Computational and reasoning skills don't always coexist?

[annoyinggirl]

Studies have found that girls do better at computational math (arithmetic) and boys excel at reasoning. However, I had always thought that computation was a branch of reasoning. Can someone explain to me how this finding could be that girls are better at computation and boys at reasoning? Doesn't computation require reasoning? And doesn't reasoning require computation? Doesn't complex math, which males dominate, require arithmetic? from http://arthurjensen.net/wp-content/...ys-and-Girls-1988-by-Arthur-Robert-Jensen.pdf
"She also states: "It is in junior high school that the sex difference in mathematics first becomes apparent. Girls excel in computation, boys on tasks requiring mathematical reasoning." She notes that, although the sex difference in mathematical reasoning is apparent by age 12,"

 

[aikismos]

Computation, per se, does not require reasoning. For instance, your calculator can compute numbers, but that doesn't mean that it has any clue why it does what it does or can explain it. Of course, the ideas of computation and reasoning can be very broad words, and in the context of human skills, can be quite complicated. Many people today associate reasoning with formal logic (e.g. Boolean functions, predicate logic), however reasoning has definitions that extend to fuzzier, complexer, and higher levels of domains. For instance, context-driven induction, deduction, and particularly abduction can be extremely sophisticated processes that can take a life-time to master, and reasoning about reasoning is another elusive cognitive process. Generally, in these psychology research projects, the definition of evidence of what constitutes computation versus reasoning is quite simplified (for evidential purposes) and is generally in line with simple tasks with easily measured outcomes.

Mathematics has two broad divisions. The first, computation, has a lot to do with recalling facts and manipulating symbols. Many of my students could record the steps of the process on telling if two linear equations were consistent or inconsistent, but when asked "what does it mean? How can this be used to understand the relationship of lines on a plane?", they would have no idea. The equation $y=2x+1$ upon substitution of an ordered pair such as $(2,5)$ yields $5=5$, but if the equation represents a line, and the ordered pair represents a point on the same plane, what does $5=5$ tell us about the relationship of the geometric objects? (In this case, the point lies on the line.)

According to one publication of the NCTM, mathematics falls into four domains: verbal, graphical, numerical, and symbolic processing. You'll note that these are fairly broad concepts, and that reasoning could be seen as fitting together with any or all of them. To highlight the difference between how a student in her early teens might have to tackle a problem, let's talk about finding roots of a quadratic equation. In fact, there's a fairly algorithmic process that is involved in finding if and where a quadratic equation crosses the abscissa (a.k.a. the x-axis). Once shown that procedure, a girl could excel at finding roots problem after problem. She neatly writes out her variables, correctly writes the formula, reduces the expressions, and arrives at the correct root.

Her classmate, a boy may have a hard time find the answers on account of poor computational skill. He may have problems isolating a variable or using the quadratic formula, however, he may in this hypothetical be able to notice that the answers he calculates are wrong, and using reasoning skills about the real number line, he might say to himself, well, I know that these roots are wrong because when I plug them in the function, I don't get ordered pairs that are of the form $(a,0)$. He may then remember that the quadratic formula says the roots have to be symmetrical about the axis of symmetry, draw that axis, and realize that the left root is consistent with the graph, but the right root is so off as to be on the wrong side of the axis. He might then use that information to go back and find his error.

Here's the difference. The girl followed an algorithm; apprised, substituted, and calculated quickly and accurately. She'd get the answer on a multiple test question immediately if they were listed. Now, if that same girl were asked what does it mean? What is the relationship between her answer and the graph, she might have no idea how it extends across the different domains. She might not know that a leading coefficient of the quadratic determines the concavity on the real number plane. She might not know that the axis of symmetry is embedded in the quadratic equation, and she might not be able to derive the quadratic formula from the general form or understand why or how the basic axioms of math apply to manipulation the equation.

Let me just say that in my experience, the typical elementary and high school math teacher tends to excel at computation, but the core of math instruction revolves around the reasoning, and that's often not explicitly taught. I've met more than one math teacher who didn't understand the relationship between the quadratic formula, and the shape of the quadratic on the plane. Recitation of a recipe with quick and accurate computation is good, but it's not the same as discovering proofs for why the recipe works. To the chagrin of some of my former students, I used to assign a multiple choice test with no computation on it whatsoever asking only conceptual questions about the signs of points, the slopes of lines, the concavity of parabolas, etc. and be greeted with groans of how the test had no "math" on it. (Yes, I'm evil like that.)

But that's the difference, then between computing and actually thinking things through. That's why so many students struggle with the ACT. It gives you a bunch of words, expects you to know their symbolic, numerical, and graphical equivalents, and then asks you which answers are possible. That's radically different than carrying out some ordinal arithmetic and picking out your results. The former is reasoning, and the latter is computation. Why do you think they don't care if one uses a calculator? Because calculators (especially those without computer algebraic systems) simply don't do proof.

 

[symbolipoint]

The distinction is between thinking and following instructions. Something about the thinking needed changes when students begin to seriously or more formally study beginning/introductory/elementary algebra. MOST students can learn this, regardless of his gender.

 

[annoyinggirl]

Computation, per se, does not require reasoning. For instance, your calculator can compute numbers, but that doesn't mean that it has any clue why it does what it does or can explain it. Of course, the ideas of computation and reasoning can be very broad words, and in the context of human skills, can be quite complicated. Many people today associate reasoning with formal logic (e.g. Boolean functions, predicate logic), however reasoning has definitions that extend to fuzzier, complexer, and higher levels of domains. For instance, context-driven induction, deduction, and particularly abduction can be extremely sophisticated processes that can take a life-time to master, and reasoning about reasoning is another elusive cognitive process. Generally, in these psychology research projects, the definition of evidence of what constitutes computation versus reasoning is quite simplified (for evidential purposes) and is generally in line with simple tasks with easily measured outcomes.

Mathematics has two broad divisions. The first, computation, has a lot to do with recalling facts and manipulating symbols. Many of my students could record the steps of the process on telling if two linear equations were consistent or inconsistent, but when asked "what does it mean? How can this be used to understand the relationship of lines on a plane?", they would have no idea. The equation $y=2x+1$ upon substitution of an ordered pair such as $(2,5)$ yields $5=5$, but if the equation represents a line, and the ordered pair represents a point on the same plane, what does $5=5$ tell us about the relationship of the geometric objects? (In this case, the point lies on the line.)

According to one publication of the NCTM, mathematics falls into four domains: verbal, graphical, numerical, and symbolic processing. You'll note that these are fairly broad concepts, and that reasoning could be seen as fitting together with any or all of them. To highlight the difference between how a student in her early teens might have to tackle a problem, let's talk about finding roots of a quadratic equation. In fact, there's a fairly algorithmic process that is involved in finding if and where a quadratic equation crosses the abscissa (a.k.a. the x-axis). Once shown that procedure, a girl could excel at finding roots problem after problem. She neatly writes out her variables, correctly writes the formula, reduces the expressions, and arrives at the correct root.

Her classmate, a boy may have a hard time find the answers on account of poor computational skill. He may have problems isolating a variable or using the quadratic formula, however, he may in this hypothetical be able to notice that the answers he calculates are wrong, and using reasoning skills about the real number line, he might say to himself, well, I know that these roots are wrong because when I plug them in the function, I don't get ordered pairs that are of the form $(a,0)$. He may then remember that the quadratic formula says the roots have to be symmetrical about the axis of symmetry, draw that axis, and realize that the left root is consistent with the graph, but the right root is so off as to be on the wrong side of the axis. He might then use that information to go back and find his error.

Here's the difference. The girl followed an algorithm; apprised, substituted, and calculated quickly and accurately. She'd get the answer on a multiple test question immediately if they were listed. Now, if that same girl were asked what does it mean? What is the relationship between her answer and the graph, she might have no idea how it extends across the different domains. She might not know that a leading coefficient of the quadratic determines the concavity on the real number plane. She might not know that the axis of symmetry is embedded in the quadratic equation, and she might not be able to derive the quadratic formula from the general form or understand why or how the basic axioms of math apply to manipulation the equation.

Let me just say that in my experience, the typical elementary and high school math teacher tends to excel at computation, but the core of math instruction revolves around the reasoning, and that's often not explicitly taught. I've met more than one math teacher who didn't understand the relationship between the quadratic formula, and the shape of the quadratic on the plane. Recitation of a recipe with quick and accurate computation is good, but it's not the same as discovering proofs for why the recipe works. To the chagrin of some of my former students, I used to assign a multiple choice test with no computation on it whatsoever asking only conceptual questions about the signs of points, the slopes of lines, the concavity of parabolas, etc. and be greeted with groans of how the test had no "math" on it. (Yes, I'm evil like that.)

But that's the difference, then between computing and actually thinking things through. That's why so many students struggle with the ACT. It gives you a bunch of words, expects you to know their symbolic, numerical, and graphical equivalents, and then asks you which answers are possible. That's radically different than carrying out some ordinal arithmetic and picking out your results. The former is reasoning, and the latter is computation. Why do you think they don't care if one uses a calculator? Because calculators (especially those without computer algebraic systems) simply don't do proof.

Thank you so much for this. This answered my question completely. One thing i am confused about, however, and perhaps this is a question for a psychologist more than a mathematician, but how could one be brilliant at understanding the relationships and meaning of numbers, but bad at plugging them in? I get that one may be good at computation but not reasoning, but how could one be good at reasoning but not computation? Reasoning is more difficult; fewer people can do. To me, it is like if you can run, then you can walk. But if you can walk, then you may or may not be able to run.

 

[pbuk]

how could one be brilliant at understanding the relationships and meaning of numbers, but bad at plugging them in? I get that one may be good at computation but not reasoning, but how could one be good at reasoning but not computation? Reasoning is more difficult; fewer people can do. To me, it is like if you can run, then you can walk. But if you can walk, then you may or may not be able to run.

Simple: mathematics in general^* has absolutely nothing to do with "the relationships and meaning of numbers".

A better analogy is that arithmetic (which is the word I would prefer to use rather than computation) is indeed like walking: some people can only walk slowly, some can sprint really fast or run a marathon. But mathematical reasoning is like swimming: it doesn't matter how fast you can run, swimming uses a whole different set of (learned and innate) skills.

^*There is a branch of mathematics called number theory that is to do with the relationships between numbers, but it has very little to do with computation.

 

[aikismos]

Thank you so much for this. This answered my question completely. One thing i am confused about, however, and perhaps this is a question for a psychologist more than a mathematician, but how could one be brilliant at understanding the relationships and meaning of numbers, but bad at plugging them in? I get that one may be good at computation but not reasoning, but how could one be good at reasoning but not computation? Reasoning is more difficult; fewer people can do. To me, it is like if you can run, then you can walk. But if you can walk, then you may or may not be able to run.

Well, @annoyinggirl (who asks really good questions so is the opposite of annoying), what you are talking about has to do with cognitive science. You see, mathematics as we know it is nothing more than software of the mind (See https://en.wikipedia.org/wiki/Embodied_cognition). Think of it this way: just as mathematics in your personal computer relies on software written for hardware to run on, so too does the discipline of mathematics (including the philosophy of mathematics) rely on the basic neural development of our mind after we are born. A perfect example of mathematical primitives built into the mind is 'subitizing' (See https://en.wikipedia.org/wiki/Subitizing). The mind is able to count so quickly because the evolution of the brain has necessitated that rapid counting actually happen at an unconscious level for small amounts. You see, counting is not a primitive ability of the brain. It has to be taught, and in fact some societies don't have numbers as you and I think of them; they count like this: 1, 2, many (and that's it). Talk about a simple number system. If they want to ensure that they have the same amount of a pile of shells, they merely pair them up until it becomes obvious if one pile has one, two, or many more. You're right that this falls into the domain of psychology, but it also falls into the domain of math at a primitive level. A good introduction into the topic of the physiological and linguistic basis of math is Where Mathematics Comes From by Lakoff and Nunez (See https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From). For those of us who are educated in computer science, we reject Platonic mathematical thinking and demand that numbers and information have to exist in a medium whether it be silicon, electrons, photons, or neurons.

The question of how one could be good at one skill and not good at another has to do with how the brain is organized. Often as conscious beings, what we think occurs seamlessly is a composite of all sorts of psychologically irreducible primitives. For instance, to reason well, one has to have good executive function (See https://en.wikipedia.org/wiki/Executive_functions) because it reasoning requires organization and planning. Proofs and problem-solving require focus, execution, and often a lot of linguistic intelligence in its fluid and crystal forms (See https://en.wikipedia.org/wiki/Fluid_and_crystallized_intelligence). Arithmetic tends to be simpler and requires less factual knowledge. For example, the closure property of the reals (See https://en.wikipedia.org/wiki/Closure_(mathematics)) pretty much ensures that if you can count to ten, understand the basics of the number system, and can memorize not too many algorithms, you number crunch in all sorts of ways (that is in a permutational manner) and are really doing the same thing. Children memorize an addition table, multiplication table, sequence of squares, and less than ten different operations (addition, subtraction, division, multiplication, exponentiation, taking a root, grouping, substituting, etc.) and can get up to algebra with very little understanding of the concepts especially philosophy involved.

I would disagree with MrAnchovy when he says "Simple: mathematics in general* has absolutely nothing to do with the relationships and meaning of numbers", but I get what he's getting at. If you were ever to take an Abstract Algebra course, what you would find is that a lot of logical thinking used in math can be applied to diverse topics besides numbers including shapes, the mapping between numbers and shapes, and other domains still. His point is that different parts of the brain are used for simple calculations as opposed to looking for tautologies and contradictions, recognizing various symmetries, organizing complex visual data, and so on. Ultimately, a good tertiary article to start understanding how the various parts of the brain have an effect on mathematics is here (https://en.wikipedia.org/wiki/Numerical_cognition). Remember that human cognitive development is predicated on neural development, and if you are unfortunate enough to have damage done to Broca's Area (See https://en.wikipedia.org/wiki/Broca's_area), then you're going to have a harder time in the verbal domain of mathematics, etc. One more example is dyscalculia, (See https://en.wikipedia.org/wiki/Dyscalculia) in which a person's neural circuitry literally makes it impossible to learn arithmetic at all. It's not to hard to imagine that on the other end of the spectrum, an autistic (think Rain Man), might have a turbo-charged set of neurons that can do arithmetic and have corresponding low quotients of intelligence in the reasoning domain.

 

[symbolipoint]

Thank you so much for this. This answered my question completely. One thing i am confused about, however, and perhaps this is a question for a psychologist more than a mathematician, but how could one be brilliant at understanding the relationships and meaning of numbers, but bad at plugging them in? I get that one may be good at computation but not reasoning, but how could one be good at reasoning but not computation? Reasoning is more difficult; fewer people can do. To me, it is like if you can run, then you can walk. But if you can walk, then you may or may not be able to run.

Both kinds need reasoning. People are usually either ALGEBRA people or GEOMETRY people. A few handle both well, but people are primarily one of these kind or the other of them.