PHILOSOPHY

The San Francisco School broadly defines mathematics as a powerful tool to decipher the patterns that abound in our physical world. We believe that all students can learn important mathematics in a coherent and well-articulated program focused on actively building new knowledge from experience and prior knowledge. Teachers create an intellectual environment where serious engagement in mathematical thinking is the norm. Students learn to take control of their own learning and become confident in their ability to tackle problems, explore new ideas and persevere when tasks are challenging. They combine factual knowledge, procedural facility, and conceptual understanding in powerful ways. This approach gives students a rich store from which they can develop mathematical insight and creativity while also appreciating the inherent beauty of mathematics itself. 

MATH BY GRADE

Preschool-Kindergarten | First | Second | Third | Fourth | Fifth | Sixth | Seventh | Eighth

OVERVIEW

Math is everywhere at The San Francisco School.  Below is an article written by Glenn Kenyon, former 7th and 8th Grade Math teacher and SFS parent.  It specifically looks at algebra but offers a great introduction to math at the school.  Enjoy.

For many, algebra was a course on x’s and y’s either in 8th grade or high school. We carried around a fat book and we knew we hit the “big time” in math education when using sophisticated terms like “distributive property” and “quadratic formula”. What we likely never knew was that we had been doing algebra since pre-school and actually enjoying it!

In fact, developing algebraic thinking is a top priority in modern elementary and middle school curriculums. Algebra is now second in importance after arithmetic. The National Council of Teachers of Mathematics (NCTM) states that algebra is a cornerstone of mathematics.  In order to “see” the algebra being learned at The San Francisco School, you need to understand that algebra has various definitions reflecting diverse human perspectives over many centuries. For example, one can define algebra as the “language of mathematics”. This includes variables, number properties and manipulating symbolic expressions and equations (see photo). This algebra is being taught explicitly in the 8th grade at The San Francisco School and is commonly referred to as Algebra I. It is being implicitly taught in the 6th and 7th grade curriculum. For example, my 7th grade has recently been working on plotting numbers on a Coordinate Plane (x and y axes) in a unit on negative numbers. Melissa’s 6th graders have been working on the concept of equivalent fractions, which in turn leads to ratios and proportions, components of the slope in a linear equation.

Another definition of algebra is generalized arithmetic. Arithmetic is at the core of elementary math both here at The San Francisco School and most other schools. But arithmetic through an algebraic lens means not only learning how to do something (procedure) but also why we do it (concepts.) For example, in Jana’s 4th grade class, students are currently studying multiplication through arrays. Arrays are grid rectangles where length is one number of squares and width is the other. The area of the rectangle is the product of those two numbers. Students use arrays to identify patterns on the multiplication chart as well as define prime and composite numbers by their factors. This model not only explains multiplication, square numbers, factors, primes and composite, but also translates directly into quadratic equation models in the 8th grade.

Algebra is also a tool to study patterns. Seeking, expressing, and generalizing patterns and rules in real world contexts are evident throughout the SFS classrooms. I recently received a packet of work from my son’s experience in Pamela’s 1st grade class. I was impressed with the breadth of activities he had been doing in mathematics. One sheet, for example, included many rows of five teddy bears he had colored. He was obviously learning about all the combinations of sums that make five, but as he explained it:. “First, I colored all the bears red…that means 5+0. Then I colored all the bears yellow for 0+5. Then I colored one bear red and 4 bears yellow for 1+4… It’s a pattern, Daddy.”

The more formal name for what he did is “combinatorics”:  the logical analysis of all possible combinations based on identifying patterns. This is precisely the type of activity first graders need to be doing to internalize addition facts from a “hands-on/minds-on” basis while at the same time exploring the patterns they find as they color. Pamela’s teddy bear coloring sheet is deceptive: you think it is about addition when it really sows the seeds that later blossom in algebra.

In fact, algebraic thinking is sprouting up throughout our school. It is not limited to the 8th grade class. In a progressive school, where the meaning of what we do is as important as the activity itself, it is helpful to remember that math class can be as progressive and challenging as any other subject.

Preschool Math

At the three- to four-year-old level, much of the math concepts we focus on include a physical component, whether it be beads, buttons, pegs or stars drawn on paper. These concepts include one-to-one correspondence; understanding of the numbers 0 to 20, both as symbols and quantity; pattern recognition as well as the ability to create new patterns; sorting and classifying objects according to attributes such as size, color and shape; simple addition and subtraction using concrete concepts; graphs; and geometry, using shapes and 3D forms.

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Kindergarten Math

In kindergarten students place value; numbers up to 1,000; the concept of estimation; an introduction to time, both the calendar and clock; as well as measurement, including height, weight, non-standard and standard baking. Again, each of these concepts is built upon to take each child to the next level at the pace that is right for them.

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First Grade Math

The First Grade math curriculum recognizes that a critical aspect of our approach is breaking down each mathematical concept into developmental steps based on observation of students as they attempt to learn a concept. The activities are designed to engage students in problem solving, apply the concepts, and make connections between mathematical ideas.

Students engage in math activities for at least 50 minutes every day. Each concept is introduced using concrete materials, such as Unifix Cubes, Pattern Blocks, Base Ten Blocks, etc. Students record information for each activity and discuss their results with the class. After a concept has been explored with these materials, students are given small workbooks to complete without concrete materials to demonstrate their mastery of the concept.  The general progression of math instruction begins with concrete representations, moves into semi abstract (e.g. a drawing of the material) and finishes with more abstract symbolic form using numbers and words.  Students who excel are given further problem-solving activities while others who have difficulty in math are given extra help and guidance.

The principal concepts taught in First Grade are:

• Number (counting, one to one correspondence, building numbers, the relationships between numbers, using numbers, etc.);
• Addition/subtraction (sums and differences practiced to 20, memorized to 10);
• Count, read, and write whole numbers to 100;
• Count by 2s, 5s, and 10s to 100;
• Place value (regrouping with manipulatives; written double digit addition and subtraction with no regrouping);
• Money (coin names and values, values of mixed coins);
• Time (hour, half-hour, quarter-hours);
• Fractions (1/2, 1/3, 1/4).

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Second Grade Math

The Second Grade math curriculum recognizes that students will build on the concepts learned in first grade with a growing sense of independence and confidence. The primary role of teaching is to create situations for students that foster the consolidation of previous math skills, such as addition and subtraction, and while introducing new concepts such as multiplication and fractions. Greater reading fluency allows second graders to read increasing complex word problems and solve them using a variety of strategies. Students learn to explain their mathematical thinking and justify their work both orally as well as written.

Second graders have a better ability to work effectively with partners or small groups. Students expand their own independent competence with the various math strands by working collaboratively in a variety of math activities. Their conversations expand the repertoires of strategies, solutions and thinking of all students. New concepts continue to be introduced using concrete materials such as pattern blocks, base ten blocks, beans, and scales.  As students move to the more abstract methods of pencil and paper, the concrete materials continue to be available to students if they choose.

The principal concepts taught in Second Grade are:

• Addition and subtraction of multi-digit numbers;
• Addition and subtraction facts up to 20, and addition and subtraction with regrouping;
• Understand and use the inverse relationship between addition and subtraction
• In-depth exploration of place value to the thousands;
• Conceptual understanding of multiplication and division;           
• Conceptual understanding of fractions and fraction equivalency;
• Describe and analyze the properties of two dimensional shapes;
• Standard and non-standard linear measurement;
• Tell time to the nearest quarter hour and know relationships of time (minutes in a hour, days in a month, weeks in a year);
• Develop and evaluate mathematical arguments.

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Third Grade Math

The Third Grade math curriculum recognizes that students need ample opportunities to work with partners or small groups and to experiment with leadership and follower roles. Therefore, learning to solve problems individually and cooperatively is a primary goal. Math activities are organized so that students have many and varied experiences with numbers and symbols as well as geometry, probability and logic. In the process, they learn to make sense of new problems in the world as they encounter them.

When working in math, students begin with the familiar and concrete and move on to new territory. A math menu is offered each month, which gives students a variety of experiences in the different strands. Advanced problem solving and other extensions are offered to those who finish early or who have mastered a concept. Students are tested in math periodically and given extra help as needed.

The principal concepts taught in Third Grade are:

• Build on the concept of multiplication during a variety of activities;
• Memorization of multiplication facts through the twelve’s;
• Multiplication with one and two digits;
• Solve simple division problems;
• Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context;
• Add and subtract simple fractions;
• Know and understand that fractions and decimals are two different representations of the same concept;
• Choose the appropriate tools and units (metric and U.S.) and estimate and measure the length, liquid volume, and weight/mass of given objects.
• Identify, describe, and classify polygons (including pentagons, hexagons, and octagons);
• Find the perimeter of a polygon with whole number side measurements;
• Identify whether common events are certain, likely, unlikely, or improbable;
• Record the possible outcomes for a simple event (e.g., tossing a coin) and systematically keep track of the outcomes when the event is repeated many times;
• Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.

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Fourth Grade Math

Many fourth graders are beginning to understand abstract concepts and apply them to new problem solving situations. The Fourth Grade math curriculum is designed to provide classroom structures which will help develop a sense of personal and social responsibility, refine their math skills, polish their work habits, and support their innate thirst for knowledge.

TERC Investigations is the core mathematics curriculum for 4th grade. The goal is to help all children understand the fundamental ideas of number and operations, geometry, data, measurement and early algebra through hands-on and collaborative investigations. Proficiency with arithmetic skills is supported throughout the program. A variety of assessment techniques including observations, students’ self-assessments, quizzes, homework and individual oral assessment are used.  Extensions or review sessions are offered to accommodate students who need further challenges or additional help.

The principal concepts taught in Fourth Grade are:

• An understanding of, and the ability to use various algorithms for the addition and subtraction of multi digit numbers;
• Mastery of multiplication facts through twelve and multiplying by two and three digits;
• Division with remainders is introduced and mastered;
• Order fractions, finding equivalent fractions, adding and subtracting fractions with common denominators;
• Identification on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places;
• Recognition that rectangles that have the same area can have different perimeters;
• Knowledge of the definitions of a right angle, an acute angle, and an obtuse angle. Understand that 90°, 180°, 270°, and 360° are associated, respectively, with 1/4, 1/2, 3/4, and full turns;
• Knowledge of the definition of different quadrilaterals (e.g., rhombus, square, rectangle, parallelogram, trapezoid);
• Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns;
• Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work;
• Estimation, mental arithmetic and problem solving strategies are also reviewed and practiced.

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Fifth Grade Math

The fifth grade year marks an important transition as some students move from concrete to abstract reasoning skills. It is a year to consolidate all the arithmetic skills and concepts learned to date and apply them in new and fascinating ways. It also the year that simple algebra concepts are learned.

Fifth grade mathematics follows two distinct yet integrated paths: exploring the ideas behind the math while continuing to hone the calculation skills needed to explore the world with numbers. TERC Investigations is the core mathematics curriculum. The goal is to help all children understand the fundamental ideas of number and operations, geometry, data, measurement and early algebra through hands-on and collaborative investigations. Proficiency with arithmetic skills is supported throughout the program. A variety of assessment techniques including observations, students’ self-assessments, quizzes, homework and individual oral assessment are used.  Extensions or review sessions are offered to accommodate students who need further challenges or additional help. As much as possible, students apply their mathematical skills to solve practical, real-life problems. During the structures unit students build bridges with paper and calculate a load-to-weight ratio (the weight of the load supported before collapse to the weight of the paper bridge) in order to compare the relative strength of each structure. This is the same rating used by civil engineers in the field.

The principal concepts taught in Fifth Grade are:

• The history of number systems, their relative merits and how they compare to our modern system;
• Estimate, round, and manipulate very large (e.g., millions) and very small (e.g., thousandths) numbers;
• Representation and interpretation of fractions, decimals and percentages;
• Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers;
• Add, subtract, multiply, and divide with decimals;
• Prime factors of all numbers through 50 and write the numbers as the product of their prime factors by using exponents to show multiples of a factor;
• Calculations involving fractions with like and unlike denominators;
• Understanding the concept of volume and use the appropriate units in common measuring systems to compute the volume of rectangular solids;
• Angle measurement, the exploration of polygons and circles, including radius, diameter and  circumference;

• Visualization and draw two-dimensional views of three-dimensional objects made from rectangular solids;

• Organization and display single-variable data in appropriate graphs and representations and explain which types of graphs are appropriate for various data sets. Strategies for checking the accuracy of calculations.

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Sixth Grade Math

The San Francisco School math program is designed to prepare students who are developmentally ready to master Algebra in 8th grade to pass into a High School level Geometry course. Furthermore, it aims to build a strong foundation for others to succeed in High School Algebra.

Sixth grade students will be using the Connected Mathematics Curriculum to explore the topics of geometry, introductory algebra, and computations with fractions, decimals and percents. Each unit of study is focused around situations that help students to learn an important set of related mathematical ideas and to become skillful at using these ideas to solve problems.  The curriculum is structured to encourage students to communicate their strategies and their reasoning to deepen their mathematical understanding. 

As suggested by the National Council of Mathematics Teachers, emphasis is put on the following three curriculum focal points: 

Number and Operations:  Developing an understanding of and fluency with multiplication and division of fractions and decimals

Number and Operations:  Connecting ratio and rate to multiplication and division

Algebra:  Writing, Interpreting, and using mathematical expressions and equations

Sixth grade students will have a total of 5 class periods of math per week (approximately 225 minutes per week). Twice a week the students will be in a "double period" math class. These are designed to facilitate more in-depth investigations as well as integrate the use of manipulatives and real world applications.

The school expects students to learn to explain their mathematical thinking and justify their work both orally as well as written. The Connected Mathematics Curriculum is built around a clear sequence of skills and concepts that will serve as a strong foundation for future work in math. In addition, there is a 10-minute daily warm-up covering a wide range of math skills for students to review or learn as needed. The teacher will guide the class through the core curriculum, but his/her role will be to facilitate student discovery more than simply teach procedures or lecture. Throughout the problem-solving investigations and following their conclusion, the class is pulled together to share insights. They communicate these discoveries in a variety of ways including oral presentations, graphs, tables, formulas and written explanations.

Working together in small groups is an integral part of the learning structure. Most of the time students will be working through the investigations with one or three other students. They will discover ideas and share them in a collaborative environment. At other times, students may be involved in more independent and introspective work.

Individual assessment of mathematical learning is crucial both for the teacher as well as the student. For the teacher, it allows customization of lessons to meet the needs of all learners. Students benefit from knowing their strengths and areas of need. There are many forms of assessment, some formal, such as unit tests or quizzes, other less formal, such as oral presentations, and others informal such as anecdotal observation of group work.

Math homework is carefully assigned to support student learning. There are two main types of homework. The first is a set of 3-5 problems a day from Monday to Thursday related to the in-class investigations. This homework either reviews key skills or previews what will be coming up in the next unit. The average students could spend up to 20 minutes daily on each homework assignment. Many students complete this work during one of the three tutorial sessions during the regular weekly class schedule.

Every two weeks all sixth graders will be assigned a Problem of the Week (POW) in addition to the unit homework. A POW is a nonstandard math word problem that requires more sustained effort to resolve. The students write a one-page essay including a summary of the problem, description of procedure, solution and reflection. Each student will receive clear instructions on how to approach this homework.

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Seventh Grade Math

The 7th Grade math program is aimed at preparing students who are developmentally ready to master Algebra in 8th grade to pass into a High School level Geometry course. Furthermore, it aims to build a strong foundation for others to succeed in High School Algebra.

The main math curriculum are units developed by the Connected Mathematics Program, based on "Principles and Standards for School Mathematics" published by the National Council of Teachers of Mathematics. The NCTM curriculum focal points are emphasized: (proportional reasoning, including geometric similarity; 3-D measurement and formulas; and linear relationships)

The curriculum is organized into units the take 4-6 weeks to complete. Each unit is focused around problem situations called investigations. This organization facilitates discovery of key mathematical concepts in a problem solving setting. Communication and mathematical reasoning are emphasized throughout this program.

Structure for Learning
Seventh grade students will have a total of 5 class periods of math per week (approximately 225 minutes per week). Twice a week the students will be in a "double period" math class. These are designed to facilitate more in-depth investigations as well as integrate the use of manipulatives and real world applications.

The SFS expects students to learn to explain their mathematically thinking and justify their work both orally as well as written. The curriculum is built around core mathematical problems that allow students to work with the mathematics embedded in the situation. In addition, there is a 10-minute daily warm-up covering a wide range of math skills for students to review or learn as needed. The teacher will guide the class through the core curriculum, but his/her role will be to facilitate student discovery more than simply teach procedures or lecture. Throughout the problem-solving investigations and following their conclusion, the class is pulled together to share insights. They communicate these discoveries in a variety of ways including oral presentations, graphs, tables, formulas and written explanations.

Working together in small groups is an integral part of the learning structure. Most of the time students will be working through the investigations with one or three other students. They will discover ideas and share them in a collaborative environment. At other times, students may be involved in more independent and introspective work.

Assessment of Student Learning
Individual assessment of mathematical learning is crucial both for the teacher as well as the student. For the teacher, it allows customization of lessons to meet the needs of all learners. Students benefit from knowing their strengths and areas of need.

There are many forms of assessment, some formal, such as unit tests or quizzes, other less formal, such as oral presentations, and others informal such as anecdotal observation of group work. Every effort will be made to schedule unit tests on double periods. If a student feels he/she needs more time, arrangements will be made to finish the test another time. Other smaller assessments may occur throughout the week. Graded student work will be sent home very two weeks with a cover letter explaining the up-coming units as well as individual comments as deemed necessary by the teacher.

Homework
Math homework is carefully assigned to support student learning . There are two main types of homework. The first is a set of 3-5 problems a day from Monday to Thursday related to the in-class investigations. The average students could spend up to 20 minutes daily on each homework assignment.  Many students complete this work during one of the three tutorial sessions during the regular weekly class schedule.

Every two weeks all seventh graders will be assigned a Problem of the Week (POW) in addition to the unit homework. A POW is a nonstandard math word problem that requires more sustained effort to resolve. The students write a one-page essay including a summary of the problem, description of procedure, solution and reflection. Each student will receive clear instructions on how to approach this homework.

Once a quarter the students will be assigned a week-long project that is related to math. This will take the place of the regular math homework as well as the POW. They will be given one class period to work on it in class, but otherwise will complete it as homework.

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Eighth Grade Math

The San Francisco School math program is designed to prepare students who are developmentally ready to master Algebra in 8th grade to pass into a High School level Geometry course. Furthermore, it aims to build a strong foundation for others to succeed in High School Algebra.

The main math curriculum is the text Algebra Connections from College Preparatory Math (CPM), based on "Principles and Standards for School Mathematics" published by the NCTM (National Council of Teachers of Mathematics). This is an Algebra I program designed to prepare students to integrate basic skills and topics with conceptual understanding and problem solving strategies. This textbook is small, compact and very accessible for 8th grade Algebra students. Students will be provided with TI-83 graphing calculators and taught to use them in graphing situations.

Structure for Learning
Eighth grade students will have a total of 6 class periods of math per week (approximately 280 minutes per week). Twice a week the students will be in a "double period" math class. These are designed to facilitate more in-depth investigations as well as integrate the use of manipulatives and real world applications.

The SFS expects students to learn to explain their mathematically thinking and justify their work both orally as well as written. The CPM Algebra Connections curriculum is built around a clear sequence of skills and concepts that eventually lead to mastery of key algebra content. In addition, there is a 10-minute daily warm-up covering a wide range of math skills for students to review or learn as needed. The teacher will guide the class through the core curriculum, but his/her role will be to facilitate student discovery more than simply teach procedures or lecture. Throughout the problem-solving investigations and following their conclusion, the class is pulled together to share insights. They communicate these discoveries in a variety of ways including oral presentations, graphs, tables, formulas and written explanations.

Working together in small groups is an integral part of the learning structure. Most of the time students will be working through the investigations with one or three other students. They will discover ideas and share them in a collaborative environment. At other times, students may be involved in more independent and introspective work.

Assessment of Student Learning
Individual assessment of mathematical learning is crucial both for the teacher as well as the student. For the teacher, it allows customization of lessons to meet the needs of all learners. Students benefit from knowing their strengths and areas of need.

There are many forms of assessment, some formal, such as unit tests or quizzes, other less formal, such as oral presentations, and others informal such as anecdotal observation of group work. Every effort will be made to schedule unit tests on double periods. If a student feels he/she needs more time, arrangements will be made to finish the test another time. Other smaller assessments may occur throughout the week. Graded student work will be sent home very two weeks with a cover letter explaining the up-coming units as well as individual comments as deemed necessary by the teacher.

Homework
Math homework is carefully assigned to support student learning . There are two main types of homework. The first is a set of 3-5 problems a day from Monday to Thursday related to the in-class investigations. This homework either reviews key skills or previews what will be coming up in the next unit. The average students could spend up to 20 minutes daily on each homework assignment.  Many students complete this work during one of the three tutorial sessions during the regular weekly class schedule.

Every two weeks all seventh graders will be assigned a Problem of the Week (POW) in addition to the unit homework. A POW is a nonstandard math word problem that requires more sustained effort to resolve. The students write a one-page essay including a summary of the problem, description of procedure, solution and reflection. Each student will receive clear instructions on how to approach this homework.

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