Developmental Stages in Mathematical Measurement and Number Sense
Learning Trajectory for Length Measurement
E1: Initial Unit Placement
Places the units from end to end. May not recognize the need for units of the same length or may not be able to measure if there are fewer units than necessary. Can use rulers with substantial guidance.
E2: Ordering and Seriation of Lengths
Orders lengths, marked from 1 to 6 units. Understands, at least intuitively, that any set of objects of different lengths can be placed in a series that is always increasing (or decreasing) in length.
E3: Unit Iteration and Inverse Relationship
Measures by repeating (iterating) a single unit and understands the need for a unit of equal length. Relates the size and number of units (inverse relationship). Can often use rulers with minimal guidance in simple situations.
E4: Accumulation, Partition, and Zero Point
Measures, knowing the need for identical units, the relationship between different units, unit partitions, the zero point in rulers, and distance accumulation. Considers the length of a bent path as the sum of its parts (not the distance between the endpoints). Starts estimating.
E5: Internal Measurement Tool Development
Has an “internal” measuring tool. Moves mentally along an object, segmenting it and counting the segments. Operates arithmetically with measurements (“connected lengths”). Subdivides a unit into at least halves. Performs accurate estimates.
E6: Perimeter Calculation and Contextual Analysis
Calculates the perimeter of a polygon, including complex cases. Can find multiple related cases of polygons with the same perimeter or area and relates these cases to each other using a logical comparison to provide evidence of an underlying pattern. Makes statements about complex sets of paths based on the perimeter and other linear aspects of objects. Can analyze length in two-dimensional and three-dimensional contexts. In the selection of units, children show well-developed ideas of precision and accuracy.
Learning Trajectory for Area Measurement
E4: Physical and Graphic Covering
Makes physical and graphic coverings. Counts the units using rows and columns. Compares areas based on the numerical result of the counting.
E5: Identifying the Square Unit
Identifies the square as the most useful unit. Counts units by tracing rows and columns.
E6: Calculation by Repeated Sums and Conservation
Divides the shape into squares with ease. Calculates by doing repeated sums in rows and columns. Shows an initial notion of conservation (maintaining the area by changing the shape).
E7: Using Formulas and Transitivity
Uses linear measures to calculate areas using multiplication. Uses the formula strategically. Identifies which figures with different shapes may have equal areas. Uses transitivity.
Learning Trajectory for Volume (Ages 0-8)
Level 3: Volume Filler
Can compare two containers by pouring one into the other (although confusion may occur at first regarding “which holds more”). Fills a container using another (smaller container) and counts the number needed to completely fill the larger container (but may not use accurately filled scoops and may not focus on quantifying the total volume or capacity). In packing situations, the child places cubes into a rectangular box to fill it. Eventually packs the entire box with cubes in an organized way. Compares objects by physically or mentally aligning; refers to at least two dimensions of objects. May be able to compare two containers using a third container and transitive reasoning.
Level 4: Volume Quantifier
Has a partial understanding of cubes as filling a space. Is able to estimate the number of scoops needed to fill. Is able to attend to both the portion of the container filled and the portion remaining unfilled. Recognizes when a container is half full. Exhibits initial spatial structuring. Packs a box neatly and completely with cubes; may count one cube at a time, while packing, to determine the total. Compares objects by physically or mentally aligning and explicitly recognizing three dimensions.
Level 5: Volume Unit Relater and Repeater
Uses simple units to fill containers, with accurate counting. Relates size and number of units explicitly; understands that fewer larger units than smaller units will be needed to fill or pack a given container. Can accurately convert units in a 1:2 ratio.
Level 6: Initial Composite 3D Structurer
Understands cubes as filling a space but does not use layers or multiplicative thinking. Moves to more accurate counting strategies. Relates the number of cubes to cubic units as measured by capacity. Given a graduated cylinder marked in cubic-inch units, the child understands that sand filled to the 10 in the cylinder would fill a box that holds ten 1-inch cubes. Begins to visualize and operate on composite units such as rows or columns (what we call a 1x1xn core). Iterates to pack the space completely, accounting for “internal/hidden” cubes. Decomposes space, allowing for accurate use of units and subunits. Recognizes when a box is half full, visualizes remaining rows or columns.
Level 7: 3D Row and Column Structurer
Is able to coordinate flexibly filling, packing, and building aspects of volume. Shows a propensity for additive comparisons (e.g., “this one has 12 more”) but may show some nascent multiplicative comparisons (e.g., “this one is four times as big”). Initially counts or computes (e.g., number of rows times number of columns) the number of cubes in one layer, and then uses addition or skip counting by layers to determine the total volume. Eventually moves to multiplication (e.g., number of cubes in a layer times number of layers).
Level 8: 3D Array Structurer
Has an abstract understanding of the rectangular prism volume formula. Shows a propensity for multiplicative comparisons, coordinates multiplicative and additive comparisons flexibly. With linear measures or other similar indications of the three dimensions, multiplicatively iterates cubes in a row, column, and/or layers to determine the volume. Constructions and drawings are not necessary. In multiple contexts, children can compute the volume of rectangular prisms from its dimensions and explain how that multiplication creates a measure of volume.
Visualization and Geometric Transformations
Level 5: Slider, Flipper, Turner
Performs slides and flips, often only horizontal and vertical, using manipulatives but guided by mental images of these motions (of turns of 45°, 90°, and 180° and flip over vertical and horizontal lines). That is, they can mentally imagine the motion and the result of it.
Level 6: Diagonal Mover
Performs diagonal slides and flips as well as all motions from previous levels. Knows a shape must be turned or flipped over an oblique line (45° orientation) to fit into a puzzle.
Level 7: Mental Mover
Predicts results of moving shapes using mental images (any direction or amount).
Learning Trajectory for Decimals
- Understand the Place Value of Numbers.
- Compare and order decimal numbers down to thousandths.
- Represent decimal numbers on the number line.
- Perform operations with decimals by estimating, rounding to natural numbers, and checking with a calculator.
- Strategically add and subtract decimal numbers down to thousandths in symbolic register without using the standard algorithm.
- Multiplication and division of decimals with appropriate models.
- Multiplication and division of decimals symbolically.
Learning Trajectory for Fraction Concepts
Level 1: Congruence Recognition Difficulties
Students have difficulties in recognizing that the parts of the whole must be congruent.
Level 2: Unit Fraction Iteration and Continuous Contexts
Students recognize that the parts could be different in form but congruent in relation to the whole. This allows them to identify and represent fractions in a continuous context, but they have difficulties with discrete contexts. They also begin to use unit fractions as an iterative unit:
- To represent proper fractions (although they have difficulties with improper fractions).
- To solve some fraction addition problems with the same denominator.
Level 3: Discrete Contexts and Inverse Relationship
Students identify and represent fractions in discrete contexts, recognizing that the groups must be equal. They also recognize that a part could be divided into other parts. When comparing fractions, they recognize that the size of a part decreases when the number of parts increases. They can use a part (not necessarily the unit fraction) as an iterative unit to represent proper fractions. They can also reconstruct the whole using any fraction as an iterative unit (continuous and discrete contexts). In addition, they use intuitive graphical representations to add/subtract fractions with different denominators.
Level 4: Equivalent Fractions and Operator Understanding
Students can solve simple arithmetic problems with the help of a guide or support. They can create equivalent fractions so that operations can be graphically represented. When they add or subtract fractions with different denominators, they understand that the parts must be congruent to join/separate, although they need a guide that allows them to choose the unit correctly. When they multiply, they understand the fraction as an operator “a/b of c/d,” and when they divide, they develop two types of reasoning:
- Division as a measure.
- Division as a partition.
Level 5: Symbolic Operation and Remainder Interpretation
Students can operate and solve arithmetic problems symbolically, identifying patterns. They can graphically justify what they do, but only in simple situations. At this level, they are able to interpret the remainder of a division of fractions.
Level 6: Graphical Justification Mastery
Students can explain operations graphically. They do not need a guide to represent fraction operations.