# Vertical Learning Trajectory

# Learning Trajectory Resources

# ongoing Assessment Project

Learn more about OGAP http://www.ogapmath.com/frameworks

OGAP is a systemic and intentional formative assessment system in mathematics grounded in the research on how students learn mathematics

# Check out LT Resources

https://www.learningtrajectories.org/

For Teachers Description of all Learning Trajectories

Learning Trajectory Use to Develop Teacher Expertise (DTE website) on Geometric thinking

Mathematics Vertical Articulation Tool (MVAT) – This tool provides support in identifying concepts aligned to the 2016 *Mathematics Standards of Learning *(SOL) that articulate across mathematics grade levels or courses. The MVAT can be used as an instructional planning tool – to identify the progression of specific content across grade levels or courses in order to connect instruction to prerequisite knowledge and future concepts.

**Vertical Articulation**

### Numbers K-8 Vertical Coherence

### Computation & Estimation K-8 Vertical Coherence

### Measurement and Geometry K-8 Vertical Coherence

### Patterns, Function and Algebra k-8 Vertical Coherence

### Prob Stat K-8 Vertical Coherence

### K-Alg 2 Vertical Coherence

# NCTM Essential Understandings

“The big ideas are mathematical statements of overarching concepts that are central to a mathematical topic and link numerous smaller mathematical ideas into coherent wholes. These smaller, more concrete ideas that are associated with each big idea are called *essential understandings*. They capture aspects of the corresponding big idea and provide evidence of its richness.”

**Number and Numeration PK-2**

Big idea 1: Number is an extension of more basic Ideas about relationships between quantities

Essential idea 1a: Quantities can be compared without assigning numerical values to them

Essential idea 1b: Physical objects are not in themselves quantities. All quantitative comparisons involve selecting particular attributes of objects or materials to compare

Essential idea 1c: The relation between one quantity and another quantity can be an equality or inequality relation.

Essential idea 1d: Two important properties of equality and order relations are conservation and transitivity

Essential understanding 1e: The equality relation between two quantities remains unchanged when one or both quantities are **decomposed into parts** and when one of the quantities is combined with another quantity to form a larger quantity.

Big idea 2: The selection of a unit makes it possible to use numbers in comparing quantities.

Essential understanding 2a: Using numbers to describe relationships between or among quantities depends on identifying a unit

Essential understanding 2b: The size of a unit determines the number of times that it must be iterated to count or measure a quantity

Essential understanding 2c: Quantities represented by numbers can be **decomposed (**or composed) into part-whole relationships

Big idea 3: Meaningful counting integrates different aspects of number and sets, such as sequence, order, one-to-one correspondence, ordinality, and cardinality

Essential understanding 3a: The number-word sequence, combined with the order inherent in the natural numbers, can be used as a foundation for counting

Essential understanding 3b: Counting includes one-to-one correspondence, regardless of the kind of objects in the set and the order in which they are counted

Essential understanding 3c: Counting strategies are based on order and hierarchical inclusion of numbers

Big idea 4: Numbers are abstract concepts

Essential understanding 4a: Patterns in the number-word sequence provide a foundation for the abstract number concept

Essential understanding 4b: The number sequence is infinite

Essential understanding 4c: Number symbols are representations of abstract mental objects

Big idea 5: **A base-ten positional number system** is an efficient way to represent numbers in writing.

Essential understanding 5a: Ten different digits can be used and sequenced to express any whole number

Essential understanding 5b: Our base-ten number system allows forming a new place-value unit by grouping ten of the previous place-value units, and this process can be iterated to obtain larger and larger place-value units

Essential understanding 5c: The value of a digit in a written numeral depends on its place, or position, in a number

Essential understanding 5d: Inherent in place value are units of different sizes.

**Addition and Subtraction PK-2**

**Addition and Subtraction PK-2**

Big Idea 1: Addition and subtraction are used to represent and solve many different kinds of problems.

Essential understanding 1a: Addition and subtraction of whole numbers are based on sequential counting with whole numbers.

Essential understanding 1b: Subtraction has an inverse relationship with addition

Essential understanding 1c: Many different problem situations can be represented by part-part-whole relationships and addition or subtraction

Essential understanding 1d: Part-part-whole relationships can be expressed by using number sentences like *a* + *b* = *c* or *c* – *b* = *a*, where *a* and *b* are parts and *c* is the whole.

Essential understanding 1e: The context of a problem situation and its interpretation can lead to different representations.

Big Idea 2: The mathematical foundations for understanding computational procedures for addition and subtraction of whole numbers are the properties of addition and place value

Essential understanding 2a: The commutative and associative properties for addition of whole numbers allow computations to be performed flexibly

Essential understanding 2b: Subtraction is not commutative or associative for whole numbers

essential understanding 2c: Place value concepts provide a convenient way to compose and decompose numbers to facilitate addition and subtraction computations

Essential understanding 2d: Properties of addition are central and justifying the correctness of computational algorithms.

**Geometry and Measurement PK-2**

Big Idea 1: A classification scheme, specifies for a space or the objects within it. The properties that are relevant to particular goals and intentions

Essential understanding 1a: Mathematical classification extends and refines everyday categorization by making more precise what we mean by “sides,” “angles,” “straightness,” and or other features that we attend to as we categorize mathematical objects.

Essential understanding 1b: We may classify the same collection of objects in different ways.

Big Idea 2: Geometry allows us to structure spaces and specify locations within them

Essential understanding 2a: To describe a location. We must provide a reference point (an origin) and independent pieces of information (often called *coordinates*) indicating distance and direction from that point

Essential understanding 2b: Geometry and measurement can specify directions, routes, and locations in the world -for example, navigation paths and spatial relations- with precision. Given a reference point and an orientation, we can label position with numbers.

Essential understanding to see geometric objects or things that exist in our minds. Many of them are idealizations of things that also exist in the physical world.

Big idea 3: We gain insight and understanding of spaces and objects within them by noting what does and does not change as we transform these spaces and objects in various ways

Essential understanding 3a: Transformations can be used to describe differences between an idealized image of an object and the way that it is positioned in space or seen by the eye

Essential understanding 3b: Under each transformation, certain properties are invariant

Big idea 4: One way to analyze and describe geometric objects, relationships among them, or the spaces that they occupy is to quantify -measure or count- one or more of their attributes.

Essential understanding 4a: Measurement can specify how much by assigning a number to such attributes as length area, volume, and angle

Essential understanding 4b: Some quantities can be compared or measured directly, others can be measured indirectly, and the measurements of some objects are computed from other measurements

Essential understanding 4c: Measurement can be performed with a variety of units. The size of the unit and the number of units in the measure are inversely related to each other

Essential understanding 4d: Objects can be decomposed and composed to facilitate their measurement.

**Multiplication and Division 3-5**

Big idea 1: Multiplication is one of two fundamental operations along with addition, which can be defined so that it is an appropriate choice for representing and solving problems in many different situations,

Essential understanding 1a: In the multiplicative expression, *A* X *B*, *A* can be defined as a *scaling factor*

Essential understanding 1b: Each multiplicative expression developed in the context of a problem situation has an accompanying explanation, and different representations and ways of reasoning about a situation can lead to different expressions or equations

Essential understanding 1c: A situation that can be represented by multiplication has an element that represents the scalar and an element that represents the quantity to which the scalar applies

Essential understanding 1d: A scalar definition of multiplication is useful in representing and solving problems beyond whole number multiplication and division

Essential understanding 1e: Division is defined by its inverse relationship with multiplication

Essential understanding 1f: Using proper terminology and understanding the division algorithm provide the basis for understanding how numbers such as the *quotient* and the *remainder* are used in division situations.

Big idea 2: The properties of multiplication and addition provide the mathematical foundation for understanding computational procedures for multiplication and division, including mental computation and estimation strategies, invented algorithms, and standard algorithms

Essential understanding 2a: The commutative and associative properties of multiplication and the distributive property of multiplication over addition ensure flexibility in computations with whole numbers and provide justifications for the sequences of computations with them

Essential understanding 2b: The right distributive property of division over addition allows computing flexibly and justifying computations with whole numbers, but there is no left distributive property of division over addition and no commutative or associative property of division of whole numbers.

Essential understanding 2c: *Order of operations* is a set of conventions that eliminates ambiguity in and multiple values for numerical expressions involving multiple or operations

Essential understanding 2d: Properties of operations on whole numbers justify written and mental computational algorithms, standard and invented.

**Rational Numbers 3-5**

Big Idea 1: Extending from whole numbers to rational numbers creates a more powerful and complicated number system

Essential understanding 1a: Rational numbers are a natural extension of the way that we use numbers

Essential understanding 1b: The rational numbers are a set of numbers that includes the whole numbers and integers as well as numbers that can be written as the quotient of two integers, *a* *b*, * *where *b *is not zero

Essential understanding 1c: The rational numbers allow us to solve problems that are not possible to solve with just whole numbers or integers.

Big Idea 2: Rational numbers have multiple interpretations and making sense of them depends on identifying the unit

Essential understanding 2a: The concept of *unit* is fundamental to the interpretation of rational numbers

Essential understanding 2b: One interpretation of a rational number is as a part-whole relationship

Essential understanding 2c: One interpretation of a rational number is as a measure

Essential understanding 2d: One interpretation of a rational number is as a quotient

Essential understanding 2e: One interpretation of a rational number is as a ratio

Essential understanding 2f: One interpretation of a rational number is as an operator

Essential understanding 2g: Whole number conceptions of *unit* become more complex when extended to rational numbers

Big Idea3: Any rational number can be represented in infinitely many in equivalent symbolic forms

Essential understanding 3a: Any rational number can be expressed as a fraction in an infinite number of ways.

Essential understanding 3b: Between any two rational numbers there are infinitely many rational numbers

Essential understanding 3c: A rational number can be expressed as a decimal

Big Idea 4: Computation with rational numbers is an extension of computation with whole numbers but introduces some new ideas and processes

Essential understanding 4a: The interpretations of the operations on rational numbers are essentially the same as those on whole numbers, but some interpretations require adaptation, and the algorithms are different

Essential understanding 4b: Estimation and mental math are more complex with rational numbers than with whole numbers.

**Algebraic Thinking 3-5**

Big Idea 1: Addition, subtraction, multiplication, and division operate under the same properties in algebra as they do in arithmetic

Essential understanding 1a: The fundamental properties of number and operations govern how operations behave and relate to one another

Essential understanding 1b: The fundamental properties are essential to computation

essential. Understanding 1c: The fundamental properties are used more explicitly in some computation strategies than in others.

Essential understanding 1d: Simplifying algebraic expressions entails decomposing quantities in insightful ways.

Essential understanding 1e: Generalizations and arithmetic can be derived from the fundamental properties.

Big Idea 2: A mathematical statement that uses an equals sign to show that two quantities are equivalent is called an equation.

Essential understanding 2a: The equals sign is a symbol that represents a relationship of equivalence

Essential understanding 2b: Equations can be reasoned about in their entirety rather than as a series of computations to execute

Essential understanding 2c: Equations can be used to represent problem situations.

Big idea 3: Variables are versatile tools that are used to describe mathematical ideas in succinct ways.

Essential understanding 3a: The meaning of variable can be interpreted in many ways.

Essential understanding 3b: A variable represents the measure or amount of an object, not the object itself

Essential understanding 3c: The same variable used more than once in the same equation must represent identical values in all instances, but different variables may represent the same value.

Essential understanding 3d: The same variable may play one or more roles within a given application problem or situation

Essential understanding 3e: A variable may represent either a discrete or a continuous quantity.

Big idea 4: Quantitative reasoning extends relationships between and among quantities to describe in generalized relationships among these quantities

Essential understanding 4a: Two quantities can relate to each other in one of three ways: (1) they can be equal, (2) one quantity can be larger than the other, or (3) one quantity can be smaller than the other

Essential understanding 4b: Known relationships between two quantities can be used as a basis for describing relationships with other quantities.

Big idea 5: Functional thinking includes generalizing relationships between covarying quantities, expressing those relationships in words, symbols, tables, or graphs, and reasoning with these various representations to analyze function behavior

Essential understanding 5a: A function is a special mathematical relationship between two sets where each element from one set called the *domain* is related uniquely to an element of the second set called the *co-domain*

Essential understanding 5b: Functions can be viewed as tools for expressing covariation between two quantities

Essential understanding 5c: In a functional relationship between two covarying quantities, a variable is said to be either *independent* or *dependent* and will represent either a discrete or continuous quantity

Essential understanding 5d: In working with functions, several important types of patterns or relationships might be observed among quantities that vary in relation to each other: recursive patterns, covariational relationships, and correspondence rules

Essential understanding 5e: Functions can be represented in a variety of forms, including words, symbols, tables, and graphs

Essential understanding 5f: Different types of functions behave in fundamentally different ways, and analyzing change or variation in function behavior is one way to capture this difference.

**Geometry and Measurement 3-5**

Big Idea 1: Transforming objects in the space they occupy in various ways while noting what does and does not change provides insight into an understanding of the objects and space

Essential understanding 1a: Transformation supplies a dynamic basis for analyzing and describing a variety of situations and relationships

Essential understanding 1b: Transformation offers a means by which to explain geometrical phenomena in ways that build on spatial intuitions.

Big Idea 2: One way to analyze and describe geometric objects, relationships among them, or the space that they occupy is to quantify, measure, or count one or more of their attributes

Essential understanding 2a: Measurement can specify “how much” by assigning a number that corresponds to a chosen unit to such attributes as length, area, volume, and angle

Essential understanding 2b: Geometry and measurement can precisely specify directions, routes, and locations in the world -for example, paths of navigation and coordinates describing spatial relationships

Essential understanding 2c: Motion is useful in coordinating measurement and generation of attributes of length, area, volume, and angle

Essential understanding 2d: Decomposing and composing objects facilitates their measurement, and using those decompositions and compositions to derive formulas clarifies relationships between quantified attributes and units of measure.

Big idea 3: A classification scheme specifies the properties of objects that are relevant to particular goals and intentions

Essential understanding 3a: Classification schemes and associated defining properties depend on the purposes and contexts envisioned for mathematical investigation and multiple classification schemes are often possible

Essential understanding 3b: Classification specifies relationships, such as equivalence and inclusion, within and between classes

Essential understanding 3c: Classification leads to investigation of criteria for particular classes of shapes, and such investigation can lead in turn to the identification of new properties and relationships among objects in the class.

**Ratios, Proportions, and Proportional Reasoning 6-8**

Essential understanding 1: Reasoning with ratios involves attending to and coordinating two quantities.

Essential understanding 2: A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit

Essential understanding 3: Forming a ratio as a measure of a real-world attribute involves isolating that attribute from other attributes and understanding the effect of changing each quantity on the attribute of interest

Essential understanding 4: A number of mathematical connections link ratios and fractions:

Ratios are often expressed in fraction notation, although ratios and fractions do not have identical meaning

Ratios are often used to make “part-part” comparisons, but fractions are not

Ratios and fractions can be thought of as overlapping sets

Ratios can often be meaningfully reinterpreted as fractions

Essential understanding 5: Ratios can be meaningfully reinterpreted as quotients

Essential understanding 6: A proportion is a relationship of equality between two ratios. In a proportion, the ratio of two quantities remains constant as the corresponding values of the quantities change.

Essential understanding 7: Proportional reasoning is complex and involves understanding that-

Equivalent ratios can be created by iterating and or partitioning a composed unit

If one quantity in a ratio is multiplied or divided by a particular factor, then the other quantity must be multiplied or divided by the same factor to maintain the proportional relationship; and

The two types of ratios -composed units and multiplicative comparisons- are related

Essential understanding 8: A rate is a set of infinitely many equivalent ratios

Essential understanding 9: Several ways of reasoning, all grounded in sense-making, can be generalized into algorithms for solving proportion problems

Essential understanding 10: Superficial cues present in the context of a problem do not provide sufficient evidence of proportional relationships between quantities.

**Expressions, Equations and Functions 6-8**

Big idea 1: Expressions are foundational for algebra. They serve as building blocks for work with equations and functions.

Essential understanding 1a: Expressions are powerful tools for exploring reasoning about and representing situations

Essential understanding 1b: Two or more expressions may be equivalent, even when their symbolic forms differ

Essential understanding 1c: A relatively small number of symbolic transformations can be applied to expressions to yield equivalent expressions

Big idea 2: Variables are tools for expressing mathematical ideas clearly and concisely. They have many different meanings depending on context and purpose.

Essential understanding 2a: Variables have many different meanings depending on context and purpose

Essential understanding 2b: Using variables permits writing expressions whose values are not known or vary under different circumstances.

Essential understanding 2c: Using variables permits representing varying quantities. This use of variables is particularly important in studying relationships between varying quantities

Big idea 3: The equals sign indicates that two expressions are equivalent. It can also be used in defining or naming a single expression or function rule.

Essential understanding 3a: The equals sign can indicate that two expressions are equivalent

Essential understanding 3b: The equals sign can be used in defining or giving a name to an expression or function rule

Essential understanding 3c: It is often important to find the value or values of a variable for which two expressions represent the same quantity

Essential understanding 3d: Finding the value(s) of a variable for which two expressions represent the same quantity is known as *solving an equation*.

Essential understanding 3e: An inequality is another way to describe a relationship between expressions; instead of showing that the values of two expressions are equal, inequalities indicates that the value of one expression is greater than (or greater than or equal to) the value of the other expression.

Essential understanding 3f: In solving an inequality, multiplying or dividing both expressions by a negative number reverses the sign (<, >, ≤, ≥) that indicates the relationship between the two expressions

Big idea 4: Functions provide a means for describing and understanding relationships between variables. They can have multiple representations -in algebraic symbols, situations, graphs, verbal descriptions, tables, and so on- and they can be classified into different families with similar patterns of change

Essential understanding 4a: Functions provide a tool for describing how variables change together.

Using a function in this way is called *modeling*. And the function is called a *model*.

Essential understanding 4b: Functions can be represented in multiple ways -in algebraic symbols, situations, graphs, verbal descriptions, tables, and so on- and these representations, and the links among them, are useful in analyzing patterns of change.

Essential understanding 4c: One important way of describing functions is by identifying the rate at which the variables change together. It is useful to group functions into *families* with similar patterns of change because these functions and the situations that they model share a certain general characteristics

Essential understanding 4d: Some representations of a function may be more useful than others, depending on how they are used

Essential 4e: Linear functions have constant rates of change.

Essential 4f: Quadratic functions are characterized by rates of change that change at a constant rate

Essential understanding 4g: In exponential growth, the rate of change increases over the domain, but in exponential decay, he rate of change decreases over the domain

Big idea 5: General algorithms exist for solving many kinds of equations. These algorithms are broadly applicable for solving a wide range of similar equations. However, for some problems or situations, alternatives to these general algorithms may be more elegant, efficient, or informative

Essential understanding 5a: A general algorithm exists for solving linear equations. This algorithm is broadly applicable and reasonably efficient

Essential understanding 5b: Linear equations can be solved by symbolic, graphical, and numerical methods. On some occasions and in some contexts, one solution method may be more elegant, efficient, or informative than another

Essential understanding 5c: Quadratic equations can be solved by using graphs and tables and by applying an algorithm that involves c*ompleting the square*. This algorithm, when expressed in a more compact form, is also known as the *quadratic formula*.

**Geometry 6-8**

Big idea 1: Behind every measurement formula lies a geometric result

Essential understanding 1a: Decomposing and rearranging provide a geometric way of both *seeing that* a measurement formula is the right one and *seeing why* it is the right one

Essential understanding 1b: In addition to decomposing and rearranging, shearing provides another geometric way of both *seeing that* a measurement formula is the right one and *seeing why* it is the right one

Big idea 2: Geometric thinking involves developing, attending to, and learning how to work with imagery

Essential understanding 2a: Geometric images provide the content in relation to which properties can be noticed. Definitions can be made, and invariances can be discerned

Essential understanding 2b: Symmetry provides a powerful way of working geometrically

Essential understanding 2c: Geometric awareness develops through practice and visualizing, diagramming, and constructing

Big idea 3: A geometric object is a mental object that, when constructed, carries with it traces of the tool or tools by which it was constructed

Essential understanding 3a: Tools provide new sources of imagery as well as specific ways of thinking about geometric objects and processes

Essential understanding 3b: Geometric thinking turns tools into objects, and in geometry the process of turning an action undertaken with a tool into an object happens over and over again.

Big Idea 4: Classifying, naming, defining, posing conjecturing, and justifying are codependent activities in geometric investigation

Essential understanding 4a: Naming is not just about nomenclature. It draws attention to the properties and objects of geometric interest

Essential understanding 4b: Definition can both generate and reflect structure: definitions are often dependent on a specific classification

Essential understanding 4c: Conjectures can emerge out of a problem-posing process that generates claims that need to be justified.

**Statistics 6-8**

Big idea 1: Distributions describe variability in data

Essential understanding 1a: Graphs and tables are useful for displaying distributions of categorical data

Understanding 1b: Numerical summaries of categorical data are useful for describing particular features of a distribution.

Essential understanding 1c: Graphs and tables are useful for displaying distributions of quantitative data.

Essential understanding 1d: Numerical summaries of quantitative data are useful for measuring the center of a distribution.

Essential understanding 1e: Numerical summaries of quantitative data are useful for measuring the amount of variability within a distribution

Essential understanding 1f: Graphs and tables based on group data are useful for displaying distributions of a quantity of quantitative data

Essential understanding 1g: The shape of a distribution influences which summary measure is most appropriate for describing the center of a distribution for quantitative data

Essential understanding 1h: Graphs and tables based on a division of the ordered data into equal sized groups are useful for displaying distributions of quantitative data

Essential understanding 1i: Some numerical summaries of quantitative data are more resistant than others to extreme data values called *outliers*.

Big idea 2: Statistics can be used to compare two or more groups of data

Essential understanding 2a: The focus of comparisons between two or more groups of data is on similarities and differences between the distributions

Essential understanding 2b: The amount of separation between two or more distributions is related to the amount of variability within them.

Big idea 3: Bivariate distributions describe patterns or trends in the covariability in data on two variables

Essential understanding 3a: Graphs and tables are useful for displaying by various distributions of data on two categorical variables

Essential understanding 3b: Conditional relative frequency distributions are useful for establishing an association between two categorical variables

Essential understanding 3c: Graphs and tables are useful for displaying bivariate distributions of data on two quantitative variables

Essential understanding 3d: A correlation coefficient is a numerical strategy of bivariate data that measures the strength of the relationship between two variables

Essential understanding 3b: When the trend in bivariate data on two quantitative variables is generally linear, a centrally located line can be used for making predictions

Big idea 4: Inferential statistics uses data in a sample selected from a population to describe features of the population

Essential understanding 4a: The sampling distribution of a statistic describes the sample-to-sample variability and values of the statistic from the multiple samples of the same size selected from the same population

Essential understanding 4b: Selecting a simple random sample from a population is a fair way to select a sample

Essential understanding 4c: The predictable pattern for the sampling distribution of a statistic based on random sampling provides a way for making inferences about a population.

**Problem Solving Strand**

3.5 solve practical problems that involve addition and subtraction with proper fractions having like denominators of 12 or less

4.5c solve single-step practical problems involving addition and subtraction with fractions and mixed numbers

4.6b solve single-step and multistep practical problems involving addition and subtraction with decimals

5.4 create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of whole numbers

5.5b create and solve single-step and multistep practical problems involving addition, subtraction, and multiplication of decimals, and create and solve single-step practical problems involving division of decimals

5.6a solve single-step and multistep practical problems involving addition and subtraction with fractions and mixed numbers

5.6b solve single-step practical problems involving multiplication of a whole number, limited to 12 or less, and a proper fraction, with models*

6.5b solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions and mixed numbers

6.5c solve multistep practical problems involving addition, subtraction, multiplication, and division of decimals

6.6b solve practical problems involving operations with integers

7.2 solve practical problems involving operations with rational numbers

7.3 solve single-step and multistep practical problems, using proportional reasoning

8.4 solve practical problems involving consumer applications

Excerpt from Battista 2004- For numerous mathematical topics, researchers have found that students’ development of conceptualizations and reasoning can be characterized in terms of levels of sophistication (e.g. Battista & Clements, 1996; Battista, & Borrow, 1998; Cobb & Wheatley, 1988; Steffe, 1992; van Hiele, 1986).To construct new knowledge and make sense of novel situations, students build on and revise their current mental structures through the processes of action, reflection, and abstraction (Battista, 2004).A set of levels for a topic (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b). A levels-model for a topic describes not only cognitive plateaus, but also what students can and cannot do, students’ conceptualizations and reasoning, cognitive obstacles that obstruct learning progress, and mental processes needed both for functioning at a level and for progressing to higher levels. The levels are derived from analysis of both the mathematics to be learned and empirical research on students’ learning of the topic.