Using Open-Ended Math Questions to Differentiate Instruction

          In this Mathematics Teacher article, teacher/consultant/author Marian Small says “too many students sit in a mathematics class where the material being taught is just not quite at the right level for them” – either too easy or too difficult. The best way to engage all students, Small believes, is asking open-ended questions. Some examples: 

• Instead of asking fifth graders to multiply 42 x 37, pose this problem: You multiply two numbers that are 5 apart. What could the product be? I hope some of you use small numbers and some larger numbers. Some students might multiply 3 x 8 while others multiply 92 x 97. 
• There are ___ students in one school and ___ in another school. Choose values that make sense to you for both blanks and tell how many there are in both schools together. 
• You multiply two numbers and the tens digit of the product is 8. What could you be multiplying? One student might multiply 40 x 2 while another multiplies 140 x 2 and the class discusses the connections. 
• A number is a lot like 50. What might it be? What’s a number that you think is very different from 50? The teacher follows up by asking what 25 and 50, for example, have in common, or how 49 and 50 are different. 
• You evaluate an algebraic expression. If you increase the value of the variable just a little, the value of the expression increases a lot. What might the expression be? 
• The answer to a question is the word quadratic. What could the question be? 

Small says there are at least four benefits to posing open-ended rather than right-answer questions: 

• More students are engaged because there will be a variety of unique answers. - Because it’s easier to be right, students’ confidence increases. 
• A variety of responses produces a richer mathematical conversation. 
• There’s the potential to change students’ beliefs about the nature of mathematics. 

 “The Power of Open-Ended Questions” by Marian Small in Mathematics Teacher: Learning & Teaching PK-12, July 2024 (Vol. 117, #7, pp. 528-529)

Please Note: This summary is reprinted with permission from issue #1046 of The Marshall Memo, an excellent resource for educators.

A Rookie Teacher Responds to Critical Feedback

          In this Mathematics Teacher article, Georgia middle-school teacher Corey Gray describes getting some blunt feedback from his principal in October of his first year. Gray thought he had taught an excellent lesson as the principal sat with her laptop at the back of the room: a clear explanation of the distributive property, students turning and talking about their strategies, then using manipulatives. But when Gray nervously took a seat in the principal’s office that afternoon, she said, “You are an amazing math teacher. However, your expectations for a one-size-fits-all type of perfection leads to your own frustration in the classroom and clouds your vision of your students’ mathematical genius.” She used the analogy of trying to put square pegs into round holes. 

          Gray was taken aback but soon realized the principal was right. His very structured classroom management and one-right-answer approach to math procedures “was creating an environment riddled with fear and fraught with comparison,” he says. “I asked for answers, rather than asking for pathways to solutions. I confirmed correct answers only, rather than affirming thinking and productive struggle. I did not take the time to truly understand, appreciate, and value my students for who they were…” Students were doing math rather than thinking about math, resulting in disappointing and inequitable outcomes. Gray’s takeaway from the conversation boiled down to three lessons: 

•  Know that you are on a journey. “This journey is not easy,” he says, “but it is necessary to develop our teaching abilities and character… Embracing this truth allows you to welcome and seek out community…” 
• No one knows everything, but surround yourself with educators who know a lot. Collectively, you and your fellow teachers need to take risks, seek out the best methods and materials, and figure out what works best for which students. 
• Don’t let perfection be the enemy of the good. “For me,” says Gray, “my desire to create the perfect educational environment for my students, anchored in problem-solving and student choice, fuels me, but at times it can become a burden because I often feel alone and burnt out… I remind myself daily to give myself grace, as I am not alone in this quest.” 

 “Teaching Is a Journey: Square Pegs, Round Holes” by Corey Gray in Mathematics Teacher: Learning & Teaching PK-12, July 2024 (Vol. 117, #7, pp. 530-532); Gray can be reached at corey.gray@uga.edu.

Please Note: This summary is reprinted with permission from issue #1046 of The Marshall Memo, an excellent resource for educators.

Tom Guskey on Solving the Problem of Inconsistent Grading

            In this Kappan article, Thomas Guskey (University of Kentucky) says the main reason for parent pushback on standards-based grading is inconsistency among teachers. Within the same school, he says, “What counts as part of the grade, what doesn’t count, and how different aspects of students’ performance are weighted in determining grades – all can be different.” Such variations in grading policies from classroom to classroom, says Guskey, are unfair and inequitable because: 

• Some students game the system, calculating and manipulating points for good grades. 
• For other students, the grading game is a mysterious puzzle they must figure out in every class, and their grades don’t reflect their actual learning. 
• When parents ask their children what grades they expect, kids often have no clue. 

“Before standards-based or competency-based grading reforms can be implemented,” says Guskey, “this inconsistency in grading must be addressed.” Here are three steps he believes schools and districts need to take: 

• Reach consensus on the purpose of grading and report cards. This involves deciding what information grades will communicate, the primary audience, and the purpose. Guskey shares these exemplars: 
• Elementary: The purpose of this report card is to describe students’ learning progress to parents and families, based on our school’s learning goals for each grade level. It is intended to inform parents and families about learning successes and to guide improvements when needed. 
• Middle/high: The purpose of this report card is to communicate with parents, families, and students about the achievement of specific learning goals. It identifies students’ current levels of performance regarding those goals, areas of strength, and areas where additional time and effort are needed. 
• The American School of Paris: The primary purpose of grading is to effectively communicate student achievement toward specific standards, at this point in time. A grade should reflect what a student knows and is able to do. Students will receive separate feedback and evaluation on their learning habits, which will not be included in the academic achievement grades. 

In the Paris school’s statement, Guskey highlights the importance of grades reporting mastery of standards, not averages (which allow early stumbles to unfairly pull down students’ grades), and separating students’ academic achievement from other areas of performance. 

• Use grading scales with 4-7 levels of performance. Guskey believes that 100-point grading scales offer “the illusion of precision” but are actually more vulnerable to teacher subjectivity and unreliability. Researchers have found that using fewer grading levels increases inter-rater reliability and reduces teacher subjectivity and variability from class to class. “Teachers with comparable knowledge and experience,” says Guskey, “are far more likely to agree when distinguishing an A level from a B level of performance than when distinguishing a 90 from an 89 using the percentage scale. The use of clear and well-defined scoring criteria, along with a limited number of grading categories, helps ensure a shared understanding among teachers and promotes more-consistent grading practices.”

• Separately report academic and non-academic student performance. “Hodgepodge” grades that combine academic attainment, progress/improvement, and other areas (effort, class participation, collaboration, responsibility, initiative, organization, self-regulation, low-stakes assessments, homework) make report cards “a confusing amalgamation that is impossible to interpret clearly and accurately,” says Guskey. He advocates separate reporting of these three categories on report cards and transcripts, which produces a more-accurate picture of academic performance, progress, and other areas. What’s more, he says, it’s less work for teachers since they don’t have to calculate an amalgam of student performance in the different domains. 

            “These steps,” Guskey concludes, “address the greatest concerns of parents and families, facilitate better communication between school and home, and ensure greater honesty, accuracy, and equity in grading.” 

“Addressing Inconsistencies in Grading Practices” by Thomas Guskey in Kappan, May 2024 (Vol. 105, #8, pp. 52-57); Guskey can be reached at Guskey@uky.edu. See Marshall Memo 962 for another Guskey article on grading.

Please Note: This summary is reprinted with permission from issue #1035 of The Marshall Memo, an excellent resource for educators.

Is Lord of the Flies For Real?

            In the second chapter of his book Humankind, Rutger Bregman raises questions about Lord of the Flies, William Golding’s 1951 novel, which has been hailed as one of the classics of the 20th century, has sold tens of millions of copies, been translated into more than 30 languages, and is still taught in classrooms around the world. The book describes how a group of British schoolboys survive a plane crash on a remote island and, after some initial attempts at civilized self-governance, paint their faces, cast off their clothes, and bully and mistreat each other. Before they are rescued, three of the boys are killed. 

            Golding, a U.K. schoolmaster, created Lord of the Flies after musing to his wife, “Wouldn’t it be a good idea to write a story about some boys on an island, showing how they would really behave?” What made the book so popular is that it skillfully dramatized a widely held belief about what humans are really like just beneath the veneer of civilization. “Man produces evil,” said Golding, “as a bee produces honey.”

            Bregman remembers reading the book as a teenager. “I turned it over and over in my mind,“ he says, “but not for a second did I think to doubt Golding’s view of human nature.” It was only years later that he read about the author’s life – his unhappiness, alcoholism, depression, about how he divided his students into gangs and encouraged them to attack one another. “I have always understood the Nazis,” Golding said, “because I am of that sort by nature.” It was from this mindset that Lord of the Flies was born. 

            Those troubling biographical details aside, did Golding’s book accurately portray the primal nature of homo sapiens? Most people believe it does, or are at least fascinated by that possibility, which explains the book’s extraordinary popularity. 

            But Bregman had his doubts, and he started looking for an actual instance of young people forced to survive away from civilization. An Internet search led him to a 1966 newspaper story of six boys who set off on a fishing trip from the South Pacific island of Tonga and were swept away by a storm and shipwrecked on a tiny island for more than a year. After more sleuthing and some good luck, Bregman tracked down the name of the captain who rescued the boys, Peter Warner, and traveled to Australia to meet him and see what happened in this real-life Lord of the Flies. 

            In months of interviews with Warner (then 90 years old) and Mano Totau, one of the surviving boys from the island (then almost 70), and more investigative work, Bregman pieced together the story. The boys, age 13 to 16, were students at St. Andrews, an Anglican boarding school on Tonga. Bored with the school’s strict routines, the teens “borrowed” a boat from a fisherman they all disliked and slipped out of the harbor one night without being seen. A storm came out of nowhere, shredding the boat’s sail and breaking the rudder. After eight days adrift, the boys spotted an island and managed to get ashore. They found it was deserted and quite inhospitable with steep cliffs and very little vegetation and spring water. 

            Far from descending into savagery and turning on each other, the boys built a hut, hollowed out tree trunks to store rainwater, cultivated a food garden, constructed chicken pens and a small gymnasium, and after many attempts, produced a spark, started a fire, and tended the flame for their entire time on the island. Working in two-person teams, the boys took turns cultivating the garden, cooking food, standing guard, and keeping a lookout for passing ships. They settled disputes by sending antagonists to opposite ends of the island until they calmed down and were willing to apologize. Days began and ended with prayer and songs, accompanied by a makeshift guitar. 

            There were many hardships. The raft the boys built to escape the island was destroyed in the crashing surf. A storm toppled a tree, demolishing their hut. One boy fell off a cliff and broke his leg and his classmates had to figure out how to fashion a splint. In a period of drought, they were all crazed with thirst. Yet the boys survived, and when they were rescued, all were in top physical condition and the broken leg had healed perfectly. 

            “This is the real-life Lord of the Flies,” says Bregman. “It’s also a story that nobody knows.” While the boys who so bravely and cooperatively survived on the island “have been consigned to obscurity, William Golding’s book is still widely read.” Building on the book’s dark premise, several reality TV shows have perpetuated the trope that human beings, left to our own devices, will behave like beasts – in the words of one contestant, “stop being polite and start getting real” – lying, cheating, provoking, antagonizing. 

            We could discount reality TV as profit-making entertainment, but studies have shown that watching Lord of the Flies-type television can make people more aggressive. “In children,” reports Bregman, “the correlation between seeing violent images and aggression in adulthood is stronger than the correlation between asbestos and cancer, or between calcium intake and bone mass.” Stories of innate barbarism also affect how people look at the world. A British study found that girls who watch a lot of reality TV more often say that being mean and telling lies are necessary to getting ahead in life.                “It’s time we told a different kind of story,” says Bregman. “The real Lord of the Flies is a story of friendship and loyalty, a story that illustrates how much stronger we are if we can lean on each other. Of course, it’s only one story. But if we’re going to make Lord of the Flies required reading for millions of teenagers, then let’s also tell them about the time real kids found themselves stranded on a deserted island.” As Peter Warner wrote in his memoir, “Life has taught me a great deal, including the lesson that you should always look for what is good and positive in people.” 

Humankind: A Hopeful History by Rutger Bregman (Little Brown & Company, 2019)

Please Note: This summary is reprinted with permission from issue #1034 of The Marshall Memo, an excellent resource for educators.

Libraries As the Heartbeat of Schools

           In this Knowledge Quest article, veteran teacher/librarian Amy Brownlee (she’s now teaching the children of some of her former students) describes the way she leverages the power of the libraries in her rural Kansas school district: 

• Getting to know learners – Brownlee learns the names of all 500 students (and how to pronounce them correctly) in the first two weeks of school, has students fill out a Getting to Know You questionnaire, pays attention to the books they request and their evolving interests – animals, trains, gymnastics (Simone Biles!), karate, soccer, swimming, biographies – and keeps track of older students’ sports events and chatter on social media. 
• Engaging students and staying relevant – Brownlee and her library paraprofessional write a new joke on the library whiteboard every two or three days and students and colleagues drop by for a chuckle. She uses music, movement, author talks, news of books that win awards, designing bookmarks, contests (was this line of poetry written by Robert Frost or Taylor Swift?), and other activities to jazz up the library, and makes sure students know about new book acquisitions. 
• A safe, inclusive space – “It is essential that our libraries welcome every learner and staff member in the school,” says Brownlee. She accomplishes this by curating a diverse collection of books and magazines (mirrors, windows, and sliding glass doors) and assuring students and colleagues that their checkout history and research interests will be kept confidential. “Learners know they are free to seek the information they want without fear of judgment or comment from others. Now more than ever our library ethics guaranteeing privacy and confidentiality are essential. The freedom to read is at the heart of what libraries provide.” 
• Giving learners a voice – Brownlee tries to accomplish this by encouraging students to request books; displaying students’ artwork, projects, and writing; showcasing connections to classroom learning; letting kids serve as library aides; and maintaining a student advisory board. 
• Tuning in on life outside the library – Brownlee praises students for their accomplishments and displays around the school, tries to attend student performances and athletic events, and chats with parents about what students are doing. 
• Leveraging the power of books as a bonding agent – She is a fan of Kylene Beers and Robert Probst’s Book, Head, and Heart (BHH) framework, which encourages students to think about books at three levels: 
• Book – What is this about? Who’s telling the story? What does the author want me to know? 
• Head – What surprised me? What does the author think I already know? What changed, challenged, or confirmed my thinking? What did I notice? 
• Heart – What did I learn about me? How will this help me to be better? 

Brownlee modeled the protocol for her students using the Shel Silverstein poem, “The Little Boy and the Old Man” and created a poster encouraging students to think about all the books they read on the three BHH dimensions. 

 “Building Bonds With Young Readers: The Power of Relationships” by Amy Brownlee in Knowledge Quest, March/April 2024 (Vol. 52, #4, pp. 16-23); Brownlee can be reached at amybrownlee93@gmail.com.

Please Note: This summary is reprinted with permission from issue #1032 of The Marshall Memo, an excellent resource for educators.

"Studenting" Behaviors That Are Essential to College Success

          In this Chronicle of Higher Education article, Emily Isaacs (Montclair State University) says too many students are arriving in college without certain proficiencies. ”The absence of those skills does not reflect on the individual’s capacity for creativity, critical thinking, or even advanced reading, writing, and problem-solving,” says Isaacs. “It doesn’t necessarily mean the student is incapable of the intellectual rigor of a college education. But their absence does make learning and college success very difficult.” 

          During the pandemic, many instructors dialed back their expectations, but now that in-person instruction is back, Isaacs says it’s time to raise the bar. She urges her colleagues to explicitly address “studenting skills,” and hopes K-12 educators, especially in high schools, will do the same. Students shouldn’t have to guess what their instructors want, which is especially important for students who enter school with any kind of disadvantage. “I see explicit instruction in writing skills, along with challenging content and assignments, as an inclusive pedagogical approach,” says Isaacs, “leveling the playing field and enabling students who are unfamiliar with the implicit rules of the ‘culture of power’ to succeed.” 

          Here’s her list of studenting skills; note their reciprocal nature, with students and instructors both having agency and responsibility: 

• Attending regularly – “Make attendance matter,” says Isaacs. “If students can learn everything they need to know without coming to class, why would they show up?” 
• Being engaged in class – Teachers need to build in active student involvement and be clear about cellphones and earbuds put away and students contributing without being called on. 
• Being savvy about study skills – These include annotating readings and using the retrieval effect and spaced review to commit important information to memory. 
• Doing homework – Out-of-class assignments need to be meaningful; if students need help managing their time, they should be pointed to counseling resources and online tools. 
• Completing assignments on time – Policies for late work and personal emergencies should be explicit up front, says Isaacs. “Students should experience consequences for late work early in the semester so they learn from their mistakes while recovery is still possible.” 
• Resisting digital distractions – Many students need to learn strategies like Pomodoro work/break scheduling, deleting certain apps, finding quiet places to work, and putting their phones out of immediate reach while they’re studying. 
• No more Lone Ranger – Students should take advantage of all available help to maintain balance and achieve – office hours, study groups, tutoring, online resources. 
• Staying healthy – Psychological distress is a real issue. Students need support coming to terms with their mental state, keeping their lives in balance, and getting counseling when they need it. 

“Teaching Students to Be Students” by Emily Isaacs in The Chronicle of Higher Education, April 12, 2024 (Vol. 70, #16, pp. 54-55); Isaacs can be reached at isaacse@montclair.edu.

Please Note: This summary is reprinted with permission from issue #1032 of The Marshall Memo, an excellent resource for educators.

Responding to Traumatized Students with Campassion (versus Empathy)

            In this Educational Leadership article, Jo Lein (Johns Hopkins University and Tula Public Schools) draws a distinction between empathy and compassion. Empathy, while good-hearted, can lead teachers to feel overwhelmed, anxious, and helpless as they share the emotional burden of students who have experienced trauma. Compassion for the same students involves an emotional connection followed by providing care to alleviate students’ suffering. Compassionate teachers are more likely to maintain their own emotional well-being and not burn out. Here are some examples: 

• Empathy response: I feel your frustration. This experience is so hard for you. Compassionate response: I understand this is challenging. Let’s work on it together. 
• Empathy response: I cannot believe you went through that. It is heartbreaking. Compassionate response: Thank you for sharing. I am here to support you through this. 
• Empathy response: I cannot believe you would do that. It is so disappointing. Compassionate response: I see you are having a tough time. Let’s discuss what happened. 
• Empathy response: I am so worried about you. This anxiety is affecting me, too. Compassionate response: I notice you’re feeling anxious. How can I help you manage it? 
• Empathy response: Your poor grades upset me. I am disappointed. Compassionate response: Your grades do not define you. Let’s figure out how to improve them. 
• Empathy response: Why don’t you care about this? It is frustrating for me. Compassionate response: I sense you are not engaged. Talk to me about that. 

Lein suggests that school leaders keep an eye out for teachers caught up in empathy stress and, in one-on-one coaching meetings, steer them toward compassionate responses. “Let’s Be Trauma-Sensitive to Teachers, Too” by Jo Lein in Educational Leadership, March 2024 (Vol. 81, #6, pp. 62-65); Lein can be reached at jo.mabee@gmail.com.

Please Note: This summary is reprinted with permission from issue #1027 of The Marshall Memo, an excellent resource for educators.