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.

Teaching Slope as Rise/Run

Our lesson today in the basic concepts of algebra course had to do with the slope concept. We had already spent some time on linear functions and investigated the notion that the rate of change of a linear function is constant – and that is what the slope of the line is. Side note: a few students were confused about the difference between a constant function and a constant rate of change – that can be a topic for future discussion here – I can’t afford to digress because as a lazy mathematician, I may never get back to the current topic – slope of a line.

So, we were going about it quite happily and figuring out our rises and our runs – focusing for today only on lines with positive slope. All of our rises were positive, an our runs were also positive. We worked with pictures like this one:

Positive slope -positive rise - positive run

Then our author did something that threw at least one student for a loop. He put this picture up as an illustration of the rise/run process for figuring out slope:

Positive slope - negative rise - negative run

My very astute and inquisitive student wondered out loud: “Shouldn’t the rise and the run both be negative?”

Well, the lazy mathematician would have to agree – the rise and the run as pictured should indeed both be negative. To have our picture match what we lazy mathematicians do, we need to have both the rise and the run as negative numbers – else we get confused.

Why the lazy mathematician?

I often tell my students that mathematicians are motivated to do the things they do out of laziness.  To avoid writing long and complicated expressions, we invent all sorts of notational devices as a sort of shorthand – a shorthand that must then be taught to others who may want to learn a little about the mathematics we develop. It is a type of code or language and if others want to communicate with us as mathematicians, I explain, then they must first learn and be comfortable with our symbols.

But mathematics encompasses more than just manipulation of symbols (apologies to all you abstract algebraists and number theorists out there – I have utmost respect for your symbolic machinations!). To me it is about establishing a world view that is concise and precise – and so the laziness comes from not so much a lack of energy or a desire to avoid hard work as from a perspective that says “Let’s try to get this done with as little effort as possible.”

So, a goal in this blog is to show you a little about how we do that – the secrets and apparent tricks we employ to make our work easier. Unlike the magician who wants to keep his tricks to himself, I want to show you the tricks and help you to avoid the traps. I will be sharing the things I find that give my students trouble with the hope that you can learn how to avoid those troubles – minimizing the effort you must expend by showing you how to avoid those blind alleys and false directions you may have traveled down before. But always remember, an occasional trip down the wrong path often leads to a surprising discovery – we’ll take that trip together and it won’t be so scary!