The video would be a nice introduction to the worksheet. I feel that it’s not necessary and the effort required to create it is not worth it, but *given* it exists, I would use it enthusiastically. If it didn’t exist, I think the worksheet would still do the job quite nicely.

What can I say? I like a nicely conceived and executed worksheet.

I spent some time last night solving the general case, and having access to that Geogebra applet was great for my motivation and exploration. I’d happily give students access to that. Is the file available for download? I want to learn how it’s constructed.

Final comment on the video: I am gobsmacked at its production quality, offer my congratulations at your skills, Dan, and would love to know how it’s done. What software, and how long does it take when you’re reasonably experienced at using that software.

Many mathematical concepts are crying out to be visualised. Geogebra and friends are great — we are so lucky to have them — but sometimes a well-produced video would be fantastic.

Linda, what a breath of fresh air it is to hear you say that.

But my mind works very methodically. So I – like David Cox – like the “start small and look for the pattern process”.

But I understand that everybody does not and in fact everybody seems to hate the “other” approach. I think that is the reason for all of this discussion. I once had a teaching assistant who taught in a completely different style than I do. I thought – this is never going to work – but actually it worked out well. Each student got at least something he could understand well.

BTW: Zeno – the questions Tony, Steve, … ask are EXACTLY the questions I asked when reading the original problem. I would never assume that everyone processes text in the same way and comes to the same conclusions and has the same pictures in their head. English is my first language and I am constantly bewildered by what people consider “completely clear”.

My one comment is a general comment. I think we should have problems we re-examine every couple of years in school. (Why do we invert and multiply, why is the area of a circle pi*r^2, this problem…?) I need “real” time to process things.

First, if you take a rectangle whose side lengths are relatively prime, you might see that the number of squares crossed is really the number of lines making up the squares crossed (except for that pesky place at the end where you cross both the horizontal and vertical line at the same time).

With that, a second rectangle, which had, say, some common factor to the side lengths could help one to see that the pesky place at the end isn’t really a special case any more than the other places where you cross at a vertical and horizontal line at the same time.

Anyway, to me, providing these two steps could allow one to see what was going on without getting embroiled in really big rectangles with lots to count. The point, I think, is to help students to look at the fundamental issues, rather than get into a fruitless counting exercise. And, while brute force can provide some information, one needs to always brought back to generalization and simplification – that’s what mathematics is about.

I like the paper (over something online), because I can draw on it as I’m thinking about the problem. Dan’s treatment has the rectangles get bigger fast, but at least for me, getting bigger didn’t provide any insight, and really made things harder to solve. It was the nature of the rectangles, not their sizes, that was important for me.

Dan, I for one would be really interested if you could write a post on this exact issue.
How do we move towards getting students using the normative (‘correct’) words? What if they come up with their own word(s)?

Also do you see this as a problem particularly relevant to Maths? Or is it education in general.

I know how offputing ‘using the correct terminology’ can be. One professor laughed at me when I said you had to flip fractions upside down if there was a divide sign. With a dismissive “don’t you mean invert’.

Tim: While I appreciate the videos that engage their math curiosity, I don’t think videos that simply circumnavigate language are helpful.

I agree, but this particular video intends to postpone language, not to circumvent it. Academic vocabulary and precise mathematical language is important and valued in my classroom. But that language should be used in the service of a problem that interests us. For instance, it’s really hard to talk about the video if you’re always referring to “the segment that connects two opposite vertices of the rectangle.” That’s where we introduce the term “diagonal.” The same goes for the rest of the language.

While I want my (minority, inner-city) high school students to succeed in math, what I want more is for them to succeed in life. For them to do that, the first step is success on the state exam needed to graduate, and then maybe the SAT. While I appreciate the videos that engage their math curiosity, I don’t think videos that simply circumnavigate language are helpful. More important would be teaching the tools to understand the problems as they are worded on (unfortunately necessary) standardized tests. In other words, mastery of math is GREAT, but if they can’t prove to society that they’ve mastered it, they’re never going to get a leg up.