Mathematics Learning Standards

Program Purpose

Mathematics provides an environment for logical, creative investigation of quantitative and relational situations. It consists of a large body of knowledge and many sub-disciplines, each of which provides an array of tools and techniques for exploration and analysis. This includes, but is not limited to: patterns of logical reasoning and inference, geometric and algebraic manipulation, and analytic, graphical, and statistical investigation of phenomena. Different sub-disciplines are especially useful for solving certain types of problems while connections between the sub-disciplines help in the understanding and solution of other types of problems.

The purpose of the mathematics program at IMSA is to help students develop and use mathematics for their own thinking, problem solving, and development as informed citizens and leaders; to help students appreciate the power, economy, elegance, and beauty of mathematical thought; to be a mathematical resource for other members of the IMSA community; and to promote IMSA and its mission by enhancing mathematical education for citizens of the state of Illinois and beyond.

Team Goals

• provide a dynamic and comprehensive curriculum informed by current thinking in the mathematical and educational communities, cognitive science, and internal experimentation, including a core base of knowledge in the mathematical and computer sciences;
• support and promote the development and dissemination of innovative pedagogy, curriculum, and assessment both internally and externally to the Academy;
• provide experiences through which students will develop and extend their ability to investigate and explore, think critically, solve problems, and apply mathematical tools of thought in a variety of situations using multiple strategies, approaches, and techniques;
• provide experiences and opportunities that will enable and encourage students to develop their mathematical potential;
• provide students with the opportunities to explore the relationships and inter-connections within the various branches of mathematics and with other disciplines;
• provide experiences and opportunities that enable students to interpret and to communicate mathematical concepts in both oral and written form;
• foster, in each student, a positive attitude toward the learning of mathematics in individual and collaborative settings and confidence as a learner and practitioner;
• foster, in each student, the acceptance and practice of ethical behavior in collaborative, individual, and assessment situations;
• provide experiences, problems, and situations that require students to thoughtfully explore mathematics using appropriate technology as extensions of the mind;
• encourage and support student participation in a variety of mathematical activities including projects, competitions and external examinations; and
• support, promote, and provide professional services to students, educators and organizations both internally and externally through such activities as publications, workshops, presentations, and committee work.

Unifying Concepts and Processes

Beginning in 1989 with the publication of the Curriculum and Evaluation Standards for School Mathematics, the National Council of Teachers of Mathematics took an historic first step by creating “a coherent vision of what it means to be mathematically literate and by creating a set of standards to guide the revision of the school mathematics curriculum and its associated evaluation toward this vision” (NCTM, 1989).

These unifying concepts and processes reflect the essence of the vision set forth in the 1989 Standards and enhanced through the NCTM’s ongoing standards process. They describe what it means to be mathematically powerful as well as mathematically literate in today’s world.

Mathematical perspective:

Seeing mathematics as a valuable means through which situations can be generalized, idealized, analyzed, and ultimately understood, is at the heart of mathematical perspective. Having a mathematical perspective involves looking for the underlying mathematical relationships in contexts which are not inherently mathematical. It also involves looking at problem situations, whether presented in mathematical or non-mathematical contexts, using a variety of mathematical lenses to bring multiple approaches to the analysis of a situation or the solving of a problem.

Mathematical power:

To have mathematical power means that one is fluent with an extensive repertoire of mathematical techniques and is able to select from among them to analyze situations and solve problems that arise in both intra- and inter-disciplinary contexts. People with mathematical power display a willingness to dig into a problem situation for which an approach is not readily apparent. They use analytical reasoning to identify potential methods for solving a particular problem, thoughtfully consider when to maintain or change an approach, follow through to a conclusion, and judge the reasonableness of results they obtain. They are able to see the similarities and differences among different problems and problem contexts, and they are able to use what they have learned in one context to suggest approaches in another.

Mathematical communication:

At the heart of mathematical communication is the power, economy of expression, elegance, and beauty that the language of mathematics affords. Mathematical communication is a two-way street; one needs to be able to express mathematical ideas clearly and accurately as well as read, hear, and understand information which is expressed in mathematical terms. Effective communication involves expressing ideas algebraically, geometrically, quantitatively, and verbally, selecting appropriate mathematical representations for an intended application and audience, and using them correctly to inform or make decisions.

Logical reasoning:
Logical reasoning provides the foundation and framework for mathematics. It is the internal consistency and reliability that logic gives to mathematics that makes this discipline one which is both beautiful and useful. In addition, the ability to reason carefully is a widely applicable habit of mind in other disciplines and in life. It is the means by which mathematical evidence can play a role in making sound choices. Mathematical reasoning involves the ability to think inductively and deductively. It builds on the use of conjecture, pattern recognition, generalization, derivation, analogy, extrapolation/ interpolation, and estimation as a means to enhance understanding. It requires knowledge of the nature, role, and necessity of convincing argument and proof and the ability to create proofs or provide counterexamples for given situations.

Mathematical transformations:
Analysis of how mathematical representations and situations respond to change constitutes the concepts of transformation in mathematics. The behavior of mathematical quantities and relationships as they are combined, transformed, or re-expressed leads to conceptual understanding of such fundamental mathematical constructs as function, rate of change, and invariance under change. Identification of those elements that are preserved, those that are not preserved, and those that emerge under various transformations can lead to deeper understanding of the characteristics of a system. Understanding how changes in one representation are reflected in other representations of the same phenomenon helps connect diverse areas of mathematical thought and helps support multiple approaches to solving a wide variety of problems.

Generalization:

Generalizing patterns and expressing the relationship between quantities is essential to mathematics. Symbols, tables, and graphs are among the ways these relationships can be expressed. Moreover, when situations are modeled using these mathematical forms of expression, the fundamental similarities and differences of the situations being modeled can be understood in new ways. Two situations which arise in very different contexts may be seen to be fundamentally the same, that is, they may be examples of a more general class of situations for which the same mathematical underpinnings apply.

Spatial orientation, navigation, and relationships:

Spatial orientation and navigation is at the heart of functioning effectively in a physical world. An understanding of geometry and space provides insight into objects which exist in one, two, and three dimensions, their characteristics, and their relationships. It is by working within physical, analytic, and synthetic systems using both static and dynamic approaches that one gains insight and understanding into the physical world.

Quantitative literacy:

Understanding of numbers, their operations, and the relationships among them is critical for functioning in an increasingly quantitative world. Much of today’s information is presented as data in some form. To make intelligent, informed decisions about issues that are based on quantitative information, citizens must be able to read, compute with, understand, interpret, and react to data. In addition, persons expecting to contribute to the general body of knowledge must be prepared to do so through research which has at its heart experimental design and the implementation, analysis, and interpretation of data. Quantitative literacy is the ability to participate as a member of society in these ways.

Learning Standards

Students studying Math at IMSA will:

1. demonstrate a disposition and propensity to use mathematics, a variety of problem solving strategies, and creative thought to solve problems.
2. reason logically in mathematical situations and understand the nature, role, and necessity of proof and counterexample in mathematical reasoning.
3. communicate clearly and accurately about mathematical relationships and results.
4. demonstrate awareness of the interconnectedness of mathematical thought in inter- and intra-disciplinary settings.
5. understand and employ the power, economy, clarity, and elegance of mathematical representations.
6. use and interpret appropriate mathematical models to represent real-world situations.
7. understand the underlying concepts and characteristics of mathematical functions and relations.
8. identify, understand, and apply the concepts of change and invariance under change.
9. understand and apply geometric relationships.
10. use data to research questions, make conjectures, inform decisions, and evaluate assertions.
11. understand and apply discrete mathematical models.
12. use technology to gain insight and obtain different perspectives on problems.

Key
IMSA Mathematics Learning Standards are cross-referenced as follows:

• IMSA’s Standards of Significant Learning [SSL-III.B]
• Common Core State Standards for Mathematics
  • Standards for Mathematical Practice [CCSSM: P1]
  • Standards for Mathematical Content [CCSSM:conceptual category code and number]
• Principles and Standards for School Mathematics [NCTM-6.1]

A. Students studying mathematics at IMSA demonstrate a disposition and propensity to use mathematics, a variety of problem solving strategies, and creative thought to solve problems by:

A.1
A.2	posing, solving, and extending both multi-step routine and multi-step unconventional problems. [SSL-I.B,III.B,III.C,IV.A,IV.C; CCSSM: P1,6,7,8; NCTM-6.2]
A.3	interpreting, generalizing , and verifying the understanding gained in the problem solving process and extending it to new settings. [SSL-I.B,II.B,III.B,III.C,IV.A,IV.C; CCSSM: P1,2,3,4,8; NCTM-6.3]
A.4	using a variety of resources and problem solving approaches. [SSL-III.B,IV.A;CCSSM: P1,7,8; NCTM-6.3]
A.5	demonstrating confidence, persistence, and reflective analysis of the effectiveness of an approach when attempting to solve a problem. [SSL-I.D,II.A,II.B,III.B,IV.A,IV.C; CCSSM: P1,2,4,8NCTM-6.2,6.4]

B. Students studying mathematics at IMSA reason logically in mathematical situations and understand the nature, role, and necessity of proof and counterexample in mathematical reasoning by:

B.1	demonstrating understanding of an axiomatic system. [SSL-I.A,III.C,IV.A; CCSSM: P3]
B.2	reasoning inductively and deductively. [SSL-I.B,III.C,IV.A; IL-9.C; CCSSM: P3; NCTM-7.2,74.]
B.3	making and testing conjectures, creating proofs, and identifying counterexamples. [SSL-III.C,IV.B; CCSSM: P3; NCTM-7.4]
B.4	enhancing inductive and deductive reasoning through the use of intuition, imagination, and other forms of reasoning. [SSL-III.C,IV.A; CCSSM: P3,7,8; NCTM-7.4]
B.5	analyzing and critiquing proofs created by themselves and others. [SSL-I.D,II.B,III.C,IV.A,IV.B,V.A; CCSSM: P3; NCTM-7.3]
B.6	understanding the role of logic in the development of mathematics and understanding the necessity of carefully proving assertions. [SSL-II.A,II.B,III.C,IV.D; CCSSM: P3,7; NCTM-7.1]

C. Students studying mathematics at IMSA communicate clearly and accurately about mathematical relationships and results by:

C.1	understanding mathematical information given in written, oral, symbolic, numeric, or graphic form and interpreting the relationship it represents. [SSL-IV.B; CCSSM: P1,4; NCTM-8.4]
C.2	accurately recording and effectively communicating using proper notation, vocabulary, and usage in a variety of modalities (written, oral, graphic, algebraic, etc.). [SSL-I.C,IV.B,V.A; CCSSM: P6; NCTM-8.1,8.2,8.4]
C.3	presenting mathematical work and results using the power of mathematical language effectively. [SSL-IV.B,V.A; CCSSM: P6; NCTM-8.2,8.4]
C.4	summarizing results in a form that is accurate, appropriate to the topic and level, and understandable to the intended audience. [SSL-I.C,IV.B,V.A; CCSSM: P6; NCTM-8.1,8.2]

D. Students studying mathematics at IMSA demonstrate awareness of the inter-connectedness of mathematical thought in inter- and intra-disciplinary settings by:

D.1	understanding that mathematics is a system of interconnected ideas. [SSL-III.B,III.C,IV.C; CCSSM: P7,8; NCTM-9.2]
D.2	recognizing the commonalties among the components and processes of the sub-disciplines of mathematics. [SSL-I.B,III.B,III.C,IV.C; CCSSM: P7; NCTM-9.1]
D.3	understanding the interaction between mathematics and culture, world history, and other disciplines. [SSL-I.B,II.A,II.B,III.B,III.C,IV.C; CCSSM: P4; NCTM-9.3]

E. Students studying mathematics at IMSA understand and employ the power, economy, clarity, and elegance of mathematical representations by:

E.1	recognizing that mathematical representations carry specific meanings and using mathematical notation correctly to enhance clarity and avoid ambiguity. [SSL-II.B,IV.B; CCSSM: P6; NCTM-10.1]
E.2	applying a variety of techniques to compare and manipulate mathematical representations. [SSL-I.A,III.B,IV.C; CCSSM: P2,4; NCTM-1.2,1.3,10.2]
E.3	recognizing the structure underlying a mathematical representation and utilizing this structure in analysis and problem solving. [SSL-III.B,IV.A,IV.C; CCSSM: P7,8; NCTM-1.1,10.2]
E.4	selecting an appropriate mathematical representation and demonstrating how it reflects the salient points of the situation it describes. [SSL-I.B,I.D,II.B,III.B,IV.A,IV.C; CCSSM: P4,6; NCTM-2.2,10.2,10.3]

F. Students studying mathematics at IMSA use and interpret appropriate mathematical models to represent real-world situations by:

F.1	choosing an appropriate representation or mathematical model for a given situation. [SSL-I.A,I.B,III.B,IV.A,IV.C; CCSSM: P4; NCTM-2.3,4.1]
F.2	understanding and explaining the relationship between the model and the given situation. [SSL-I.B,III.A,III.B,IV.A,IV.B,IV.C; CCSSM: P4]
F.3	analyzing and explaining how variations in the situation will affect the model and how parametric changes in the model would be reflected in the situation it describes. [SSL-I.B,I.D,III.A,III.B,IV.A,IV.B,IV.C; CCSSM: P4]
F.4	interpreting mathematical results in terms of the situation modeled. [SSL-I.D,III.B,IV.A,IV.C; CCSSM: P4; NCTM-9.3]
F.5	understand the notion of chance and the use of probabilistic models to quantify and analyze chance. [SSL-IV.A; CCSSM:S-CP5; NCTM-5.4]

G. Students studying mathematics at IMSA understand the underlying concepts and characteristics of mathematical functions and relations by:

G.1	demonstrating fundamental recognition and analysis of relations and functions and their characteristics. [SSL-I.A,IV.C; CCSSM: P7,F-IF1,2; NCTM-2.1]
G.2	developing and using a toolbox of prototypical functions (linear, exponential, logarithmic, polynomial, rational, trigonometric, etc.). [SSL-I.A,III.B,IV.C; CCSSM:F-IF1,2,3,7, F-BF2,4,5, F-LE4, F-TF1,2,3,4; NCTM-2.2]
G.3	evaluating and manipulating functions, creating multiple representations for a single function. [SSL-I.A,IV.C; CCSSM: F-IF4,5,8, F-TF8,9; NCTM-2.2]
G.4	applying operations and transformations to functions and demonstrating how changes in one representation of a function affect other representations of that function. [SSL-I.A,III.B,IV.C; CCSSM: F-IF4,5, F-BF3, F-TF3,4,6; NCTM-2.2,2.3]
G.5	modeling a given situation or data with an appropriate function, using the model to make predictions. [SSL-I.B,III.B,IV.A; CCSSM: P4, F-BF1,2, F-LE2,5 F-TF5; NCTM-2.3,3.2]
G.6	Understanding the strengths and limitations of the function as a model for a particular situation or data and adjusting the model in response to these strengths and limitations. [SSL-I.D,III.B,IV.A,IV.B; CCSSM: P4, F-LE1, F-TF6,7]

H. Students studying mathematics at IMSA understand and apply the concepts of change and invariance under change by:

H.1	identifying, describing, and measuring various patterns of change. [SSL-I.C,IV.A,IV.B; CCSSM: P7, G-CO2-8, G-SRT1-3, G-C3,4, G-GMD4; NCTM-2.3,4.1]
H.2	applying limiting processes in graphical, numerical, and symbolic situations. [SSL-I.A,III.B; CCSSM: P8NCTM-4.1,4.2]
H.3	applying concepts of change to problem situations using approximate or analytic methods as appropriate. [SSL-I.B,I.D,III.B; CCSSM: P4; NCTM-2.3,4.1,4.2]

I. Students studying mathematics at IMSA understand and apply geometric relationships by:

I.1	analyzing spatial relationships from both static and dynamic perspectives. [SSL-III.B; CCSSM: P1, G-CO2-8, G-SRT1-3, G-C3,4, G-GMD4; NCTM-3.4]
I.2	identifying, classifying, and using characteristics of two- and three-dimensional objects. [SSL-IV.A,IV.C; CCSSM: P7, G-CO8-13, G-SRT4-6, G-C1,2; NCTM-3.1]
I.3	selecting and using appropriate geometric relationships, properties, formulas, tools, and units when working in a geometric context. [SSL-II.A,III.B,IV.A,IV.C; CCSSM: P6, G-CO12,13, G-C3,5; NCTM-3.3,4.1,4.2]
I.4	modeling situations geometrically in two and three dimensions to formulate, describe, and solve problems. [SSL-I.B,III.B,IV.A,IV.C; CCSSM: P4, G-SRT5, G-GPE1-3, G-GMD1-3, G-MG1-3; NCTM-3.2]
I.5	performing and describing geometric transformations. [SSL-I.A,IV.B; CCSSM: P2, G-CO2-8; NCTM-3.3]
I.6	solving problems involving coordinate (analytic) geometry. [SSL-I.A; CCSSM: P2, G-PE1-7; NCTM-3.2]
I.7	recognizing that geometry is an effective context for the study of deductive systems. [CCSSM: P3, G-CO9-11; NCTM-3.1elaboration]

J. Students studying mathematics at IMSA use data to research questions, inform decisions, and evaluate assertions by:

J.1	creating and implementing a valid design for research or an investigation. [SSL-I.C,I.D,III.B,IV.A,IV.C; CCSSM: S-IC1,2; NCTM-5.1]
J.2	identifying, selecting, and using appropriate statistical and graphical tools to analyze data in a variety of contexts. [SSL-I.A,III.B,IV.A; CCSSM: S-ID1,2; NCTM-5.1]
J.3	interpreting data and presenting it in such a way so as to make the information contained therein more readily evident. [SSL-I.B; CCSSM: S-ID1,2,6,7; NCTM-5.2]
J.4	carefully critiquing data, the way it was collected, its presentation, and the conclusions drawn from it. [SSL-I.B,I.C,I.D,II.A,IV.A,V.A; CCSSM: S-IC6; NCTM-5.3,8.3]

K. Students studying mathematics at IMSA understand and apply discrete mathematical models by:

K.1	using matrices, sequences, and their operations to model phenomena. [SSL-I.A,I.C,III.B; CCSSM: P4, N-Q2, A-SSE1 ; NCTM-2.1,2.2]
K.2	analyzing and interpreting situations using recursive thinking and inductive reasoning. [SSL-I.A,I.B,IV.A; CCSSM: P3, F-IF3, F-BF1,2; NCTM-2.1,2.2,7.3,7.4]
K.3	creating and interpreting directed graphs and networks. [SSL-III.B,IV.C; NCTM-3.2]
K.4	demonstrating an understanding of basic counting principles and the situations under which they may be applied. [SSL-I.A,I.D,III.B; NCTM-5.4]
K.5	understanding and using probability as a way to measure chance events. [SSL-I.A,III.B; CCSSM: S-CP1-3,5-9, S-MD5; NCTM-5.4]

L. Students studying mathematics at IMSA use technology to gain insight and obtain different perspectives on problems by:

L.1	deciding whether to use technology, selecting an appropriate technology for a given situation, and understanding the limitation of the technology. [SSL-I.D,II.A,III.A; CCSSM: P5]
L.2	using technology to facilitate doing, exploring, and understanding of mathematics. [SSL-II.A,III.A,IV.A; CCSSM: P5; NCTM-6.1]
L.3	judging the reasonableness of information and answers given by technology. [SSL-III.A,IV.A; CCSSM: P5; NCTM-4.2]

  Correlations to Other Standards

IMSA’s Standards of Significant Learning

IMSA’s Mathematics Learning Standards

I. Developing the Tools of Thought

A. Develop automaticity in skills, concepts, and processes that support and enable complex thought.	B.1, E.2, F.1, G.1-4, H.2, I.5-6, J.2, K.1-2, K.4-5
B. Construct questions which further understanding, forge connections, and deepen meaning.	A.1-3, B.2, D.2-3, E.4, F.1-3, G.5, H.3, J.3-4, K.2
C. Precisely observe phenomena and accurately record findings.	C.2, C.4, H.1, I.4, J.1, J.4, K.1
D. Evaluate the soundness and relevance of information and reasoning.	A.5, B.5, E.4, F.2-4, G.6, H.3, J.1, J.4, K.4, L.1

II. Thinking About Thinking

A. Identify unexamined cultural, historical, and personal assumptions and misconceptions that impede and skew inquiry.	A.5, B.6, D.3, I.3, J.4, L.1-2
B. Find and analyze ambiguities inherent within any set of textual, social, physical, or theoretical circumstances.	A.3, A.5, B.5-6, D.3, E.1, E.4

III. Extending and Integrating Thought

A. Use appropriate technologies as extensions of the mind.	A.1, F.2-3, L.1-3
B. Recognize, pursue, and explain substantive connections within and among areas of knowledge.	A.2-5, D.1-3, E.2-4, F.1-4, G.2, G.4-6, H.2-3, I.1, I.3-4, J.1-2, K.1, K.3-5
C. Recreate the beautiful conceptions that give coherence to structures of thought.	A.1-3, B.1-6, D.1-3

IV. Expressing and Evaluating Constructs

A. Construct and support judgments based on evidence.	A.1-5, B.1-5, E.3-4, F.1-5, G.5-6, H.1-4, J.1-2, J.4, K.2, L.2-3
B. Write and speak with power, economy, and elegance.	B.3, B.5, C.1-4, E.1, F.1-3, H.1, I.5
C. Identify and characterize the composing elements of dynamic and organic wholes, structures, and systems.	A.5, D.1-3, E.2-4, F.1-4, G.1-4, G.6, I.2-4, K.3
D. Develop an aesthetic awareness and capability.	B.6

V. Thinking and Acting with Others

A. Identify, understand, and accept the rights and responsibilities of belonging to a diverse community	B.5, C.2, C.3, C.4, J.4
B. Make reasoned decisions which reflect ethical standards, and act in accordance with those decisions.
C. Establish and commit to a personal wellness lifestyle in the development of the whole self.