Monthly Archives: April 2007

Educational Philosophy and Technology Vision

It’s a near-universal truth among educators that critical thinking is important in school.  I was excited and felt vindicated when I learned this.  I had been training my own kids since they could talk by asking them, whenever possible, “How do you know that?”  As the son of an Apollo test engineer, and as a former journalist and software developer (with the callused thumbs from endless test/adjust/test cycles of debugging to prove it), I had learned many times that assumptions are a sure way to wander off track.  (Did you know that “error” and “err” come from the Latin verb meaning “to stray”?)

For example, you have a bug in a program which is supposed to multiply two numbers.  It works fine when both numbers are positive, but when both numbers are negative, you keep getting a negative (rather than positive) product.  You “know” that everything’s okay in the input module of the program, since you just wrote it a few minutes ago, so you keep looking for the bug in the processing module, where the multiplication takes place.  You spend hours testing and re-testing.  But you could have saved yourself some trouble by exploring the input module.  When you eventually do, you discover that any negative signs in the second factor are being erroneously and reliably dropped before the two factors are passed to the processing module. An easy fix, and a hard lesson.

I was horrified when I quickly recognized a pattern of non-critical thinking in my students: no quick scans for reasonableness, no checking their work through inverting the process, no checking for dropped negative signs or decimal points.  Those are examples of problems that show up in written work.  There are similar problems with oral work, such as shouting out answers that are nonsense because the speaker doesn’t filter his thoughts.

I would say, “Be sure to check your work,” and they would mollify me by glancing at their scribbles to see if everything looked okay.  But it wasn’t only laziness.  They were exhibiting what educators recognize as overconfidence, defined as “an overestimate of the likelihood of the correctness of a judgment” (Sternberg & Williams, 2002, p. 318).

Critical thinking, with its emphasis on testing ideas against alternatives and meta-cognition, fits naturally into constructivism, described by Jonassen as the following: “Learners begin constructing their own simple mental models to explain their worlds, and with experience, support, and more reflection, their mental models become increasingly complex.”  This is in comparison to behavioral theories of learning, which “focus primarily on changes in observable behavior rather than on internal mental processes”  and to sociocultural theory, whose major premise is that “a person’s intrapersonal, or internal processes have their roots in interactions with others” (Sternberg & Williams).  Rather than considering how inner growth might be expressed internally (behaviorist) or how external growth might influence internal development (cognitivist), constructivist theory sees growth as a series of internal loops feeding into each other.  “How do I know the quotient of x and y is 2?  Well, I know that 2y=x.  How do I know that?  Etc. And are they 2 in all cases?  What about if y=0?”  The student needs to have constructed a valid mental model of division, of the meaning of variables, of the undefined nature of some number when it’s divided by 0, and so on.

You can see how the proper use of technology exploits and confirms constructivist learning theories.  The simplest example is an assignment to develop a product – web site, podcast, digital movie, Powerpoint presentation.  The student has to master the concept in order to present it coherently to the consumer.  “I need to tell my reader about suspension bridges, so first I’d better define what a suspension bridge is.  And this will need to be in the context of bridges in general.  So we’ll need a history of bridges, with several juicy stories about bridge collapses.  Plus a bit of physics of bridges and the challenges they address and solve.”

Also, after developing one or two products using the same media, the developer will begin to appreciate the need to present information multiple ways, to accommodate multiple learning styles.  A web site might have the construction of a suspension bridge demonstrated in a Flash movie, explained in text, illustrated in a drawing, maybe even described in audio.  By being forced to describe the same concept using multiple media, the developer begins to construct a complete three-dimensional, full-color mental picture of the idea he is attempting to teach.

Finally, the use of technology as a means to an end forces the media developer to answer the question, “How do I know that?” Each claim – a bridge stays up because of this, a bridge falls when this happens, steel bridges are stronger than brick bridges but not as strong as steel-reinforced concrete bridges – must be defended and explained coherently.  The consumer and the developer of the media project both benefit from the constraints of critical thinking and the inherent mental construction taking place.


  1. Sternberg, R.J., & Williams, W.M. (2002). Educational Psychology. Boston: Allyn and Bacon.
  2. Jonassen, D.H., Howland, J., Moore, J., & Marra, R.M. (2003). Learning to Solve Problems with Technology: A Constructivist Perspective (Second Edition). New Jersey: Merrill Prentice Hall.


Review: “A Brief History of American K-12 Mathematics Education in the 20th Century”

As related in detail by David Klein, the history of mathematics standards in the United States is the story of a pendulum swinging back and forth for the last hundred years.  The priority – placed at various locations along the spectrum of practical skills vs. intellectualism – has shifted repeatedly since 1925, when William Heard Kilpatrick argued for discovery learning and utilitarianism in his book Foundations of Method.

Klein traces the ebb and flow of “progressive” education in its various forms since Kilpatrick (who considered mathematics “harmful rather than helpful to the kind of thinking necessary for ordinary living”).  The Life Adjustment Movement of the 1940s advocated a paternalistic approach to education, actively avoiding stigmatization of the majority of secondary-school students who were intellectually incapable of algebra and beyond.  After dormancy during the Sputnik and New Math eras, progressivism emerged again in the 1970s as the Open Education Movement, in which children were free to choose what they wanted to learn and what they didn’t.  (Hint: most of them preferred making Jello and cookies to studying math.)

Klein uses the repeated failures of the various progressive movements as the backdrop for the growth of national math standards, inspired by An Agenda for Action (published by  the National Council of Teachers of Mathematics in 1980) and by A Nation at Risk (written by a commission appointed by the U.S. Secretary of Education).  The combined effect of these two documents was that the public collectively woke up to the sorry state of math education and began supporting the development of standards.  In the wake of this attention, National Council of Teachers of Mathematics (NCTM)  developed several standards-like documents – which Klein describes as instruments for promoting NCTM’s agenda,  characterized by Klein as pro-calculator and anti-calculus.

Against the anti-intellectual tide of the NCTM pressure, California and other states adopted rigorous standards in the 1990s.  Still, various localities (for example the LAUSD and El Paso schools) worked with NCTM-aligned programs, some homegrown and others developed by textbook publishers such as McDougal Littell.

Klein pits the National Council of Teachers of Mathematics against organized groups of informed parents and university mathematicians, who were concerned about the documented poor results of public math education in the 1990s.  It’s a fascinating and complicated story, with no obvious villain and enough ineptitude and corruption to go around.

1. Klein, D. (2005). A Brief History of American K-12 Mathematics Education in the 20th Century. In Royer, J.M. (Ed.), Mathematical Cognition. Information Age Publishing, Charlotte, NC.