Mathematics

Our students engage in Mathematics, not just arithmetic. While our students develop fluency with numbers and variables and develop mastery of the traditional algorithms during their time at Speyer, we challenge our students to see Mathematics as more than just memorizing procedures and performing calculations. Once students find an answer to a problem, they ask themselves how they know their answer is correct--but that's just the first question! They also ask, among many other questions, whether they could have been more efficient in solving the problem and if their answer is the only possible one. At Speyer, Mathematics is much more than just finding an answer to a problem; instead, we see Mathematics as the way we reason about--and pose--problems. Through the development of explicit problem solving and metacognitive strategies, as well as through the way we invite and consider multiple potential solution methods and answers, we instill in our students a mathematician's habits of mind: exploring, conjecturing, proving, estimating, checking, planning, abstracting, and generalizing.

We recognize that every student who arrives at Speyer is unique. Each student has a different history and relationship with Mathematics, coming to Speyer with differences in natural ability, prior knowledge, and the experiences and opportunities they've had related to Mathematics. Despite these differences, we believe that every student at Speyer can achieve at high levels, and we provide all students an accelerated curricular experience in every grade. To support our students, we form small groups for Mathematics instruction so that each student can receive increased teacher attention and feedback on their independent work; as students make progress, and recognize their own progress, they grow in their self-confidence as capable and competent mathematicians.

As each student develops into a capable and competent mathematician, we also approach Mathematics as a social activity. Students regularly have the opportunity to complete work in groups, allowing students to learn from and build off of each other's work. Every student is empowered to feel like they can provide meaningful contributions to group work; at the same time, students come to the realization that many problems require the collective expertise of a group whose individual members each bring their own unique strengths and perspectives.

Our goal is to ensure each student feels both supported and appropriately challenged, and we recognize that some students are ready to tackle more advanced Mathematics than their same-age peers. We place no ceilings on our students; we continuously monitor the growth of each student, and as students demonstrate mastery with the content being taught, our faculty members collaborate to create individualized and small group learning plans consisting of enrichment or increased acceleration. Our students are actively learning and being challenged every day, and there is always more to learn.

Though Mathematics is already worth studying, and is best studied, for its own sake as a distinct, pleasurable, and intellectually rich subject, Mathematics also has a practical use meriting its study and granting it a privileged place among other subjects: Mathematics provides a shared language and a toolkit practitioners of all fields can use to represent, to understand, and to better the world we live in.

Detailed divisional and grade-level descriptions of Speyer’s math curriculum can be found below.

LOWER SCHOOL MATHEMATICS

In the Lower School, math periods are combined across both home rooms, which allows for five teachers (the four grade-level teachers and our Lower School Math Specialist) to lead small-group instruction allowing for greater differentiation. The accelerated math curriculum incorporates Singapore math along with the “Math in the City” but, overall, focuses on material designed by our own math faculty to meet the needs of our gifted learners.

As our students progress from grade to grade, our goal to expose students to a variety of techniques to add versatility in their mathematical thinking opposed to applying algorithms. This enables students to focus on conceptual understanding beyond basic computation.  In addition to strengthening students’ facility in making calculations, we emphasize decoding and solving complex word problems. Students develop the essential skill of reading a problem and determining what calculations are required to determine a solution.

Problem solving and perseverance are two constant themes in our Lower School mathematics instruction. There is an internal joy that can be found by solving a difficult puzzle or exploring a situation. The ability to persevere with a task that at first seems unattainable is one of the many ways we try to challenge our students.

In addition to their day-to-day math instruction, students are offered the opportunity to participate in a variety of math competitions and events throughout the school year, including the Noetic Math and Continental Math League competitions and the Mathematical Olympiads for Elementary and Middle Schools.

An example of our Lower School math curriculum and the skills and concepts explored by grade level are below. Please note that these lists are not the full scope of the math curriculum for each grade — they are highlights of the minimum level of mastery for each skill. If a student demonstrates mastery of ability beyond these concepts, our faculty works to ensure their learning potential is supported.

Lower School Math Highlights:

  • Each Kindergarten student at Speyer begins their mathematical journey learning to represent numbers beyond quantities they can measure. 

  • By the end of 4th grade a student will show competence with rational numbers, all four operations, and geometry. 

  • Exceptionally advanced students are offered courses in algebra, number theory, and combinatorics.

  • Patterns
    —     Understand that patterns consist of repeating units
    — Be able to name patterns using letters (i.e., AB, ABBC)
    — Continue patterns after identifying units

    Whole Numbers
    —     Identify and write the numbers 1 to 20
    — Find combinations to make the numbers 2 to 10
    — Relate number bonds to addition equations
    — Find the sum of two numbers by making 10s

    Shapes and Solids
    —     Recognize, describe, compare and sort 2-D and 3-D shapes
    — Compose and decompose 2-D and 3-D shapes

    Decomposition and Subtraction
    — Relate subtraction to taking away
    — Relate subtraction to taking apart
    — Relate number bonds to an unknown part

    Measurement
    —     Measure length and use linear units
    — Develop strategies for measuring the length of an object
    — Use multiple nonstandard units to quantify length
    — Understand the two hands of a clock and what they represent
    — Read digital and analog clocks to the nearest 15 minutes

    Numbers to 100
    —     Use place-value models to represent numbers to 100
    — Read, write in words, standard and expanded notation, identify place values of digits, and compare and order numbers within 100
    — Count and identify 1 more than, 1 less than, 10 more than, 10 less than a number within 100
    — Add/subtract numbers within 100.
    — Count by 10s within 100.

    Money
    —     Identify and relate coin values (penny, nickel, dime, quarter)
    — Count and make simple coin combinations

  • Place Value
    — Identify and name numbers up to 100
    — Regroup numbers when more than 10 items are in each place
    — Jump by 10s and 1s on the Hundred Square
    — Work with 100s, 10s, and 1s
    — Compare a pair of numbers by their place values

    Addition
    — Add two numbers by adding their place values
    — Understand that addition can be considered “counting up” and can be solved in any order

    Subtraction
    — Compare numbers by using subtraction to find the difference
    — Understand that subtraction can be thought of as “taking away”, “taking apart” or “counting back”
    — Understand that subtraction can be approached as finding the missing number of a sum

    Foundations of Multiplication
    — Connect addition and multiplication
    — Represent multiplication as skip counting, arrays, and picture models

    Foundations of Division
    —     Use sharing and grouping to divide
    — Understand that division is an unknown multiplication sentence
    — Connect division and subtraction

    Measurement
    Money
    —     Identify the value of pennies, nickels, dimes, quarters, and dollar coins
    — Skip count by 1, 5, 10, 25
    — Understand that money values can be written using the $ or ¢ sign and a decimal point when necessary, can be made using different coin combinations, and that American coins are not the same as coins in other countries

    Time
    —     Identify numbers on an analog clock face have different values depending on which hand is pointed to it
    — A.M. refers to the hours between midnight and noon
    — P.M. refers to the hours between noon and midnight

    Measurement
    —     Measure length and use linear units
    — Develop strategies for measuring the length of an object
    — Use multiple nonstandard units to quantify length

  • Whole Numbers
    —     Understand and use place value (no limit)
    — Perform operations with multiples of 10
    — Round, order, and compare numbers
    — Regroup numbers when more than 10 items are in each place
    — Identify number patterns

    Addition Strategies and Algorithm
    — Represent their math thinking
    — Add multi-digit numbers
    — Represent addition with the standard algorithm
    — Use appropriate mental math strategies

    Subtraction Strategies and Algorithm
    —     Subtract multi-digit numbers
    — Represent subtraction with the standard algorithm
    — Regroup when subtracting
    — Use appropriate mental math strategies

    Word Problems
    —     Represent the unknown of a word problem using Part-Part-Whole or Comparison bar models
    — Solve multi-step word problems using Bar Model Strategies

    Multiplication strategies and algorithm
    — Relate addition and arrays to multiplication
    — Multiply single-digit numbers
    — Use appropriate mental math strategies

    Division
    — Use sharing and grouping to divide
    — Relate division to multiplication
    — Use appropriate mental math strategies
    — Fluent dividing numbers within 100

    Measuring Length
    — Estimate lengths using units of inches, feet, centimeters, and meters
    — Measure to determine how much longer one object is than another

    Capacity and Mass
    — Measure and estimate mass
    — Estimate and calculate capacity

    Time
    — Tell time to the nearest minute
    — Find the duration of a time interval
    — Find the starting or ending time, given a time and the interval

    Geometry
    — Identify, describe and categorize common 2-D shapes, including faces of 3-D shapes
    — Identify common 2-D shapes within compound shapes, combine shapes to form common shapes
    — Describe and classify common 3-D shapes according to the number and shape of faces, edges and vertices

    Fractions
    —     Recognize halves and fourths
    — Recognize, write, name, and illustrate fractions of a whole
    — Find the fraction with the same denominator to make a whole with another fraction
    — Compare and order unit fractions

    Money
    —     Count Combinations of money to $100.00
    — Use decimal notation for money
    — Add and subtract money

  • Place Value
    —     Understand and use place value for numbers up to 1,000,000
    — Perform operations with multiples of 10
    — Round, order and compare numbers up to 1,000,000
    — Regroup numbers when more than 10 items are in each place

    Addition and Subtraction
    — Add numbers up to 100,000
    — Subtract numbers up to 100,000

    Multiplication and Division
    —     Multiply three-digit numbers by one-digit numbers
    — Divide three-digit numbers by a one-digit divisor
    — Solve a division problem that has a remainder
    — Solve multi-step word problems

    Problem Solving
    —     Represent the various ways to represent an unknown word problem
    — Solve multi-step word problems using the Bar Model Strategies

    Fractions
    —     Understand fractions as numbers
    — Express equivalence of fractions
    — Use equivalent fractions to add and subtract fractions
    — Extend knowledge of multiplication with fractions

    Measurement
    —     Find the duration of a time interval
    — Find the starting time or ending time, given a time and the interval
    — Convert units of time
    — Measure length, weight, mass and capacity
    — Convert units of measurement
    — Add and subtract measurements in compound units

    Geometry
    —     Find the perimeter and area of polygons using appropriate units
    — Understand and use units of area, such as square centimeter and square inch
    — Identify acute, obtuse, and right angles and relate 90, 180, 270 and 360 with quarter, half, three-quarter, and whole turn
    — Identify perpendicular and parallel lines
    — Reason with shapes and their attributes

    Money
    —     Add and subtract money within $100.00

  • Whole Numbers and Mathematical Expressions
    —     Perform operations with multiples of 10, 100, and 1,000
    — Simplify expressions with parentheses and exponents
    — Use the order of operations
    — Find the greatest common factor and least common multiple of two numbers
    — Identify a number as prime or composite

    Multiplication and Division
    —     Multiply a three-digit by a two-digit number
    — Divide a four-digit number by a one-digit number
    — Effectively model and solve multi-step word problems

    Fractions
    —     Represent equivalent fractions
    — Add and subtract mixed numbers
    — Apply and extend previous understandings of multiplication to fractions
    — Apply and extend previous understandings of division to fractions

    Data and Statistics
    —     Collect, organize, and interpret data in a table
    — Create a table from data in a tally chart and a bar graph
    — Make, read, and interpret line graphs
    — Find the mean, median, mode, and range of a data set
    — Interpret which measure of central tendency gives the most accurate understanding of the data set

    Decimals
    —     Compare, order, and round decimals (up to thousandths)
    — Convert fractions to terminating decimals and vice versa
    — Add, subtract, multiply, and divide decimals
    — Solve word problems involving decimals

    Geometry
    — Find the area, perimeter, and unknown sides of rectangular shapes
    — Find the area, perimeter, and unknown sides of composite rectilinear shapes
    — Count unit cubes in solid figures
    — Find the volume of rectangular prisms
    — Identify angles up to 360°
    — Name and relate triangles and quadrilaterals based on their traits

MIDDLE SCHOOL MATHEMATICS

Middle School Mathematics at Speyer is guided by three simple principles:

  • Technical proficiency is indispensable, but active reasoning is supreme.

  • A mathematics course must challenge the intellect, inspire imaginative thinking, and nurture mature self-confidence.

  • Mathematics confers huge pragmatic benefits, but the best reason to do mathematics is for pleasure.

Speyer Middle School courses in mathematics are accelerated above nominal grade level by at least one year. Courses are customized by our teachers to meet the needs of our gifted learners. In addition to our extensive collection of proprietary materials, we draw from published texts oriented toward gifted students, intended for high school and college courses, or based on problems from local and international Mathematics competitions, including the Art of Problem Solving series.

All Speyer graduates receive at least one full year of high school or higher level instruction. Fifth and Sixth Grade courses condense what would normally be considered Middle School Mathematics into a two-year sequence. Our Seventh and Eighth Grade mathematics courses are distinct from most Middle School curricula in their emphasis on topics such as mathematical proof, number theory, and combinatorics. Exceptionally accelerated students have the opportunity to study College Algebra, Precalculus, Calculus, and other college-level material.

Speyer Middle School also offers two afterschool math teams: a junior varsity team for students in Grades 5-7 and a varsity team for students in Eighth Grade (students in other grades may be invited to participate with the varsity team on a case-by-case basis). Members of our Middle School math teams represent Speyer in several prestigious competitions, including AMC 8 and AMC 10, MATHCOUNTS Manhattan and New York State MATHCOUNTS Competition, Math Olympiads for Elementary and Middle Schools, and New York Math League. In the past, Speyer has hosted Girls’ Adventures in Math, a contest originated in 2016 by Speyer in cooperation with Math-M-Addicts.

Recent Team Highlights:

  • First among all independent schools at both MATHCOUNTS Manhattan and New York State MATHCOUNTS Competition in 2017, 2019, and 2020. One qualifier for MATHCOUNTS National Competition in 2020.

  • Multiple qualifiers for the American Invitational Mathematics Examination (AIME) in 2018 and 2020, requiring scores in the top 2.5% nationally on AMC 10 competing against students up to 10th grade.

  • Girls’ Adventures in Math (GAIM): multiple medal-winning teams in both divisions.

Examples of our Middle School math curriculum and highlights of the skills and concepts explored in each course are below. Please note that these list are not the full scope of mathematic curriculum for each grade; if a student demonstrates mastery of ability beyond these concepts, our faculty works to ensure their learning potential is supported.

Middle School Math Highlights:

  • By the end of 8th grade, every Speyer student completes at least one and a half years of high-school math, including a rigorous course in Algebra 1.

  • Most Speyer middle schoolers complete two or three years of high-school math, including proof-centered Geometry, Algebra 2, and Trigonometry.

  • Exceptionally advanced students are offered courses in college-level math, including Calculus.

  • Fifth Grade courses in Accelerated Middle School Mathematics and Enriched Middle School Mathematics synthesize and generalize the material learned in the Lower School, provide broad exposure to diverse areas of Mathematics, and incorporate both informal and formal early algebraic thinking and reasoning. Topics of study include number systems and theory, multiplicative thinking and proportional reasoning, two- and three-dimensional geometry, and spatial reasoning, discrete probability and descriptive statistics, and linear equations in one variable. For students already demonstrating mastery in our accelerated curriculum, Speyer occasionally offers a fifth grade Pre-Algebra and Problem Solving courses that serves the needs of profoundly math-gifted students who show a personal drive to study Mathematics at an exceptional pace.

    Accelerated and Enriched Mathematics Course Highlights
    —     Operations with Natural Numbers and Decimals
    — Operations with Negative Numbers
    —     Number Theory and Exponentiation
    — Representations of Rational Numbers
    — Percents, Ratios, and Rates
    — Exposure to Geometry
    — Probability and Statistics

    Pre-Algebra and Problem Solving Course Highlights
    —     Integers and Rational Number Operations
    — One Variable Linear Equations
    — Irrational Numbers in Triangles
    — Operations with Polynomial and Rational Expressions
    — Ratios, Percents, and Similarity
    — Transformations and Complex Numbers in Rectangular Form

  • All Sixth Grade courses serve as courses in Pre-Algebra. Sixth grade students take either Accelerated Pre-Algebra or Enriched Pre-Algebra, and sixth grade students particularly gifted in Mathematics have the opportunity to study much, or even all, of a high school Algebra 1 course. Sixth Grade students revisit and extend previously encountered ideas using the language of algebra; draw connections among geometric, arithmetic and algebraic concepts; and begin working with constructs from Algebra 1 including functions and systems of equations. Toward the end of the year, some students may also study more advanced algebraic and problem solving concepts and theories.

    Accelerated Pre-Algebra Course Highlights
    —     Forms and Operations of Rational Numbers
    — Scientific Notation and Unit Conversions
    — One-Variable Linear Equations
    — Graphs and Equations of Lines
    — Transformations and Area
    — Topics in Probability and Statistics

    Enriched Pre-Algebra Course Highlights
    — Linear Equations in One or More Variables
    — Scientific Notation, Units, and Rate Conversions
    — Graphs and Equations of Lines
    — Irrational Numbers and Triangles
    — Similarity Transformations, Shapes, and Solids
    — Topics in Descriptive Statistics
    — Topics in Discrete Probability

    Algebra and Problem Solving Course Highlights

    — Rules of Divisibility and Number Bases
    — Linear Equations
    — Multiplying Polynomials
    — Number Theory Basics
    — Exponents and Exponential Equations
    — Modular Arithmetic
    — Linear Functions
    — Linear Systems
    — Inequalities
    — Geometry and Measurement
    — Factoring Polynomials
    — Quadratic Equations

  • In Seventh Grade, a Speyer student gains a more formal understanding of algebra and different function families by following one of three curricula: Introduction to Algebra; Algebra 1; or Algebra 1 and 2.

    Introduction to Algebra Course Highlights
    —     Rational Number Operations
    — Exponents and Irrational Numbers
    — Linear Equations in One or More Variables
    — Linear Functions
    — Introduction to Nonlinear Functions
    — Nonlinear Functions
    — Exponential Functions
    — Comparing Linear, Quadratic, and Exponential Functions

    Algebra 1 Course Highlights
    —     Linear Equations and Inequalities
    — Linear Functions and Modeling
    — Exponents and Irrational Numbers
    — Quadratic Functions
    — Rational Expressions and Complex Number Arithmetic
    — Exponential Functions
    — Piecewise Functions
    — Comparing Data Distributions

    Algebra 1 and 2 Course Highlights
    —     Linear Inequalities in One- and Two-Variables
    — Exponents and Irrational Numbers
    — Quadratic Functions
    — Polynomial and Rational Equations
    — Exponential Functions
    — Inverse Functions
    — Sequences
    — Putting Functions Together
    — Statistics on the Graphing Calculator

  • In Eighth Grade, a Speyer student develops their visualization skills and spatial thinking through a course in Bridge to Geometry, Euclidean Geometry, or Euclidean Geometry and Trigonometry.

    Bridge to Geometry Course Highlights
    —     Linear Inequalities in One- and Two-Variables
    — Quadratics: Parabolas and Circles
    — Families of Functions
    — Statistics on the Graphing
    —Calculator
    — Constructions with Compass and Straightedge
    — Transformations
    — Introduction to Trigonometry
    — Area, Surface Area, and Volume

    Euclidean Geometry Course Highlights
    —     Proofs in Symbolic Logic
    — Geometry Proofs with Angle and Triangle Congruency
    — Isometries
    — Parallelism and Quadrilaterals
    — Dilations and Similarity Transformations
    — Introduction to Trigonometry
    — Properties of Circles
    — Shapes and Solids

    Euclidean Geometry and Trigonometry Course Highlights
    —     Angle, Segment, and Triangle Congruency
    — Parallelism and Quadrilaterals
    — Similar Triangles
    — Right Triangle Trigonometry
    — Complex Numbers
    — Unit Circle Trigonometry
    — Circles and Solids
    — Compass and Straightedge Constructions
    — Transformations