Principles of Mathematics at HIS
This is the final installment of the two-part conversation with Lower School Principal John Heffron and Curriculum Director Chris Bezsylko about the math curriculum and the new, primary resource in the lower school, Investigations in Numbers, Data, and Space.
This conversation was framed around the National Council of Teachers of Mathematics (NCTM) Principles for School Mathematics: equity, curriculum, teaching, learning, assessment, and technology. Part I of this series focused on the first three principles, this final installment focuses on learning, assessment, and technology.
The Learning Principle states that research has solidly established the importance of conceptual understanding in becoming proficient in a subject. When students understand mathematics, they are able to use their knowledge flexibly. They combine factual knowledge, procedural facility, and conceptual understanding in powerful ways. How does our new Investigations program foster this type of understanding?
Mr. H: “One of the benefits of Investigations is its powerful focus on building conceptual understanding within our students. In the 21st Century, getting the “right” answer is not enough. Today, any of us can quickly find the right answer to a simple math problem by pulling up a calculator on our smart phone. What is needed in primary school learning is a deeper understanding of how and why the math works. In higher education and for the jobs of tomorrow, this deeper understanding will help our children to be innovators and have the ingrained skills to think critically, to solve complex problems that we don’t yet know about and to reason their way forward, instead of regurgitating pat answers that are quickly forgotten.
We want our students to be able to take their conceptual understanding and apply it to unusual or unfamiliar situations. The standardized tests that our students take have a deliberate focus on word problems and mathematical situations embedded into some sort of context. Without conceptual understanding and problem-solving skills, students are often lost – it’s been thrown at them in a way that they’re not used to seeing, and so they aren’t sure what to do. With conceptual understanding and classroom experience solving problems in multiple contexts and using multiple strategies, students are able to reason their way through the problem and come to an accurate answer or solution, though their approach may be different from how you or I would have tackled the problem! Not only is this okay, it’s encouraged!”
The Assessment Principle:
The Assessment Principle states teachers should be continually gathering information about their students through questions, interviews, writing tasks, and other means. They can then make appropriate decisions about such matters as reviewing material, reteaching a difficult concept, or providing something more or different for students who are struggling or need enrichment. To be consistent with the Learning Principle, assessments should focus on understanding as well as procedural skills. How do our teachers assess students in Mathematics?
Mr. H: “Assessments can be divided into two categories: formal and informal. Formal assessments for our students might be the CPAA or the ERB/CTP4 or the End-of-Unit assessments that are embedded in the Investigations program. While it goes without saying that these formal assessments are important, they actually make up a small percentage of our total assessment practice. Most assessment is informal — the minute-by-minute observations that teachers make of students in action: tackling a problem, collaborating with peers, playing a math game, creating, writing, talking about the math, justifying their solution paths, asking questions. Add to this the quizzes and writing samples and homework and classwork that students are continually doing and you end up with lots of formal and informal data that a teacher uses to make instructional decisions on what concepts and skills have been mastered, by whom, and to what extent; who needs reteaching, who needs more practice, and who needs to be challenged. What is so nice about our Investigations program, as witnessed on much of the homework, is that it asks students to not only write down the correct answer but also to show how they got to that answer – to explain their thinking or draw a picture that shows what the numbers and symbols and variables represent.”
The Technology Principle: Calculators and computers are reshaping the mathematical landscape. The technology principle states that students can learn mathematics more deeply with appropriate and responsible use of technology. What role does technology play in math education at HIS?
Mr. B: “Everyone over the age of thirty recognizes that technology has had a significant impact on all aspects of our lives. This is no different in academics. As the NCTM technology principle states, technology is a tool that must be used appropriately and responsibly. Technology cannot replace a teacher. As educators, our role is to make prudent decisions about when and how to use technology.
Technology supports our mathematics program in a variety of ways. Web resources such as NCTM’s Lessons and Resources pages provide access to a wide variety of student, teacher, and parent tools. In fact, NCTM’s Family Resources has pages dedicated to helping your child succeed in math, helping your child with homework, and even a section called Math Education Today that aims to deliver an important message to all of us – Math is taught differently now. The information on the NCTM website supports what families will find on the For Families pages of the Investigations website.
There are an ever-growing number of Apps out there that are dedicated to math. While well respected organizations like Common Sense Media and family resources such as Parents magazine publish recommendations on all types of Math Apps, scientific research on the educational value of Apps is virtually nonexistent. Apps that focus on simple tasks such as counting, sorting, or doing math facts may provide additional skills practice but they don’t provide a foundation for concepts like scaling and fractions. We will be sharing some of our top Math App recommendations through our Friday Folders and class communications as we continue to find Apps that extend student learning in a variety of meaningful ways.
At HIS technology is a tool, one of many we empower our students and teachers with. As a part of our math program, we recognize that technology makes information accessible to all members of our community, and it provides multiple means to support skills practice and engage students in higher-order learning. We also recognize that technology is advancing rapidly and while software and Apps may change, great teaching depends on a strong relationship between the student and teacher and a commitment to provide each student with the tools they need for continued growth and development.”