I think I might know the answer to that one:

– People have a general tendency to mistake what they see (the artefact) with the real thing (all the circumstancial forces that make it work). I believe it is a natural result of ignorance.

Some examples:

– People mistake having a facebook-page with being present in social media, when it is really the interaction there that really matters. But it is the page that is most clearly visible.

– People mistake a plan document with planning. A document is just an artefact, and not necessarily worthwhile, as many business plans are a testament to. Planning (continuously thinking about future alternatives and contingencies), on the other hand, is immensely worthwhile. Alas, the document is what outside people actually see.

– People mistake badges, with rewards. They see players getting real exited about getting a badge, and conclude that the badge is what triggers it. In fact, it is overcoming a challenge that is rewarding, and the badge is just a symbolic representation of that. You can see this fallacy play out in tons and tons of websites that try to use game mechanics to motivate users/customers, by handing out worthless badges (i.e. badges handed out indiscriminately, after no challenge at all).

– People mistake mathematics with symbols, when it is actually a way of thinking. Since you can’t “see” a “way of thinking”.

I hope this was helpful to someone.

Nice article, btw.

Cheers,

Magne Matre Gåsland

]]>What I object to is a widespread emphasis on preschool and early-elementary children mastering abstract, symbolic manipulation by rote.

I think my problem with convincing homeschoolers to postpone math drill is similar to the problem you have convincing potential sponsors that your games teach mathematical thinking. It’s such a strong part of most people’s cultural understanding of mathematics that “real math” is school workbook math.

]]>I succeeded with my kids using the low-tech method of telling lots of little stories, and having them make up math stories of their own. Over and over, in informal settings like during the drive into town for soccer practice, for years and years. When the math symbols in a workbook were too abstract, I set the book aside and put the same problem into a story setting. I think kids naturally think in terms of characters, settings, plot, and analogies. It’s only after they’ve been bludgeoned by school math that they learn to fear story problems.

The combination of thinking about math in the context of story, of making up math stories, and of figuring out ways to come up with answers — all this made for a great foundation for the eventual introduction of symbols and school math. As you wrote in your book, “the skills come largely for free.”

I’ve tried to spread this idea to other homeschoolers (see, for example, this blog post), but so many people are primarily worried about how quickly their children can master the math facts. The idea of delaying symbolic manipulation in order to “just talk” doesn’t get much traction…

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