Visual Problem Solving

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I just mentioned (and capitalized) Visual Problem Solving. Now I should be responsible and define my new term.

What I call Visual Problem Solving is a three-step process in which someone studies their objectives, visually examines the situation to plan an approach, and then follows their plan to accomplish the objectives.

As a math teacher at community college, almost every student I encounter has well-developed Visual Problem Solving skills. This is not surprising, for real life provides non-stop opportunities to develop all related skills. Every time someone plans an efficient route through a store to buy a few items or successfully navigates a tricky merge in traffic they have done more complex problem solving than any of my classwork's math problems.

What many students lack is the ability to turn a text problem into a visual problem. Here is an example from the level of math I teach, which would stump most students when provided only as text but would not be very difficult if I also provided a picture.

A stained glass window of area 6 square feet is shaped like a semicircle on a rectangle. The window's height is three times its width. What is the width of the window?

Again this behavior among students is not surprising. Real life is providing increasingly fewer opportunities before college that practice visualizing parameters described in text. People are used to thinking in pictures, and have never encountered the difference between that and thinking in symbolic structures.

This is very important socially! Better jobs in the adult workplace are full of problems described in text that need to be visualized. As popular society includes fewer games, instruction manuals, and educational tasks that require children and teens to visualize situations described by text, those parents who "fill in the gap" do a whole lot towards helping their children aim for better jobs in later life.

As I teach remedial math I sometimes ask students to write about their work, or to do a portfolio assignment that includes an essay. Then I get to see an interesting connection: the students who have trouble with order of operations are the same students who cannot write a structured paragraph.

That is worth rephrasing. The structure of text provides meaning beyond the content of the words. Some students get to college without learning those rules of structure for math and English, and those who have the lack tend to lack it in both.

Now, all of my students can speak well. They speak with decent grammar and their oral presentations of work are acceptable if not good. But many fall apart when dealing with the written word. When those write, they just put down what words feel right. They do keep the first sentence of a paragraph in mind throughout the paragraph, just as in math they do not keep a ( in mind until they see a ). I have even met a few cases so extreme that a sentence's structure is not kept in mind through the sentence; the math equivalent might be a fraction bar.

This produces a deep pedagogical problem. Those students who arrive in my classes not understanding that the very structure of text includes meaning have no concept of structure to build upon. They only know algorithms. They have mastered many kinds of problem-solving, but only kinds that require knowing The Steps To Do. Since the remedial math I teach is all about structure, not algorithms, these students get quickly frustrated. It genuinely feels to them as if the math, or the textbook, or my teaching is somehow Not Fair.

We learn the meaning inherent in structure by seeing lots of examples, by unconsciously comparing and contrasting, and thus by slowly gaining familiarity with the structures. This requires a lot more effort and time than learning a few algorithms.

Part of my role as a math teacher is to try to explain all this. Students benefit from remedial math because they gain experience with structures that have inherent meaning, not because they acquire a handful of additional algorithms.

This essay builds off a discussion I had with Cathy Miner last fall. She gets credit for many of these ideas, as well as the opening joke.