Mathematics

Aspen Academy is committed to being on the cutting edge of education.  Bringing in award winning educators and innovative curriculum from around the world provides our students with the opportunities to become Renaissance boys and girls. 

One of the leaps we have taken to become an exemplary model school is to implement an international mathematics curriculum that propels children to their highest performance level in deductive and logical reasoning, mathematical algorithms, and contextual problem solving skills. 

Students will gain the foundational knowledge in mathematics that reaches far beyond the traditional bulimic approach to the subject area, in which students complete a series of workbook exercises and then regurgitate the information for an assessment. Instead, students are immersed in an educational journey, in which he or she believes that the solution is around the next corner, or the one after that, while he or she gathers the knowledge to continue the hunt. Once students have demonstrated mastery of the content, they are presented “Challenges by Choice” within that content area. These project-based, minds-on activities seek to move students from concrete and pictorial comprehension to abstract, experiential and project based inquiry learning, while cultivating their interests and ability levels. As students navigate through their individualized mathematics curriculum, they not only learn mathematics, but more importantly how mathematics is relevant and meaningful in the real world. 

In this overview, you will have the opportunity to understand the philosophy, methods, curriculum and contextual applications that comprise the Aspen Mathematics Academy.

ASPEN ACADEMY’S MATHEMATICS STANDARDS

After learning the results from the 2003 Trends in International Mathematics and Science Study that indicated United States students had continued a rapidly spiraling downward performance in math and science, Aspen Academy made a bold move. 

We chose to move our students toward the highest international mathematics and science standards in the world. Our model for foundational mathematics was established in Singapore where students have placed in the number one position in the world for the last decade. 

We added to that international model and standard, a commitment to real-life application of mathematics within our top-tier level contextual projects for advanced math students.  Aspen Academy is creating a mathematics program for which there is no equal.

We have taken the best program in the world and have added to it.

• The TIMSS Advanced Study (Trends in Mathematics and Science Study), measures how students are doing in math and science around the globe.
• American results have been poor. In the most recent Advanced survey, students from only two countries scored lower than U.S. school kids: Cyprus and South Africa

ASPEN ACADEMY’S MATHEMATICS CURRICULUM

Foundational Level:
Aspen Academy uses Singapore Mathematics Workbooks, Textbooks, Manipulatives and Enrichments for the foundational stage of mathematical unit study.

Pictorial and Abstract Levels:
Following is an example of programs and curriculum used at the pictorial and abstract levels. More resources will be added based on the needs of the students and the math content being examined.

• Miquon Mathematics Program
• Key to Mathematics Program
• Math Olympiad Program (E and M Division)
• Interactive Mathematics Program (Grade 9-12)
• Junior Achievement Stock Exchange and Financial Literacy Program
• Interact
• Illuminations
• SAT Prep (Grades 6-12)
• Rocky Mountain Talent Search
• Marilyn Burns Mathematics
• Essentials
• Brain Math I and II
• I-Excel
• Math Express
• Math Counts

THE MATH WORKSHOP

Aspen Academy employs the workshop model in its reading, writing and mathematics academic instructional programs. This workshop model is the single most effective class management system in promoting students’ opportunities to achieve their highest academic potentials.

In the Math Workshop model, students understand:

Mathematics Is Thinking
Mathematicians explore and investigate an often uncertain world and attempt to explain what they see. Sometimes a single explanation or answer works, but more often, patterns and relationships emerge that lead to several different conclusions. Logic helps mathematicians justify their methods and determine the validity of their solutions as they try to make sense of the unknown.

Mathematics Is Meta-Cognitive
Proficient mathematicians are meta-cognitive. They think about their own thinking while they do mathematics. They know when their solutions make sense and when they do not.  They can identify the demands placed on them by a particular problem and the methods they need to use to solve that problem. They can also identify when and why the demands of a problem are unclear to them, using a variety of strategies to understand what they need to do.

Mathematics Is Complex
Mathematics is a diverse domain that encompasses quantitative, spatial, and logical knowledge.  The pattern and order found throughout mathematics unify these diverse concepts and help people to construct relationships between and among mathematical ideas. While there is great diversity in the types of skills and thinking necessary to do mathematics, pattern and order create a common thread to help individuals make sense of a complex world.

Mathematics Is Language
Mathematics is an abstract, universal language that provides a way for individuals to communicate their thinking. Sets of symbols and carefully defined terms allow them to describe relationships and to order the world. Visual aids, such as diagrams, graphs, and manipulatives enhance the ability to communicate and justify mathematical ideas to others.
 

THINKING STRATEGIES FOR MATHEMATICIANS

Aspen Academy has applied the cognitive thinking strategies to mathematics, adapting them to the problem-solving processes that mathematicians employ. As students use these strategies across content areas, they learn to use them much more readily and purposefully.

Students as Mathematicians Learn to:

1.  Monitor for Meaning

• check to make sure answers are reasonable
• use manipulatives/charts/diagrams to help them make sense of the problem
• understand that others will build meaning in different ways and solve problems with different problem-solving strategies
• write what makes sense to them
• check their work in many ways (e.g., working backwards, redoing problems)
• agree/disagree with solutions and ideas
• “think aloud” about what’s going on in their head as they work through a problem
• are meta-cognitive, continually ask themselves if each step makes sense
• discuss problems with others and write about their problem-solving process to clarify their thinking and make problems clearer
• use accurate math vocabulary and show their work in clear, concise forms so others can follow their thinking without asking questions

2.  Activate, Utilize and Build Background Knowledge (Schema)

• use current understandings as first steps in the problem-solving process
• use their number sense to understand a problem
• add to schema by trying more challenging problems and hearing from others about different problem-solving methods
• build understanding based on prior knowledge of math concepts and develop purpose based on prior knowledge
• use their prior knowledge to generalize about similar problems and to choose problemsolving strategies
• develop their own problems

3.  Ask Questions

• ask questions (e.g., Could it be this? What happens if? How else could I do this? Have I seen this problem before? What does this mean?) before, during, and after doing a math problem
• test theories/answers/their hypothesis by using different approaches to a problem
• question others to understand their own process and to clarify problems
• extend their own thinking by asking themselves questions for which they don’t have answers

4.  Draw Inferences

• predict, generalize and estimate
• make problem-solving decisions based on their conceptual understanding of math concepts
• compose (like writers) by drawing pictures, using charts, and creating equations
• solve problems in different ways and support their methods through proof, number sentences, pictures, charts and graphs use reasoning and make connections throughout the problem-solving process
• conjecture (i.e., infer based on evidence)
• use patterns (i.e., consistencies) and relationships to generalize and infer what comes next in the problem-solving process

5.  Determine Importance

• look for patterns and relationships
• identify and use key words to build an understanding of the problem
• gather text information from graphs, charts, and tables
• decide what information is relevant to a problem and what information is irrelevant

6.  Creating Sensory Images

• use mental pictures/models of shapes, numbers, and processes to build understanding of concepts and problems and to experiment with ideas
• use concrete models/manipulatives to build understanding and visualize problems
• visually represent thinking through drawings, pictures, graphs, and charts
• picture story problems like a movie in the mind to help understand the problem
• visualize concepts in their head (e.g., parallel lines, fractions)

7.  Synthesizing Information

• generalize from patterns they observe
• generalize in words, equations, charts, and graphs to retell or synthesize
• synthesize math concepts when they use them in real-life applications
• use deductive reasoning (e.g., reach conclusions based on known information)

8.  Problem Solving

• listen to others’ strategies and adjust their own
• use estimation to determine if their answer is reasonable
• use trial and error to build thinking
• cross check by using more than one way to do a problem (e.g., check subtraction by adding)
• use tools (e.g., manipulatives, graphs, calculators) to enhance meaning

 

HOW DO WE TEACH MATHEMATICS EFFECTIVELY?

Educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding (Ginsberg, 1989; Kamii, 1985; Kamii, 1989; Yackel, et al, 1990). To understand what they learn, they must be active learners who enact for themselves the verbs that permeate the mathematics curriculum. This happens most readily when students work in groups, engage in discussion, make presentations, and take charge of their own learning (Everybody Counts, 1989).

Problem solving, rather than being a distinct topic to be covered, becomes the context in which all concepts and procedures are learned. When they truly solve problems, students engage in authentic tasks that they do not know how to solve in advance. Engaged in problem solving, students construct understanding through the processes of assimilation and accommodation with prior knowledge.

Effective math instruction can vary greatly depending on the setting and grade level. However, there are critical components to math instruction at all grade levels: Foundational Instruction, Projects, Math Tools, Flexible Grouping and Communication.

1.  Projects
Authentic projects integrate the students’ conceptual knowledge with their procedural knowledge, allowing them to apply this combined knowledge to real-world problems. When students prepare for a class camping trip, they might make predictions about the cost of the items on their grocery list. After shopping, they would compare their estimations to the actual cost of the supplies. Using the grocery store receipts, they would be able to create an accurate budget for their next trip.

2.  Math Tools
When students use tangible objects to construct the meaning of complex mathematical concepts, they develop a deeper understanding of the content that allows them to apply these concepts accurately in a variety of new settings. Too often, learners are simply taught math facts, without learning the tools to apply those facts. For instance, they learn the multiplication tables without being given the opportunity to really explore the meaning of multiplication. Math tools help students develop the conceptual understanding that is sometimes omitted from the curriculum. Hands-on materials do not lose their value in the upper grades. In fact, as concepts become more complex, these materials can help to accommodate the various learning styles of students so that they are more likely to
internalize the concepts.

3.  Flexible Grouping
Flexible grouping allows learners to work independently, with their peers, one-on-one with the teacher or in small groups with the teacher. Rather than assigning mathematica problems solely on the basis of students’ competency with a given concept, flexible groups allow students to articulate their own understanding of the content and to teach each other. Working with flexible groups also allows the teacher to provide individual feedback and direction. Working one-on-one or with a small group of students, the teacher can evaluate students’ conceptual and procedural abilities. The teacher is able to provide timely, individualized instruction and direct students toward manipulatives and exercises that will help them build their skills. Through conferences, the teacher can also evaluate whether or not a student understands and can apply a given mathematical concept. If a student is struggling, the teacher is available to provide immediate support and reinforcement. These informal assessments provide a window into the progress of the entire class, shaping instruction for future lessons, activities, and projects.

4.  Communication
Effective math instruction includes complex, open-ended problems in which multiple approaches to the problem, if not multiple answers, are possible. Teachers must ask complex questions to extend student thinking, and students must articulate their findings and the logic behind their processes. Small-group work and writing assignments give students this opportunity to clarify their own thinking and to synthesize what they have learned.
 

CREATING A MATH WORKSHOP

The idea of creating a content-focused workshop environment is nothing new. Artists and craftspeople have been using a workshop frame for years. When students are able to learn about math in a carefully established workshop classroom, they are given the time, instruction and reflection needed to fully develop their math know-how. The three major components of a literacy workshop are also evident in a math workshop:

1.  Crafting Session
This is the time in the workshop where students and their teacher explore the question “What do successful mathematicians think and do?” A crafting session is the time for explicit, precise mathematical instruction in math procedures, concepts, strategies, and skills.

2.  Composing as Mathematicians
During this portion of a math workshop, students put their new math learning to work, building the stamina necessary to take on more challenging math thinking. To make this time operate effectively, students need to have access to “in the moment” math tasks (assigned as a direct result of the crafting session) and “ongoing” math work (long-term math projects). Support for students’ learning can be offered through conferring and collaboration with others in small groups (teacher-guided as well as peer-guided).

3.  Reflecting
As is true of any classroom workshop, math time needs to end with students answering the question, “What do we now know about math and about ourselves as mathematicians that we didn’t know before?” This time needs to be more than simply sharing work. It should be a time for students to examine the many ways their thinking has evolved and become clearer. Math reflections can be done orally, as in a share circle or pair share, or in writing, as in an entry in a math notebook.
 

ASSESSMENT

Assessment should reflect the mathematics that all students need to know and be able to do. 
Teachers develop assessment strategies that allow students to display their developing math competencies and to document their progress towards meeting international standards.

Assessment should enhance mathematics learning. 
Teachers develop multiple sources of assessment that strengthen student learning while providing information to guide the teachers’ instruction. Such assessments include conferences, journals, tests, discussions, and projects.

Assessment should promote equitable opportunity for each child to reach their individual academic potential.
Teachers develop equitable practices and to accommodate students with special needs or talents.

Assessment should be an open process.
Teachers establish criteria, or rubrics, for assignments and communicate their expected standard of performance to their students.

Assessment should promote valid inferences about mathematics learning.
Teachers gather adequate and relevant evidence from multiple sources and to make valid inferences about student learning.

Assessment should be a coherent process.
Teachers develop strategies for planning assessment, gathering evidence, interpreting evidence, and using the results as an ongoing part of the math instruction.

Application of mathematical knowledge in problem-solving situations
Teachers develop instructional strategies, such as teaching through problem solving and using a solid framework, that help students become proficient problem solvers.

Communication of reasoning
Teachers develop questioning techniques that probe student thinking. Teachers learn instructional strategies that help students explain their thinking and justify their answers.

Development of meaningful mathematical knowledge
Teachers increase their own understanding of the math content of the international standards — number sense, patterns and algebra, data analysis including statistics and probability, geometry, measurement, and computation. They model lessons that focus on conceptual understanding and the meaningful development of mathematical procedures.
 

REFERENCES

Singapore Mathematics

Ginsberg, H. P. 1989. Children’s Arithmetic: How They Learn It and How You Teach It. 2nd ed.
Austin, TX: Pro Ed.

Hensee, D. 2002. Reworking the Workshop: Math and Science Reform in the Primary Grades.
Portsmouth, NH: Heinemann

Hiebert, J. and P. Lefevre. 1986. “Conceptual and Procedural Knowledge in Mathematics: An
Introductory Analysis.” in J. Heibert, ed. Conceptual and Procedural Knowledge: The Case of
Mathematics. Old Tappan, NJ: Macmillan.

Kamii, C. 1985. Young Children Reinvent Arithmetic: Implications of Piaget’s Theory. New York:
Teachers College Press.

Kamii, C. 1989. Young Children Continue to Reinvent Arithmetic, 2nd Grade. New York: Teachers
College Press.

National Research Council. 1989. Everybody Counts: A Report to the Nation on the Future of
Mathematics Education. Washington, D.C.: National Academy Press.

Yackel, P. Cobb, T. Wood, G. H. Wheatley, and G. Merkel. 1990. “The Importance of Social
Interaction in Children’s Construction of Mathematical Knowledge.” in T. J. Cooney, ed. Teaching
and Learning Mathematics in the 1990’s. Reston, VA: National Council of Teachers of Mathematics.

Workshop content provided by the PEBC. Aspen Academy teachers are trained by national
trainers from the PEBC in workshop development.
 

Click here for Aspen Academy Stages of Mathematical Development Within Each Unit of Study document.