My first post about “Duke is Missing” focused on wondering. I think of wondering as a good part of *launching* the PoW. Then comes *exploring*… we love to know about all the thinking that goes into working on the problem. Check out this submission to the “Duke is Missing” problem:

i got that dukes body is 24 inches because his tail (18 inches) and his head (6 inches) sum is 24 inches which is equall to his body. so his tail is 6 inches (his head) plus 12 (half his body).

For this one, I keep wondering, “How did you think to choose 24 inches for the body?” As a reader, I’m really curious! I also think the submitter probably did some good thinking, and without writing down what they thought, they can’t get feedback on it, and they might not be able to use their good thinking discover more math, wonder more, or think about how to solve a similar problem.

Some good questions to get to the next level of explanation are:

- What strategy did I use?
- What made me think of that strategy?
- How did I come up with that number — was it a good guess? an adjustment of a previous guess?

What would you ask this student to learn more about their thinking process?

How could writing about thinking help the student improve their math & problem-solving skills?

**Some “Duke is Missing” links** in case you are interested:

- The problem [requires a Math Forum PoW Membership].
- Information about accessing “Duke is Missing” (and all our current PoWs) for two weeks with a free Math Forum trial account.
- Information about becoming a Math Forum Problems of the Week Member. Compare prices – consider starting with a $25 membership giving you access to all of this year’s Current PoWs — and now you can create 36 student logins as well!

[...] We read a lot of explanations submitted by students, and since we rarely get to talk to those students, we hope that their written explanations will include descriptions of the decisions they made. Partly we’re really curious about student thinking, wondering things like: How did they get there? Why did they decide to do it that way? Did they make any mistakes and change their minds? Partly we want to know if the kids know the answers to those questions. (Max also posted about this on Wednesday in Explaining How You Solved “Duke is Missing”.) [...]

Because there was discrepancy from one teacher to another about order of operations in

algebra, I paid particular attenton to the discovery of the word: viniculum in the Websters

Dictionary (1999). The viniculum is used over the term that should be worked “first” and

this seemed to be the answer to the problem of order of operations. Why then, is the

term never referred to in the Beginning Algebra books printed by Elayne Martin-Gay? I can only speak for this book and one by Angel, an Introduction to Intermediate Algebra, because those are the only two authors I have read. We had to skip chapters because there was not enough time in the semester to cover the whole book. Rather than split the book into to A and B semester sections, some of the most important parts were just skipped. I feel this is sabatoge of the learning process, since the Order of Operations problem is not new and is not unique to one math discipline. The Math Forum returns no results for the use of the viniculum for “Order of Operations” in algebra and this is really a surprise!

I’d never heard of the vinculum before, so was excited to learn about it. I searched Dr. Math (http://mathforum.org/dr.math) and found the following links:

A history of the vinculum term: http://mathforum.org/dr.math/faq/faq.terms.html

An overview of grouping symbols: http://mathforum.org/library/drmath/view/58406.html

A long conversation about the vinculum in fractions, order of operations, and division: http://mathforum.org/library/drmath/view/75038.html

It sounds like you have some questions about how order of operations is taught. It would be wonderful if you were able to ask your question in our “Teacher to Teacher” Q&A service. Our volunteer answerers would love to think about vinculums (vinculae?) and order of operations with you! http://mathforum.org/t2t/ask/