Conceptual counting test

Roughly a year ago Edmund Harriss asked me to respond to his post regarding what he called “Mathematical Scales”, that is–

I am in love with this idea of training, taking someone who has proved incredibly able in an area and taking them back to the most basic ideas. I started to wonder what the equivalent might be for mathematics.

Here is a test of counting in the natural numbers. The difficulty increases as the test goes on.

1. If 3 were dropped from the counting sequence 0, 1, 2, 3, 4, 5, … what would 2 + 2 equal?

2. The Yukis were located in modern day California and used a base 4 and a base 8 system. This came about from a natural form of body counting — how?

3. Subitizing is counting without having to enumerate; for small quantities this means a “a feeling of immediately knowing how many items lie within the visual scene” and for larger quantities this becomes an estimate. For example, it is possible to note how many triangles are depicted

without having to stop and mentally think “one, two, three” but rather “three” straightaway.

Explain how subitizing and the Roman numeral system are related.

4. Describe how counting and the commutive property of addition ($x + y = y + x$) are related.

5. Early in a child’s development counting tends to be just “recite the numbers in order.” Later they understand that counting enumerates a set. For example, one principle is that the last number counted equals the number of items. Name three more such principles.

6. Count from 1 to 10 (where each number represents our usual base 10 understanding of 1 to 10) in base $\pi$.

7. The number of sense of babies is tested by checking for essentially, level of surprise. In an experiment of Karen Wynn, one puppet was moved behind a curtain followed by a second puppet; the curtain opened to reveal either two or three puppets. The babies were more startled by three puppets than two, essentially thinking that two was the “correct” solution to 1 + 1.

In a different experiment by Xu and Carey, babies were startled by the situation depicted below

but not by this one:

Explain the ramifications for the number sense of babies.

[Diagram from The Number Sense by Stanislas Dehaene.]

8. Explain the following quote:

As Russell also helpfully pointed out, one of the reasons it took so long to discover a “definition” of the natural numbers, is that the names of the natural numbers — e.g. “3″ — are used both as adjectives and nouns, and this distinction was not sooner realized.
– From Smullyan and Fitting’s Set Theory and the Continuum Problem

9. Here is a typical (although not the only) formulation of the Peano axioms:

I. 0 is a natural number.
II. If $n$ is a natural number, so is $n^+$.
III. For any natural number $n$, $n^+$ does not equal 0.
IV. For any $n$ and $m$, if $n^+ = m^+$ then $n = m$.
V. Mathematical induction: For any set $A$, these conditions are sufficient to show $A$ contains every natural number. a.) $0 \in A$. b.) For every natural number n, $n \in A$ implies $n^+ \in A$.

What happens if the statement “0 is a natural number” is omitted?

10. Demonstrate why “every set of natural numbers has a smallest member” can substitute for the Principle of Induction in the usual Peano axioms.

11. The (admittedly radical) philosophy of ultrafinitism denies that the infinite set of natural numbers exists. Give one possible justification for this philosophy.

12. Number can be defined without sets or classes.

$\exists_0 x F x \leftrightarrow \neg \exists x F x$
$\exists_1 x F x \leftrightarrow \exists x [F x \wedge \exists_0y(Fy \wedge y \neq x)]$
$\exists_2 x F x \leftrightarrow \exists x [F x \wedge \exists_1y(Fy \wedge y \neq x)]$

What is the major downside of this formulation?

13. The now-standard formulation of natural numbers in set theory (due to John von Neumann) is $0=\O$, $1 = \{\O, \{\O\}\}$, $2 = \{\O, \{\O\}, \{\O, \{\O\}\}\}$ and so on. Give an alternate formulation.

14. Design a set of axioms (without referring to previously formulated counting axioms) so it produces the sequence 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, …

12 Responses

1. WHAT A POST!

This stuff is superb. Just lovely lovely math.

2. Ive used lessons like these when I used to teach years ago, and I found them on the shared web database for panaboards. An interactive whiteboard is a great classroom tool , and with shared lessons such as these it really gets kids interested and involved!!

3. Most of this isn’t actually math at all, it is semantics. I am not saying that some of these are not worthy of classroom discussion, some would be good openers to “wake up their brains” but I would hope that the students could recognize the difference between mathematical reasoning and questions that are simply open to interpretation.

4. Your fractions test is dead on target with what “mathematical” understanding questions should look like. In fact, these are the types of questions that good teachers ask and that all teachers should be asking as students progress through fractions. The fraction questions (the majority of them at least) are derived from straight forward mathematical truths that the student should (must) experience in their journey through fractions to be successful. On the other hand, most of the questions above have no clear answer and are open to interpretation and some are just plain philosophical. They can be good discussion openers but these two tests are distinctly different in personality.

My only argument with your fractions test, and this is my argument with concept inventories in general, is with how they tend to be used. Some teachers unfortunately get the impression that the answers to these questions are the things to teach when actually they are the things that the student must get. What I mean by this is that in my experience, students who are successful with fractions (and any math topic) achieve that success through copious amounts of practice with fractions followed by these types of questions midway through, which they should by now have strong inklings for what is going on. Unfortunately, some teachers teach to these questions directly and spoil the whole thing. Mathematics is practice and theory, not just one nor just the other. Your fractions questions are only good (and they are dead on) when they are used in the practice of working with fractions, not in place of. And as we know, it takes a lot of practice to do justice to your fraction questions.

5. I think I’m with Robert. I loved the fractions concepts, but this doesn’t move me much. (Commenting so I can click to ‘follow comments box.)

• This thing’s a bit harder than the fraction test — how many of these do you actually know (or think you know) the answer to? This could be the reason for the different reactions.

#12, for instance, suggests to me some very interesting discussions on Robinson arithmetic and first-order logic, but if you have no idea what the answer is it would, well, not move you much.

• I don’t think it is a matter of “harder”, it is a matter of these questions being open to interpretation and philosophical. In fact, most of the questions before #9 (except #6) are entirely conjecture and many are psychological, which is even worse than conjecture. It is very strange that this goes on for eight questions and then all of the sudden jumps into formal set theory. Believe me, there is very little connecting elementary notions of numbers and arithmetic to formal set theory.

Here are my answers and I do not pretend to be an expert in formal mathematics. I am unsure of #10, 12 and 13, but if I saw the answer (if there is an answer) I would probably understand it (barely).

1. Depends on what you mean. If you are simply shifting the “labels” to the left then the number “3” is now labeled “4” and the number 4 is now labeled “5” and so on, so 2 + 2 would be “5”. If you mean the set of non negative integers no longer contains the number 3 then the set is no longer closed under addition (or counting) but 2 + 2 is still 4.

2. The only thing I could come up with is that Yukis have no thumbs, I am sure there are a number of correct answers to this question.

3. This question is incorrect. Subtilizing is a visual field thing, roman numerals are, even though odd, still quite regular and syntactical.

4. You can start at 4 and count 6 forward or start at 6 and count 4 forward and still reach 10.

5. Each subsequent number is one plus the previous number (toddler induction). No mater how many items there are in a set, you can add another one and make a better set (though they still think infinity is a number). You can split the items up into separate sets and still count the same total if you count across all the sets. This isn’t even a math question by the way, more of a teacher/parent question.

6. Assuming I read the problem correctly, this would be very complicated… http://en.wikipedia.org/wiki/Non-integer_representation and http://en.wikipedia.org/wiki/Golden_ratio_base.

7. I don’t think Wynn’s experiment signifies “number” any more than I think knowing things fall down signifies a model of gravity. If we see something “floating” we are amazed, not because we understand the laws of gravity, but because we never see average things do that (float). The first set of slides (with individual blinds) represent something just “disappearing” while the second set of slides (with one big blind) can represent something more complex than just “disappearing”. I mean, if you put a snake and a mouse behind the single blind and reveal that there is now only a (fat) snake left, something happened to the mouse. I have written a lot more about why Dehane’s notions (like this one) are essentially a plot to sell a book, like the theories in “Chariots of the Gods”. If our sense of “number” was as innate as what these experiments suggest then this blog wouldn’t exist.

8. We took numbers for granted.

9. You lose the requirements for inductive proof. By the way, Peano started with 1 not 0, so as long as the “starting” statement is there, whether it starts at 1 or at 0, the development is valid.

10. I think this would follow recursion (which is basically induction).

11. This could go in any direction because is philosophical, not mathematical.

12. It is not constructive.

13. I am not sure what this question is asking? Do you mean 0 = {0}, 1 = {0,1}. 2 = {0,1,2}, …

14. a_0 = {0}, a_n+ = {{a_n}, {{a_n}, n+}}

• I think you will be more satisfied on some of these when you see the answers (I perhaps should have insisted that the justification on #11 is mathematical — only a very small number of the already small number of constructivists believe it, but those few are research mathematicians and so have a mathematical argument; I know of two arguments, one very technical, one not.)

#7 suggests to me maybe you’re thinking of a different author? I only pulled the diagram out of The Number Sense because it is a little clearer than the original research paper, but let me mention anyway the book’s main contention is simply that there is a.) an approximate number sense and b.) people arrange their mental number line spatially. Both a.) and b.) have been tested many times and are hardly magic bullets for education; on the contrary, they can actively sabotage efforts in formal mathematics.

For example, one well-established effect is people from cultures who read left to right will tend to look to the right when they add and look to the left when they subtract. This makes teaching -3 – 7 = -10 an act of fighting against the original mental number line.

I do recall one company taking subitizing and trying to cash in with a product (especially given the recent study showing stronger approximate number sense in childhood correlates with high math achievement later) but the notion that one could possibly “train” one’s approximate sense and that it can help with anything in particular needs a lot more study.

6. in my response to #5…

“No mater how many items there are in a set, you can add another one and make a better set”

That was supposed to be “bigger” set, not “better”.:)

7. [...] Conceptual counting test [...]

8. Problem 1
5.

Problem 2
Since I know nothing about the Yukis, I can’t make any sort of educated guess about how their counting system came about. All I can do is tell a story plausible enough that someone gullible might believe it.

The Yukis have two systems of body counting: one for small numbers, and one for large ones.

For small numbers, they use the the right thumb to point sequentially at the fingers of the left hand, starting with the index finger. Every time the pinky is pointed at, a finger of the right hand is raised, starting with the index finger. In this way, numbers up to twenty can be enumerated.

For large numbers, they do the same thing, but using the fingers of the feet instead of the fingers of the left hand (the thumbs of the feet are, of course, excluded). This allows them to count up to forty.

Problem 3
I’m pretty sure any (small enough) number can be written in Roman numerals without any numeral appearing more than three times in a row. This might help readers process numbers faster by subitizing each group of repeated numerals.

Wikipedia gives MDCCCCX and MCMLIV as two examples of years written in Roman numerals, and the four Cs in the first one did throw me a bit. So maybe there really is something to this!

Problem 4
In the language of Problem 5, the commutative property can be stated as follows.

Say you have a set of oranges and a set of bananas, and you want to count all the pieces of fruit. One way is to count the bananas starting at the number after the number of oranges. Another way is to count the oranges starting at the number after the number of bananas. Both of these ways give the same answer.

Problem 5
I think the ‘principle’ given in the question is really a definition: it defines the phrase “number of items.” A more detailed form of the definition might be, If an adult asks you for the “number of items” in a set, count the items and respond with the last number counted. With that said, here are some actual principles.

1) There is a precise relationship between the amount-feeling that set gives you and the number of items in the set.

Amount-feeling ↔ Number
@ ↔ one
@ @ ↔ two
@ @ @ ↔ three
@ @ @ … ↔ a number after three

2) The order in which you count the items in a set does not affect the number of items.

3) If two sets have the same number of items, you can pair each item from one set with an item from the other set, and there will be no items left unpaired.

Problem 6

In base π…

One is between 0.3 and 0.4
Two is between 0.6 and 0.7
Three is between 0.9 and 1.0
Four is between 1.2 and 1.3
Five is between 1.5 and 1.6

Problem 7
In light of Wynn’s experiment, one way to explain Xu and Carey’s experiment is to say that babies expect one plus one to be two when both ones are the same kind of thing, but have no expectation when the two ones are different kinds of things.

Another possible explanation is that babies expect one plus one to be less than or equal to two.

Problem 8
I think it was probably known for a long time that if you do a calculation with sets of sheep, and you do the same calculation with identically numbered sets of pebbles, the answer-sets will have the same numbers.

It may have been this knowledge that led to the key step in the invention of arithmetic: the realization that you can forget about the sets entirely, and do calculations with the numbers themselves.

An analogy: it was probably known for a long time that if you mix red paint with yellow paint, you get orange paint, and if you mix red powder with yellow powder, you get orange powder. At some point, someone may have realized that you can forget about the paints and powders entirely, and simply say that red mixed with yellow gives orange.

Problem 9
If statement I is omitted, we have no idea what the symbol 0 means, so statements III and V make no sense. The remaining statements, II and IV, are satisfied by any “successor system” that can be broken down into a (possibly empty, not necessarily finite) set of “chains,” each of which looks like either the natural numbers or the integers.

I’m finding it hard to give a more precise characterization of the systems described by axioms II and IV, perhaps because any “description” other than the axioms themselves must by nature be somewhat vague and suggestive.

Problem 10
I can’t think of a way to define the phrase “smallest member,” or even the relationship “smaller than,” without implicitly invoking the principle of mathematical induction.

For example, we might say that a is smaller than b if and only if the list

a
a+
a++
a+++
a++++
.
.
.

contains b. But what is that list, exactly? I appear to be defining it inductively…

Problem 11
A computational justification. Because our lives and resources are finite, we cannot do any computation whose result depends on whether or not the set of natural numbers is infinite. Therefore, a mathematician who does not believe the set of natural numbers is infinite will make exactly the same predictions about the results of calculations as a mathematician who does… and the former mathematician’s beliefs are, perhaps, a little simpler.

A mathematical justification. We can start with any axioms we want, so there is nothing wrong with using axioms which do not imply that the set of natural numbers is infinite.

Problem 12
I don’t know what these symbols mean. (This may be taken as either a rejection of the question or an answer to it.)

Problem 13
0 = empty
1 = {0}
2 = {1}
3 = {2}
.
.
.

Problem 14
I don’t understand what it means for a set of axioms to produce a sequence of symbols. I also don’t understand what meaning, if any, I am supposed to ascribe to the symbols you wrote.