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.