I wrote recently about my issues with learning goals. Suffice to say, I think that picking a very specific learning goal for a lesson and focusing all of the teacher and student effort on meeting that goal creates an illusion of learning. Information cycles in and out of students’ short term memory without any incentive for long-term retention, it breaks content into silos that prevent connections and transferable learning, and sends negative messages about what it means to do math.

I’m also captivated by the idea of a spiraled curriculum, but moving away from a unit-centered approach to teaching means I need to reevaluate some of my assumptions about how students learn. This post is my attempt to do that, in the context of learning goals for a specific lesson.

For context, I’m at a new school this year. My classes are small, which gives me big advantages on the formative assessment end. I’m teaching Algebra-II and two different flavors of Pre-Calc, which means much of the content I’m teaching is either review or building directly off of previous content. I also teach 90 minute blocks, which is another influence on my approach this year.

Here is my central premise:

**Rather than thinking about all students meeting a concrete, measurable learning goal — something they can do or produce — by the end of the lesson, I want to think about students meeting a competence threshold. A lesson is purposeful when, by the end of the lesson, students are clear on where that learning fits in a progression, and can articulate where they are relative to criteria for understanding.**

More specifically, I’m thinking about this topic in the context of different purposes lessons can serve in a learning progression. I’ve been reading Elizabeth, and her take on How People Learn. I’ll quote her:

Specifically,

HPLadvocates:

STAGE 1–a hands-on introductory taskdesigned to uncover & organize prior knowledge. In this stage, collaborative activity provides an occasion for exploratory talk so that students can uncover and begin to organize their existing knowledge;

STAGE 2–initial provision of a new expert model,with scaffolding & metacognitive practices woven together. The goal here is to help students bring their new ideas and knowledge into clearer focus so that they can reach the next level. Here again, collaborative activity can provide a setting in which to externalize mental processes and to negotiate understanding, although often, this can be a good place to offer some direct instruction;

STAGE 3– whatHPLrefers to as “‘deliberate practice’ with metacognitive self-monitoring.”Here the idea is to use cooperative learning structures to create a place of practice in which learners can work within a clearly defined structure in which they can advance through the 3 stages of fluency (effortful -> relatively effortless -> automatic)

STAGE 4–working through a transfer task (or tasks)to apply and extend their new knowledge in new and non-routine contexts.

I think this is a great way of thinking about teaching that is applicable to both a unit-focused approach and a spiraling approach to learning. Interpreting this in the context of the competence threshold, there are three different types of lessons or activities that I ask my students to engage in.

**Topic Introduction**

This lesson moves through stages 1 and 2, introducing a concept and provisioning an expert model. This is a great example of a time when it can be beneficial to withhold a learning goal at the beginning of a lesson in order to promote student thinking. Students are introduced to a new concept through an activity that ideally both engages students in novel thinking and motivates the purpose of a new piece of knowledge. Achieving a competence threshold for this activity means that, by the end of the activity, students are clear on the purpose of the learning, have been introduced to an expert model, and have an opportunity to evaluate their understanding of the expert model.

**Practice**

A practice task incorporates stages 2 and 3, provisioning an expert model again to reinforce learning and engaging students in deliberate practice. The learning goal should be clear fairly quickly, although it may be worthwhile to incorporate what might normally be thought of as multiple learning goals into one practice lesson in order to interleave practice and have students more authentically draw on concepts. While an introduction lesson may only allow for an initial understanding of a topic, a practice lesson allows for students to calibrate their performance to an external benchmark and get feedback that moves their learning forward by applying a concept in a number of contexts. A competence threshold is reached when students can successfully compare their performance to criteria for understanding and use that feedback to move their learning forward.

**Transfer**

This is stage 4, a task that asks students to apply previous knowledge in a new context. The learning goal is to solve a problem that synthesizes a number of concepts, rather than explicitly focusing on the individual topics. This can take a variety of forms, but the common thread is that the problem is non-routine from the students’ perspective, yet draws on relevant concepts that can be reinforced and strengthened. A competence threshold is reached when students successfully apply the relevant concept or concepts to the transfer task and connect the new situation to prior knowledge.

**Things I Like About This Model**

- A competence threshold is student-centered rather than teacher centered. I am using effective learning goals if each of my students is meeting certain criteria for understanding, not if I write it on the board at the beginning of class and refer to it multiple times.
- A competence threshold acknowledges that it may not be realistic for every student to be in the same place by the end of a lesson or activity, but validates their learning if they are moving forward in understanding.
- A competence threshold separates introducing a topic from practicing it. I am becoming more and more convinced that these two need to be separated — there’s nothing wrong with practice immediately after teaching a topic, but it is essential that deliberate practice is spaced as well.
- A competence threshold sets a benchmark for a successful spiraled lesson, and helps give me something concrete to evaluate whether students are learning, without just making excuses that I’m spiraling and they’ll learn it eventually.

**Some Examples**

**Expressions & Equations in Algebra-II
**My Algebra-II students struggle with many topics — I teach juniors who, at least compared to their peers here, are a bit behind in math., But starting the year with endless review makes me a bit queasy. I could spend the first 2-3 days diving deep intro equations and expressions, but I’ll inevitably still be running into misconceptions as I want to move on, and that seems like an unlikely way to reinforce algebraic ideas that I want them to retain throughout the course. Instead, several days a week, I’m giving students a task toward the start of class that looks at a different perspective on expressions & equations. This task from Illustrative Mathematics

or this task from Five Triangles

or the Desmos lesson Central Park, have been great examples. Each provides multiple opportunities for practice, emphasizes useful algebraic ideas, and has been an opportunity for growth for my students. At the same time, each task serves as formative assessment to identify further areas for growth.

I am under no illusion that I am doing magical things for students algebraic understanding through these tasks. But the lens of a competence threshold helps me to evaluate whether these tasks are successful in learning forward — and I think that reinforcing these ideas over time, pushing students to look at the structure of expressions and equations and apply that knowledge in different contexts, will do far more than three days practicing en masse before moving on and leaving these topics behind.

**Quadratics in Pre-Calculus
**In one of my Pre-Calc classes, we start the year with function transformations before moving into rational functions and some other more advanced topics. One area that isn’t explicitly in my curriculum but I wanted to emphasize was quadratics. They reinforce lots of algebraic structure that will come up over and over again, and are good practice for thinking about concrete functions in addition to all of our work with function transformations.

Vertex form is pretty easy — it dovetails with function transformations, and I introduced it during that part of the unit. Then, again during that unit, I did a task around graphing functions in factored form and the advantages and disadvantages of factored form compared with vertex form. (Sorry, no file — it was on whiteboards in small groups). Then we spent a day focusing on graphing quadratics, looking at properties of quadratics, and making connections between representations. The fact that the different representations we were using had been spaced out over a week or more made a big difference, I think, in students competence and retention from this lesson.

But I didn’t want to stop there. I have a bunch more application tasks for quadratics that are worth doing. For one lesson in particular, almost a week later, I did Alex Overwijk’s Serial Position Curve activity to reinforce a number of quadratics concepts and provide an opportunity to transfer some knowledge about what different forms of quadratics mean in context — what might the vertex form look like? What might the standard form look like? What about factored form? What’s weird? What would happen if we had more people? More words?

It was a great activity, and took up the first half of class. After that, we were about to jump into function composition, so I used an activity that I believe is from Kate but can’t find a link to right now. I had students write a function to find the side length of a square given its perimeter, then a function to find the area of a square given its side length, then a function to find the area of a square given its perimeter. I used this to illustrate what mathematicians mean when one function is composed with another, and a situation where it might be useful.

With a bit of practice on whiteboards to wrap up class, I’ve got students leaving with a decent idea of what function composition is, ready to come to class the next day ready to dive into practice. I’ve deepened understanding of quadratics along the way. And I have a chance to respond to what I’ve learned about students’ strengths and weaknesses with function composition and what moves will be most productive the next day.

All of this is stuff that sounds great to me normally — the reason thinking about a competence threshold is useful is that it gives me a way of looking back at these lessons and being more critical and productive than “oh they had some good mixed practice I think they learned some stuff”.

**In Summary**

This learning is messier than what I’ve done before — but if I’m being honest with myself, what I’ve done before hasn’t worked particularly well. I think that this approach matches my experience, is consistent what I know about how the brain works, and maybe most importantly, provides a change of pace during class that my students appreciate and helps them stay engaged.

I’m hoping to use this lens going forward to look at my forays into spiraling curriculum more critically — it’s easy to try something new and cherry pick evidence that it’s working. I’m hoping a competence threshold perspective will help me examine my own practice more critically and learn from it at the same time.

howardat58To lighten things up a bit, consider this:

f(x)=a.sqr(x)+b.x+c

g(p,q,r)=p.q+r

f(x)=g(g(a,x,b),x,c)

Simpler to write f(x)=(ax+b)x+c

and it has the function of a function structure, but not explicitly.

Now do it for a cubic !