# Is there a Piagetian age at which proofs can be comprehended?

I am wondering if there is literature on the developmental age (pre-adolescent?, adolescent?) at which the notion of a "proof" can be understood? I am less interested in mathematical proofs and more interested in "water-tight arguments"—at what age can students understand what constitutes a logically correct, almost legal argument, impervious to objections?

(My own view is that this is in fact what constitutes a mathematical proof: a "water-tight argument" for the appropriately educated audience. But let's leave that aside.)

Certainly this capacity is present in U.S. 7th or 8th-grade (age: ~12-13 yrs old), when (typically) two-column Euclidean geometry proofs are taught. But I have sensed the capacity may be (nascently?) present even at the U.S. 4th or 5th-grade levels (~9-10 yrs old), in my occasional forays into that domain.

Added. With the help of keywords from the useful comments, I found a 2003 MS Thesis literature review of the topic:

Kate Mansi. "Reasoning and Geometric Proof in Mathematics Education: A Review of the Literature." North Carolina State Univ. Masters Thesis. 2003. (PDF download link)

Abstract. Through a comparison of the theories of Piaget and van Hiele, I discuss how students acquire mathematical and geometric reasoning skills and how this relates to their readiness to produce formal proofs. I then discuss research findings, which indicate that students are not typically at a high enough van Hiele level to be successful with proof by the time they get to high school.

[p.16:] Piaget (1987) claims that students progress through three levels (PL) in the development of their justification and proof skills. Unlike van Hiele’s levels of geometric understanding, Piaget’s levels for proof and justification coincide with the biological development of the student.

Piaget, J. Possibility and necessity. Vol 2. The role of necessity in cognitive development. Minneapolis: University of Minnesota Press. 1987.

• I do not have an answer for this, but I'm thinking Tall (of Tall and Vinner) might be a person whose work is worth looking into (google scholar)... – Benjamin Dickman Apr 18 '15 at 3:13
• No answer, but I think that it depends on what one is trying to prove. Make is interesting enough and within the experience of the kids, and maybe the age goes way younger. Also, playing the game of definitions probably helps. (Team A gives a word, say chair, to Team B. Team B gives a definition. Team A gives either an example that fits the given definition but which Team B would agree is not a chair, or shows a chair that does not fit the definition. Team B improves. Repeat.) I got this from my friend, Maria Droujkova, at naturalmath.com. – Sue VanHattum Apr 18 '15 at 14:36
• Certainly this capacity is present in U.S. 7th or 8th-grade (age: ~12-13 yrs old), when (typically) two-column Euclidean geometry proofs are taught. I think this is extremely over-optimistic. Some kids at that age are ready to handle proofs, but by no means all. Many people reach adulthood without ever developing the abilities you're talking about, despite systematic instruction. – Ben Crowell Apr 19 '15 at 15:40
• "Certainly this capacity is present in U.S. 7th or 8th-grade" Can we really assume that this is true for most students just because "it is taught" ? – futurebird Apr 19 '15 at 21:50
• Thinking (way) back to when I had to do two-column proofs (in the first semester of the10th grade), I don't remember being taught that these are intended to be water-tight arguments. They were more like combinatorial exercises or puzzles, where we try to fit together the few theorems that we had memorized in a way that matches the statement that we're supposed to prove. (The second semester was way better, an honors course where we actually learned some geometry and we were expected to think about the geometry, not just concatenate theorems.) – Andreas Blass Apr 19 '15 at 23:20

The simplest answer to this question is

Proofs can be comprehended at the same age rules can be comprehended.

Now, I must say why this is the case.

Although it may seem that the difficulty in answering this question is identifying what is meant by "comprehending" with regards to "proofs", the actual difficulty is in the word "proof" itself. As a human behavior "proof" is not well defined. To be more precise, there are plenty of human behaviors that can be clearly defined based on an an observable sequence of events e.g. the behavior "push a button" can be recorded and identified without the intervention of any human being. The problem with the behavior "give a proof" is that, except in the most technical circumstances, we are unable to say what constitutes the behavior of "give a proof" much less to say how we might go about observing or measuring this behavior.

(Note, it has been the goal of mathematicians since the dawn of mathematics to say clearly and exactly just what constitutes the behavior "give a proof" and so far our best efforts are found in Hilbert's formalism, but there is a huge difference between proof as argument and proof as formalism.)

Certainly, there is some agreement among humans as to what arguments are, but when we look at them closely, we are hard pressed to say why one argument "is better" or "more water-tight" than another, except to say that "we know it when we see/hear/feel/sense it."

Suppose we ignore all forms of classical Rhetoric with regards to a definition of argument. Historically, with the elimination of Rhetoric from an argument, we are left only with its "logical structure" and, it has been believed, that from this logical structure alone we can judge whether an argument is or is not "water-tight". Furthermore, we can use this judgement to say whether an argument is "good" or "bad" based on their respective use of Logic. The problem now becomes: by what standards are we to judge one logical argument as being "good" or "bad"? In other words, what rules are there to making a logical argument and how do we know when we've followed those rules and whether those rules are the "right" rules or not?

It is a common, and dangerous, misconception that mathematical statements are statements of fact; that the statement "Two plus two equals four." is like the statement "I have two pennies in my pocket." That this IS NOT THE CASE is just one of the many barriers to addressing your very important question.

The rules we use in mathematics are conventions. All that mathematics "is" is part of the statements of convention in a language. The conventions are specified by giving rules for the use of some part of the language. In arithmetic, the rules we give are the conventions for a part of a language playing the role of a number, of addition of numbers, of multiplication of numbers etc. For example, what makes a sign of a language a natural number is not its shape, color, or sound but rather the rules we follow when using that sign e.g. a sign of a language is a natural number variable if it follows the following rules:

x    x
-  ---
0   Sx


In fact, these rules characterize what it means to play the role of zero, the role of successor, and the role of variable all at the same time. Outside of the question "does this sign of this language follow these rules" there is no way to say whether a sign like $y$ or $z$ is on its own a "numeral variable". (As a side note, a natural number is a sign which results from the elimination of $x$ via the two rules given.)

I've said all of this so that I can conclude with the simplest answer to your question:

Proofs can be comprehended at the same age rules can be comprehended.

Now, thus far I've been limiting myself to math because there is a distinct difference between a mathematical proof, which is only and solely tested based on whether we have or have not followed a certain collection of conventional rules, and proof in the broader "legal" or "unobjectionable" or "logically correct" sense that you hinted at in your question.

The toughest thing to accept is this:

There is no experiment which can test whether a rule is "right" or "wrong".

This sounds counter intuitive at first, but the only thing we can do with a rule is to test whether or not it has been followed in a specific situation. So you might make a rule in your classroom like "Students must raise their hands before they speak." and sure enough you will observe that some students raise their hands before they speak, others speak without raising their hands, and some raise their hands and wait to be called on before they speak, but none of these observations tests whether the rule "Students must raise their hands before they speak." is "good" or "bad" or "right" or "wrong". All of these features which we ascribe to rules, specifically rules from science, are not based on any external observation, rather we choose, as individuals and a community, to say "This rule is good" or "This rule is bad".

In addressing this question we say that there is nothing which makes an argument "logical" rather we decide as individuals and as a community that "An argument which follows these rules is 'logical', i.e. 'good', and an argument that follows these rules is 'illogical' i.e. 'bad'."

So even in the more general case, when considering arguments of a less mathematical nature, the only thing required for a student to judge whether an argument is logical or not ("good" or "bad") is whether they can say whether or not a certain set of rules has or has not been followed i.e.

Arguments can be comprehended at the same age that rules can be comprehended.

The difficulty then becomes "What is the most efficient way to familiarize students with the conventional rules of our society?" and then there is the more difficult question of "Which rules should we teacher as 'the conventional rules' and why?"