The Four Operations Through Fractions And Decimals Directly Connect Elementary Math To Algebra And Beyond
The principal foundation for students to be ready for pre-algebra is the full concept-procedure integration of the four operations through fractions and decimals and automaticity with the 400 math facts. Why do students need the four operations through fractions and decimals to be ready for pre-algebra and beyond? Because the major fundamental for elementary grades mathematics is addition, subtraction, multiplication, and division through fractions and decimals when we know how much we have; and the major fundamental for algebra is also addition, subtraction, multiplication, and division through fractions and decimals, but in algebra, we may not know how much we have, because in algebra, there are variable expressions and variable equations.
All students need the complete concept-procedure integration of the four operations through fractions and decimals because much of the mathematical structure of algebra IS based on the four operations through fractions and decimals: the order of operations to simplify variable expressions and the inverse order of operations to solve variable equations.
To cover this voluminous and essential foundation in elementary grades mathematics, I have created two seminars: Making Math Real: The 4 Operations and the 400 Math Facts (4 Ops) and Making Math Real: Fractions, Decimals, and Advanced Place Value (FDAP). At 12 days each, these are the two longest seminars the Making Math Real Institute offers, and since they provide the critical foundation for all future mathematics, they are the only two large-scale (seminars of 10 or more days) institute courses offered once per year, every year.
In the appropriate developmental sequence, I intend for Making Math Real: The 4 Operations and the 400 Math Facts to precede Making Math Real: Fractions, Decimals, and Advanced Place Value because much of the content in FDAP is a direct extension and application of the 4 Ops content:
MMRI Recommended Course Sequence Guide for Elementary and Middle School
To see the complete MMRI RCSG, click here.
- 4 Ops provides the concepts of place and place value for ones, tens, and hundreds; FDAP continues with advanced concepts and applications of place and place value for the thousands through the billions
- 4 Ops provides complete concept-procedure integration for addition, subtraction, multiplication, and division through multi-digit operands with renaming in addition and subtraction, and through double-digit operators for multiplication and long division; FDAP applies all concept-procedure integration of the four operations with whole numbers to addition, subtraction, multiplication, and division for fractions and decimals
- 4 Ops provides the structures to develop automaticity for the 400 math facts*; FDAP applies all math facts to develop concept-procedure integration for the four operations with fractions and decimals. In particular, automaticity with the multiplication and division facts directly supports the abilities to factor products, prime factor products, mentally find the greatest common factor of two products, mentally find the least common multiple of two products, generate equivalent fractions, simplify equivalent fractions, rename two or more fractions as equivalent fractions with least common denominators, transform mixed fractions to improper fractions/improper fractions to mixed fractions, and cross simplify fractions in multiplication – all of which are presented in FDAP.
* The 100 multiplication facts are only taught in the 3-day Making Math Real: The 9 Lines Intensive, and NOT the 4 Ops course. Consequently, it is strongly recommended to take the 9 Lines course prior to 4 Ops and FDAP
In Addition to all the Content, Math Comprehension is a Focus of the 4 Operations and 400 Math Facts Course
Making Math Real: The 4 Operations and the 400 Math Facts course presents the most essential foundational 1st through 5th grade content and cognitive developments necessary to prepare students for all future mathematics. Embedded within the teaching of the content, this course emphasizes the development of math comprehension as the basis for the learning of all mathematics, because the definition of success in math is the confidence that comes from the deep understanding of “I know that I know”, rather than “I remember what to do”.
Concept-Procedure Integration Develops Comprehension
All of the operations content in the 4 Ops course is based on concept-procedure integration, which means the concept of the mathematics and its respective procedure(s) are integrated as one. The teaching of concept-procedure integration will be fully covered throughout the 4 Ops course.
Concept-procedure integration is vastly different from the way math has been traditionally and currently taught: as a series of disconnected procedural commands intended to program students to remember what to do rather than develop the comprehension of what they are doing. Often, the phrase “Just do…” is the sum total of the teaching. Some representative examples of disconnected “Just do…” procedural commands:
- For renaming (regrouping) in addition: “Just carry the 1
- For subtracting integers: “Just do the opposite”
- For increasing or decreasing decimals by powers of 10: “Just move the decimal to the left or right”
- For division with fractions: “Just keep, change, flip” or “Just invert and multiply”.
All of these procedural commands are an educational “hope and a prayer” that students can remember what to do and are completely disassociated from the actual mathematics, and even worse, are frequently mathematically incorrect. For example, “Just carry the 1”: it’s not a “1”, it’s a ten (or it could be a hundred, a thousand, etc.); “Just move the decimal”: the decimal is a symbolic wall that separates whole units from parts of a unit, and therefore, can never be moved, rather, it’s the numbers in their respective places that increase or decrease in relation to the decimal point (the “wall”).
Active Working Memory = Math Comprehension
There is no math comprehension with procedural commands, and consequently, students must rely on memorizing procedural steps rather than building knowledge of correct mathematics. The major failure with using procedural commands is even deeper and more critical. Teaching math procedures disassociated from comprehension means students’ working memory for math is never activated. Working memory is the perceptual understanding of what we are currently doing, and we need and use working memory for every active experience (as opposed to receptive experiences, such as watching TV), spanning from basic and/or habituated experiences to extremely complex and challenging ones. Activating and sustaining working memory throughout an activity is how we maintain conscious awareness of what we are doing. Sustaining working memory enables us to complete a task without making mistakes, or if we make mistakes, noticing and self-correcting any error(s) if and when they occur.
Active working memory in math is the perceptual experience of comprehension and knowing what we are doing throughout problem solving. Math is like performing surgery. Just as the surgeon cannot afford to make a mistake during a procedure, so, too, must a person doing math sustain a clear and robust working memory picture throughout problem solving. In math, even though all of the other details within the problem have been completed correctly, just one detail mistake within that problem will make for an incorrect solution. In math, we have to bat 1,000, or be 100% accurate from the free-throw line if we are to complete a problem correctly. The better we understand and know what we are doing, the stronger our working memory picture, and the better we can manage all the details in our problem solving.
Losing our working memory picture during an activity is extremely risky. The moment we lose our working memory picture we become perceptually blind because we no longer can perceive what we are doing, and are now eminently capable of making mistakes without noticing them. Frequently, these mistakes can be silly ones about which we “should have known better”, and may ask ourselves, “How could I have done that?”
Activating and sustaining working memory is an executive function and requires significant cognitive effort, therefore, there are numerous factors that can contribute to making us lose our working memory picture. Some of these factors can be multitasking, becoming distracted, going on autopilot, getting confused, overloading working memory, or following procedural commands disassociated from comprehension.
Working Memory: One of the Great Developmental Gifts of Math
One of the major developmental purposes of the successful math experience across the grades is the ever-increasing ability to activate and sustain working memory for longer and more challenging periods of problem solving. This powerful gift of the successful math experience means, if we can activate and sustain working memory in math, then we can transfer that ability to activate and sustain working memory for almost any task. The ability to activate and sustain working memory provides a key exit ticket from school for becoming successful independent adults. We need a strong working memory to navigate the challenges of being an adult as this helps us manage the details of all that we do including our jobs, our responsibilities, and our households. Developing working memory is one of the most significant contributions we can give to our students to foster their future success as independent adults, and having a sustained, successful math experience is one of the best pathways to do so.