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.

A Low-Floor/High-Ceiling Math Problem for Multiple Grade Levels

            In this Mathematics Teacher article, Stephen Caviness (University of Virginia) shares what he calls a quintessential low-floor/high-ceiling math task – one that is accessible to every student and has multiple, creative solutions to explore. “Such tasks,” he says, “support students in embracing uncertainty, engaging in productive struggle, and expressing creativity in doing mathematics.”

            He’s had success using this task – Close to 1,000 – with students from second grade to college level: 

 Using the digits 1 to 9 exactly once each, place a digit in each box to make the sum as close to 1,000 as possible. 

Caviness encourages students to start filling the boxes without a strategy to see what happens – perhaps 123 + 456 + 789. Common mistakes are using a digit more than once and using zero, which is not an option. “From here,” he says, “students will have to think about place value as they make small iterations, narrowing in on sums closer and closer to 1,000. Ultimately, the closest that one can get to 1,000 is 999, but there are many ways to get there.” 

            As students find correct answers, Caviness suggests a number of ways the task can be extended to get students thinking about the math in deeper ways:

• Why do we regroup? What does it mean to “carry the one”?
• Which sum is furthest from 1,000? What would be the “wrongest” answer?
• What if you were allowed to use a zero?
• Are there any values as close to 1,000 as 999?
• Why can’t we add to 1,001?
• What patterns do you notice about our closest solutions?
• How many ways are there to add to 999? (There are 180.) 

“Close to 1,000” by Stephen Caviness in Mathematics Teacher, July 2026 (Vol. 119, #7, pp. 520-523); Caviness can be reached at slc9th@virginia.edu. For an extensive collection of non-routine math problems, see Stella’s Stunners, compiled and organized by Ohio teacher Rudd Crawford.

Please Note: This summary is reprinted with permission from issue #1148 of The Marshall Memo, an excellent resource for educators.

How a Children's Book Can Build Empathy and Understanding

            In this Council Chronicle article, teacher/author Priten Soundar-Shah says she reads the same children’s picturebook, Fish Is Fish by Leo Lionni, to every class she teaches – middle school, summer camp, college, adult learners. She reads the book with sing-song expression, holding up the illustrations page by page. “Some people look at me like I’ve completely lost my mind,” she says. “But some are intrigued. They want to know where I’m going with this.” 

            Here’s the book’s story. A minnow and tadpole become best friends. The tadpole grows into a frog and leaves the pond to explore the world. When he returns, he tells the minnow about the amazing things he’s seen – birds, cows, people. But his friend can only imagine these unknown things as fish-shaped versions. Birds are fish with wings, cows are fish with udders, people are fish walking upright. “The fish’s imagination is constrained entirely by what it already knows,” says Soundar-Shah. “It imagines these things, but it’s limited by its own experiences. So, it imagines things that are slightly incorrect.”

            The frog’s perception is also distorted. “He’s excited,” says Soundar-Shah. “He genuinely wants to communicate. And yet, because he doesn’t know what he doesn’t know, because he has his own lived experiences that he’s drawing from, he can’t even conceptualize what might not make sense to the fish. He doesn’t know what the gaps are.”

            This becomes a theme as Soundar-Shah and her students explore new material. “We’re going to read something,” she says. “Our intuition might be to immediately disagree. But remember, we might be fish in this scenario. We really want to listen carefully to make sure we’re understanding the whole context.” When students are inclined to think an idea is completely ridiculous, she asks, “Are we being the frog? Are we being the fish? Have we conveyed things as accurately and clearly as possible to someone who doesn’t have our lived experiences?”

            Fish Is Fish becomes a way to approach difficult conversations. “It’s a reminder that understanding across difference requires both humility about what we might be missing and generosity toward what others are trying to communicate,” says Soundar-Shah. “In a moment when so many of us are struggling with how to hold space for genuine dialogue in our classrooms, a five-minute children’s book might be exactly the kind of disarming, humble starting point we need.” 

 “Why I Teach the Same Picturebook to Every Class I Teach” by Priten Soundar-Shah in Council Chronicle, Spring 2026 (Vol. 35, #3, pp. 27-28)

Please Note: This summary is reprinted with permission from issue #1148 of The Marshall Memo, an excellent resource for educators.

Helping Insructional Coaches Do Their Best Work

            In this article in The Learning Professional, Kimberly Richardson (Hampton City Schools, Virginia) says that instructional coaches “are often expected to navigate complex leadership responsibilities with limited preparation, unclear expectations, and inconsistent support.” To address these common issues, Richardson suggests that districts:

• Clarify the purpose of coaching. For coaches to maximize the time they spend working directly with teachers on systemwide priorities, it needs to be clear that they should not be doing substitute coverage, coordinating testing, monitoring duties, providing logistical support, or playing an indirect role in teacher evaluation. “Teachers are more likely to engage meaningfully with coaching,” says Richardon, “when leaders consistently communicate that coaches are support partners rather than evaluators or managers.”

• Develop human-centered leadership skills. Just because coaches were outstanding teachers and skillfully implemented their curriculum doesn’t mean they know how to support adult learning, navigate resistance, manage difficult conversations, or deal with colleagues’ stress and burnout. There needs to be intentional training and support around building trust with teachers, communicating clearly, navigating conflict, and creating psychological safety so teachers will address shortcomings and grow professionally. New coaches should get thoughtful instruction in the role, observe effective coaching, and work with a mentor.

• Orchestrate the conditions for effective coaching. Districts need to carve out time for coaches to reflect, plan, collaborate, and problem-solve with each other, while protecting them from being asked to do work that is extraneous to their mission. “Supporting coaches is leadership work,” Richardson concludes. “If schools expect coaches to cultivate growth in others, leaders must create systems that cultivate growth in coaches as well.” 

 “Setting Coaches Up for Success: How Leaders Can Help Coaches Thrive” by Kimberly Richardson in The Learning Professional, June 2026 (Vol. 47, #3, pp. 28-31)

Please Note: This summary is reprinted with permission from issue #1146 of The Marshall Memo, an excellent resource for educators.

Can Real-Time Teacher Coaching Be Effective?

            In this article in The Learning Professional, consultant/author Jacobé Bell says that for instructional coaches to have a positive impact on teaching and learning, they need to:

• Build a trusting relationship with each teacher;
• Have a shared commitment to student success;
• Create low-stakes spaces to practice;
• Provide timely support when teachers are trying something new. 

Bell believes that under the right conditions, coaching can sometimes happen while teachers are working with students. “The coach’s intent is not to take over, but to make teaching practices visible and more transferable during live instruction,” she says. “When done well, active coaching creates a scaffold that supports teachers as they try on new practices, building their confidence and competence until they can execute those moves independently.” She describes two approaches:

• Preplanned in-the-moment coaching – In a planning meeting, the coach and teacher identify possible challenges in a lesson and plan how they’ll collaborate – perhaps tag-team teaching, a short demonstration or segment taught by the coach, prompts at a particular point in the lesson, the coach monitoring 4-5 students’ work and giving the teacher real-time feedback.

• Spontaneous in-the-moment coaching – These coaching interventions are responsive, based on what the coach sees during the lesson, says Bell, and “are often subtle, quick, and highly situational” – for example, the coach makes a suggestion (“This might be a good time for students to turn and talk”), uses an agreed-upon signal (a circling gesture to suggest that the teacher should circulate and look at students’ work), modeling an instructional move (when a student gives a vague answer, saying, “What in the text makes you say that?”), and redirecting students (“I see that Table 3 is using whisper voices”). 

            Moments like these can make teachers feel like they have a supportive thought partner as they build their teaching repertoire in real time, says Bell. “Jumping in is most effective when the coach asks the teacher ahead of time how they prefer the coach to do so. Without this clarity, in-the-moment coaching can feel intrusive or undermining, and not all teachers are comfortable with support that is clearly visible to students.”

            In that vein, Bell suggests that coaches ask themselves three questions to decide when and how to get involved during a class:

• Is the need urgent? “If yes,” she says, “I use whisper, signal, or reset strategies. If no, I let it play out.”
• Is the skill easily teachable? “If yes, I model the skill quickly or script the next line for the teacher to say. If no, I let it play out.”
• Will it stall the lesson? Should it be saved for the after-lesson debrief? “If yes, I take notes and let it play out. If no, I whisper, signal, or use strategies.” 

[See my 2015 Kappan article, “Should Supervisors Intervene During Classroom Visits?” summarized in Memo 606. K.M.] 

 “Coaching in the Moment: Real-Time Practices That Accelerate Learning” by Jacobé Bell in The Learning Professional, June 2026 (Vol. 47, #3, pp. 18-23)

Please Note: This summary is reprinted with permission from issue #1146 of The Marshall Memo, an excellent resource for educators.