Some Reflections: How Assessment Impacts Curriculum
Some Foundational Ideas: Assessments are how I communicate to my students “These are the important mathematical ideas of my course – you are responsible for them”. When I tinker with my assessments, there is collateral damage to my curriculum (the order that I present mathematical ideas) and my lessons/activities (the depth with which we explore mathematical ideas). It all has to be aligned.
This post is building up to a realization I had earlier today. It comes from two key ideas I stole from Standards Based Grading:
1) I dissected my course into discrete concepts and skills that I could assess individually. When my students see my tests, each page is a separate skill and goes into the gradebook as a separate grade. This gives me a way to isolate particular skills (such as solving an algebra equation, or performing integer arithmetic) away from other mathematical ideas that build on these (finding missing angles with parallel lines, or finding the slope of a line given two points). This makes remediation easier, but it also makes it more explicit to my students which skills are ‘foundational’ and are needed to solve more complex problems
2) I assess certain skills multiple times. If my class still struggles with integer arithmetic (-2 + 5, etc), that skill appears on later assessments as its own page. Because each page is designated as a separate skill, students are aware of the fact that this is the explicit ‘algebra’ page. I also include this page when it’s the building-block for a skill we’ve been working on recently (for example: when I teach distance and midpoint in the coordinate plane, I also reassess on integer arithmetic because you need integers in order to do distance and midpoint calculations).
I’m realizing that these choices have fundamentally impacted some of my curriculum choices. Here’s what I mean:
Typical Situation from Last Year (Before SBG): I assess basic algebra skills at the beginning of the year, including integer operations (-3 + 4) and solving algebraic equations (2x – 14 = 26). For the purpose of this post, let’s say the skill of choice is solving algebra equations (two other foundational skill students usually take time to master at the beginning of the year are integer arithmetic and drawing geometric figures).
My test has several skills on it so the grade is more of a ‘summary’ than an itemized analysis – we lose information in a purely numerical grade. Because of this, many of my students get an ‘acceptable’ grade on my test (for some students, a 61% is acceptable), so they stuff it in their backpack and don’t think of it again – they passed, so it doesn’t bother them that they missed every single algebra question. However, I as the teacher can see that most of my class doesn’t know their algebra, even if each individual student doesn’t really care that they don’t know their algebra (remember: they still passed my test, so they’ve moved on to think about other things). I need to figure out a way to revisit algebra so my students realize that they need this skill for work we’re going to do later. Therefore, I adjust my curriculum so that algebra magically reappears a few weeks later in a different context, forcing my students to again confront the fact that they don’t understand this skill. So we spend a few more days on algebra ‘wrapped’ in a geometry concept, and then several problems like this appear on the test at the end of the unit. This gives me a chance to stealthily remediate and reassess their algebra skills without it seeming like we haven’t moved forward in the curriculum. One of the most unmotivating factors in curriculum is to linger on a topic for too long, which is why I need to create the illusion that this is actually a ‘new skill’ and we’re moving forward with our year.
So, we test again. Several of the problems on the test are these algebra problems ‘wrapped’ in a geometric context. After the test, more of my students understand algebra but still not as many as I would like. So I repeat this process. Before long, half of my curriculum has some sort of algebra component because I know that’s how long it will take for me to stealthily remediate and teach this skill.
More after the break below…
Real-Life Examples of This from My 1st Year:
Last year, at any given time, about one-third of my tests were old skills ‘wrapped’ in a new geometry context. When designing the test, the assumption was never “My students will use their skills with (triangles/bisecting angles/quadrilaterals) to solve these problems” – or, as I’m thinking this year, “These problems will give me an accurate picture of how well a student understands (triangles/similarity/congruent)”. Instead, the assumption was always: These problems are an excuse to make my students do algebra because they still need to learn it and these problems will force them to do so.
Reflection 1: This is fundamentally dishonest – I’m ‘tricking’ my students into learning algebra by making it reappear throughout the whole year. Because it’s a trick, when my students do start to understand, they rarely (if ever) realize that its because of their weak foundation. I would try to tell them “You need to learn your algebra! Once you know algebra, everything else will click!”, but they would usually respond with “Just teach me what I need to learn for the test!” (which is the whole reason I put algebra on my future tests in the first place).
Reflection 2: This practice kept the cognitive demand of my classroom at a continually low level. An example: I’m teaching triangle congruency and how to write congruency statements. Then I throw in algebra problems because I need to hit algebra again because they didn’t master it the first time. Now my students are struggling with mastering the new skill (triangle congruency) and the old skill (solving algebra equations). If I want to be fair to my students, I need to keep my expectations simple and concrete: if they can just solve these problems completely and correctly, that’s enough to show that they’ve mastered both of these skills. My idea of a higher-level question was one that incorporated several procedure skills rather than one that required a deeper exploration of a singular skill. Again: this is because in any particular unit, I’m usually teaching to one or two new skills and one or two previous skills that my students never mastered.
To Summarize: Last year, because of how I graded and how I assessed, I was adjusting my curriculum to account for when students failed a particular skill and still needed to master it. I remember having this thought a lot: “I want to do problems like (this) because most of the class failed these types of problems on the last assessment, so we better hit them again”. I was using my curriculum to solve an assessment problem.
Now That I do SBG Grading: Separating my skills makes it completely apparent when a student doesn’t understand a particular skill, which encourages more immediate attention. They can’t hide it anymore. It also makes remediation on that skill meaningful since they know it will appear on later assessments anyway, which means there is always the incentive to raise their grade. Students don’t like getting the same questions wrong week after week after week, which is part of the motivation to work on skills that we first learned at the beginning of the year. And since I make it apparent that these skills build on each other because of how I layer my assessments, my students now appreciate how a weak foundation can affect everything else we do in my class. Separating and layering my assessments makes this conversation more meaningful: “Whoa… I see it now. Integers are everywhere!” (real comment from a student earlier today, which may be another catalyst for this post).
Back to Curriculum: There will always be the problem of “How do you motivate students to go back and master a skill that they need to know?” Last year, I solved that problem by adjusting my curriculum so students were forced to face these skills that they didn’t know. This year: that motivation is built into the very foundation of how I assess. I don’t need to think ‘what problems do I need to talk about so my students will be forced to learn (this)’. Instead, I can rely on the very nature of my assessments to make it apparent when a student doesn’t understand something and needs to remediate.
And just to be clear: I still keep these types of problems in my curriculum and still expect my students to solve them. But my mentality towards them is different – I now approach them as a way to apply conceptual knowledge rather than the catalyst to revisit algebra for a few days. They appear on my assessments embedded in a concept rather than the primary focus of the assessment.
Lastly, I think this also explains some of the disappointment I felt with some of my lessons from last year. For example, I remember my unit on Quadrilaterals being excellent last year – my students were very successful with pretty much everything we did. But this year, it was very mediocre. I’ve been trying to figure out why that was, when it hit me: my unit on Quadrilaterals last year was almost entirely grounded in algebra problems (like the parallelogram one in the document above) and in integer arithmetic (here are 4 points – use slope and distance to determine what kind of shape this is). And last year, this was the unit where both of these concepts finally clicked for most of my classes, which is why it was so successful. But this year, when I taught these same lessons, my students saw straight through these problems and recognized them for what they were: an excuse to do algebra and an excuse to do slope & distance problems. And since I didn’t raise the level of rigor beyond “Connect all these different skills together”, my students felt like they were spinning their wheels. And they realized it because they had learned to see through my dishonest curriculum tricks which my SBG assessment system has brutally exposed.
So… lots of stuff in this post. I’m still exploring this idea of how assessment choices affect curriculum choices. Any ideas or comments are absolutely appreciated in the comments.