What is the Purpose of Mathematics?

        In this article in Mathematics Teacher: Learning & Teaching PK-12, Lucy Watson (Belmont University) and Christopher Bonnesen and Jeremy Strayer (Middle Tennessee State University) describe a common dilemma for math teachers: what do you say when students ask, Why do I need to know that? Some teachers point to practical, real-life applications in science, technology, engineering, and math education. Others extoll the beauty and wonder of mathematics. What teachers say might reveal one of three views of the nature of mathematics: 

        - It’s a set of facts, rules, and tools that need to be memorized; 

        - It’s a static body of knowledge bound by discovered truths that never change; 

        - It’s a dynamic, problem-driven discipline defined by creativity, inquiry, and openness to revision. Students taught by a teacher holding each view will learn mathematics quite differently, and will likely be exposed to distinct teaching methods, from rote lectures to discussions and hands-on projects.          

        Watson, Bonnesen, and Strayer believe there hasn’t been enough guidance for math teachers on exactly what the nature of mathematics is, leaving the field wide open to a variety of rationales – and probably some pretty dull teaching. Drawing on several guiding documents in the field, the authors suggest this five-point view of the nature of mathematics: 

        • Mathematics is a product of the exploration of structure and patterns. 

        • Mathematics uses multiple strategies and multiple representations to make claims. 

        • Mathematics is critiqued and verified by people within particular cultures through justification or             proof that is communicated to oneself and others. 

        • Mathematics is refined over time as cultures interact and change. 

        • Mathematics is worthwhile, beautiful, often useful, and can be produced by each and every                         person. 

         The authors believe that as students grapple with high-quality math problems, teachers should get them thinking about this broader view of the nature of mathematics, asking students about purpose before, during, and after solving the problems. Watson, Bonnesen, and Strayer suggest repeating this meaning-seeking activity at intervals through the grades – perhaps with a unit on counting in kindergarten, equivalent fractions in third grade, area relationships in middle school, and absolute value in high school. If this occurs, say the authors, teachers will less frequently hear the question, Why do I need to know that? 

 “The Nature of Mathematics: Let’s Talk About It” by Lucy Watson, Christopher Bonnesen, and Jeremy Strayer in Mathematics Teacher: Learning & Teaching PK-12, May 2021 (Vol. 114, #5, pp. 552-561); the authors can be reached at Lucy.watson@belmont.edu, ctb4d@mtmail.mtsu.edu, and jeremy.strayer@mtsu.edu.

Please Note: This summary is reprinted with permission from issue #888 of The Marshall Memo, an excellent resource for educators.

"Learning Loss" - Wrong and Right Solutions

        In this online article, Harvey Silver and Jay McTighe worry that “lost learning” is an unfortunate way to define the challenge schools face as they reopen for in-person instruction. By framing the challenge as instructional time lost, there’s a tendency to think the solution is rapidly covering the curriculum that students missed – which has two downsides. “At the classroom level,” say Silver and McTighe, “this solution could take the form of cutting out any of those time-consuming learning activities such as discussions, debates, hands-on science investigations, art creation, and authentic performance tasks and projects” – instead “trying to blitz through lots of factual information.” 

        Rather than focusing on the content that wasn’t covered during remote and hybrid instruction, they propose two more-productive approaches: 

        • Prioritizing the curriculum on outcomes that matter the most – A simple but effective way to accomplish this is preceding the title of each curriculum unit with the words, A study in… Several examples: 

        - The calendar – A study in systems 

        - Linear equations – A study in mathematical modeling 

        - Media literacy – A study in critical thinking 

        - Any sport – A study in technique 

        - Argumentation – A study in craftsmanship 

Preceding a unit title with those three words, say Silver and McTighe, “establishes a conceptual lens to focus learning on transferable ideas, rather than isolated facts or discrete skills.” 

        It’s also helpful to frame the unit around Essential Questions. For the five units above, here are some possibilities: 

        - How is the calendar a system? What makes a system a system? 

        - How can mathematics model or represent change? What are the limits of a mathematical model? 

        - Can I trust this source? How do I know what to believe in what I read, hear, and view? 

        - Why does technique matter? How can I achieve maximum power without losing control? 

        - What makes an argument convincing? How do you craft a persuasive argument? 

Well-framed Essential Questions are open-ended, stimulate thinking, discussions, and debate, and raise additional questions. 

        • Engaging learners in deeper learning that will endure – “To learn deeply,” say Silver and McTighe, “students need to interact with content, e.g., by linking new information with prior knowledge, wrestling with questions and problems, considering different points of view, and trying to apply their learning to novel situations.” The most important skills are comparing, conceptualizing, reading for understanding, predicting and hypothesizing, perspective-taking, and exercising empathy. 

        A kindergarten example: challenging students to predict how high they can stack blocks before a tower falls down, then having them try different hypotheses and see what works best, and note the success factors. “This focus on cause and effect will become a yearlong inquiry for students,” say Silver and McTighe, “as they learn to use it to examine scientific phenomena, characters’ behavior in stories, and even their own attitudes and motivations as learners.” (The full article, linked below, includes a middle-school unit on genetically modified food and a high-school unit comparing the educational philosophies of Booker T. Washington and W.E.B. DuBois.) 

        This two-part approach to curriculum is not just “a stopgap measure tied to current anxieties about learning loss,” conclude Silver and McTighe: “Framing content around big ideas and actively engaging students in powerful forms of thinking is good practice – in any year, under any conditions.” 

“Learning Loss: Are We Defining the Problem Correctly?” by Harvey Silver and Jay McTighe on McTighe’s website, May 7, 2021; McTighe can be reached at jmctigh@aol.com.

Please Note: This summary is reprinted with permission from issue #886 of The Marshall Memo, an excellent resource for educators.

Online Groups for School Leaders

         In this article on The Main Idea website, Jenn David-Lang says school leaders are hungry for professional development, but receive less than other educators – which may explain some of the attrition we’re seeing among administrators. While schools were closed during the pandemic, David-Lang had an idea: why not involve groups of administrators in Masterminds, an online version of accountable, results-focused teacher PLCs? The term Masterminds was coined almost 100 years ago by author Napoleon Hill, but has only recently found its way into the world of K-12 schools. 

        Here’s how David-Lang and her colleague Mitch Center have implemented the concept. They’re running several year-long Mastermind groups, each with about eight school- and district-based leaders from varied locations (“from Baltimore to Bangkok,” says David-Lang). Groups meet twice a month via Zoom to learn new ideas, share strategies, solve problems, and support one another. The one-hour meetings go quickly, following this structure: 

    • Check-ins – Everyone briefly shares a struggle or a win. “Getting an inside view of how everyone is doing and what is going on at each other’s schools builds trust,” says David-Lang; “principals are rarely given space to share how they’re honestly doing without the need to put on their ‘principal face.’” 

        • Goal sharing – In two-person breakout rooms, members report on a goal they committed to in a shared Google Doc at the end of the previous session. This provides continuity from meeting to meeting and keeps people accountable for actions to which they have committed. 

         • New content – David-Lang and Center share a one-page summary of ideas or research on their screens and everyone reads it silently. A recent example: a synthesis of five mindset shifts described in a recent book on unconscious bias by Sarah Fiarman and Tracey Benson. David-Lang and Center then facilitate a discussion of the ideas, sometimes regrouping into two breakout rooms, or participants fill out a shared graphic organizer. 

         • Think tank – One member presents a real-life dilemma, including the background and context of the problem (one example: dealing with a new assistant principal who is not garnering respect from colleagues). Other members ask clarifying questions, and then the presenter remains silent while the rest of the team discusses the issue and suggests possible solutions. Finally, the presenter recaps those ideas and thinks out loud about the ones that seem most likely to work. 

        • One Big Thing (OBT) – In the Chat area, there’s a link to a shared Google Sheet with a row for each member, and they write their biggest takeaways from the session. This makes available to everyone the collective learning from the reading, discussion, and problem-solving. This segment was inspired by John Dewey’s insight that true understanding comes not from doing, but from reflecting on what’s been done. 

        • Committing to a goal. Each session ends with each member writing a commitment for specific action, to be reviewed at the beginning of the next meeting. 

        Reflecting on a year of leading Mastermind groups, David-Lang looked up the criteria for effective professional development compiled by Linda Darling-Hammond and colleagues. It turned out that her groups were meeting every one of them:

• Focused on content;
• Incorporating active learning; 
• Supporting collaboration; 
• Using models of effective practice; 
• Providing coaching and expert support; 
• Offering opportunities for feedback and reflection; 
• Sustained over time. 

 “While my co-facilitator and I have coached school leaders individually,” says David-Lang, “we were immediately struck by the exponential power of coaching that comes from all members sharing their own learned strategies and diverse perspectives… It is the collective wisdom, energy, and passion that truly distinguishes Masterminds from other forms of PD for educational leaders.” 

         While the sessions have been particularly valuable during the disruptions of the pandemic, David-Lang believes they should continue to be an important forum in the new normal. 

 “Masterminds: When PL Meets PLC” by Jenn David-Lang, The Main Idea, May 2021; David-Lang can be reached at Jenn@TheMainIdea.net. 

 Please note: This summary was reprinted with permission from issue # 886 of the Marshall Memo, an excellent resource for educators.

Four Key Teacher Roles in a Personalized Classroom

           In this Elementary School Journal article, Penny Bishop, John Downes, Steven Netcoh, Katy Farber, Jessica DeMink-Carthew, Tricia Brown, and Rachel Mark (University of Vermont) report on their interviews with a number of elementary and middle-school teachers on personalized learning.  The researchers define personalization "as an approach that encourages partnership between individual students and teachers in the design of learning that emerges from students' interests, questions, needs, and preferences to foster self-directed learning."  Assessments may take the form of portfolios of student work, authentic performance tasks, and exhibitions of learning in which students demonstrate their skills and understandings.

          Teachers in the study described the shift from running adult-centered classrooms to supporting students as they brought their interests, needs, and different levels of readiness to the classroom.  One teacher drew a distinction between personalization and individualization, the latter being about getting all students "to arrive at the same spot through different means."

          Synthesizing what they learned from interviews, the researchers identified four roles teachers played in personalized classrooms:

• Empowerer - Teachers sought to increase students' independence and ownership of learning by letting them lead, offering choices, enabling students to work at their own pace and level, increasing student talk, and learning with and from students.

• Scout - Teachers often needed to seek out resources to support students and figure out next steps in their learning progressions.  This involved ascertaining students' interests, aligning the curriculum with those interests, curating digital and material resources, and connecting students with helpful people inside and outside the school.  "We can't offer everything," said one teachers, "but it's not our job to offer everything.  It's our job to explain how to navigate the world."

• Scaffolder - Teachers constantly worked to ensure that students engaged productively in learning.  This involved structuring routines, time, and learning experiences, fading the support when students didn't need as much, modeling possible approaches, and asking questions.  "Okay," said one teacher to her students, "you have your team leaders. You have your roles.  You can do it.  Sign up on the board if you need my help." She then "floated" around the room.

• Assessor - Teachers said it was important to distinguish between assessment and evaluation (with the latter, offering lots of narrative feedback to students,) provide ongoing formative assessment (a lot of over-the-shoulder checking for understanding and redirection) and be clear about learning targets and rubrics posted around the room.

"Teacher Roles in Personalized Learning Environments" by Penny Bishop, John Downes, Steven Netcoh, Katy Farber, Jessica DeMink-Carthew, Tricia Brown, and Rachel Mark in Elementary School Journal, December 2020 (Vol. 121,#2, pp. 311-336); Bishop can be reached at Penny.Bishop@uvm.edu.

(Please Note: The summary above is reprinted with permission from issue #871 of 

The Marshall Memo, an excellent resource for educators.)

Addressing Students' Unfinished Learning

            In this article in Mathematics Teacher: Learning and Teaching PK-12, Cathy Martin (Denver Public Schools) says many students have entered the 2020-21 school year with "unfinished learning" from interrupted instruction in the spring - "prerequisite skills and concepts that are essential for student engagement in grade-level content that students do not have yet."

            Some pars of the previous year's curriculum are more important to success this year than others.  Martin believes the best mindset for addressing the 2020-21 school year is not remediation, but accelerating unfinished learning.  There's a key difference between the two, she says "Remediation is based on a mistaken belief that students need to master everything they missed before they are able to engage in grade-level content.  Thus, remediation focuses on students' learning gaps from a deficit-based mindset and then drills students on isolated skills and topics that have little connection with current grade-level content." This backwards-looking approach results in deceleration and widening achievement gaps.

            Acceleration, by contrast, "prepares students for success in the present - this week on this content, "addressing incomplete understanding in the context of the current grade's standards, and treating students with an asset-based mindset.  The two key steps: first, selecting "just in time" skills and concepts relevant to current units, with clear connections between the previous year's curriculum and 2020-21 content and skills.  Second, giving informal, teacher-created just-in-time assessment tasks that tell how far instruction has to "back up" to fill in gaps in skills and knowledge.  Then teachers can launch instruction that catches students up and prepares them for successful grade-level work.

"Accelerating Unfinished Learning" by Cathy Martin in Mathematics Teacher: Learning and Teaching PK-12, October 2020 (Vol 113, #10, pp.774-76); Martin is at cathymartin90@gmail.com . 

(Please Note: The summary above is reprinted with permission from issue #859 of 

The Marshall Memo, an excellent resource for educators.)

                

Online Maker Resources

 

In this School Library Journal article, Virginia school librarian Idamae Craddock suggests 

ten STEMmaker resources that can be used for remote learning.

 

- CoBuildAtHome - a wealth of online maker activities, a Facebook group, and suggestions from luminaries in the maker world - appropriate for all ages.

- Community Science Workshop Network - engaging and not technologically complicated; has low-cost projects using inexpensive or recycled materials.

- Science Friday – This site’s maker challenge has everything from “The Many Uses of Mucus” to “Fossilize Me” – upper elementary and older.

- San Francisco Exploratorium - this amazing museum's site investigates everything from skateboarding engineering to using Orea cookies to explore plate tectonics - upper elementary and older.

- MakerEd – A compilation of projects and learning approaches designed for educators and parents. 

- MakeCode – This Microsoft program has physical and virtual coding and app development – upper elementary through middle school.

- Scratch – An archive of hundreds of math and visual arts activities with examples and stories to help build classroom community and support curriculum content – elementary through middle school.

- Algodoo – This free download allows students to create, alter, and run engineering simulations, changing gravity, adding gears, planes, ropes, and wheels to see how they will interact – middle school and older. 

- Blockscad – A simple block coding program to make 3-D objects and teach math concepts – middle school and older. 

- Google Experiments – An archive of experiments  that require little or no equipment, entertaining and with firm curriculum foundations – middle school and older. 

“On-Screen and Hands-On” by Idamae Craddock in School Library Journal, October 2020 (Vol. 66, #10, p. 17) 

(Please Note: The summary above is reprinted with permission from issue #857 of 

The Marshall Memo, an excellent resource for educators.) 

Teaching Students to be Discerning with Evidence

         "Typically, and for too many years, elementary social studies lessons have consisted of a single story," say Muffet Trout (University of St. Thomas) and Jeff Sambs (St. Paul, Minnesota teacher) in this article in Social Studies for the Young Learner." They describe the very different depictions of Christopher Columbus that Sambs encountered as a student.  His 5th grade teacher portrayed Columbus as a hero (In 1492, Columbus sailed the ocean blue...) , but a decade later a college professor said Columbus was a villain who stole and subjugated.  Both instructors, say Trout and Sambs, "had missed the opportunity to help their students think with more complexity." leaving them unskilled in the key social studies competency of being able to "read, reconstruct, and interpret the past" (National Curriculum Standards for Social Studies). It also left them unprepared for the key civic duty of deliberation - being able to discuss, listen and come to a fair (not purely self-interested) resolution.

        As a rookie elementary teacher, Sambs was determined to do better.  He asked his fifth graders to look at the events of 1492 and 1493 through the lens of the European explorers and then from the point of view of the Indigenous people of the Caribbean.  His students were able to do this quite well, and over the next 18 years, Sambs developed what he calls the Evidence on the U approach.  The goal has been to support deep and complex student thinking and gradually place much of the responsibility for learning on students.

        Here's how it works.  Groups of students examine a wide variety of paintings, original source documents, texts, and artifacts and debate where to place them on a U-shaped graphic, with evidence tending toward one point of view (for example, Columbus as a hero) on one side, evidence supporting the opposite viewpoint somewhere on the other, evidence that's more complex at the bottom.  "We have found that the highest quality of conversation happens in that in-between spot," say Trout and Sambs, "when resources do not reflect an extreme position.  Students begin to focus on their justification, causing them to examine closely and think analytically, requisite skills for engaging in complex deliberation." Using a U-shaped continuum rather than a straight line emphasizes the complexity of evidence and pushes back on the idea that the middle is where the truth always lies.  For the final assessment, students are presented with new documents or artifacts and asked where they belong on the U-graphic, with grades based on how well they justify their decisions.

        As a follow-up, Sambs set up student chairs in his classroom in a large U and has students think through a provocative issue (for example, whether students should be required to wear uniforms), develop an argument somewhere on the continuum, and then sit in the location in the U corresponding to their argument.  Students are then asked to shift to a new location and make the argument from that vantage point.

        Sambs has found he can use the U graphic in several other curriculum areas, including when students write persuasive essays, explore current events, and present inquiry findings.  He's also found it helpful for discussing issues where there's a continuum from one extreme to the other - for example:

• From colonizer to colonized - Guiding question: How does the process of colonization influence specific populations?
• From us to other - Guiding questions: How do we view people who are different from ourselves? In what ways are they different? How do we behave toward someone we see as "different"?
• From aggressive to passive response to a conflict - Guiding question: What are the costs and benefits to being aggressive, assertive, or passive when handling a conflict?
• From individual to community - Guiding question: How do you balance your rights as an individual with your responsibility to others?
• From private to public - Guiding question: What are some examples of personal freedoms (e.g., saying what you want) that are limited by public needs (e.g., safety, privacy, personal respect)?

"A Teaching Strategy to Strengthen Habits of Deliberation: The 'Evidence on the U' Graphic" by Muffet Trout and Jeff Sambs is in the Social Studies and the Young Learner, September/October 2020 (Vol. 33 #1, pp. 17-21); the authors can be reached at trout@stthomas.edu and jeff.sambs@spps.org. 

(Please Note: The summary above is reprinted with permission from issue #856 of 

The Marshall Memo, an excellent resource for educators.)