The number first geometric measure second approach is discussed here in terms of natural numbers. We consider the case of length first.
We are dealing only with natural numbers. No fractions and no signs. We have a number line with the natural numbers marked at 0, 1, 2, etc.
The point sets we consider are point sets of natural numbers. We define the length from i to i’, to be 1. Here i’ is the successor of i, so i+1. So the length of the point set consisting of i and i’ is 1.
We consider bounded point sets, meaning that there is some k so that if i is in the point set then i is less than or equal to k. Call such a point set B.
We take each pair of natural numbers i,j in a point set B such that j = i’. Call them a successor pair. For each successor pair in B we count one unit of length. We call this contribution a successor pair contribution to length. The sum of the successor pair contributions is defined as the length of B.
So if B is the point set consisting of the points 2,3 and 5, then B has one successor pair 2 and 3, and so has a single successor pair contribution to length. The length of B is the sum of such lengths, so the length of B is 1. The singlet point 5 gives no contribution to the length of B.
If we had a point set C consisting of the single point 5, then C would have length 0.
The point set D consisting of 2,3,5 and 6, has two successor pairs and thus a length of 2.
When we do natural numbers first and define length in this way for bounded point sets, we get a well defined concept of length. We do not say length of line segments along a line gives us our concept of natural numbers. We instead develop natural numbers first from the Peano Axioms or some other way and then define lengths of natural number sets using a procedure such as the above.
The properties of length so defined then can be developed in theorems. This definition was done on the fly, so the work in developing the theory was not done prior to this post. This can be made a project.
Exercise: Define a finite gapless set of natural numbers as a set that contains the predecessors of each element in the set, except one (the start), and the successor of each element except for one (the end). Prove a finite gapless set is bounded. Prove that the length of a finite gapless point set is the end minus the start.
Exercise: For a given bounded point set B, a maximal gapless point set in B is one that has no predecessor in B and no successor in B. Prove at least one such gapless point set exists for each bounded point set B. Prove that the length of B is the sum of the lengths of the maximal gapless point sets it contains.
Exercise: Show that using gapless point sets that are not maximal and summing their length results in a length less than or equal to B and state conditions that the inequality is strict.
Example, the point set 2,3,4,5 where one takes the two gapless sets 2,3 and 4,5. These sum to a length of 2 as opposed to the length of 3 already assigned.
Metaexercise: Do any of the above proofs require using mathematical induction?
Metaexercise: Are these proofs too hard for students at low grade levels?
Let’s assume the answer to both metaexercises is yes. What does that imply about instruction that mingles the concept of natural number and length of numberline bounded sets without any attempt to separate out number concept from length concept? Such an approach also likely does not do much on defining gapless sets in terms of successor and predecessor and of decompositions of a bounded point set into gapless sets and maximal gapless sets. All that is skipped over.
Metaexercise: Accepting the statements of the prior paragraph, do such lesson plans assume the students know these concepts without being told them? The assume a gapless set is understood even without mentioning a definition of such or an example? If a gapless set is a key concept of length of natural number sets, and it is not defined and no examples given, what sort of concept of length do students acquire when instructed in number lines? Do students at low grade levels have an idea of the length of a natural number set that contains a single element such as the element 5 by itself? Would such students be puzzled if asked to tell what the length of such a set is? Would they be stumped to prove the length of a singleton set is 0? Without a definition as above, they could not do it? If students were asked this and led by self discovery and projects to define length of natural number point sets in a consistent way, could they do it? Would they of necessity arrive at a definition that gave length 0 to a singleton point set?
Has this simple type of formal length measure for natural number point sets ever been used in instruction in K-12? For pre-service teachers in college?
has many hits but it does not seem to give a concept of length of bounded natural number point sets as above.
The search “natural number point set” length
returns no results.
I would appreciate anyone knowing a reference to such a simplified concept of length of natural number point sets but which still states definitions and proves theorems.
Assuming no such approach exists for instruction of K-12 students or pre-service teachers, we have to ask why not?
Metaexercise: Why is length not defined for bounded natural number point sets in teaching K-12 students or pre-service teachers? Is the reason because the Peano Axioms are not taught to either group, except in rare cases?
The failure to ground instruction in number to K-12 students and pre-service teachers then results in a failure to have a formal concept of length of natural number point sets.
Metaexercise: To teach a formal concept of length of natural number point sets we would have to teach the Peano Axioms to students and pre-service teachers?
Metaexercise: In the current instruction, are students given a confused concept of length?
Metaexercise: Is the confused concept of length students are given for line segments of natural number point sets then confusing them further when fractions are justified as in Euler by reference to line segments that don’t start or end on a natural number? They don’t understand length of natural number point sets. Then they are told that the existence of fractions is tied to line segments that don’t start or end on a natural number. Probably with less explicit articulation than just given. They are then led to believe that the procedures for fractions are derived from this pseudo definition. Then they conclude that they don’t understand math because they can’t get it.
Pedagogical Content Knowledge is defined as knowledge of how students learn concepts, procedures, etc.
Metaexercise: We can’t even start to ask the right questions of how students form a concept of length of natural number point sets until we ourselves know these concepts in a formal way? We would never think to ask the above questions until we had formulated the concepts as formal mathematical concepts? That includes, definitions, examples, theorems and proofs? So PCK is logically dependent on not just random subject matter knowledge, but on foundations of math knowledge? So teachers and researchers in PCK need to understand math foundations?
Metaexercise: Isn’t this what Lance Rips, Jennifer Asmuth, etc. were saying in their papers on Peano Axioms, mathematical induction etc? These are referenced elsewhere on this blog.
Lance and Rips state in the conclusion to their paper on page 394:
We’ve argued elsewhere (Rips, Bloomfield, & Asmuth, 2006) that math induction is central to knowledge of mathematics: It seems unlikely that people could have correct concepts of number and other key math objects without a grip on induction. Basic rules of arithmetic, such as the associative and commutative laws of addition and multiplication, are naturally proved via math induction. If this is right, then it’s a crucial issue how people come to terms with it.
“Other key math objects” would include the concept of length of natural number point sets? As well as decompositions of natural number point sets into maximal gapless sets? Gapless sets are another natural number concept that is key to understanding the use of natural numbers in geometry?
Metaexercise: How well do students understand the extension of these concepts to rational number point sets if they think the justification of rational number point sets is from geometry, i.e. backwards?
If the lesson plan is set up properly, it flows logically and naturally. The proper concepts are short and easily explained. They hang together logically. Less work and time is needed to learn them. A few examples and some challenging exercises.
When a muddle of concepts is taught, then the students can’t absorb them. So they have to practice. In effect, they are learning the procedures without understanding the concepts properly, since they are not taught them properly.
With the wrong concepts at the heart of the lesson, or a muddle of concepts, then the students need more and more time and applications to fix the correct procedures. They are not learning the correct concepts, so they have to have more time to learn the procedures by rote. This is what actually happens in actual classrooms.
This is because the starting concepts of natural numbers from the Peano Axioms are not taught. Geometric measures are not taught as functions from point sets to numbers. Thus geometric measures such as length and area are confused muddles as well. Those designing the curriculum are as confused and as muddled as the students they produce. But that outcome was not foreordained. It came from choosing to turn away from math foundations. Math foundations are the demuddled explanations of elementary math.
When we use math foundations to design the curriculum it is shorter and easier to learn. Weaker students can learn the correct concepts because they 1) make sense individually 2) suggest new insights they would not have thought of 3) fill gaps in knowledge 4) build on themselves in a consistent way.
The total time to learn is shorter when the correct concepts from math foundations are taught. However, we have to focus on the path from natural numbers to rational numbers. This means the Peano Axioms, successor function, definition of addition, mathematical induction and proofs using it. The hardest part of this is proof by mathematical induction.
When we teach a muddle, we end up teaching procedures by rote. This then requires lots of practice time to get by heart. When we try to teach muddled concepts as our main focus, we may not have enough drill time and so the students may not even learn the procedures well. This is what actually happened when reform math started in 1990 according to its critics.
Concept teaching only works if one uses mathematically correct concepts. The mathematically correct concepts for elementary math are foundations of math. Grassmann, Dedekind and Peano worked out the mathematically correct concepts for the natural numbers. As we saw, Euler, like modern K-12 and pre-service teaching did not have these concepts and he ends up ignoring his own statements and just using rules and procedures that are not explained or justified from his supposed definitions or foundations. This means Euler students have to memorize the rules for algebra and give up trying to understand how lengths of line segments leads somehow to the rules for adding fractions.
When we use the muddled concepts, then we have to add more and more time for procedural drills and rote learning. This is the typical government way. They start wrong and then bloat the budget with more resources.
Elementary math taught in terms of the concepts of foundations of math is the only way to teach concepts because that has been proven by mathematicians. That is what foundations of math is, the only way to teach the concepts of elementary math that is consistent and valid. Other ways are muddles. The muddles lead to bloated curriculum of more and more projects, homework, etc. to give time to learn procedures by rote.
The way to unbloat the curriculum, shorten homework and restore childhood is to teach the foundations of math. This has to focus on the road from the Peano Axioms to the rules for rational numbers. When this foundation is set, other applications like length of natural number point sets follow easily. We can then get the insight that an isolated natural number has zero length as a result of a consistent and logical definition of length of natural number point sets.
This approaches gives students and pre-service teachers a senes of mastery of the material of elementary math. They understand the concepts of elementary math when they understand them from the point of view of math foundations. Muddled approaches destroy confidence and make them think they can’t understand how to go from pseudo definitions such as in Euler 1765 to the rules for fractions and so on.
The current system teaches students they can’t understand math because the math they are taught is not understandable. Parts and links are left out in Euler 1765. Those were put in during the 19th century by Grassmann, Dedekind and Peano. But the schools ignore that, so they are teaching the Euler 1765 procedural math that contains conceptual gaps already found and filled by Grassmann, Dedekind and Peano.
The result is that students are sure to experience the gaps in the Euler procedural approach and then think that they can’t understand math because they don’t see how Euler’s procedural rules follow from his incomplete or incorrect foundation statements. That is what is perpetuated in the current system. As it fails, it demands more resources and becomes ever more bloated in time and money.
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