Keys to Effective Teaching of Algebra

             In this article in Mathematics Teacher, Jill Newton (Purdue University) says that when she returned to a classroom during a sabbatical, she encountered many of the same challenges she’d experienced in her years as a high-school math teacher – plus some new ones: enforcing the school’s cellphone policy, a high rate of student absenteeism, and the pressure of standardized testing.

            As she re-immersed herself in teaching algebra, Newton compiled a list of what she wished she’d known as a new teacher. “I loved my students,” she says, “supported them, and did what I was taught to do, but as they say, When you know better, you can do better.” Here are her suggestions to her younger self – and other teachers of algebra: 

• Help students see the bigger picture of algebra – conceptual understanding and connections across algebraic procedures and topics. This is more helpful, says Newton, than breaking down algebraic procedures to the smallest steps.
• Discuss the uses of variables with students – for example, unknown, constant, argument, parameter, label, and varying quantity. Discussing conceptions of variables, Newton has found, really enhances students’ understanding of algebra.
• Give students opportunities to investigate strategies and make decisions about selecting methods for solving. “Learning when and why to use an algebraic procedure is just as important as learning how to use it,” she says.
• Facilitate students’ thinking about taking algebraic objects apart, putting them back together, and transforming them. Thinking about “doing” and “undoing” in algebra helps develop students’ procedural fluency and conceptual understanding.
• Begin with the idea of function families, and work with students to summarize their properties and compare and contrast different function families. This helps students make sense of algebra.
• Introduce the many ways to represent algebraic ideas, make connections among them, and make decisions about their uses. Focusing on multiple representations and building representational fluency builds students’ understanding of algebra.
• Engage students in exploring the power of algebra to investigate real-world issues and develop possible solutions.
• Model correct algebraic language, connecting ideas through words, and avoid imprecise terms. This helps students participate in mathematical discourse.
• Discuss the history of algebraic ideas, stories about important mathematicians, and the contributions of female and non-European practitioners. This sparks students’ interest in and understanding of algebra.
• Be frank about algebra’s historical role stratifying students in schools and its gatekeeper role for post-secondary opportunities. “I wish I had viewed algebra teaching and learning through a more-critical lens,” says Newton, “and asked more questions about who benefited and who was disadvantaged by school policies related to algebra.” 

“10 Things I Wish I Had Known About Teaching Algebra” by Jill Newton in Mathematics Teacher, September 2026 (Vol. 119, #9, pp. 742-747); Newton is at janewton@purdue.edu.

Please Note: This summary is reprinted with permission from issue #1155 of The Marshall Memo, an excellent resource for educators.

Co-Constructing "Thinking Notes" in Building Thinking Classrooms

            In this article in Mathematics Teacher, Texas math educators Priya Pierre-Louis, Laura Godard, and Margaret Pearce describe Thinking Notes as an adaptation to Peter Liljedahl’s Building Thinking Classrooms process. Here’s what led to the idea.

            In the middle of a BTC class, with students working standing up in groups of three solving a challenging calculus problem at erasable whiteboards, the teacher noticed an error being made by all the groups. Getting the attention of the class, the teacher provided an explanation that got students back on track.

            But the teacher (and colleagues in similar situations) noticed that some students “felt uneasy moving on without a detailed, teacher-guided, written synthesis,” say Pierre-Louis, Godard, and Pearce. “They valued collaborative discovery but wanted reassurance from the teacher about the precision of their understanding.”

            That’s what sparked the idea of Thinking Notes. “Instead of providing prewritten notes,” say the authors, “we co-create summaries with the class after problem-solving sessions. These notes are not ‘examples and answers’ but rather artifacts of our collective reasoning” – including graphs and diagrams, insights, generalizations, and examples and non-examples that surfaced during the all-class discussion.

            The Thinking Notes are captured in the class’s digital notebook, say the authors, forming “a bridge between the spontaneity of thinking tasks and the students’ desire for precise records for future reference.” At each of these “pause points” in a lesson, “students independently decide whether to record notes in their notebooks, often adding to the student-created notes they have already written during group work.”

            The Thinking Notes provide “a shared, vetted synthesis that students can draw on as their understanding continues to develop,” say Pierre-Louis, Godard, and Pearce. “While still grounded in student language and reasoning, Thinking Notes benefit from teacher facilitation and tend to be more conceptually coherent, mathematically precise, and reusable as reference for future tasks, assessments, and revision.”

            The authors report that this adaptation of Building Thinking Classrooms has resulted in students being more confident with subsequent tasks: “For some students, the obstacle was not a lack of ideas but uncertainty about whether their thinking was ‘correct’ enough to share publicly. The Thinking Notes phase reduced this risk by positioning ideas as provisional and open to refinement. As a result, several students who were hesitant to speak during group work began volunteering contributions during notes creation, knowing their ideas would be examined and revised collectively, rather than evaluated individually.” Quieter students contributed more at the whiteboards and in all-class discussions.

            Thinking Notes, conclude Pierre-Louis, Godard, and Pearce, “have had a profound impact on our classroom culture. Students report feeling empowered because they are the authors of the mathematical ideas recorded: their language, reasoning, and representations shape the notes. At the same time, students experience a sense of security through teacher facilitation as their collective thinking is affirmed, refined, and validated for mathematical accuracy. This combination allows students to take intellectual risks during exploration while trusting that misconceptions will be surfaced and addressed during synthesis. This process has helped us, as teachers, slow down and honor moments of deep reasoning before moving to the next task.” 

“Thinking Notes: Co-Constructed Reflection” by Priya Pierre-Louis, Laura Godard, and Margaret Pearce in Mathematics Teacher, September 2026 (Vol. 119, #9, pp. 748-751); Pierre-Louis can be reached at pierrelouis@gmail.com.

Please Note: This summary is reprinted with permission from issue #1155 of The Marshall Memo, an excellent resource for educators.

What Do Students Want in History and Civics Classroom Discussions?

            Running a good classroom discussion is “challenging work,” say Andrew Schiera (University of Colorado/Boulder) and Andrew del Calvo (Rutgers University) in this article in Social Education, especially on controversial topics. They share suggestions from a diverse group of high-school students on how social studies teachers can make discussions lively, engaging, and safe: 

• A big question – The focus of discussion needs to be open-ended, students said, not pointed toward a single correct answer or a binary agree/disagree choice. Students liked being asked to formulate an initial opinion and then modify it in light of evidence and contrary opinions. In short, say Schiera and del Calvo, students appreciated “an authentic, shared problem space so that they could inquire deeply, individually, and collectively.”
• Facilitation – Students wanted teachers to provoke discussion with questions and follow-up comments that sparked deeper thinking. They also appreciated teachers clarifying when things got confusing and refocusing when students were straying from the topic or “going back and forth on the same stuff.”
• Multiple perspectives – Students said they wanted to hear all sides of historical and civics debates, perhaps engaging in small-group discussions to elicit and understand different points of view. But students cautioned against “both-sides-ism” and hoped that clear moral choices could emerge from discussions.
• Participation – Students had seen too many discussions involving “mostly just like three people,” and asked teachers to ensure broad and equitable participation, including: having students sit in a circle; establishing and monitoring air-time norms; inviting new voices; cold-calling; using thumb polls; and assigning participation grades. At the same time, students recognized that there were times when an intense dialogue should be allowed to continue, not cutting it off and letting others take the discussion off track. 
• The teacher’s role – Students suggested that teachers go beyond mere facilitation, encouraging them to chime in as co-inquirers – without dominating the discussion or conveying that their opinion was correct. Students were aware that a teacher being too opinionated or seeming to push a political agenda could spark parental outrage. 
• Psychological safety – Students mentioned several ways to ensure that discussions felt emotionally comfortable while being intellectually open: setting norms about respectful engagement; thoughtfully introducing difficult topics; reassuring students about fear of saying something “wrong;” allowing students time to think; not trying to orchestrate a single conclusion; not interrupting and shutting students down; and using small-group turn-and-talk when appropriate.
• Connecting past, present, and lived experience – Students appreciated delving into primary-source material but wanted historical events to be connected to the present day and their own lived experience. “Overall,” say Schiera and del Calvo, “students wanted discussions that connected historical inquiry to contemporary life while providing enough structure to help them navigate those connections in meaningful ways.” 

“‘Setting the Conversation on Fire’: What Youth Want from Social Studies Conversations” by Andrew Schiera and Andrew del Calvo in Social Education, September 2026 (Vol. 90, #4, pp. 216-220); the authors are at AJ.Schiera@colorado.edu and andrew.delcalvo@gse.rutgers.edu.

Please Note: This summary is reprinted with permission from issue #1151 of The Marshall Memo, an excellent resource for educators.