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exposition, Handbook, language of math, math, representations, understanding math

Conceptual blending

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This post uses MathJax.  If you see formulas in unrendered TeX, try refreshing the screen.

A conceptual blend is a structure in your brain that connects two concepts by associating part of one with part of another.  Conceptual blending is a major tool used by our brain to understand the world.

The concept of conceptual blend includes special cases, such as representations, images and conceptual metaphors, that math educators have used for years to understand how mathematics is communicated and how it is learned.  The Wikipedia article is a good starting place for understanding conceptual blending. 

In this post I will illustrate some of the ways conceptual blending is used to understand a function of the sort you meet with in freshman calculus.  I omit the connections with programs, which I will discuss in a separate post.

A particular function

Consider the function $h(t)=4-(t-2)^2$. You may think of this function in many ways.

FORMULA:

$h(t)$ is defined by the formula $4-(t-2)^2$.

  • The formula encapsulates a particular computation of the value of $h$ at a given value $t$.
  • The formula defines the function, which is a stronger statement than saying it represents the function.
  • The formula is in standard algebraic notation. (See Note 1)
  • To use the formula requires one of these:
    • Understand and use the rules of algebra
    • Use a calculator
    • Use an algebraic programming language. 
  • Other formulas could be used, for example $4t-t^2$.
    • That formula encapsulates a different computation of the value of $h$.

TREE: 

$h(t)$ is also defined by this tree (right).
  • The tree makes explicit the computation needed to evaluate the function.
  • The form of the tree is based on a convention, almost universal in computing science, that the last operation performed (the root) is placed at the top and that evaluation is done from bottom to top.
  • Both formula and tree require knowledge of conventions.
  • The blending of formula and tree matches some of the symbols in the formula with nodes in the tree, but the parentheses do not appear in the tree because they are not necessary by the bottom-up convention.
  • Other formulas correspond to other trees.  In other words, conceptually, each tree captures not only everything about the function, but everything about a particular computation of the function.
  • More about trees in these posts:

GRAPH:

$h(t)$ is represented by its graph (right). (See note 2.)

  • This is the graph as visual image, not the graph as a set of ordered pairs.
  • The blending of graph and formula associates each point on the (blue) graph with the value of the formula at the number on the x-axis directly underneath the point.
  • In contrast to the formula, the graph does not define the function because it is a physical picture that is only approximate.
  • But the formula does represent the function.  (This is "represents" in the sense of cognitive psychology, but not in the mathematical sense.)
  • The blending requires familiarity with the conventions concerning graphs of functions. 
  • It sets into operation the vision machinery of your brain, which is remarkably elaborate and powerful.
    • Your visual machinery allows you to see instantly that the maximum of the curve occurs at about $t=2$. 
  • The blending leaves out many things.
    • For one, the graph does not show the whole function.  (That's another reason why the graph does not define the function.)
    • Nor does it make it obvious that the rest of the graph goes off to negative infinity in both directions, whereas that formula does make that obvious (if you understand algebraic notation).  

GEOMETRIC

The graph of $h(t)$ is the parabola with vertex $(2,4)$, directrix $x=2$, and focus $(2,\frac{3}{4})$. 

  • The blending with the graph makes the parabola identical with the graph.
  • This tells you immediately (if you know enough about parabolas!) that the maximum is at $(2,4)$ (because the directrix is vertical).
  • Knowing where the focus and directrix are enables you to mechanically construct a drawing of the parabola using a pins, string, T-square and pencil.  (In the age of computers, do you care?)

HEIGHT:

$h(t)$ gives the height of a certain projectile going straight up and down over time.

  • The blending of height and graph lets you see instantly (using your visual machinery) how high the projectile goes. 
  • The blending of formula and height allows you to determing the projectile's velocity at any point by taking the derivative of the function.
  • A student may easily be confused into thinking that the path of the projectile is a parabola like the graph shown.  Such a student has misunderstood the blending.

KINETIC:

You may understand $h(t)$ kinetically in various ways.

  • You can visualize moving along the graph from left to right, going, reaching the maximum, then starting down.
    • This calls on your experience of going over a hill. 
    • You are feeling this with the help of mirror neurons.
  • As you imagine traversing the graph, you feel it getting less and less steep until it is briefly level at the maximum, then it gets steeper and steeper going down.
    • This gives you a physical understanding of how the derivative represents the slope.
    • You may have seen teachers swooping with their hand up one side and down the other to illustrate this.
  • You can kinetically blend the movement of the projectile (see height above) with the graph of the function.
    • As it goes up (with $t$ increasing) the projectile starts fast but begins to slow down.
    • Then it is briefly stationery at $t=2$ and then starts to go down.
    • You can associate these feelings with riding in an elevator.
      • Yes, the elevator is not a projectile, so this blending is inaccurate in detail.
    • This gives you a kinetic understanding of how the derivative gives the velocity and the second derivative gives the acceleration.

OBJECT:

The function $h(t)$ is a mathematical object.

  • Usually the mental picture of function-as-object consists of thinking of the function as a set of ordered pairs $\Gamma(h):=\{(t,4-(t-2)^2)|t\in\mathbb{R}\}$. 
  • Sometimes you have to specify domain and codomain, but not usually in calculus problems, where conventions tell you they are both the set of real numbers.
  • The blend object and graph identifies each point on the graph with an element of $\Gamma(h)$.
  • When you give a formal proof, you usually revert to a dry-bones mode and think of math objects as inert and timeless, so that the proof does not mention change or causation.
    • The mathematical object $h(t)$ is a particular set of ordered pairs. 
    • It just sits there.
    • When reasoning about something like this, implication statements work like they are supposed to in math: no causation, just picking apart a bunch of dead things. (See Note 3).
    • I did not say that math objects are inert and timeless, I said you think of them that way.  This post is not about Platonism or formalism. What math objects "really are" is irrelevant to understanding understanding math [sic].

DEFINITION

A definition of the concept of function provides a way of thinking about the function.

  • One definition is simply to specify a mathematical object corresponding to a function: A set of ordered pairs satisfying the property that no two distinct ordered pairs have the same second coordinate, along with a specification of the codomain if that is necessary.
  • A concept can have many different definitions.
    • A group is usually defined as a set with a binary operation, an inverse operation, and an identity with specific properties.  But it can be defined as a set with a ternary operation, as well.
    • A partition of a set is a set of subsets of a set with certain properties. An equivalence relation is a relation on a set with certain properties.  But a partition is an equivalence relation and an equivalence relation is a partition.  You have just picked different primitives to spell out the definition. 
    • If you are a beginner at doing proofs, you may focus on the particular primitive objects in the definition to the exclusion of other objects and properties that may be more important for your current purposes.
      • For example, the definition of $h(t)$ does not mention continuity, differentiability, parabola, and other such things.
      • The definition of group doesn't mention that it has linear representations.

SPECIFICATION

A function can be given as a specification, such as this:

If $t$ is a real number, then $h(t)$ is a real number, whose value is obtained by subtracting $2$ from $t$, squaring the result, and then subtracting that result from $4$.

  • This tells you everything you need to know to use the function $h$.
  • It does not tell you what it is as a mathematical object: It is only a description of how to use the notation $h(t)$.

Notes

1. Formulas can be give in other notations, in particular Polish and Reverse Polish notation. Some forms of these notations don't need parentheses.

2. There are various ways to give a pictorial image of the function.  The usual way to do this is presenting the graph as shown above.  But you can also show its cograph and its endograph, which are other ways of representing a function pictorially.  They  are particularly useful for finite and discrete functions. You can find lots of detail in these posts and Mathematica notebooks:

3. See How to understand conditionals in the abstractmath article on conditionals.

References

  1. Conceptual blending (Wikipedia)
  2. Conceptual metaphors (Wikipedia)
  3. Definitions (abstractmath)
  4. Embodied cognition (Wikipedia)
  5. Handbook of mathematical discourse (see articles on conceptual blend, mental representation, representation, and metaphor)
  6. Images and Metaphors (article in abstractmath)
  7. Links to G&G posts on representations
  8. Metaphors in Computing Science (previous post)
  9. Mirror neurons (Wikipedia)
  10. Representations and models (article in abstractmath)
  11. Representations II: dry bones (article in abstractmath)
  12. The transition to formal thinking in mathematics, David Tall, 2010
  13. What is the object of the encapsulation of a process? Tall et al., 2000.

 

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Making visible the abstraction in algebraic notation

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Handmade Exp Tree.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Algebraic notation

Algebraic notation contains a hidden abstract structure coded by apparently arbitrary conventions that many college calculus students don't understand completely. This very simple example shows one of the ways in which calc students may be confused:

  1. $x+2y$
  2. $(x+2)y$

Students often mean to express formula 2 when they write something like \[x\!\!+\!\!2\,\,\,\,y\] (with a space).  This is a perfectly natural way to write it. But it is against the rules, I presume because in handwriting it is not clear when you mean a space and when you don't. 

Formula 1 can also be written as $x+(2y)$, and if it were usually written that way students (I predict) would be less confused.   Always writing it this way would exacerbate the clutter of parentheses but would allow a simple rule:

Evaluate every expression inside parentheses first, starting with the innermost.

Using trees for algebra

Writing algebraic expressions as a tree (as in computing science)

  • makes it obvious what gets evaluated first
  • uses no parentheses at all.

An example of using the tree of an expression to do calculations is available in Expressions.nb (requires Mathematica) and Expressions.cdf (requires CDF player only) on my Mathematica website.  I could imagine using tree expressions instead of standard notation as the normal way of doing things. That would require working on Ipads or some such and would take a big amount of investment in software making it intuitive and easy to use.  No, I am not going to embark on such an adventure, but I think it ought to be attempted.  (Brett Victor has many ideas like this.)

Transforming algebraic notation into trees

The two manipulable diagrams below show the algebraic notation being transformed into tree form.  I expect that this will make the abstract structure more concrete for many students and I encourage others to show it to their students.  Note that the tree form makes everything explicit.>

After I return from a ten-day trip I will explore the possibility of making the expression-to-tree transformer turn the expression into an evaluable tree as in Expressions.nb and Expressions.cdf.  In the I hope not to distant future students should have access to many transformers that morph expressions from one form into another.  Such transformers are much more politically correct than Optimus Prime.

Offloading chunking and Computable algebraic expressions in tree form are earlier posts related to this post.

 

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Syntax Trees in Mathematicians’ Brains

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Understanding the quadratic formula

In my last post I wrote about how a student’s pattern recognition mechanism can go awry in applying the quadratic formula.

The template for the quadratic formula says that the solution of a quadratic equation of the form ${ax^2+bx+c=0}$ is given by the formula

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

When you ask students to solve ${a+bx+cx^2=0}$ some may write

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

instead of

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2c}$

That’s because they have memorized the template in terms of the letters ${a}$, ${b}$ and ${c}$ instead of in terms of their structural meaning — $ {a}$ is the coefficient of the quadratic term, ${c}$ is the constant term, etc.

The problem occurs because there is a clash between the occurrences of the letters “a”, “b”, and “c” in the template and in the equation to solve. But maybe the confusion would occur anyway, just because of the ordering of the coefficients. As I asked in the previous post, what happens if students are asked to solve $ {3+5x+2x^2=0}$ after having learned the quadratic formula in terms of ${ax^2+bx+c=0}$? Some may make the same kind of mistake, getting ${x=-1}$ and ${x=-\frac{2}{3}}$ instead of $ {x=-1}$ and $ {x=-\frac{3}{2}}$. Has anyone ever investigated this sort of thing?

People do pattern recognition remarkably well, but how they do it is mysterious. Just as mistakes in speech may give the linguist a clue as to how the brain processes language, students’ mistakes may tell us something about how pattern recognition works in parsing symbolic statements as well as perhaps suggesting ways to teach them the correct understanding of the quadratic formula.

Syntactic Structure

“Structural meaning” refers to the syntactic structure of a mathematical expression such as ${3+5x+2x^2}$. It can be represented as a tree:

(1)

This is more or less the way a program compiler or interpreter for some language would represent the polynomial. I believe it corresponds pretty well to the organization of the quadratic-polynomial parser in a mathematician’s brain. This is not surprising: The compiler writer would have to have in mind the correct understanding of how polynomials are evaluated in order to write a correct compiler.

Linguists represent English sentences with syntax trees, too. This is a deep and complicated subject, but the kind of tree they would use to represent a sentence such as “My cousin saw a large ship” would look like this:

Parsing by mathematicians

Presumably a mathematician has constructed a parser that builds a structure in their brain corresponding to a quadratic polynomial using the same mechanisms that as a child they learned to parse sentences in their native language. The mathematician learned this mostly unconsciously, just as a child learns a language. In any case it shouldn’t be surprising that the mathematicians’s syntax tree for the polynomial is similar to the compiler’s.

Students who are not yet skilled in algebra have presumably constructed incorrect syntax trees, just as young children do for their native language.

Lots of theoretical work has been done on human parsing of natural language. Parsing mathematical symbolism to be compiled into a computer program is well understood. You can get a start on both of these by reading the Wikipedia articles on parsing and on syntax trees.

There are papers on students’ misunderstandings of mathematical notation. Two articles I recently turned up in a Google search are:

Both of these papers talk specifically about the syntax of mathematical expressions. I know I have read other such papers in the past, as well.

What I have not found is any study of how the trained mathematician parses mathematical expression.

For one thing, for my parsing of the expression $ {3+5x+2x^2}$, the branching is wrong in (1). I think of ${3+5x+2x^2}$ as “Take 3 and add $ {5x}$ to it and then add ${2x^2}$ to that”, which would require the shape of the tree to be like this:

I am saying this from introspection, which is dangerous!

Of course, a compiler may group it that way, too, although my dim recollection of the little bit I understand about compilers is that they tend to group it as in (1) because they read the expression from left to right.

This difference in compiling is well-understood.  Another difference is that the expression could be compiled using addition as an operator on a list, in this case a list of length 3.  I don’t visualize quadratics that way but I certainly understand that it is equivalent to the tree in Diagram (1).  Maybe some mathematicians do think that way.

But these observations indicate what might be learned about mathematicians’ understanding of mathematical expressions if linguists and mathematicians got together to study human parsing of expressions by trained mathematicians.

Some educational constructivists argue against the idea that there is only one correct way to understand a mathematical expression.  To have many metaphors for thinking about math is great, but I believe we want uniformity of understanding of the symbolism, at least in the narrow sense of parsing, so that we can communicate dependably.  It would be really neat if we discovered deep differences in parsing among mathematicians.  It would also be neat if we discovered that mathematicians parsed in generally the same way!

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