In this chapter (13) in
Building Thinking Classrooms in Mathematics, Peter Liljedahl (Simon Fraser University) says he and his research colleagues frequently ask students this question:
So, you just finished a unit on ---. Was that unit just one big topic, or was it a collection of a bunch of smaller topics?
“I have never asked a question that is so predictive of student performance on a unit test,” says Liljedahl. Typically, about 15 percent of students answer that the unit was made up of subtopics and can name and describe those chunks; those students score above 90 percent on the upcoming test. Students who know there are subtopics but can’t fully describe them score between 75 and 90 percent on the unit test. And students who say the unit is one big topic score below 75 percent.
Why the big difference? Because students who know the subtopics of the unit have a grasp of the content similar to the teacher’s and can see specific areas where they are doing well and others where they have work to do. This is a key insight about formative assessments, says Liljedahl: “Information communicated from a teacher to a student who sees the topic as one big unit will only inform that student of what it is that they can do; but because they don’t have a clear picture of the whole unit and all its subtopics, they cannot see what is still left to learn.”
The missing piece in many classrooms, he believes, is finding a way “to help students see mathematical topics as collections of subtopics, sections, and/or special cases the way teachers do, and use this knowledge to inform themselves about what it is they can and cannot yet do.” The analogy in navigating on land and sea is knowing where you are and where you are going. For students, “where they are is what they understand, know, and/or are able to do. And where they are going, within the scope of a unit of study, is what they have not yet learned, don’t yet understand, and/or are not yet able to do.”
To accomplish this, Liljedahl says, we need to give students a navigation instrument with the subtopics of a unit, including specific examples of what they are expected to learn in each one. After a lot of trial and error, he and his colleagues came up with a grid that looked like this for a unit on fractions, with examples of fractions problems.
***
Fractions Basic Intermediate Advanced
Add and subtract 1/5 + 3/5 1/4 + 3/8 3/5 + 1/7
proper fractions
Add and subtract
mixed fractions
Multiply and divide
proper fractions
Multiply and divide
mixed fractions
Solve order of
operation tasks with
proper and mixed
fractions
Solve contextual
problems involving
fractions
Estimate solutions for
problems involving
fractions
***
Linking specific questions to the outcomes of each row “turned out to be vital,” says Liljedahl. “Although the language in the left-hand column is clear to us, students needed to see specific questions to fully understand what many of the outcomes meant.” This was especially important in the primary grades, where students’ reading proficiency was still developing, but was important right through high school.
The real power of navigation instruments comes when students have taken an end-of-unit review assessment prior to the final test. Students compare their answers to correct answers and mark each one on the navigation grid with these symbols:
- A check if it was correct;
- S if it was mostly correct but there was a silly mistake;
- H if it was answered correctly with help from the teacher or a classmate;
- G if it was answered correctly with a collaborative group;
- X if it was attempted and answered incorrectly;
- N if it was not attempted.
Having students do this after an interim assessment and then use the results to study for the final test, Liljedahl and his colleagues saw “astonishing results:” 50 to 70 percent of students saw immediate improvement of 10-15 percent; knowing where they were and where they were going was all they needed to improve. “I mean, now I know exactly what I need to work on,” said one student. “I finally get what we are doing,” said another. A third: “Are you kidding me? This is great. I know what we are doing now.” This was especially helpful for low-achieving students; they made significant progress by focusing on the basic-level questions.
Why didn’t all students improve? Some of them (about 15 percent) already knew what the subtopics were, so the navigation grid was redundant information and produced no improvement. Another subgroup really didn’t care about their learning or their grade. They already knew where they were (in the lower achievement range) and didn’t have any ambition to improve. “That is not to say they couldn’t be helped,” says Liljedahl. “Just not in this way.”
There was a third group of students who didn’t benefit from getting specific information on their practice test: students who were achieving at a B level, and thought that was good enough. “Hey, I got a B,” said one student, “without doing anything. Why would I want to put in a bunch of work to try to get an A?” Another: “A B is good enough for my mom.” A third: “I’m not one to go the extra mile.”
Isn’t it enough for teachers to give students written feedback on their quizzes and tests? For students who understand the details of curriculum units, yes, but for the 85 percent of students who don’t, says Liljedahl, this is not enough; they need to know where they are and where they are going, in detail.
Why the categories Basic, Intermediate, Advanced? Liljedahl and his colleagues started with Easy, Medium, Hard, and students found those were clearest. But teachers preferred Basic/Intermediate/Advanced, and students had no difficulty with it, so that’s what was chosen. Another option considered was Novice, Emergent, Expert, but the researchers realized that those labels describe the abilities of the students rather than the complexity of the tasks.
What about students who see the three levels and are happy to do just the Basic level? This is a problem, says Liljedahl, “but the problem is with the students, not with the navigation instrument. And for this reason, the solution lies not in the instrument, but within the students.” The teacher’s challenge is working on students’ basic motivation so they care about learning.
Does splitting up each curriculum unit into subtopics and levels of complexity keep students from seeing the bigger picture of mathematics? “This is a very good question,” says Liljedahl. “We were concerned about this as well.” But it turns out that for students to see math as a connected whole, they must first see the subcomponents. This was especially important for students who answered the initial question saying that the unit was one big topic: “They needed to see the distinctions to see the connections.”
Doesn’t stating the learning goal at the beginning of a lesson (as many teachers are required to do) take care of students understanding what they’re doing? “In theory, yes,” says Liljedahl. “In reality, however, it doesn’t.” Students need to see the detail and dive into assessing their own work and taking responsibility for improving it.
Isn’t this the same as self-assessments that students are sometimes asked to do? The difference, says Liljedahl, is that most assessments ask students for their opinion of their abilities. Here, students are looking at their actual achievement. He and his colleagues found that students took the data seriously – and most of them rolled up their sleeves and went about improving their learning.
How can teachers know if they’re doing a good job helping students know where they are and where they’re going? At the end of a unit, suggests Liljedahl, have students make a review test on which they will get 100 percent. If they can do this, they know what they know. Then ask them to make a review test on which they will get only 50 percent. If they can do that, they know what they know and what they don’t know.
“How We Use Formative Assessment in a Thinking Classroom” – Chapter 13 of
Building Thinking Classrooms in Mathematics by Peter Liljedahl (Corwin, 2021); Liljedahl can be reached at
liljedahl@sfu.ca; see Memo 992 for a summary of chapters 1-3 of the book, Memo 1013 for a summary of chapter 5.
Please Note: This summary is reprinted with permission from issue #1070 of The Marshall Memo, an excellent resource for educators.
