Dr. Cottingham was appointed Interim Upper School Director in July 2021, and her Pingry experience includes three years as Upper School Academic Dean and a semester as Interim English Department Chair.
There are two primary goals of the elementary mathematics program. The first is development of strong number sense, or a child’s ability to use and understand numbers. The second is building resilience, which is perseverance when a concept is challenging or a first glance solution strategy is unsuccessful. Children with strong number sense think flexibly about numbers, make efficient mental calculations, estimate reasonably, and identify important mathematical relationships such as part-whole and place value. These skills are nurtured at Pingry through instruction in arithmetic, spatial reasoning, problem solving, and recording. Because young children’s brains are as different as their fingerprints, elementary students are encouraged to sophisticate their mathematical thinking at the individualized pace that matches their cognitive development. Activities are designed to allow students at every level to experiment with a variety of problem solving strategies and determine which ones work best for them.
The curriculum reflects grade-level standards established by the National Council of Teachers of Mathematics.
Beginning in Kindergarten, students are introduced to a way of thinking about mathematics that emphasizes strategy development, communication, and collaboration. Daily routines include recognizing combinations to 16, calendar math, collecting weather data, answering survey questions, and finding ways to represent the elapsed number of school days.
Counting is the foundation of arithmetic and is extremely important in Kindergarten. Students count sets that are naturally interesting to them: fingers, hands, people, pockets, days, and objects that relate either to the season or a topic of classroom study. A great deal of the counting in Kindergarten is done orally or mentally, but the children also learn to record their counting strategies on paper in order to explain their thinking. Kindergarteners use pictures, tally marks, numerals, and words to represent both fixed and accumulating quantities. Skip counting by 2s, 5s and 10s is also introduced in Kindergarten, first through active play with body movements and grouping objects, and later with tools such as the 100 chart.
Adding and subtracting are both introduced in Kindergarten as combining, separating, and comparing. Combining is putting groups together. Separating is removing part of a group. Comparing is finding the difference between two groups. The children first explore combinations to 10 and solve simple equations using pictures, fingers, and objects. As their understanding of the concepts deepens, they work on combinations up to 20 and begin to develop strategies for solving problems by drawing math diagrams and writing equations that represent the entire scenario.
Estimation is another component of number sense that is introduced in Kindergarten. Students learn to make estimates of quantity, weight, and measurement before they actually count, weigh, or measure their materials. Comparing their estimates to precise values guides students toward making more reasonable estimates as the year progresses.
Kindergarteners work extensively with patterns throughout the school year. They create, compare and extend visual and numerical patterns, both on paper and using manipulatives such as pattern blocks or interlocking cubes. Students also shrink, record and predict patterns. Data analysis is another important Kindergarten topic. Students collect, record, and represent data in a variety of ways all year. They also analyze data sets in order to solve problems.
In geometry, Kindergarteners identify, name, and explore relationships among 2-dimensional and 3-dimensional shapes, then combine them to make new shapes or cover a given area. They also measure in non-standard units, and write comparison statements of relative size. Throughout the school year, students continually question, estimate, and explain their reasoning in a comforting environment where math feels like play. Incorporation of art, reading, and active games makes the Kindergarten mathematics curriculum meaningful and enjoyable for the youngest students.
In first grade, students use coins, objects, number lines and 100 charts to count, combine, compare, and sequence numbers to 100. Daily routines include estimation, weather data analysis, time telling, money math, and calendar math.
In addition to their study of combinations and fact families to 20, first graders write equations, find the total of several single-digit addends, and fortify their strategies for solving increasingly complex addition and subtraction story problems. The children learn to draw math diagrams, such as pictures and tables, and matching equations to represent mathematical scenarios. First graders also skip count, forward and backward, by 2s, 3s, 4s, 5s and 10s.
An important concept that is introduced in Grade 1 is money math. The children identify and classify coins by appearance and value, then group and separate them to reinforce their understanding of counting on and counting back.
First graders interpret data that they have collected and categorized, and they begin to analyze data in quantitative terms. The children also create and extend increasingly complex patterns, including geometric patterns involving size, shape, and orientation. This leads them to their study of 2D and 3D geometry, which goes beyond naming and classifying shapes to solving covering, filling, relative size, and capacity problems. Then, through hands on experimentation, first graders discover language for describing weight, capacity, and length and use standard units to measure and compare them.
The first grade curriculum is extremely flexible, to accommodate the developmental stages of a range of learners. The children progress at their own pace from concrete, to pictorial, to abstract representations of mathematical scenarios. They also play a variety of engaging math games, which allows them to practice their math skills in a fun and social setting.
The second grade mathematics program balances daily skills practice with problem solving and hands-on exploration. Students begin to apply their knowledge of counting, basic addition and subtraction facts, and logical reasoning to increasingly sophisticated activities. Second graders learn to interpret mathematical language, choose appropriate solution strategies, and clearly record their thinking in diagrams, equations, tables, and words. Students often work in pairs or groups, sharing solution strategies and playing games. Explaining their reasoning to their peers cements the children’s understanding of their own strategies, and seeing a variety of solutions helps every student to build problem solving skills by internalizing the best of what they see presented to the group or class.
Second graders work toward mastery of addition and subtraction facts to 20. They use basic facts and landmark numbers, like multiples of 5 and 10, to help them solve addition and subtraction problems within 100. Students draw diagrams and write equations to represent combining, separating, and comparing scenarios. These scenarios can be modeled with manipulative materials like tiles or coins, drawn in pictures, counted on a 100 chart, or computed mentally. Second graders work extensively with coins. They count on by 1s, 5s, 10s and 25s to evaluate coin combinations, and they use coins to practice trading and making change. This naturally leads them to deeper understanding of place value to 100. Making the connection between skip counting and repeated addition prepares second graders to study multiplication, and they routinely practice number patterns and skip counting through 12s. They also build and record array models. Second graders explore topics in 2D and 3D geometry such as points, lines, polygons, and symmetry. They write and compare fractions representing equal parts or members of a group, and they perform data analysis and measurement of length, weight, and capacity. Daily classroom routines include time, temperature, and calendar math, estimation and prediction, mental math exercises, skip counting, and solving logic and discrete math problems.
In Grade 3, students continue to strengthen their number line fluency. They discover and predict patterns in the multiplication tables to 12, and model multiplication with arrays. Constructing and deconstructing arrays helps the children to recognize factor pairs and the relationship between multiplication and division, which prepares them to write and solve story problems involving grouping and partitioning. To parallel this study, third graders explore the effects of halving and doubling the terms in multiplication equations, and they begin extensive work with an application of the distributive property called “break-apart multiplication.” Grade 3 geometry combines visual-spatial reasoning with data analysis and linear measurement. The need for standard units is discovered by the children as they collect and analyze measurement data in a variety of non-standard units. Similar data is then recorded and evaluated in English or metric units. The emphasis in the children’s data interpretation is on making predictions that are based on qualities of either the set of data being studied, the manner in which it was collected, or both.
Third graders compare, combine and measure the area and perimeter of shapes in units and half-units. They also explore geometric transformations - slides, flips, and turns - to determine congruence. Addition and subtraction strategies become more sophisticated in Grade 3. Students learn to manipulate landmark numbers in the hundreds and thousands, and to calculate positive and negative net change. When calculating net change, the children use subtraction to cancel addition, and net change is also identified on graphs. Third graders use fractions and mixed numbers to build wholes from fractional parts and to solve sharing problems. The children gain familiarity with equivalent fractions, such as 1/6 + 1/3 = 1/2. They also recognize the embedded division problem in fraction notation, and use calculators to observe fraction and decimal equivalence. Daily routines include estimation and mental math, creation of equations to represent the date, probability discussion, and thinking flexibly about problem solving by exploring multiple ways to arrive at a numerically correct solution.
Fourth graders focus on developing their number sense in order to deepen their understanding of arithmetic and personalize their problem solving strategies. They work toward fluency with factor pairs and multiples, and learn to identify factors of numbers up to three digits. The children classify prime, composite, and square numbers according to the attributes of their array(s), and they solve complex problems by breaking them into manageable components like familiar multiplication relationships.
Fourth graders apply their knowledge of factors to division as they gain understanding of how division can represent either sharing or partitioning. The children also learn how to correctly express and label remainders, depending on the situation. Landmark numbers in Grade 4 include multiples of 10, 100, and 1000. Students learn how to quickly multiply by these numbers, and how to identify them in place value problems. Geometry enriches several topics of study throughout the Grade 4 year. Array modeling leads students to in-depth study of area and perimeter and to their study of fractions. The children partition rectangular areas into halves, fourths, eighths, thirds, sixths, and twelfths. Using both relative areas and numerical reasoning, they compare and order fractions and identify equivalent fractions. Estimation is encouraged when an unfamiliar fraction is encountered. For example, 12/25 is approximately ½. Fourth graders learn about the sides, vertices and angles of polygons, and how polygons are put together to form solid shapes with faces, corners and edges. Their solid geometry work extends to finding the volume of boxes. In statistics, the students advance from noticing individual features of data to describing the overall shape of the distribution and realizing how that yields usable information. They learn to express what is typical, what kind of landmark the median is, and how to use that information to compare data sets. Daily problem-solving routines include mental math and estimation, careful interpretation of mathematical language, drawing diagrams and writing equations, and simplifying arithmetic by deliberate manipulation of the terms in complex equations.
The Grade 5 curriculum builds on the foundation laid in earlier grades and emphasizes number sense, cooperative learning, critical thinking, and a variety of problem solving approaches. Through carefully planned problems, hands-on activities, and fun games, students make discoveries and deepen their understanding of topics rather than memorize procedural algorithms. Students are challenged to solve current and relevant real-world problems that are appropriate for a variety of ability levels. The children’s study of a collection of pre-algebra topics is enriched by STEAM challenges, in-depth explorations of pi, the Fibonacci Sequence, the Golden Rectangle, and Pascal’s Triangle, to name a few. In addition, the math and technology departments work together throughout the year to seamlessly infuse technology into the curriculum.
The fifth grade final project, titled Think Big, is to design and build a large, mathematically proportional 3-D model of a small object. For this project, students are introduced to basic engineering terms and the product design process. In order to complete their own prototypes, the children must apply many of the math skills that they have learned during the school year, including measurement, ratio and proportion, and geometry. The projects are celebrated at the end of the school year with a slide show and an exhibit for Grade 5 families.
Students spend the majority of the Grade 5 year studying the structures of integers and rational numbers. Early in the year, the children explore various topics in number theory such as factors, multiples, prime and composite numbers and prime factorization. Constructing and deconstructing whole numbers leads fifth graders to understanding of common factors and multiples, and to their in-depth study of rational numbers. Initially, students identify, construct and order benchmark fractions such as halves, thirds, tenths, and their equivalents. These benchmark fractions become their bases of comparison for positive and negative rational numbers in the fraction, decimal, or percent forms. Building rate tables and calculating unit rates prepare fifth graders to convert fractions to decimals and percentages and vice versa.
As the year progresses, rather than memorizing algorithms for fraction and decimal operations, fifth graders investigate realistic scenarios that require them to combine and partition rational numbers thoughtfully. Emphasis is placed on carefully choosing both the appropriate operation and the form of each rational number. Toward the end of the school year, fifth graders review area and perimeter. Through estimation, graphing, and experimentation with orientation and overlay, students discover formulas for calculating the areas of triangles, parallelograms, and other polygons. They extend this work to calculating surface area and volume of 3-dimensional shapes. Finally, fifth graders study measurement and data analysis and an introduction to variable expressions and equations. Throughout the school year, fifth graders work collaboratively to solve complex problems relevant to their classroom studies, which prepares them for the rigors of middle school math and beyond.