“Nature is written in mathematical language.” Galileo
Mathematics is indeed a critical language for understanding the world, and so mathematics is integral to the curriculum at Plymouth Meeting Friends School. In addition to daily dedicated math instruction, children in all grades take part in mathematical encounters and challenges throughout the day, including learning about one-to-one correspondence while taking attendance using clothespins, exploring patterns with the calendar, graphing birthdays using cupcakes, planning circus seating, using Cartesian coordinates to enlarge pre-Columbian designs, and measuring an acre in the field. In every grade at PMFS, projects like these build students’ mathematical curiosity, imagination, and problem-solving abilities.
The Math in Focus curriculum, used in PMFS classrooms from Kindergarten through Sixth Grade, overlaps with the school’s broader philosophy of active, experiential, and project-based learning. This curriculum uses a proven teaching approach that builds from concrete hands-on learning, to pictorial representations of concepts, to abstract understanding. The Math in Focus curriculum is built around the acclaimed Singapore math curriculum, and it is fully aligned with Common Core standards. This curriculum enables our students to become strategic, confident, and persevering mathematical problem-solvers who are well prepared for math beyond PMFS.
The Math in Focus curriculum places particularly strong emphasis on three areas of mathematical understanding and proficiency:
- Number and Operations: Students gain a deep comprehension of place value, the number line, fractions and decimals, and a strong grasp of the four mathematical operations, enabling them to handle numbers and operations with skill and confidence. The goal is to go well beyond procedural skill to ensure that students build a thorough understanding of the meaning of their mathematical work.
- Model Drawing and Problem Solving: From Second through Sixth Grades, bar modeling is used as a tool in mathematical problem solving, enabling students to visualize complex relationships and tackle difficult problems, as well as giving them a solid foundation in algebraic thinking. (An example of bar modeling from the Third Grade curriculum is given below.)
- Mental Math: The curriculum gives greater emphasis to students’ mental fluency with numbers than most math programs, encouraging flexible strategies for mental calculations and estimates, which leads students to have greater ease and confidence in all their math work.
The following example, taken from the Third Grade Math in Focus textbook, demonstrates the benefit of using bar modeling to solve word problems.
A bookstore had 4 shelves of books. Each shelf had 116 books. The store owner sold 382 books. How many books were left?
While the bar model does not solve the equation for a learner, such a pictoral representation enables students to tangibly represent the known and unknown parts of a whole and to make sense of the relationships among the numbers given in a word problem. It helps students break down a complex problem into manageable chunks, all part of algebraic thinking.
The pentagon above shows the framework of the Math in Focus curriculum, which is focused on developing students’ mathematical problem-solving abilities and the practices that contribute to building those abilities. The curriculum is designed to develop and strengthen:
- attitudes such as interest, confidence, and perseverance, that support students’ mathematical work;
- metacognition, the awareness of students’ own thinking and learning processes;
- processes such as reasoning, communicating ideas, modeling, and applications;
- concepts including numerical, algebraic, geometrical, statistical, probabilistic, and analytical concepts;
- skills, such as numerical calculation, algebraic manipulation, and data analysis.
Kindergarteners are becoming aware of the connection between “math” and their everyday lives; explorations stimulate their math reasoning and create opportunities for critical thinking. Students develop confidence and communication skills, using not only the Math in Focus program but also its companion Everyday Counts Calendar Math. While students engage with the fun and nurturing scaffolding of Calendar Math, they also use math manipulatives and hands-on activities in the program’s Investigate-Discover-Explore-Apply Math in Focus progression to build a critical core understanding of numbers, number sense, shapes, and patterns. Counting on and back, ordering, comparing, classifying and sorting, and doing addition and subtraction stories in context help Kindergartners develop deep number sense to 100. Additionally, work on non-standard units of measurement, ordinal numbers, positional vocabulary, and coins are not only engaging but also lay the critical groundwork for math thinking throughout school years and beyond.
Mathematics in Primary centers on building a core understanding of numbers, shapes and patterns. This includes learning about measurement, time, money, place value, relationships between numbers, and how they can be manipulated to solve basic and critical thinking problems. This underlying knowledge is critical to a child’s growth in mathematical thinking. It is the basis upon which all other math skills are built. While First Grade math instruction centers on the Math in Focus curriculum, it is supplemented and extended by daily math experiences and investigations which include using calendars, making predictions, estimating, sorting, counting, organizing and interpreting data in tables and graphs, and solving real-life problems. Embedded in each math experience is the opportunity for students to develop math reasoning through collaboration and debate, thus allowing for a more profound, shared mathematical understanding. To develop their number sense, First Graders explore numbers to 120 through number bonds, adding and subtracting, counting by multiples, doubling, and making equal groups. First Grade students enjoy challenging themselves with math games using dice, wheels and math tiles.
Mathematics in Primary centers on building a core understanding of numbers, shapes and patterns. This includes learning about measurement, time, money, place value, relationships between numbers, and how they can be manipulated to solve basic and critical thinking problems. This underlying knowledge is critical to a child’s growth in mathematical thinking. It is the basis upon which all other math skills are built. While Second Grade math instruction centers on the Math in Focus curriculum, it is supplemented and extended by daily math experiences and investigations which include using calendars, making predictions, estimating, sorting, counting, organizing and interpreting data in tables and graphs, and solving real-life problems. Embedded in each math experience is the opportunity for students to develop math reasoning through collaboration and debate, thus allowing for a more profound, shared mathematical understanding. Second Graders explore, add, and subtract numbers to 1000; learn to use bar models; multiply and divide; measure in metric and conventional units; practice mental math and estimation; count money; tell time; add simple fractions; read picture graphs and investigate geometry.
Third Graders are ready to build on their foundational understanding of addition, subtraction, multiplication, and division. Third Graders learn how to solve problems independently by drawing on an explicit repertoire of strategies and also learn to work collaboratively to solve challenging problems. They spend significant time working with numbers up to 10,000 and figuring out how to visualize and then solve multistep problems; bar modeling, for example, becomes a valuable tool for Third Graders. Another example of Third Graders’ progression from the concrete to the abstract occurs when working with fractions. Students initially find equivalent fractions by folding paper strips and using number lines, so that they conceptually understand how and why multiplication and division are used to find equivalent fractions. While learning multiplication tables to speed their problem solving, students continue to learn that mathematicians value the process of problem solving and work with persistence to find answers. Third Graders apply such learning as they approach problems involving money, metric measurement, and time. While math lessons follow the Third Grade Math in Focus curriculum, they also benefit from teacher-developed materials, daily math experiences, and hands-on activities.
In the Fourth Grade study of Mathematics, students build tenacity as they approach a variety of topics according to the Math in Focus model and through a real-world context for problem solving. Students learn to calculate time span through “graveyard math” on our historic campus, and to use their mathematical skills in preparing for the Fourth Grade Circus. (How many chairs per row will fit in the gym? How many tickets can we sell?). Students use estimation theory as they extend their number sense to 1,000,000. After reviewing large number multi-step addition and subtraction problems, Fourth graders work on 2-digit multiplication and division, addition and subtraction with decimals and fractions, and concrete and conceptual relationships between decimals and fractions. Work with tables and graphs, data and probability, and two-dimensional geometry complete Fourth Grade mathematics. Each area allows students to continue to explore their role as risk-takers and problem solvers, moving forward in comfort and confidence, as they affirm and concretize their understanding of previously learned concepts.
The three components of the Fifth Grade math curriculum—computation, concepts and confidence—spring from the Math In Focus curriculum and its consistent lesson structure that builds understanding with concrete-pictorial-abstract learning. Fifth Graders develop a broad repertoire of different problem-solving strategies, as well as habits of mind necessary for productive work which include patience, persistence, intellectual risk-taking, flexibility, and accuracy. Students learn to organize their thoughts logically through active listening and articulation of ideas and arguments. With number sense to 10,000,000, Fifth Graders multiply and divide with 2-3-digit numbers, fractions, mixed numbers, and decimals; they solve equations and evaluate inequalities; they use bar models and other strategies to solve complex pre-algebraic word problems. They practice geometry and graphing; a combined math/art project—enlarging an Aztec design using Cartesian coordinates—solidifies these skills, which are also used in mural- and map-making. Maintaining a positive attitude toward math and building students’ confidence in their abilities as mathematicians are other goals.
The Sixth Grade math program stresses versatility and math literacy. Centered on the middle school Course 1 of the Math in Focus program, students develop a deep understanding through real-world application of mathematical concepts and class discussion; they articulate their own processes and selection of strategies in problem-solving. Coursework is supplemented with hands-on activities, interdisciplinary projects, math history, and thematic units, often drawn from the companion text How Math Works. Sixth Grade work includes rational numbers, fractions, decimals and percent, ratios and rates, geometry and area, and measurement. Much of this work reinforces algebraic thinking and skills; Sixth Graders study formal algebraic expressions, equations and inequalities, tables and graphs, as well as data set work with statistical reasoning with mean, median, and mode. The spiral structure of the Sixth Grade curriculum strengthens students’ metacognitive ability to consider both concrete and abstract applications of mathematics, to consider and express multiple approaches in problem solving, and to understand the relevance of mathematics in all areas of the curriculum. Sixth Graders leave PMFS ready and excited for the opportunities and challenge of mathematics in their future academic endeavours.