Using Student Interviews to Probe Kids' Mathematical Thinking

            In this article in Mathematics Teacher, Nicora Placa (Hunter College) remembers, as a new teacher, looking over a student’s shoulder and spotting this problem: 1/3 + 1/4 = 2/7 Here was the ensuing conversation: 

    - Teacher: I see you added the numerators and denominators. Are those the same size pieces?   

    - Student: Yes? 

    - Teacher: Are you sure those pieces are the same size? 

    - Student: Um. No? 

    - Teacher: Good! So, what do we do if they aren’t the same size? 

    - Student: Um… 

    - Teacher: I’ll give you a hint. We worked on it yesterday. We need to find… 

    - Student: The same size? 

    - Teacher: Yes. We need to find common… 

    - Student: Ummmmmm… denominators? 

    - Teacher: Yes. Very good! We need to find common denominators. Why don’t you review the notes from yesterday on how to find the common denominator and then redo this? 

“I did this with the best of intentions,” says Placa, “thinking I was helping the student. I truly believed that with some prompting or a hint, they would remember what to do. I did not understand that I was dragging them through a solution path that made sense to me instead of trying to understand how they were thinking about the task… I was not hearing all the interesting ways students thought about the problem, and I misunderstood their ideas.” 

            A little later in her career, Placa learned the value of student interviews and began to approach conversations with students in a different way – even when their answers were correct. “When conducting these interviews, I began listening to students with the goal of making sense of what they were doing,” she says. “It was eye-opening. I was able to learn about the different strategies that students brought to the problem and, in turn, change my responses in the classroom… I became fascinated by all the ways students thought about problems and started to build on these conceptions to design instruction.” 

            Using this approach, here’s how Placa would handle the conversation with a student who had written this incorrect solution: 1/3 + 1/4 = 2/7 

    - Teacher: Can you tell me how you got your answer? 

    - Student: Is it wrong? 

    - Teacher: I don’t know. Why don’t you explain it to me, and we will try to figure it out together? 

    - Student: Well, here you have one of three things, and here you are adding one out of four things, so basically you now have two of seven things. 

     - Teacher: Interesting. Can you try using these manipulatives or a drawing to show me another way to solve it? 

“With this change in questioning,” says Placa, “I could see how this solution made sense to students if they thought of a fraction as two distinct whole numbers and not as a quantity itself. Listening to students’ thinking made me rethink the ‘out of’ fractions language I was using when introducing fractions and whether I was sufficiently allowing students to explore a variety of models. I revisited activities that explored the concept of fractions as a quantity before I tried to address the addition of fractions.”

            When she became a math coach, Placa became an advocate of student interviews, and Let’s ask a kid! became her mantra. She encourages teachers to anticipate different ways students might solve challenging problems, interview individual students outside of regular class time, and choose effective questions to probe kids about their solutions. She counsels teachers to avoid responses like: 

    - That’s right!

    - You know that if you just…

    - Remember what we did in class last week…

    - And --- is just another way to say ---.

    - Do you mean…? 

Instead, she helps teachers use interview questions like these:

• General probing questions: 
• What did you notice? 
• Why did you write (or draw) that?
• You wrote ---. Why? How did that help you?
• I noticed that you stopped what you were doing just now (or erased or crossed something out). What were you thinking?
• I don’t know what you mean by that. Can you explain? 
• Questions about alternative solution paths:
• Can you solve it in a different way? Tell me about it.
• Can you use a picture (or tool) to represent your thinking? Show me.
• Another student said the solution was ---. Do you agree or disagree? Why?
• Questions about explaining and justifying solutions:
• How do you know?
• How did you figure that out?
• How sure are you of your answer? Why?
• Can you justify your work with these manipulatives?
• Is there another way to justify your work? What is it? 

“When teachers carefully listen to and make sense of students’ thinking,” says Placa, “they can design instruction that is tailored to students’ current understanding. Through student interviews, coaches can help teachers to develop these skills in one-on-one situations and then transfer them to their work in the classroom.” Some good questions for teacher team meetings:

- What did you notice about the student’s thinking?

- In what ways do the student’s explanations make sense?

- What different conceptions do you notice the student has?

- What types of questions help uncover the student’s thinking?

- What questions are less helpful?

- What instructional moves might be helpful if we notice these ideas in our classrooms?

- What implications does this have as we plan instruction going forward? 

 “Let’s Ask a Kid! Conducting Student Interviews” by Nicora Placa in Mathematics Teacher: Learning & Teaching PK-12, December 2024 (Vol. 117, #12, pp. 900-906); Placa can be reached at np798@hunter.cuny.edu.

Please Note: This summary is reprinted with permission from issue #1067 of The Marshall Memo, an excellent resource for educators.

Centering Hope in the Social Studies Curriculum

            In this article in Social Education, Li-Ching Ho (University of Wisconsin/Madison) and Keith Barton (Indiana University) worry that the world’s problems – political division, mistrust of leaders, racism, poverty, hunger, disease, war, refugees, climate change – may lead K-12 students to be pessimistic, even despairing, about the future. “To counter this,” say Ho and Barton, social studies educators “must provide students a sense of hope – a belief that a better world is possible, and that human action makes a difference.” 

            Implicit in history, geography, and civics classes, they believe, is a focus on hope: “Embedded within the curriculum is an assumption that by providing young people with the necessary skills, knowledge, and dispositions, they will be positioned to help their communities address societal issues and imagine a different and better future.” But Ho and Barton wonder if this emphasis has been explicit enough: “Even when dealing with potentially hopeful content – such as successful social movements – we may fail to highlight its relevance for today.” They believe social studies should embed hopeful content about possibilities, goals, and pathways at a pragmatic and visionary level.

• Pragmatic hope – This curriculum strand would embody the belief that a better future can be attained through strategies that are currently available, with a focus on making a positive difference in people’s lives today. Students might study:
• Possibilities – Successful social movements such as women’s suffrage, civil rights, farmworkers, industrial safety, LGBTQ rights; 
• Goals – Ways to target specific areas of progress in a reasonable timeframe – for example, reducing childhood hunger; 
• Pathways – Understanding strategies that bring about social change – for example, a coalition that made helped restore the Louisiana coastline.
• Visionary hope – “At the core of visionary hope,” say Ho and Barton, “is a belief that the world can be very different than it is.” Looking far beyond present-day realities, students might engage in big-picture thinking about an ideal future, moving beyond conditions that are taken as givens today:
• Possibilities – Students might study how the Harlem Renaissance affected art and culture in the U.S. 
• Goals – Students might consider grand ideals that people have held throughout history – for example, less-exploitative economic arrangements, more-equitable gender relations, greater harmony between people and nature.
• Pathways – “Utopian has gotten a bad reputation as synonymous with ‘impossible,’” say the authors. “Visionary hope must engage students in thinking about how to achieve a different society. Helping students think through how to get from here to there is a corrective to feelings of inevitability.” Women’s suffrage is a good example. 

“Without a deeper and more-complete understanding of visionary goals,” say Ho and Barton, “students may fail to see what it would mean to apply them. And without exploring the rationales behind such goals, students may abandon their beliefs in the face of public opinion or self-interest.” 

            The authors add three cautionary notes. First, they say, “Centering hope does not mean simplifying or romanticizing social change, whether past or present. Social movements are complicated, and their intricacies cannot be ignored in a misguided attempt to make them more inspirational.” The U.S. civil rights movement wasn’t linear or straightforward, nor has it been completely successful. A hope-oriented curriculum needs to delve into such complexities lest we leave students with “false hope.”

            Second, teachers need to make good decisions on the case studies they use to teach about hope – and in the current political climate, some choices will be controversial. “Teaching for hope does not make potential controversies go away,” say Ho and Barton; “dedicated teachers still have to be ready to defend their choices.”

            Finally, the authors believe hope must be central to the curriculum, not an occasional add-on. “Occasional hopeful examples will have limited impact on students’ imagination,” they say. “Centering hope requires consistently and systematically studying hopeful prospects for addressing many different social issues, at a variety of scales and in different settings.” 

            “Despair is central to the thoughts and feelings of many people these days, whether young people or adults,” conclude Ho and Barton. “Schools must counter this sense of despair – not romantically, not simplistically, but forcefully. This means focusing our efforts to provide a foundation for both pragmatic and visionary hope: realistic and successful struggles to improve the world, and the idealistic visions that guide and motivate effective action.” 

 “Centering Hope in Social Studies Education” by Li-Ching Ho and Keith Barton in Social Education, November/December 2024 (Vol. 88, #6, pp. 334-340); the authors can be reached at liching.ho@wisc.edu and kcbarton@iu.edu.

Please Note: This summary is reprinted with permission from issue #1067 of The Marshall Memo, an excellent resource for educators.

A High School Student Schools His Father

            In this Kappan article, teacher/instructional coach/writer Steven Goldman says that for four years, he commuted to his school in Cambridge, Massachusetts with his teenage son Theo, who attended the school. Since the drive took 60-90 minutes, they had plenty of time to talk (and listen to his son’s music selections). Goldman says Theo’s observations about the school have been the best professional development of his career. A few examples: 

• Teachers slow to return students’ work – Theo says teachers shouldn’t ask students to take on new tasks until they’ve given feedback on the previous assignment. 
• Student bravado – Theo told his history teacher that he’d blown off studying for a test when in fact he had put in the time. The alternative to lying, he told his father, was “to admit that I’m [expletive] stupid.” 
• Lesson plan “menus” with fun “dessert” options – “All this means,” said Theo, is that I will never get to do anything interesting because I work slowly.”
• The “dyslexia tax” – Theo has a mild learning disability and some executive function challenges, and says he is often unfairly marked down on tests. “School is a minefield for kids who do not fit our stereotypes of the ‘good student,’” says Goldman. 
• Seen as lazy – “I can’t begin to count the number of times that teachers have assumed that he wasn’t trying or didn’t really care based on small mistakes that are a real challenge for him to avoid,” says Goldman. “Believing that you know a student well enough to judge them for inadequate ‘effort’ is arrogance. Unfortunately, it is something I know I did often as a teacher. Theo helped me see that.” 
• Unhelpful teacher judgments – One wrote “Good grief” in the margin for a spelling error, another took points off for an assignment left at home. “Feedback should be about how someone can improve,” says Goldman, “not about making them feel like they aren’t measuring up.”
• Student support – Four years ago when he moved from middle school to high school, Theo commented on a difference he noticed in the educators: “The real difference is that some teachers are on your side and some really aren’t.” 

            “I don’t know of a better definition of what makes someone a good teacher,” says Goldman. “If someone is on your side, both of you are working toward the same goal. Being on a student’s side is not simply giving help or being sensitive to who they are. It is less about whether they measure up to your standards and more about conveying your belief in their capability of achieving their own. It is a mindset that allows the vulnerability necessary for learning to happen.” 

            Theo went off to college this year and Goldman is making the commute alone. “I feel pretty certain that he will never choose education as a career,” he says, “which is fine. But he’s been a teacher, nonetheless. We all learn so much from our children. I feel lucky that one of the things I learned from mine was how to be a better teacher.” 

“The Person Who Taught Me the Most About Teaching Just Graduated from High School” by Steven Goldman in Kappan, December 2024/January 2025 (Vol. 106, #4, p. 48); Goldman can be reached at arthurstevengoldman@gmail.com.

Please Note: This summary is reprinted with permission from issue #1067 of The Marshall Memo, an excellent resource for educators.

Peter Liljedahl on "Thinking Classrooms"

            In this Kappan interview with Kathleen Vail, Peter Liljedahl (Simon Fraser University, Canada) discusses the details of Building Thinking Classrooms. Some key points:

• The difficulty of getting kids thinking – Twenty years ago, at the beginning of Liljedahl’s research, a middle-school math teacher asked him for help teaching complex problem-solving (she was getting students ready for impending changes in Canada’s curriculum expectations). Working together for a week, says Liljedahl, “it was disaster after disaster after disaster.” Why? Because even though students seemed to be working hard and the teacher was running the class well, the standard I do, we do, you do lesson structure resulted in most students mimicking the teacher, stalling, faking, or spinning their wheels. Very few of them were actually thinking, so it was difficult for them to solve challenging problems.
• Problems with standard math lessons – Observing math classes around the world, Liljedahl noticed that almost everywhere, lessons began with a teacher explanation, then students tried a problem, the teacher clarified, then there was independent work and an assessment. Another universal feature, even in classes trying various innovations: the teacher was standing and writing on a vertical surface, students were sitting down and writing on horizontal surfaces. Everywhere he went, Liljedahl noticed that only about 20 percent of students were thinking (about 20 percent of the time) and there was generally a low level of intellectual engagement.
• Restructuring classroom norms – He realized that to change this dynamic – which has been a feature of math classes for almost two centuries – the usual lesson pattern had to be shaken up. He proposed a radically different structure within the four walls of classrooms and the standard bell schedule:
• Giving students an engaging thinking task they couldn’t solve by mimicking or faking;
• Getting students started with only a brief launching introduction from the teacher;
• Having students work in randomly selected groups of three (two in the primary grades);
• Having groups work standing up, using one marker to write on erasable whiteboards;
• The teacher circulating and giving hints, support, and additional challenges as needed. 

“This is a massive revision to the way a classroom normally functions,” says Liljedahl. “In the institutional, normative structure of school, the teacher says, ‘Let me show you how to do it. Now you do it.’ That sets up mimicking. In a thinking classroom, the teacher says, ‘I’m going to give you a task. You’re going to have to think about it, so I’m not going to do it first.’” 

• Why visibly randomized groups – Trying out various strategies, Liljedahl and his colleagues found that even if students were told the groups were chosen at random, they assumed the teacher had formed each group with a “smart” student who was expected to do the real work. In this case, or when students were allowed to choose their own groupmates, most were unlikely to offer an idea because they thought their role was to follow. But when students saw that groups were truly random (using playing cards or another visibly random process), the researchers found that within three weeks, 100 percent of students were likely or highly likely to offer an idea. Truly random grouping told students that the teacher thought they were capable of making a real contribution and they did.
• Why erasable surfaces – With whiteboards (or glass surfaces), students were more likely to get started with ideas knowing they could erase mistakes. With flip chart paper, on the other hand, students thought, “We don’t know what the answer is yet, so we can’t put anything down.” Without knowing that their work was perfect, many students didn’t write anything, and there was a negative feedback loop on problem-solving and creativity.
• Why vertical surfaces – Liljedahl and his colleagues tried a number of configurations and found that having students work standing up was by far the most effective. When students were sitting down facing each other, even if they were using small erasable whiteboards, the work was oriented toward the student who was writing, making that person the leader. With a vertical surface, everyone in the group was facing the work, creating a more collaborative dynamic. Each group’s work was also visible to students around the classroom, enhancing “knowledge mobility” – students could see common errors and good ideas spread from group to group. And the teacher was more aware of what was going on and could push in with just-in-time interventions, versus waiting till the quiz on Friday to see who understood.
• Students feeling visible – “It turns out that when students are sitting,” says Liljedahl, “they feel anonymous. The further they sit from the teacher, the more anonymous they feel. When students feel anonymous, they are more likely to disengage. And the more anonymous they feel, the more likely they are to disengage. Standing up took away that anonymity, and not in a way that made students feel exposed. I’m not anonymous, and I’m not invisible. If I’m not invisible, I’m less likely to disengage. This was a huge shift.” 
• The teacher’s role – The first order of business is choosing tasks wisely, then launching the lesson in a way that reminds students of key skills but leaves the real work to each group, then cruising around the room intervening strategically. “This group needs a hint,” says Liljedahl. “This group needs to talk to another group. This group needs a little bit of direct instruction on this, and this group needs an extension.” Then in the last one-third of the lesson, the class sits down to consolidate learning, students take notes “for my future forgetful self,” and everyone does a quick formative assessment that tells the teacher the level of mastery. 
• Curriculum coverage and innovation – “We cover tremendous amounts of content,” says Liljedahl. “Because the kids are thinking, so much learning can happen. When they’re not thinking, everything is difficult. It takes a long time and they don’t retain it… Imagine how hard teaching and learning is in a space like that, versus where 93% of students are thinking for 100% of the time. Just think about how much more learning can happen in those spaces.” He also believes that other pedagogical innovations are much more likely to get traction in Thinking Classrooms because the radical changes in classroom structure get students collaborating, engaging with the content, and thinking. 

“Building Thinking Classrooms in Mathematics: A Conversation with Peter Liljedahl” by Kathleen Vail in Kappan, December 2024/January 2025 (Vol. 106, #4, pp. 32-35); for other articles on Thinking Classrooms, see Memos 976, 992, 1013, and 1052. Liljedahl can be reached at peter@buildingthinkingclassrooms.com.

Please Note: This summary is reprinted with permission from issue #1066 of The Marshall Memo, an excellent resource for educators.

Books About Social Activism for Young Readers

            In this article in Social Studies and the Young Learner, Iowa preschool teacher Taylor Marsho and five colleagues recommend books that several teachers used to explore issues of injustice, affirm students’ identities, and link activism to students’ artwork. In the course of this curriculum unit, teachers asked students:

• How can you help a friend with a problem they are facing?
• “Big ideas need big plans.” What is something you would like to do to make our classroom better?
• What can you do to treat others the way they want to be treated?
• When you try to make big, important changes in the world, sometimes people will think differently of you. How can you keep going to make change? 

Here are the books, which sparked lively discussions and student artwork on the questions:

• All the Way to the Top by Annette Bay Pimentel about Jennifer Keelan-Chaffins’s fight for the Americans with Disabilities Act;
• Mary Wears What She Wants by Keith Negley about women’s rights activist Mary Edwards Walker;
• Kamala and Maya’s Big Day by Meena Harris about how the young Kamala Harris and her sister Maya Harris advocated for transforming an empty apartment courtyard into a playground;
• The Youngest Marcher by Cynthia Levinson about how nine-year-old Audrey Faye Hendricks protested segregation laws in Alabama. 

“Promoting Student Activism Through Children’s Literature and Social Justice Art” by Taylor Marsho, Ashtyn Riley, Deidra Rudd, Morgan Schmidt, Sunah Chung, and Sarah Montgomery in Social Studies for the Young Learner, November/December 2024 (Vol. 37, #2, pp. 5-10); Marsho can be reached at taylor.marsho16@gmail.com.

Please Note: This summary is reprinted with permission from issue #1065 of The Marshall Memo, an excellent resource for educators.

I Am On My Way!

 Posted on Facebook by S. Bear Bergman

            My feed today on BlueSky included a post from Icelandic writer Hildur Knútsdóttir, who saw the original exchange on a Reykjavik Facebook page - someone posted to say that it appeared a cygnet was frozen to the ice and dying. As people worried online about how or whether it would be possible to help, naturalist Kerstin Langerberger replied to the post, saying: “I am on my way with the necessary equipment.” Langenberger brought a friend, some thermoses of warm water, and a surfboard in case the ice failed - the necessary equipment - and thawed then freed the baby swan, which promptly flew off.

            There are a number of things I appreciate about this story, but it’s Langenberger’s statement that resonates for me. That is what I aspire to, to see injury or difficulty or something gone to trouble and decide: I am on my way with the necessary equipment. But also, to hold on to the understanding that sometimes the necessary equipment is the resolve to try and a friend who will help, and that sometimes I am the necessary equipment - the friend who shows up to make myself useful under the direction of an expert.

            There are a lot of moments these days that I find it very, very hard to do the next thing, or indeed to do anything. The world is so, so difficult. I often feel useless or overwhelmed or exhausted or just really fucking sad. But I am going to try starting the days by saying, with all the starch that I imagine a naturalist willing to chance the ice in darkness to save a baby bird might possess: I am on my way with the necessary equipment.

Using Word Games and Puzzles in Elementary Classrooms

            In this article in The Reading Teacher, Mark Lauterbach (Brooklyn College) and Marcy Zipke (Providence College) say that playing online word games and puzzles helps elementary students recognize and manipulate phonemes, words, and phrases – a.k.a. metalinguistic awareness. This is an important component of skilled reading; research shows that paying attention to the details of language – phonology, orthography, morphology, semantics, and syntax – improves students’ decoding, spelling, and comprehension. 

            “Developing students’ interest and motivation for uncovering language conventions is an important part of a teacher’s job,” say Lauterbach and Zipke. “Word games and puzzles are uniquely suited for this, in that they can be solved individually, or in groups of any size. They can be tailored to specific student needs, or in support of the curriculum. Additionally, some students find them particularly engaging.” Playing with word puzzles and games also makes students’ thinking about language less rigid and more fun. 

            Lauterbach and Zipke experimented with the use of three New York Times games – Wordle, Spelling Bee, and Connections – with elementary students. Brief descriptions:

• Wordle – The goal is to find a 5-letter mystery word in six or fewer tries. With each guess, the correct letters turn green (right letter, right position) or yellow (right letter, wrong position).
• Spelling Bee – This is a puzzle with seven spaces for letters, six of them encircling one target letter. The goal is to make as many words using the central letter and as many of the other letters as possible. Letters can be used more than once, with extra points for words that use all the letters.
• Connections – The reader is presented with 16 seemingly disparate words and asked to group them into four categories – for example: break, holiday, leave, and recess (time off); holy, wholly, holey, and holi (sound the same, different spellings and meanings); ink, range, lack, and old (colors with their initial letters missing). Four mistakes are allowed. 

With elementary students, the authors found it was helpful to start by modeling playing the game and thinking out loud about different strategies – in Wordle, for example, what letter to start with, how to proceed with a green or yellow letter, strategy with double letters, and so on. Then the class plays the game together, with “gentle feedback” as they proceed, scaffolding, and gradual release of responsibility.

            Where might word games fit into the school day? Some possibilities: in the 15 minutes before lunch; during a brain break; during indoor recess or free time; sending them home as homework; or creating a puzzle center for choice time. For resources, students might use dictionaries, word walls, and brainstorming friends, with Google and Siri off limits. An additional activity might be taking advantage of websites that allow users to create their own versions of Wordle, Spelling Bee, and Connections. And there are plenty of technology-free options, including putting letters on the board and challenging students to make as many words as they can from the letters. 

            “These puzzles and games are an opportunity to create engagement around activities that promote metalinguistic awareness,” conclude Lauterbach and Zipke. “However, as engaging and useful as these puzzles and games are, they are in no way systematic or comprehensive enough to replace the scope and sequence of a research-based reading curriculum.” 

“Wordling with Elementary Students: Developing Discrete Literacy Skills Through Puzzles and Word Games” by Mark Lauterbach and Marcy Zipke in The Reading Teacher, November/December 2024 (Vol. 78, #3, pp. 195-201); the authors can be reached at mlauterbach@brooklyn.edu and mzipke@providence.edu.

Please Note: This summary is reprinted with permission from issue #1064 of The Marshall Memo, an excellent resource for educators.