note.
I would say yes…I’m talking about how objectives that “give away the truth” deny students the chance to explore, do, and talk math. I think it’s extremely important that teachers have clear objectives. Every practice we do as teachers should have clear and intentional objectives. Without objectives, we have no way to measure the effectiveness of our instructional choices as teachers. It’s our practice of how we communicate and frame those objectives to students that has my feathers ruffled. I’ve seen many well-intentioned teachers derail even the most engaging learning opportunities with this practice. As you said, not all objectives are created equally.
Now you have me wondering more about how to evaluate/revise/refine an objective to make them better. Thoughts?
I’m wondering if some sort of protocol for teachers would help: given some explicitly defined objectives, what revisions and improvements could be made to move towards a more “inviting”, as you call it, objective? For example, given the objective: “Students will be able to find that the formula for calculating the area of a parallelogram is the same as for that of a rectangle by decomposing a parallelogram into a triangle and quadrilateral that can be recomposed into a rectangle” (I know, that is a ridiculous objective, but you get the point), what could be substituted, removed, reworded, to open up the students’ experiences with this content? Better (less helpful) objectives: “Students will investigate the relationship between finding the area of a parallelogram and the area of a rectangle” or “Students will investigate area”. I think IM did a good job with their objectives in Open Up. A good start is to do what they did and think about how the objective statement for the teacher might be different than the “student facing” statement. I don’t think many teachers have thought about objectives this way and I think it could really support this conversation.
Yeah. What does that protocol look like?
What I’m wondering about the template:
Having a teacher facing objective on the protocol that also includes the “how” in the language. For example: “Students will investigate the area of parallelograms using decomposition/rearrangement (how). Through discourse and guided discussion, students will formalize their thinking to conclude that A=bh holds true for all parallelograms, not just rectangles.” In other words, “Students will exclaim ‘Eureka!’ upon discovering (how) that the area formulas for all parallelograms is b*h.”
The ultimate goal is that near the end of the lesson, students tell us the objective.
Then, backward planning from there, teachers start to deconstruct that objective into less and less helpful language and structure learning experiences for those “sub-objectives”.
So using your example, I might do:
1. Quick warm-up that has them comparing areas of two quads that invites simple decomposition/rearrangement.
2. Conclude warm up with: “Today, we will investigate more about the areas for quads like this.” This is the student facing objective that comes out after 5-10 minutes of the math lesson and is referenced throughout the following investigation.
3. Investigate the areas of 1-2 rectangles on graph paper. Then investigate the areas of 2-3 non-rectangular parallelograms on graph paper.
4. Share student work and facilitate discussion of strategies.
5. Then with several student work samples displayed ask something like “What are you noticing about finding the areas of all the parallelograms we’ve explore today?”
6. Hopefully there’s the “Eureka!” moment. Formalize that their conclusion in student friendly terms in writing. And then teacher can formalize that into the lesson objective before the lesson closure/cool-down/exit-ticket.
To me, this is how we invite, engage, and bring joy into the classroom using objectives. Rather than make mandates and squish student engagement.
Your thoughts? If that jives with you, what would need to go on the protocol so teachers are successful at planning an engaging lesson that aligns with the objectives?