Peter Liljedahl on Giving Students a "Navigation Instrument"

            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.

David Brooks on Young People Becoming the Best Versions of Themselves

            In this New York Times column, David Brooks says he believes that for individuals, character is destiny, and for a healthy society, moral formation is essential. At a recent meeting hosted by the Making Caring Common project at Harvard, Brooks took note of some key ideas for teachers, parents, and “anyone who wants to build a society in which it is easier to be good”: 

• A communitarian ethos – A common belief today, says Richard Weissbourd, faculty director of Making Caring Common, is that young people’s ultimate goal is individual achievement and happiness, versus the common good and caring for others. “Schools that focus on moral education,” says Brooks, “stand athwart that tide. They have a sense of moral mission, that who you become is more important than what career track you pursue… They have rituals to mark transitions. They have retreats and group travel so that people can see one another before the makeup goes on.”
• Moral skill-building – “Treating people well involves practicing certain skills, which can be taught just as the skills of carpentry and tennis can be taught,” says Brooks. They include:
• The skills of understanding – listening well, eliciting people’s life stories so we accurately see them and they feel seen; 
• The skills of consideration and treating people well – offering criticism with care, asking for and offering forgiveness, breaking up with someone without crushing their hearts. 

Brooks fears that many young people aren’t learning these skills. 

• Exemplars – “Admiration is one of the most powerful moral emotions,” he says. Nelson Mandela revered Mahatma Gandhi; Susan B. Anthony and Elizabeth Cady Standon admired each other. Kids need to study examples of true greatness. In the words of Warren Buffett, “Tell me who your heroes are, and I’ll tell you how you’re going to turn out.” 
• Moral traditions – “We are the lucky inheritors of many rich and varied moral traditions,” says Brooks. “Schools can teach these traditions and students can decide which seem true to them. People become their best selves as they begin to embody the values of a specific moral tradition.”
• Self-confrontation – Everyone has core faults they must confront and conquer, says Brooks. Dwight Eisenhower had a terrible temper; some people are egotistical, judgmental, or people pleasers. Parents and schools can help young people to acknowledge and try to fix their shortcomings. 
• Public service – “Community service, whether it’s feeding the poor, sitting with the homeless, or championing a cause, is not just to make society better,” says Brooks; “it is done to usher a transformation in the person doing the service.” This kind of service fosters emotional understanding – “the ability to be made indignant by injustice, outraged by cruelty, to know how to gracefully do things with people, not for people. That kind of knowledge comes through direct contact with the problems.” 

 “The Character-Building Tool Kit” by David Brooks in The New York Times, January 10, 2025; Brooks can be reached at dabrooks@nytimes.com.

Please Note: This summary is reprinted with permission from issue #1070 of The Marshall Memo, an excellent resource for educators.

Recommended Comic Graphic Novels for the Elementary Grades

            In this School Library Journal feature, Brigid Alverson suggests these children’s books with witty dialogue and silly plots: 

• Detective Beans and the Case of the Missing Hat by Li Chen, grade 1-4
• Rocket and Groot: Tales of Terror by Amanda Deibert, illustrated by Leo Trinidad, grade 2
• Schnozzer and Tatertoes: Take a Hike! and Schnozzer and Tatertoes: Shoot the Moon! by Rick Stromoski, grade 2-5
• Sophie: Jurassic Bark and Sophie: Frankenstein’s Hound by Brian Anderson, grade 3-6
• Barkham Asylum by Yehudi Mercado, grade 3-7 
• Bunny vs. Monkey by Jamie Smart, grade 3-7 
• The Unpetables and The Unpetables Book Two: Unpetable in the City by Dennis Messner, grade 4-7
• It’s Jeff! by Kelly Thompson, illustrated by Gurihiru, grade 4-7 

 “Critter Comfort” by Brigid Alverson in School Library Journal, January 2025 (Vol. 71, #1, pp. 32-35)

Please Note: This summary is reprinted with permission from issue #1069 of The Marshall Memo, an excellent resource for educators.