Always Use a Title

Teaching college algebra this semester – two sections – we adopted a new textbook since I last taught it so took the opportunity to mix things up a bit – couple of changes -

No More Powerpoint

Well, not promising to be totally and permanently off, but so far, all has been done without powerpoint – keynote actually – biggest noticeable difference is classroom air – much less distance between me and students – could that be a bad thing?

Second change is -

Moderately Flipped

Not officially – haven’t told them – haven’t used that term except here – but I make the handout available marked up with my voice-over notes – upload to Sophia – link in D2L (classroom web presence) – also email to class – voila! ready for tomorrow’s class – I hope – we’ll see -

Anyway I’m ready -

A Rant from Class

Giving an algebra final exam today – it’s almost like the first day of class – here’s the problem on the test that is causing the most angst:

The proceeds from your college talent show totaled $2475. All the tickets were sold so you know that 600 people bought tickets. The cost per ticket for students was $3.25 and for non-students was $4.75. You need to figure out how many students attended the show. Use n to represent the number of students who attended. Write expressions in terms of n to represent the number of non-students who attended, the proceeds from student tickets sold and the proceeds from non-student tickets sold.

Knowing full well from my experience that this is really difficult for my students, I gave them what I thought was a nice clue as to how to organize the information:

1-Number of students who attended n

2-Number of non-students who attended _______

3-Proceeds from student tickets ($) _________

4-Proceeds from non-student tickets ($) _________

It was as though we had not spent a single hour on problems like this, although I know they did work many of them in class, for homework, and (presumably) studying for this exam.

No one knew how to write the expressions for 2-4. Crazy.

It occurs to me that I have no idea how to make them learn this – I try again and again and they still can’t seem to get the idea that the letter n stands for a number that we don’t know, and that all of the other numbers depend on n.

If I ask them to tell me how many non-students attended the show if 100 students attended, they can do it.

If I again ask that same question but change the number to 101, they can still do it.

They can do it no matter what number I give them for n. But once I ask them to generalize, it’s like their brain melts and they have no idea what to do – even though they just did it with actual numbers 4 or 5 times!

So, they cannot generalize – and this is the final exam! We spent the entire course generalizing – and they still can’t do it -

Makes me wonder if they will ever be able to do it -

They’ve taken algebra courses before – mostly in high school – probably since 7th grade. For 7 years they have had this topic – generalize – and still can’t do it.

If I decided to become a downhill skier, and after 7 years still couldn’t figure out how to put my skis on, I think I would give up.

that analogy isn’t really right – it’s more like, I knew how to put my skis on at Vail, but when we went to Keystone, why that’s a different place so I don’t know how to put skis on there – someone has to help me!

Maybe they need to give up.

Because if they keep showing up, we will keep trying to shove it into their skulls and they will continue to not get it -

I’m sure they’ve had better teachers than me in some past experience with basic algebra – and still they are here, trying again to learn the same things that have not been able to learn since 7th grade -

What is wrong with us?

A Friendly Chat

Had one of those today with a colleague – he wanted to know if I had an undergraduate real analysis book – something covering the subject of Dedekind cut – hadn’t thought about that for some time -

My teacher’s name was Dorothy Smith and she drove a convertible Alfa Romeo – it was the early ’80′s, so chill.

Anyway, I was sad to find that I no longer had the book – it was finite math and it was the first course I took for credit after repeating all of the calculus series to prove myself to the graduate school. It was only after I earned A’s in all of those courses that the math department would accept me on probation for graduate study.

So I wish I still had the book – and I wish I could talk to Dorothy Smith – it was she who had the most confidence in me and encouraged me to keep studying mathematics.

Antivenin? For Real?

I can’t read through my local daily newspaper without running across more than one misspelling, typo, or grammar error. I guess the copy editor was fired and not replaced some time ago. Anyway, this morning I was reading an article about the recent flooding in Louisiana and came across what I thought was a misspelling of antivenom – the word I saw was antivenin. Couldn’t be right – could it?

So, I consulted my handy Dictionary.com and was really surprised to learn that I had been doing it wrong all my life! (Sidenote: I should get partial credit for only having 2 letters wrong out of 9 – let’s call it a C+.)

My students do the same thing – well, not all of them, but there is a sizable number who express surprise at being told that they are misapplying a concept, and swear that’s how they were told to do it. They stubbornly cling to their false knowledge, and I spend a lot of time unteaching them things they have mislearned!

I think about what motivated me to consult the dictionary – after all, my experience told me to expect the newspaper to be wrong, so why did I doubt myself in this case? I remembered another time when I was in a similar situation – that time the word was temblor and it was referring to an earthquake. I thought it should have been tremor – so I looked it up and was surprised to find that I was wrong. If I was wrong once, I could be wrong again – so now I know the correct word for snakebite antitoxin.

I want my students to consult their mathematical dictionary – I want them to be curious about their process and question themselves if something they are doing doesn’t make sense – after all, it could be right but it could also be wrong. Either way, what they learn will more likely become part of their permanent mathematical memory.

Four Door Monty

OK, so in my regular Monty post I promised an answer to the 4 door Monty question and here it is.

Simply put, the only way you can win if you decide to use the switching strategy is if you first select the wrong door. Selecting the right door on your first choice will guarantee that you lose because you are switching after the host reveals one of the three doors you did not select. So, the first probability is just the probability of selecting a losing door from the four original doors,  3/4. Assuming you have selected a losing door, when the host opens one of the remaining doors, offering you the opportunity to switch, there are now two doors remaining, and only one is a winner. Your probability of selecting the winner now is 1/2. Thus, the combined probability is (3/4)(1/2)=3/8.

This is better than the probability of selecting a winner if you stick to your original choice – that probability is 2/8, clearly less than 3/8!

So, the interesting part of this is that it seems like the advantage gained by switching decreases as the number of doors increases – more on this later. I’m too lazy to do any more on this today!

Bucket

Check out this little guy – it took him a few minutes to figure out exactly how to maneuver the shovel and get a little sand into the bucket. First he struggled with which hand to use, then he found a fork and started using both hands – ambidextrous?

Is this one for the anyqs

Let’s Make a Deal

I love this problem – even though it is just an exercise – it’s still kind of fun to work out.

It is also known as The Monty Hall and was floating around a while back in the early 1990′s on Marilyn vos Savant’s web page. Most of my current students have little or no idea who Monty Hall is and so I have fun explaining to them the whole idea behind the game show, Let’s Make a Deal.

Anyway, readers of this blog are probably familiar with the show and the problem, but for completeness sake, here it is in a nutshell.

You are on a game show where there are 3 doors. Behind two of the doors is a mule, and behind the third door is a Cadillac. You choose a door. The host then gives you an opportunity to switch your choice after opening one of the three doors – not the one you chose, of course – and revealing what is behind it. The question is, should you switch or keep to your original choice?

You have no doubt already clicked on the link above and checked out the solution, so you know that in the long run it is actually to your advantage to switch after the host reveals what is behind one of the unopened doors. But I have another question for you: What if there are 4 doors, and after you select one, the host reveals what is behind one of the remaining 3 doors – not the one you selected, of course – and gives you the option of switching. What should you do? And what is the exact probability of winning or losing in any case?

I’ll post an answer soon – unless you beat me to it! No Googling, now!

Should I Repair or Rebuy?

I bought a top of the line front loading washer two years ago. It was on sale for $500.00. What a deal – the regular price was $899.00. Things were humming along nicely, laundry was getting done – until one morning I heard loud noises from the basement and ran down the steps to find the washer hopping across the floor. The load was out of balance, which is very bad for any type of washer, but most especially bad for a front loader.

The vibration was really severe and it resulted in some pretty serious damage to the drum – it came disconnected from the body of the washer, so it sort of hung there.

I called the repair guys. Since this is a foreign made washer (sorry all you “buy American” people – it was a good deal and the people I bought it from are Americans – also, the company that services it is American, so I guess I can rationalize it that way. . .). Worst case scenario was replace the drum for $387.00.

Got me thinking – what is my top line cut-off? How much before I say “I may as well just buy a new washer?”

So, I needed to know some things. Like, what is the yearly rate of depreciation for a washer?  How many years is a washer likely to last? Should I use the price I actually paid or the original price of $899.00 in figuring out the depreciation?

I, of course, went to Google and found this Depreciation Guide:

depreciation_cp-1-2

Apparently my washer has a useful lifetime of 8 years and depreciates by 12.5% per year. So, after two years what is it worth?

I’m just a little lazy today and would like you to figure this out for me. Should I buy new or repair?

By the way, the actual repair cost was a mere $167.00. Does this change your answer?

Why Can’t Students Use Prior Knowledge – Redux

OK, I get it – here is a snapshot of what I saw along with my comments on the work:

Solve by graphing -

Upon closer inspection, I saw that the only real error in the student’s work was a sign error in solving the second equation for y. She wanted to have the two equations in the familiar form y=mx+b, and then find three points for each line. That was her method – and if she hadn’t made that one error, her solution would have been completely correct. I was thinking “Gee, if she were lazy like me, she would not have first manipulated the given equations into slope/intercept form, maybe not made that mistake, and then would have had the correct graph and solution.”  But for what she did, her solution was actually correct.

So, now as I think about it, I see that this student is actually a great lazy mathematician – she has a method that works for her – she can’t use extra energy to apply different methods for each situation – she can find a solution quite nicely, thank you,  just doing it her way. And I should not have a problem with that – she found some points that were on each line – plotted those points – drew the lines – and gave her answer.

Then I noticed that she used the same x values for each line – WOW! What a great economy of time that works out to be – she doesn’t need to try and think of x values that give nice y values – she is happy to just plug away and plot points. Nice.

Why Can’t Students Use Prior Knowledge in Novel Situations?

In my intermediate algebra class, we spent almost 2 weeks studying lines, slopes, and intercepts. Students were instructed on several techniques for graphing lines: plotting random points, using the x- and y-intercepts, and finally using the y-intercept and two other points. So, students came away thinking that these are three different methods of graphing a line when actually they are the same thing – using points to draw lines. And they end up doing a lot of calculations that only encourage arithmetic errors of many types.

They were then asked to solve a system of equations graphically.  One student very painstakingly expressed each of the given equations in the slope/intercept form and THEN substituted values for x to calculate y. She didn’t get two things, at least, that I see.

First, expressing each equation in slope-intercept form makes it unnecessary to then plot points and get the graph. Once you know the slope and the y-interept it is easy to draw the line.

Second, if your plan is to choose a value for x and then calculate y, there is no reason to use the slope-intercept form to begin with. Just plug and chug in the given equation.

So, what could I have done differently to make sure that my students made that connection? Because I’m pretty lazy and I’m always looking for ways to make problems easier – I assume my students are the same way. But the fact is they have a lot of energy but expend it most inefficiently in their problem-solving.