What We Teach
Mathematical Structures, Ideas, and Practices that Promote
a Deep Understanding of Elementary Mathematics.
Dale Oliver
Humboldt State University
June, 2004
There are several different resources that mathematics faculty can use to decide which content areas to include in their courses for prospective teachers (see, for example, the MET document, the NCTM Principles and Standards for School Mathematics, or your state academic content standards.) Most sources imply or explicitly list a collection of understandings that are desirable for teachers. Not surprisingly, these lists include topics in Number, Operations, Algebra, Functions, Geometry, Measurement, Data Analysis, Statistics, and Probability. Most resources also imply or implicitly state the importance of the quality of these understandings by describing the cohesiveness, depth, and flexibility of mathematical knowledge that is required to teach well. It is this quality of understanding that I want to address here.
I like how Liping Ma, in Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States (1999), distinguishes the quality of the understandings of teachers about mathematics through discussion of the “basic” and the “fundamental” points of view. When elementary mathematics is viewed as “basic” mathematics, then the mathematics can be described as a collection of procedures. When elementary mathematics is viewed as “fundamental”, then the mathematics can be described as elementary (the beginning of the discipline), primary (containing the rudiments of advanced mathematics) and foundational (supporting future learning). Supporting the “fundamental” point of view, Ma argues that elementary teachers require a profound understanding of fundamental mathematics that has breadth, depth, and thoroughness.
How can mathematics faculty promote such an understanding, particularly one which has breadth, depth, and thoroughness? Recommendations 3 and 4 of the CBMS document on the Mathematical Preparation of Teachers give some general guidelines. One model for meeting the spirit of these guidelines requires math faculty who teach prospective elementary teachers to deliberately include the following features in their courses:
- Organizing schemes (Mathematical structures and/or developmental sequences with connections to school mathematics) that promote breadth of understanding.
- Study of key mathematical ideas and concepts from multiple points of view that promote depth of understanding.
- Opportunities for prospective teachers to develop the habits of mind of a mathematical thinker that promote thoroughness of understanding.
What follows are examples of these three features in a given content area.
Number and Operations (Number Sense)
A. Organizing Schemes
- A diagram of nested subsets of the Real numbers. I use the diagram of nested subsets several times during our course on number sense and algebraic thinking. I use it for discussing types of numbers, closure of operations, identities, inverses, solutions to equations, and children’s development of number sense.
- Definitions of “Number Sense” and “Numeracy”.
What is number Sense?
“Much of school mathematics depends on numbers, which are used to count, compute, measure, and estimate. Mathematical activity belongs here that centers primarily on the development of number concepts, on computation with numbers (addition, subtraction, multiplication, division, finding powers and roots, etc.), on numerations (systems for writing numbers, including base ten, fractions, negative numbers, rational numbers, percents, scientific notation, etc.), and on estimation. At higher levels this strand includes the study of prime and composite numbers, of irrational numbers and their approximation by rationals, of real numbers, and of complex numbers.”
Mathematics Framework for California Public Schools, K-12, 1999
“Like common sense, number sense produces good and useful results with the least amount of effort. It is not mindlessly mechanical, but flexible and synthetic in attitude. It evolves from concrete experience and takes shape in oral, written, and symbolic expressions.”
National Research Council, 1989
A person who has Number Sense has some combination of
- an ability to represent number in a variety of ways,
- an awareness of the relationships between numbers,
- a knowledge of the effects of operations,
- an ability to interpret and use numbers in real world counting and measurement.
NCTM Addenda Series, 1994
- Connections to State Standards.
Number sets through the grade levelsCA StandardsOperations through the grade levels
K: Counting Numbers through 301. Counting Numbers though 100
2. Counting numbers through 1,000
Fractions – halves, thirds, sixths, eighths, twelfths.
Decimals as related to money.
3.Counting numbers through 10,000
Development of Fractions done in 2nd grade
Decimals to the hundredths.
4.Counting numbers through 1,000,000
Decimals and fractions through hundredths
Concepts of negative integers (Number line, counting, temperature, owing)
- Decimals number from thousandths to millions
Develop positive and negative integers
- Positive and negative fractions and decimals
Pi
- All of the above
1. Addition facts to 20 (and corresponding differences)
Addition and subtraction problems with a one-digit and a two-digit number.
2. Sums and differences of 3-digit numbers
Mental Arithmetic for sums and differences of 2-digit numbers
Model and solve simple multiplication and division.
3. Multiplication tables through 10 x 10.
Sums and differences of 4 digit numbers.
Multiply and divide a multi-digit number by 1-digit number.
Add, Subtract, Multiply, and Divide with money.
Add and subtract simple fractions.
4. Estimate and Compute the sums or difference of whole numbers and positive decimals (2 places).
Multiply or Divide a multi-digit number by a 2-digit number.
Add and subtract negative numbers.
5. Add, subtract, multiply, and divide decimals and negative numbers.
Proficient with multi-digit multiplication and division.
Add, subtract, multiply, and divide simple fractions
6.Calculate (Add, subtract, multiply, and divide with all types of rational numbers.
- All of the above. Use exponents and roots when working with rational numbers.
B. Key concepts from multiple viewpoints
Here are two examples of concepts that I emphasize under “Number and Operations”. Whenever we are working on any of the “multiple approaches” for a concept, I conduct a classroom discussion connecting the activity or approach to that concept.
- Place value
We study the concept of place value in various ways throughout the course, including
- Studying Egyptian and Mayan numeration systems,
- Doing arithmetic base 5,
- Analyzing algorithms for whole-number operations,
- Extending whole number operations to decimal forms of rational numbers,
- Developing Scientific Notation.
- Binary Operations
We study the concept of binary operations in various ways throughout the course, including
- Analyzing the 4-color game (see the Sunday afternoon presentation) to see binary operations as functions,
- Studying different conceptual models and word problems for operations over the whole numbers, integers, and rationals,
- Investigating compositions of functions, particularly of isometries.
C. Opportunities for developing habits of mind.
In the CBMS document, one of the items on the list of understandings for elementary teachers is, “understanding how any number represented by a finite or repeating decimal is rational, and conversely.” The activity started by Stuart Moskowitz in the laboratory class of the workshop is intended to get prospective teachers to thoroughly investigate a part of this understanding.
Geometry and Measurement
A. Organizing Schemes
1. The van Hiele Levels of Geometric Thinking.[1]
Dina and Pierre van Hiele are two Dutch educators who were concerned about the difficulties that their students were having in geometry. This concern motivated their research aimed at understanding students’ levels of geometric thinking to determine the kinds of instruction that can best help students.
The five levels that are described below are not age-dependent, but, instead, are related more to the experiences students have had. The levels are sequential; that is, students must pass through the levels in order as their understanding increases. The descriptions of the levels are in terms of “students” – and remember that we are all students in some sense.
Level 0 – Visualization
Students recognize shapes by their global, holistic appearance.
Students at level 0 think about shapes in terms of what they resemble and are able to sort shapes into groups that “seem to be alike.” For example, a student at this level might describe a triangle as a “clown’s hat.” The student, however, might not recognize the same triangle if it is rotated so that it “stands on its point.”
Level 1 – Analysis
Students observe the component parts of figures (e.g., a parallelogram has opposite sides that are parallel) but are unable to explain the relationships between properties within a shape or among shapes.
Student at level 1 are able to understand that all shapes in a group such as parallelograms have the same properties, and they can describe those properties.
Level 2 – Informal deduction (relationships)
Students deduce properties of figures and express interrelationships both within and between figures.
Students at level 2 are able to notice relationships between properties and to understand informal deductive discussions about shapes and their properties.
Level 3 – Formal deduction
Students can create formal deductive proofs.
Students at level 3 think about relationships between properties of shapes and also understand relationships between axions, definitions, theorems, corollaries, and postulates. At this level, students are able to “work with abstract statements about geometric properties and make conclusions based more on logic than intuition” (Van de Walle).
Level 4 – Rigor
Students rigorously compare different axiomatic systems.
Students at this level think about deductive axiomatic systems of geometry. This is the level that college mathematics majors think about Geometry.
2. A connection to the State Standards
In general, most elementary school students are at levels 0 or 1; some middle school students are at level 2. The CA standards are written to begin the transition from levels 0 and 1 to level 2 as early as 5th grade “Students identify, describe, draw and classify properties of, and relationships between, plane and solid geometric figures.” (5th grade, standard 2 under Geometry and Measurement) This emphasis on relationships is magnified in the 6th and 7th grade standards.
Interestingly, the sixth National Assessment of Educational Progress report (1997) states that “most of the students at all three grade levels (fourth, eight, and twelfth) appear to be performing at the ‘holistic’ level (level 0) of the van Heile levels of geometric thought.”
3. Definitions related to measurement
What is Measurement?
Measurement is the process of attaching numbers to certain qualities of objects and events.
Counting involves discrete objectsMeasuring involves continuous properties
(How many?) (How much?)
What is Measurable?
A measurable attribute of an object or event is a characteristic that can be quantified by comparing it to a unit.
LengthArea Volume Capacity WeightMass
TemperatureTimeAnglesPositionDistanceSpeed
AccelerationPressurePitchEnergyEtc.
What is the Process of Measurement?
Choose an appropriate unit for the attribute being measured;
Compare the attribute of the object or event to the unit;
Obtain a measurement (always an approximation) — a number AND a unit.
What is a common instructional sequence?
- Develop the meaning of the attribute being measured through activities involving perception and direct comparison. (Which is taller?)
- Children begin to measure using arbitrary or nonstandard units. (How tall?)
- Children measure and estimate using standard units. (How tall in Feet and Inches.)
Accuracy of Measurement
“To apply care to”
When a person puts a lot of care into a measurement, the result is likely to be accurate.
Precision of Measurement
“to cut off in front”
The smallest unit of measure used to express an approximate value (the exact value is “cut off” there).
B. Key concepts from multiple viewpoints
Here are two examples of concepts that I emphasize under “Geometry and Measurement”. Whenever we are working on any of the “multiple approaches” for a concept, I conduct a classroom discussion connecting the activity or approach to that concept.
1. Similarity
We study the concept of similarity in various ways throughout the course, including
- Investigating similar triangles in the context of parallel lines and circles,
- Conducting remote measurement activities (Heights of Trees),
- Exploring generalizations of the Pythagorean Theorem,
- Studying size transformations and scalings in a line, plane, or space.
- The role of the Unit of measure
We study the concept of unit in various ways throughout the course, including
- Use of non-standard and standard units in many different contexts.
- Conversion between one unit of measure and another unit of measure
- Dimensional analysis in solving problems.
- Extension of the concept to the meaning of the standard deviation
C. Opportunities for developing habits of mind.
I use a collection of investigations in Key Curriculum Press’ Exploring Geometry with the Geometer’s Sketchpad (1993). My favorites are related to the topic, “devising area formulas for triangles, parallelograms, and trapezoids; knowing the formula for the area of a circle (CBMS),” particularly one in which students test formulas for various quadrilaterals that involve either the length of a mid-segment or the length of the diagonals.
In addition, I have students work on selected investigations on the Geoboard. (See the next page.)
Geoboard Activities
Consider the square area bounded by 4 adjacent pegs to be 1 square unit.
1. Find the area of this triangle. See if you can find more than one way to calculate the area.
2. Make shapes on your geoboard that can be made with only one geoband which does not cross itself anywhere. For each shape, calculate its area, record the shape and its area on your geoboard paper. Make a chart for your shapes, listing the number of pegs each shape touches (T), the number of pegs each totally enclosed in the interior of each shape (I) and the area of the shape (A). Look for a formula that describes the algebraic relationship among these three variables. (Hint -- find a systematic way to organize your data.)
TI
A
3. How many triangular regions of different area can you create on your geoboard where all three vertices of the triangle are pegs of the board?
4. How many noncongruent triangles can you make on your geoboard where all three vertices of the triangle are pegs of the board?
5. How many distinct lengths of line segments can you make on your geoboard where both endpoints of the segment are pegs of the board?
6. How many different size squares can you make on your geoboard? Find the area of each.
Extensions: Repeat questions 5 and 6 for an n by n geoboard.
Algebra and Functions
A. Organizing Schemes
- Algebra in School mathematics
What is “Algebra and Functions”, Pre-K through Grade 12?
“This strand involves two closely related subjects. Functions are rules that assign to each element in an initial set an element in a second set….they are encountered informally in the elementary grades and grow in prominence and importance with the student’s increasing grasp of algebra in the higher grades. …
Algebra…starts informally. It appears initially in its proper form in the third grade as “generalized arithmetic.” In later grades algebra is the vital tool needed for solving equations and inequalities and using them as mathematical models of real situations.”
Mathematics Framework for California Public Schools, K-12, 1999
“Instructional programs from prekindergarten through grade 12 should enable all students to
- understand patterns, relations, and functions;
- analyze change in various contexts.
- represent and analyze mathematical situations and structures using algebraic symbols
- use mathematical models to represent and understand quantitative relationships”
Principles and Standards for School Mathematics, NCTM, 2000
Sample tasks that are algebraic
The ideas for these samples came from the NCTM Principles and Standards for School Mathematics.
Primary Grades – Front and Back Cards
A primary teacher had prepared two groups of cards for her students. In the first group, the number on the front and back of each card differed by 1. In the second group, these numbers differed by 2.
The teacher showed the students a card with 12 written on it and explained, "On the back of this card, I've written another number." She turned the card over to show the number 13. Then she showed the students a second card with 15 on the front and 16 on the back. As she continued showing the students the cards, each time she asked the students, "What do you think will be on the back?" Soon the students figured out that she was adding 1 to the number on the front to get the number on the back of the card.
Then the teacher brought out a second set of cards. These were also numbered front and back, but the numbers differed by 2, for example, 33 and 35, 46 and 48, 22 and 24. Again, the teacher showed the students a sample card and continued with other cards, encouraging them to predict what number was on the back of each card. Soon the students figured out that them numbers on the backs of the cards were 2 more than the numbers on the fronts.
When the set of cards was exhausted, the students wanted to play again. "But," said the teacher, "we can't do that until I make another set of cards." One student spoke up, "You don't have to do that, we can just flip the cards over. The cards will all be minus 2."
As a follow-up to the discussion, this teacher could have described what was on each group of cards in a more algebraic manner. The numbers on the backs of the cards in the first group could be named as "front number plus 1" and the second as "front number plus 2." Following the student's suggestion, if the cards in the second group were flipped over, the numbers on the backs could then be described as "front number minus 2." Such activities, together with the discussions and analysis that follow them, build a foundation for understanding the inverse relationship.