This is a continuation of the discussion of procedural v conceptual in the context of Khan Academy videos. It also brings in the Keith Devlin initiated debate on multiplication as repeated addition for natural numbers as opposed to an operation that is introduced directly in the field axioms of a field like the real field.
Define K-12-AA to stand for K-12 arithmetic and algebra that is taught in schools. Constructive real analysis, recursive analysis and computable analysis are subsets of standard math. The content of K-12-AA is typically within constructive, recursive, computable and other similar approaches.
If we define procedural math as that having an algorithm that obeys some specified requirements, then K-12-AA will typically comply with it. So K-12-AA is all procedural in this way of thinking. The dispute over procedural v conceptual teaching of math can’t be about this then.
Let’s Play Math tried to give Keith Devlin a lifeline recently by pointing out that repeated addition with variable increments is distinct from repeated addition with constant increments. It was not worded that way but it amounts to that. Thus if multiplication is defined as repeated addition with constant increments, then it is distinct from unrestricted repeated addition with varying increments.
However, this was not really what Devlin was saying. If he was, he would have done so. What Devlin was arguing for was that multiplication not be taught in a constructive way with a new definition for each new type of numbers starting with natural numbers, then rational numbers and then real numbers. Devlin though this was too difficult. So Devlin said just start with real numbers and make multiplication an axiomatic operation.
However, actual teaching of addition and multiplication does start with natural numbers, then goes to rational numbers and then goes to some sort of real number concept such as infinite decimals.
Some concepts of multiplication are argued to be distinct from repeated addition. Examples cited are usually scaling or some sort of area argument. This brings us to procedural v conceptual.
In math, we have rational numbers as a mechanical definition of an ordered pair (m,n) obeying specified rules for addition, multiplication, etc.
In math, we also have measure theory in the sense of Jordan measure, Lebesgue measure or general measures. For example, we might require a measure to be finite additive in the sense that for finite disjoint sets, the sum of the measures of each set equals that of the total set. For point sets on a plane represented as ordered pairs (x,y), we could require that if we multiplied all the points in a set by k, then the area of the point set would be k time bigger. We might call this a finite linear measure.
Suppose we start from natural numbers and we then construction rational numbers. We then definite linear measures on rational point sets in order to define their area, at least for some cases. (We stop at rational numbers because reals are more complicated in a constructive sense.)
So in order to have a measure on point sets for more than just natural numbers, we use rational numbers as point sets and apply our area measure to these point sets.
Here we have distinct concepts working together. We have rational numbers defined first in a purely mechanical way. We then use those for point sets in the plane. We then define a simplified area measure on those point sets. The area measure is finite linear additive.
We have procedures for each distinct object. We have procedures for rational numbers to be added or multiplied. We have procedures for point sets in the plane using rational numbers. We have procedures for the area of points sets of rational numbers.
Now we contrast this approach with taking the concept of rational number, point sets of rational numbers, and finite linear measures on rational point sets and mixing them together. The latter is what is actually done in K-12-AA currently.
A major part of the talking past each other in the Keith Devlin repeated addition debate was that concepts of number, point sets, and measure theory on point sets were mixed together. This led to a confused debate which was unable to converge as a consequence. It still leaves a legacy of confusion.
Now to procedural v conceptual videos. If we define procedural math as being the subset of standard math that is computable or constructive in some sense, then practically all of K-12-AA math is procedural. If however, we consider procedures bound to concepts in an object oriented way, then the distinct of procedural and conceptual makes sense.
In the context of multiplication, multiplication of rational numbers is a concept that has procedures. Rational number point sets are a concept with procedures. Area measures that are finite linear are a concept that has procedures, once we commit to a definition of area. These are distinct concepts with distinct bound procedures.
When we take the procedures of area measures and say those are what multiplication of numbers means we are creating a confusion by taking the procedures of area of rational number point sets and saying those are procedures that belong to the concept of rational numbers. This is what is actually done in K-12-AA. This is where blind proceduralism is the enemy of conceptual understanding.
To teach multiplication, area measure procedures are used as if they define multiplication. In fact, area measure procedures use pre-defined concepts of number and number procedures to define area procedures. To say the area procedure then defines multiplication of numbers as a concept or procedure is to create a confusion and commingle concepts. This is blind proceduralism. It creates lasting confusion. We can see that since it infects the participants in the repeated addition debate including professors.
Spaghetti Proceduralism ignores the class structure of procedures and concepts.
Rational numbers are a class with procedures.
Rational point sets in the plane are a class that inherits from rational numbers as a class.
Area of rational point sets in a plane inherits from rational point sets and thus from rational numbers.
Multiplication by rational numbers in the context of area of sets of rational numbers does not define multiplication. It is defining or implementing area of rational number point sets in the plane. It does not change the base class of rational numbers. It is an application of that class, but it does not define rational number multiplication.
Let’s Play Math linked to the following exhibit with the title “Multiplication Models”.
(That exhibit does not restrict itself to rational number point sets in the plane, but we shall impose that restriction for simplicity.)
This is an example of confusing the class hierarchy given above for rational numbers, rational number point sets in the plane and area of rational point sets. That type of application is commingled with other applications of number concepts as if these were all parallel number concepts instead of an aggregation of varying concepts.
Indexing using natural numbers is a very distinct concept from area of point sets of rational numbers in a plane. One can argue that natural numbers and all their procedures are always about indexing. They are about keeping your place in the count, i.e. an index location. Thus multiplication of natural numbers is a technique for indexing a location and dealing with index locations. These are distinct procedures from multiplication rational numbers as a mechanical procedure. This is distinct from area procedures for point sets of rational numbers in the plane.
Spaghetti proceduralism is to throw any procedures together and say all that counts is the procedure. It then tends to confuse the binding of procedures to concepts. So procedures for area of point sets in the plane of rational numbers are confused with the definition of multiplication of rational numbers. Using multiplication of natural numbers to index locations in a list is confused with area in a plane.
This Spaghetti Proceduralism makes a muddle of procedures and concepts. It tries to level all procedures as equal and just say learn each procedure as on the same level with any other procedure. We can call this Procedural Levelism.
As opposed to Sphaghetti Proceduralism is a class hierarchy approach. Natural numbers are defined as a class with procedures. The class is a way to define a concept. Rational numbers inherit from the Natural Number class. Rational Point Sets inherit from the Rational Number Class. Area of rational point sets in the plane inherits from the Rational Point Set class. Here we have a distinct concept hierarchy in which distinct concepts have procedures bound to them.
When we use the concept hierarchy approach, we don’t confuse applications of multiplication by a descendant class such as area of point sets of rational numbers with the definition of multiplication of rational numbers.
The commingling of the concept of rational number and area is presumably thousands of years old. When rational number as a formal concept and measure theory came along, the disentangling of these concepts for K-12 education never happened. Thus it persists. It is not the fault of any person to have been taught in the context of this confusion. Working our way out is the important thing.
New Math Done Right is based on applying foundations of math and logic in teaching elementary math in schools. This was the idea of New Math. Grassmann was a teacher when he developed his inductive approach in 1861. There is no going back from a better understanding of the concepts. Leaving commingled procedural lessons for elementary students that we know mix different concepts is not a good solution. These confusions eventually add up to stop progress.
Teaching the procedure without clarification when we know we are mixing up concepts is bad proceduralism. Teaching procedures where we clearly link the procedure to the concept is good proceduralism. In good proceduralism we distinguish concepts and we distinguish a procedure that is part of the concept from the use of a procedure from another concept. So area of rational point sets is distinguished from multiplication of rational numbers.
Recent references on this topic
Some older ones
I may add some additional links.
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