Written by Sarah Stecher published 11 months ago

You’ve just finished an EFFL lesson. Your debrief was lively, multiple students shared their approaches, you summarized the key ideas in the QuickNotes, and students worked on a few Check Your Understanding problems. What’s next?

We would all probably agree that students need additional practice to solidify the ideas learned in the lesson. This is generally seen as the purpose of homework. But oftentimes when we think of practice, we think about practicing a sport or instrument, which often involves repeating the same movements over and over to attain muscle memory, making it so you don’t have to think.

I have found it to be more helpful to talk about this next stage of learning as opportunities to build fluency. NCTM defines procedural fluency in their position statement as $\text{\textquotedblleft}$the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another.$\text{\textquotedblright}$Note that this definition involves being able to transfer understanding to a different problem or context, which requires having flexible, conceptual understanding, not just being able to execute a set of procedures. Fluency is not about speed or even automaticity, but about flexibility.

This is where the homework comes in. Homework presents additional opportunities for students to check their understanding, but it also should challenge them to explore new ideas related to the topic. This is the chance to dig deeper into some special cases, ask a more open-ended question with multiple solutions, explore underlying structures of problems, and learn to strategically choose a problem solving method. Appropriate scaffolds are given that help students notice key features of the problem, but they should never eliminate the thinking or proceduralize something that we want students to understand conceptually.

Students need to be given these opportunities to explore and deepen their understanding without the pressure of a grade. Students should expect to see things they have not seen before instead of immediately classifying something unfamiliar as something hard, as students instinctually do.

Another common issue is assuming that once students build intuition about the *why* behind a formula during the in-class activity, they will be able to immediately use the generalized formula and make connections to the underlying ideas each time. This is not true. In our experience, students need multiple opportunities to build intuition and make sense of new ideas. In the same position statement mentioned above, the National Council for Teachers of Mathematics (NCTM) states that $\text{\textquotedblleft}$effective teaching practices provide experiences that help students to connect procedures with the underlying concepts and provide students with opportunities to rehearse or practice strategies and to justify their procedures.$\text{\textquotedblright}$
For example, perhaps the lesson is about writing equations for horizontal and vertical lines. In the QuickNotes, we generalize that horizontal lines have the form y=___ and vertical lines have the form x=____. But we can’t assume that students really understand this yet, or at least not fully.

A homework question could have students plot multiple points on a vertical line, look at their ordered pairs, discuss the change in x-values and y-values separately and then describe the similarities among the ordered pairs by saying that x is always ___, there is no change in the x-values. They could also use this problem string to explain why the slope of this line is undefined. Homework questions are not just used to ask $\text{\textquotedblleft}$do you know this?$\text{\textquotedblright}$ but also $\text{\textquotedblleft}$why does this make sense?$\text{\textquotedblright}$ Assuming that all conceptual learning takes place in class and homework is solely for building automaticity is not realistic or even best for students’ learning.

In the Math Medic curriculum, we generally give two quizzes each unit and then an end-of-unit test. Quizzes are meant to assess understanding and give feedback to both the students and the teacher about how the student is progressing. Quizzes represent the middle ground between low-stake homework assignments and a more formal end-of-unit assessment. A quiz helps focus students on the most important ideas of the lessons. Imagine a funnel. At the top of the funnel are all the ideas from a unit. At the bottom of the funnel are the most important ideas of the unit, or biggest take-aways, that appear on the test. The quiz is somewhere in the middle of the funnel. The quiz may not assess every concept or skill that was taught, but it should assess the more important ideas that will also appear on the test. When we write a quiz, we look at each lesson and ask ourselves, $\text{\textquotedblleft}$What are the 2-3 most important skills or ideas from this lesson?$\text{\textquotedblright}$Then we write quiz questions to assess those things. Due to time and space constraints, when we write a test, we try to identify the ONE most important skill or idea from each lesson. The quizzes help communicate to students what the most important ideas are and provide them with feedback on how they’re doing with them. Test questions are the most formal and generally have the least scaffolding. They should assess both conceptual understanding and procedural fluency, which we’ve already shown are both related to flexible thinking. To ease in grading, we may tweak the structure of the question so there is only one correct answer. For example, suppose we wanted to test students’ understanding of dependent, consistent, and inconsistent systems. Here’s what some questions could look like: Note how instead of asking the question in the same way each time, we instead mix up our verbs. A homework question might have students generate their own problem that meets a condition (Example 1) or look at all three cases in one problem (Example 2) so students can notice structure.

The quiz question (Example 3) gets at the same idea but has students thinking more with respect to the graph features instead of simply as algebraic scaling. Note that while there is only one correct answer for the slope, the y-intercept can be anything except -3. This is an important idea for students to graph as they connect the algebraic features of a system with the graphical features. Nevertheless, this question is still easy to grade, even though students are asked to explain.

The test question (Example 4) focuses on just one of the three types of systems, since time and space is limited. There is only one correct answer but students could use either an algebraic or graphical strategy to solve. Note also how the language is more formal and little scaffolding is given.

When comparing the kinds of questions we ask in the activity, the Check Your Understanding, homework, quizzes, and tests, it can be helpful to look at a few different characteristics. Below, you can see how these five structures compare in terms of formality, scaffolding, content specificity, and difficulty. Note that the way we are using $\text{\textquotedblleft}$difficulty$\text{\textquotedblright}$ here is related to the amount of effort required, specifically with respect to the $\text{\textquotedblleft}$niceness$\text{\textquotedblright}$ of the numbers and the algebraic skills required. This is different from $\text{\textquotedblleft}$complexity$\text{\textquotedblright}$ which refers to the higher order thinking skills that a question addresses. We are using difficulty here to mean $\text{\textquotedblleft}$working hard$\text{\textquotedblright}$ not $\text{\textquotedblleft}$thinking deeply.$\text{\textquotedblright}$

Asking good questions that support the conceptual learning students are doing in class is critical to students’ success. This is why we made the Math Medic Assessment Platform with ready-to-use homework assignments, quizzes, and tests with the ability to edit to use with your students. If you are interested, sign up to receive updates and information about how your school can use the Math Medic Assessment Platform for the '22-'23 school year.