Trouble with Semantics - The Obelus, or Division Symbol: ÷

We've all seen it, and we all know what it means... right?  Commonly referred to as the division symbol, I'm talking about the obelus, ÷.

Now we're used to seeing the obelus in elementary mathematics problems of the form:

No surprises thus far, I hope.  We use this symbol almost interchangeably with the idea of division and it even exists as the symbol on our calculators for the division operation.  When we punch the button on the calculator, though, it will generally display as a solidus (or slash, "/").  The interesting thing to note here is that we treat these as interchangeable symbols, and there should be no reason to think otherwise.

Recently, however, a discussion came on a forum that I frequent focusing on the potential ambiguity of the obelus.  More specifically, a problem was posed.  To what does the following expression reduce?

Too simple, right?  We just use our order of operations, PEMDAS (or BEMDAS), so that it reduces as follows:

Well, here's the interesting part.  We are making the assumption that the obelus is used exactly like the solidus and that it is representing the basic division operation.  To look at why this assumption might not be founded in truth, let's consider the origins of the obelus.

A friend of mine named Adrian, after doing some reaching, found an old German book called Teutsche Algebra by Johann H. Rahn, dated 1659.  It's entirely in German, but the beauty of mathematics is that it's a universal language.  Without being able to read any of the words, I was able to decipher many of the symbols.  It starts with an elementary introduction to the notation to be used in the book, which includes addition, subtraction, multiplication, exponentiation, and of course division.  Interestingly the symbol for exponentiation was a swirl:

At any rate, the obelus is introduced to represent division on page 8.  Previously, the obelus had been used to mark ancient manuscripts that were believed to be corrupt.  As it is used in Teutsche Algebra, however, was the first time it is used in a more modern sense.  There are pages of sample uses for division and all of them are analogous to how it used today.  This is the first known use of the obelus to represent division.

Things get interesting on page 76, however, when the following assertion is made:

For clarity, and since this is provided without context, this can be likened to the following, which is more relate-able:

Now, this clearly doesn't obey the aforementioned order of operation rules, which suggests something about the nature of the obelus as a symbol for division.  As presented here, it is a line operation that divides the expression entirely (without need for parenthesis or bracketing), pun intended.

When presented with  this information, my first instinct was to defend the original answer to the problem above as a more modern interpretation of the obelus symbol.  My thought was that the use of symbols and their inherent interpretations evolve over time.  I considered that parenthesis were not used as a form of grouping, so the inherent system is slightly devolved from what it is today.  Hell, as I mentioned, this text even uses a swirl to indicate exponentiation, and we clearly don't use a swirl anymore.

I was content with this way of looking at the situation until that same friend located a more recent book called A First Book in Algebra, by Wallace C. Boyden, dated 1895.  Adrian's Google-fu is very strong, apparently.  In this text, the following is presented:

Which is clearly aligned with the usage in Teutsche Algebra.  Now, I don't know where this wild-goose-chase will take me next, but it seems that when the obelus is used as a line operation, it splits the line evenly into numerator and denominator, which is removed from my previous assumptions.  Returning to the original problem, then, yields:

So, which is it?  288 or 2?

My TI-84 thinks it's 288, as does Wolfram Alpha.  Now, before you start arguing that Wolfram is the infallible machine that we often assume, check out this result.  My intuition even says that it has to be 288, but my (and mostly Adrian's) research leaves me wondering.  What do you guys think?

I know one thing is for certain... for my own sanity, I will not be using the obelus in my future classrooms.  There's just one thing that bothers me, though:

Um...what does that button do?

Bonus Points: There is a calculation error on page 15 of Teutsche Algebra.  Can you find it?

Last Blog of The Semester (In Review)

Looking back at my blog posts this semester, it seems like there were a few topics of interest, namely:

• Remembering Names
• Getting new students after Week 4
• Classroom Management
• Time Management
• Stressing out

It seems like I had been using my blog as a source of venting, which is all well and fine, but I predict that the last point would be less prominent if I took some time to focus on the positives in the experience.  That may be easier now, when I'm basically finished for the semester, as many things didn't seem all too positive at the time.

The issue with remembering names seems silly now.  I know all of their names extremely well, and now I think I have some confidence in myself to remember names in the future.  It sure seems like it'd be easier, anyways.  It also seems silly that I was stressing out over the new student in class as well.  Everything was super smooth after the first few weeks, and now they are just like anyone else.  We even got classes mixed around after the first trimester, and the transition was smooth and stress-free.

Classroom management and time management are still big concerns but I feel infinitely more comfortable with each of them at this point.  I feel like I'm way more comfortable in front of the class in general, which is good news to consider.  I think classroom management has become easier because students have a clearer idea of what I expect of them in class and I feel they respect me a bit more.  Time management is not something I'm yet completely comfortable with - I tend to want to run class right up until the end of the hour, but we always end up needing those last few minutes for housekeeping sorts of things.  I'm hoping it gets better at the high school level (or at least can be better)

I believe that stressing out is the one constant that'll be there for a while, though.  I'm feeling incredibly de-stressed at the moment, since I got two major project finished.  It feels like the semester is basically over, and it feels nice.  However, I know that a whole new line of stresses are ready to meet me on the other side of winter break.  Hooray!

Lastly, I'm gonna miss the kids in my class once next week ends.  I'm not sure about the level of reciprocity, but I love the little buggers.  It's weird to think that I will likely not see them again after this placement.

See you on the other side of 2011!

:)

Last few weeks of the semester.

Boy, that snuck up on me in a hurry!

I'm trying to make a mad dash to get as much of my portfolio finished as I can before class on Tuesday.  It has quite literally gone untouched for several weeks leading up until today.  I have been so overwhelmingly busy with everything else that it felt nice to just relax and be lazy over the long weekend.  Getting back into gear today was quite the battle of wills.  I actually cleaned my room as a form of prolonging the inevitable.

Regardless, I got 5 portions of my portfolio finished tonight, and it feels pretty damn good.  If only there weren't a significant chunk to go!  I'm not looking forward to doing the Outside Resources or Equity Project portion, so I'll just put those off a little longer... :)

As for teaching in the classroom, I feel like my role has been significantly diminished.  It sounds like I will go back to doing the opening questions like I did for the first half of the semester, which seems so long ago!  I still have two tests to grade tomorrow in class, but after that, my unit is completely finished.  It feels so weird to not have to lesson plan or stress out about grading tests anymore.  Weird, but good!

I honestly don't know if there is something else I need to be doing in the classroom.  I suppose I will wander around and help out where I'm needed.  I now know how Mr. Martin must've felt during my unit - and it makes sense that he was always looking for something to do.  I feel like the last few weeks is going to be boring!

As for my unit as a whole, I feel like it went pretty well.  If I'm considering how my students did, their test scores indicate that my feelings are accurate.  There were a few young ladies who completely bombed both tests, but I have a hard time feeling bad for them.  I approached each of them to see if I could offer additional help, and gave them opportunities to earn back some points on their tests, but neither of them took me up on either offer.  You can only lead a horse to water, right?

In other news, I hope to be hearing from Allendale about my student teaching placement for next semester this week.  Just to have official word will be nice.

:)

Last Week of my Unit

Wow, that went by very quickly!

Tomorrow and Tuesday are my last two days of the unit, and it feels very good to know that I won't have to do any more planning or stressing out about my students and I can focus on the other stuff I need to finish for the COE.  I still need to grade the test that I gave out on Friday, so that is my plan for Monday night.

Frankly, it seems that there isn't anything else exciting going on.  I got my classroom management plan done, so that's one less thing to stress out about.  The only two things left on my plate are the Multi-Genre project for my literacy class and the Placement Portfolio.  That one will actually take a bit of stress yet... Thanksgiving break should give me the time I need.

In other news, it sounds like there might be a secondary teaching placement opening in my old high school.  I have quite a few friends in the right places that it could work out nicely.  I'm going to do a bit of probing to see what I need to do to initiate the right 'first contact' and so that I can express interest before the position is filled.

Hm... what else is going on?  Well, I ran into an issue with the test that I gave on Friday.  There was nothing wrong with the test, per se, but there were a few people who needed the whole hour to take the test.  This is not unexpected, but what was unexpected was the inability of the rest of the students to refrain from talking so as to not distract those still taking the test.  It was downright disrespectful, and I'm disappointed that even with a very explicit reasoning that students are still testing, they could muster no sympathy.  It's partially my fault for not having something specific for them to work on, but I did give options for those who finished early, and they refused them.

Ugh, it was frustrating.  If my unit wasn't concluding, I would definitely make it a point that such actions would not be tolerated.

Well, I guess this is a short blog post this week.  Not much else to share.

The Busiest Week so Far!

The week started off with a bang - I gave students my first ever assigned tests on Monday.  I was pretty nervous to see how they'd do, which I figured would reflect on how well I'd taught.  I know that there are so many other factors that this would be an invalid assertion, but it's tough to avoid thinking that way.

So, I spent a lot of my downtime throughout the week slowly hacking away at grading their tests, which was quite a novel experience.  I've heard quite a few strategies for grading tests, but I opted to just grade a page at a time in an attempt to maximize consistency.  I hit a snag on page 2 when I noticed that everyone was struggling with one question.  It was a pretty difficult question, so I gave them the benefit of the doubt and made the question extra credit, so as to not factor into the total number of points possible.  Interestingly, nobody got the question correct.  The right call I'd say, and an indication to do a bit of reteaching.  More on this later.

We had a bit of a wild week, thanks to a professional development day on Wednesday, my absence from class on Tuesday for seminar, and our CT being out to take care of a sick kid on Thursday.

On Thursday, we continued learning about writing rules, and then equations, for simple linear relationships between two variables.  I was pleasantly surprised to see the students behave as well as they did, considering the presence of a substitute teacher in the room, but there seemed to be mass confusion on the topic discussed.  We didn't get through much after a long list of housekeeping things, so I decided to rethink and stretch the lesson for another day.

That night, I created handouts for the students which could be used a reference to look back on writing rules and equations, as well as provide an opportunity to practice writing their own rules.  Having finished grading their tests as well, I created a re-test of sorts, which I dubbed the Test 2 Recovery, since I intended it to be completed at home and not in class.

Friday went better than Thursday, but was still not great.  One student had one of those educational baby dolls that liked to cry at random intervals.  Fun!  Ignoring this the best we could, I let the class try to come up with their own rules and equations on the worksheet I gave them.  They were all over writing equations, but the rules (with words) were tough for them - I'm thinking it's more of a writing issue than an understanding issue.  But we must push on... we are going to be writing more complex equations this coming week.

I concluded the day on Friday by giving them their tests back.  We spent a bit of time reviewing the questions that they most struggled with, but they were being a bit rowdy.  I used the line, "I would love to continue going over this as soon as the volume level gets back under control. I'm reviewing this stuff for you all - I already know it.  I just assumed you'd want to know how to do this for the Test 2 Recovery which I'll be giving you to do over the weekend."

That seemed to work.  And now I just need to see how many people actually turn them in, and I get to have fun grading stuff again.  Interesting side note - I was pleasantly surprised to see one of the quiet, blend-in kids get a 96% for the class high.  Go him.

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Other than that, this past week/end has been obnoxiously busy.  I haven't left my house since Friday, and I've been writing at my computer for most of it.

Weeeeeeeeeeeeeeeeeee!

Week... Err... I don't even know anymore

I'm so far into the semester that I have no idea how many weeks we've been going for, and I'm not terribly looking forward to the next few of them.  In the next 3 weeks, I have to write my classroom management plan, my placement field folio, and a teaching literacy philosophy, while make progress on my placement portfolio and completing a working prototype of my super secret project.

I hope my hair can handle all of pulling it will be asked to endure...

My teaching placement has been going well, for the most part.  Aside from them going much slower than I'd like, and having to teach a level of mathematics that is not my preference, I think I have been learning a ton about myself as a teacher.  For one, I don't do enough (in my inexperienced opinion) collecting of their work for a grade.  I will be giving a text/quiz on Monday, and I realize that I haven't yet graded them on anything I've given them for this investigation.

It's not as if I haven't been giving them assignments, but in this school, homework completion rates are significantly lower than ideal.  It sounds terrible to sell them short, but from experience, only about 10% of my students will actually do any homework assigned to them.  At one point, I had a plan to check over their homework from the previous day using our CPS remote system, but we only got one question in before class ended.  My plan was to hold them accountable the following day, but my cooperating teacher suggested that I just push on to the next section.

I need to get better at that... I think.  Right?

I had them turn in some 'Exit Slips' to gauge their understanding, which I intended to give a participation score to, which I still may do yet, but I have been told that grading exit slips should be avoided.  Gah!

---

Otherwise, last week I had one day where I fell flat on my face, followed the next day by one of the better lessons given in that classroom this year.  My cooperating teacher said that he was actually pretty engaged in the lesson, having distracted him from the work that he was trying to do off in the corner.  If I can just get rid of the bad days, then we're in business!

This week should go by pretty quickly... We have a test tomorrow, professional development for the College of Ed on Tuesday, and our school has no students on Wednesday (the jury is still out on whether I need to drive in that day).

I think I'm going to spend the next 30 minutes or so depressingly writing down everything I need to get done in the next few weeks.  Ugh.

It has begun! ...err, my unit, that is.

Yep, I have officially begun teaching my unit...

The first day went very well - much better than expected.  That is not to say that I thought I was going to do terribly, I was just shocked at how well my students responded to my first day.  We started off with a short warm-up activity that served almost as pre-test to the unit, which is: interpreting tables and graphs in order to make informed decisions.  In addition to that, though, it allowed students to mentally prepare for the lesson... I gave two short table/graph interpretation problems and told them to think about why these problems might be difficult for them.  I was more interested in that than any answer they might give, so I made that specifically clear when I gave them the handout.

Sure, there was some resistance by a few students, but most of them participated in some regard.

Then, I had them turn the paper over, and we focused on the lesson, which was essentially teaching them how to address the types of problems that they had on their handout.  Class involvement here was probably the best it's been all year.  I don't know if it's just because I left all the questions open, or what, but they were legitimately engaged!

After the lesson concluded, I had them return to their warm up worksheets and do a quick reflection on what they wrote to start the hour.  I instructed them to make a note on the sheet if they felt more comfortable with the problems or if they still felt confused by them.  I got some nice responses from these.  Then, that night, I went through and wrote a short note to each of them, either asking them some extension-type questions (that they aren't expected to formally answer), or to invite them to talk with me personally, or a tip that might help them out.

I think they appreciated that personal touch, and I could tell that the couple of students who didn't turn one in were disappointed that they didn't get a note from me.

---

My second day wasn't as eventful or well done.  They had a Friday remote quiz, which took much longer than expected.  Then, I gave them time to discuss their homework with another student in class, and then the opportunity to review one question (of their choosing) as a whole.  Sadly, when we ran out of time at this point, and I very nearly ran over.

Because of my poor time management, I didn't collect anything from them.  I feel pretty bad about that, but I'll see if I can get something tomorrow (Monday).

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In all, I think it went well.  I just need to note the time and be more wary when I don't have any more of it.

I'm more nervous for tomorrow than I was last week.  Also, I'm already behind.

:)