Developing Enduring Understandings

Conceptual Math is a signature program of The Summit’s Lower School which builds a conceptual mathematic foundation that endures for a lifetime. The math curriculum developed at The Summit exceeds national standards for each grade level by drawing upon research-based pedagogy, years of successful experience from specially trained faculty and globally recognized best practices.

Teachers are not limited to using a single textbook or textbook series for the curriculum. Materials-rich classrooms are equipped with many hands-on learning tools to physically demonstrate the mathematic principles.

Promethean SmartBoards and wireless student-response clickers help turn every classroom into a 21st Century learning environment in order to give faculty resources that allow them to monitor and tailor instruction to each child’s needs.

Students learn to communicate and reason mathematically. They develop problem-solving skills and confidence in their abilities.

Many math programs are geared to teaching children how to solve a particular problem. Often, children go through school learning to solve a type of problem but don’t really understand why learning to solve that type of problem is important or why the method taught to solve the problem works. As they move into higher levels of math in middle through upper school, they lack a coherent, conceptual framework for math. So they struggle and say, “I just can’t do math.” Yes, they can. These children just lack the conceptual framework to understand what math is all about. That’s why The Summit’s Conceptual Math program is so successful. Children in our Lower School first are given an understanding of the conceptual framework. Then as they move forward, they can easily connect the higher level concepts with the foundational ones they learned in the Lower School.

Key aspects of the Lower School Conceptual Math Program include:

Method of Progression

Explicit instruction deliberately follows a natural progression of conceptualizing math from concrete, to pictorial, to abstract thinking. Recent research suggests that students who are taught within this framework score higher on standardized math tests. “Teachers plan for the progression,” says Math Specialist Julia Almaguer. “Students begin exploring concepts by working with concrete manipulatives. For example, students may manipulate beads that represent ones, tens and hundreds in order to understand the place value in the multiplication algorithm. The physical manipulation develops the fundamental, concrete understanding. Then students draw pictures to relate pictorial understanding to concrete understanding. When that’s mastered, they move onto the abstract level. That’s the goal. A lot of students can successfully dive right into the abstraction but don’t fully understand the concept. In the long term, developing concrete understanding first will serve students best.”

Enduring Understanding

Every unit within the curriculum includes an enduring understanding of the concept as a goal. Essential questions that provoke thought about the enduring understanding serve as catalysts for exploration. “These are big ideas, the big takeaways that you hope the students will remember five years from now,” says Mrs. Almaguer. Lower School Director Mike Johnson offers an example.  “We lay the foundation for an important enduring understanding around measurement early in the first grade year.  Students hold in hand a one inch by one inch square so that they can feel the length of an inch – they literally get to “sense” that unit of measure.  They can then use cubes to discover the length of a pencil or how many inches are in a foot.  As they move deeper into their understanding of measurement this process allows them to develop a proportional understanding of various units of measure.”

Algebraic Thinking

This cognitive process involves identifying and analyzing patterns in order to develop generalizations. Research indicates that students are better able to understand algebra if they have mastered the general properties of numbers. Developing the ability to generalize understanding about patterns gives children confidence, which contributes to a more positive attitude. Foundational skills for algebraic thinking are implicit in the Lower School lesson plans and taught with intention. “About 10 foundational skills can be taught in an age appropriate way to lay a foundation for algebra," says Mrs. Almaguer. “Categorization, ordering, patterning, balance and equality, and symbolic representations are foundations for algebra. Algebra is deliberately included in lessons.”

Frequent Assessments

Pre-assessment and post-assessment testing of units of study identify how much students know as they progress through the curriculum. This testing allows teachers and their assistants to personalize instruction. Combined with the knowledge of how to teach to a variety of learning styles and an understanding of each child’s interests, frequent assessments help teachers determine what methods will help each student meet, and often exceed, goals identified for each academic year. Because of the data gained through frequent assessment and the Lower School’s use of differentiated instruction, students are divided into skill groups so one group can review a strand of the curriculum while another group may begin something new.

Mathematical Writing

The act of writing about math requires students to organize, clarify, and reflect on their own understanding of mathematical concepts. Teachers use writing assignments to help students think more deeply and clearly about what they are learning. Writing assignments fall into three categories: keeping journals, solving math problems and explaining mathematical ideas. “Each of these categories focuses the students on their math learning in a different way, and each provides useful information for assessing their progress,” says Mrs. Almaguer. Math journals are important repositories of student mathematical thinking. The written product provides our teachers with a road map to a student’s understanding which can then be broadened, and self-articulated misconceptions which can be addressed.

21st Century Math

A culminating experience for the Math program is the fourth grade 21st Century Math class. In each of the three trimesters, students are immersed in an in-depth exploration of economics, a Science, Technology, Engineering, Art and Math (STEAM) experience and plan the "Party of the 21st Century."

  • In the economics unit, students learn about the factors of production and the law of supply and demand. Students conduct market research to develop a good or service and develop a business plan. Then the fourth graders create products which they “sell” at Market Day. In the process, they make change when selling their goods, keep a ledger sheet of income and expenses, and calculate percentage for taxes.
  • Students act as planners for the "Party of the 21st Century." Collaborating in small groups, students must calculate every detail of the cost of the party – guest list, invitations, schedule, floor plan and decorations, entertainment and menu – within a given budget. They pitch their proposals in "Shark Tank" fashion to a Summit “celebrity” panel. One party is chosen to be the year-end fourth grade class celebration.
  • Students also explore connections between math and art. They design kente cloth to describe themselves through patterns and symbols, use geometry in the creation of origami and discover the Rule of Thirds in photography and famous paintings.

Skilled Faculty

Lower School teachers are experts at meeting each student where he or she is and supplying targeted instruction to engender growth. Our program is devised to provide classroom teachers with Educational Assistants whose help allows flexible, needs based groupings within classrooms. Additionally, a specially trained math specialist and assistant math coach help faculty identify additional classroom or technological resources. They also work directly with students individually or in groups to develop better foundational understanding or leap forward to advanced practice.