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Secondary Mathematics

The State Board of Education adopted the Core Standards in July 2010. Since that time, the decision was made to craft a set of PA Core Standards in English Language Arts and Mathematics. A group of Pennsylvania educators created a draft set of PA Core Standards. These new standards mirror the content and rigor of Common Core, but reflect the organization and design of the PA Academic Standards.

The PK-12 PA Core Standards for Mathematics cannot be viewed or addressed in isolation, as each standard depends upon or may lead into multiple standards across grades; thus, it is imperative that educators are familiar with both the standards that come before and those that follow a particular grade level. These revised standards reflect instructional shifts that cannot occur without the integrated emphasis on content and practice. Standards are overarching statements of what a proficient math student should know and be able to do. The Pennsylvania Assessment Anchors and Eligible Content closely align with the revised standards and are an invaluable source for greater detail.

What it Means for Students?

Key Points in Mathematics:

  • The standards stress both procedural skills and conceptual understanding to ensure students are learning and applying the critical information they need to succeed at higher levels.
  • K–5 standards, which provide students with a solid foundation in whole numbers, addition, subtraction, multiplication, division, fractions, and decimals, help young students build the foundation to successfully apply more demanding math concepts and procedures, and move into application. They also provide detailed guidance to teachers on how to navigate their way through topics such as fractions, negative numbers, and geometry, and do so by maintaining a continuous progression from grade to grade.
  • Having built a strong foundation at K–5, students can do hands-on learning in geometry, algebra, and probability and statistics. Students who have mastered the content and skills through the seventh grade will be well-prepared for algebra in grade 8.
  • High school standards emphasize applying mathematical ways of thinking to real world issues and challenges.

Teachers shall expect that students know and can apply the concepts and skills expressed at the preceding level. Consequently, previous learning is reinforced but not re-taught. Students who achieve these mathematical standards will be able to communicate mathematically.

What Does that Look Like In the Classroom?

K to 12 Standards for Mathematical Practice

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

In Upper Perkiomen School District, the new PA Core State Standards for Math mean deeper, more meaningful learning for our students.

How is this different from what students have done in the past?

The new standards will ask students to think more conceptually, and to think deeper and even more thoroughly about what they're learning. The new standards go beyond basic memorization to help students truly understand what they are learning.

Previously, students might have been given a word problem that required them to memorize a formula to calculate something like "area."

For example:

Under the old PA Standards, students might have been asked the follow:

  • 6 x 6
  • Mrs. Brown's class has a rabbit pen that is 6 feet long by 6 feet wide. How much room do the rabbits have to run around?

Under the new Common Core Standards, a word problem will more likely look like this:

Mrs. Brown's class has 24 feet of fencing to build a pen for rabbits. They want the rabbits to have as much room as possible. How long will each of the sides be?

The first problem simply requires a calculation: "Area" is defined by calculating length times width (for a rectangle or square.) In this case, simply multiply 6 x 6. The second problem requires the same calculation, but requires much more in depth thinking.

What is some of the knowledge and what are some of the questions that students need to know and answer to solve the problem?

  • What "perimeter" means (how many sides/combinations should they examine - should the pen be a rectangle, an oval, etc. and which would give the rabbits the most space)
  • How do you calculate the area of various shapes?

In order to solve this problem, the student may have to experiment with several formulas and concepts. They have to work through the problem. This process is referred to in educational circles as "productive struggle" or "brain sweat." This is the process of taking prior knowledge and working through the problem individually and then in small groups, during which time a teacher circulates throughout the class to help guide thinking pathways and conversations.

The workforce demands that children go beyond rote memorization and be able to solve real world problems. The programs selected by Upper Perkiomen School District staff and administration are aligned to the PA Core Content and Practices, create a cohesive K to 12 math program, provide "productive struggle", and challenge students at higher levels while providing more intensive help for students who need more time to learn.

To ensure that students were prepared to master these standards, the mathematics departments adopted the College Preparatory Mathematics (CPM) at the middle school and Carnegie Math program for math courses 9-12 during the most recent department curriculum review.

Middle Level Program Resource: CPM

CPM Educational Program, a California non-profit corporation, has provided problem-based instructional materials and professional development for teachers since its inception in 1989. “College Preparatory Mathematics (CPM)” was originally an Eisenhower-funded grant program. CPM teaching strategies focus on how students best learn and retain mathematics. Teaching strategies rely on the recommendations of the National Council of Teachers of Mathematics, and are based solidly on the methodological research in teaching mathematics. The research-based principles that guide the course are:

  • Students should engage in problem-based lessons structured around a core idea.
  • Guided by a knowledgeable teacher, students should interact in groups to foster mathematical discourse.
  • Practice with concepts and procedures should be spaced over time; that is, mastery comes over time.

CPM’s mission is to empower mathematics students and teachers through exemplary curriculum, professional development, and leadership. We recognize and foster teacher expertise and leadership in mathematics education. We engage all students in learning mathematics through problem solving, reasoning, and communication.

CPM envisions a world where mathematics is viewed as intriguing and useful, and is appreciated by all; where powerful mathematical thinking is an essential, universal, and desirable trait; and where people are empowered by mathematical problem-solving and reasoning to solve the world’s problems.

To learn more about CPM, please , click on the this CPM Video.

High School Level Program Resource: Carnegie Math

Carnegie Learning’s instructional approach is based upon the collective knowledge of our cognitive learning scientists, master practitioners, and ongoing research initiatives. Not only is it aligned with the most current math education standards, it’s based on a scientific understanding of how people learn, and a real world understanding of how to apply that science to conceptual math understanding as well as deeper learning skills like creativity, collaboration, critical thinking and communication. It’s not just smart, it’s practical.

At its core, the instructional approach is based on three simple, yet critical components.

ENGAGE: Materials draw students in, activating their prior knowledge and experiences, presenting real-world examples, and facilitating collaborative classroom activities that generate curiosity and plant the seeds for deeper learning.

DEVELOP: Rigorous and challenging opportunities for group and independent learning, helping teachers identify where an individual student needs additional support and how best to adjust instruction to make progress and build confidence.

DEMONSTRATE: Students are expected to bring it all together and show what they know. Ongoing formative assessment underlies the entire learning experience, driving real-time adjustments, insights, and measurements in a way that is invisible to the student.

Carnegie Learning combines consumable textbooks, intelligent 1-to-1 math tutoring software, and transformative professional learning and data analysis services into a comprehensive and cohesive learning solution.

In our high school math classrooms, students will have access to not only the textbooks, but also to a digital resource know as Mathia.

MATHia coaches and adapts, combining a sophisticated model of math skills and collecting literally thousands of data points as students work through multi-part — not simple multiple choice or single answer — problems. That’s not frustrating, inefficient math practice—that’s targeted, empowering math learning.

Like a human tutor, MATHia re-phrases questions, re-directs the student, and hones in on the parts of the problem that are proving difficult for the student. Hints are customized to address the individual student, understanding that there are often multiple ways to do the math correctly.








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