Reply to End 2.0 MCPS math curriculum. List complaints about specific problems.

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[quote=Anonymous][quote=Anonymous]Here are some examples of the quality of story problem writing without a published textbook. (Algebra 1, unit 1, topic 3 packet, pp 25-29)

[quote]
Sabrina brings a Hershey’s chocolate bar to her 3rd period math class. She snaps the bar in half and gives one half to her BFF Yarely. Then everyone sees it and wants a piece. Sabrina is good at saying “no” but Yarely is not. She decides to divide her half of the bar in half and give that half to Steven. Then John attacks Steven, wanting half of his piece. Then, Isaiah grabs John’s piece, rips it in half and eats it. How much of the chocolate bar does Isaiah get?

At 11:00 a.m., Sally leaves a serving of yogurt on a table. The number of bacteria in the yogurt at that time is 1. The number of bacteria in Sally’s yogurt increases by a factor of 4 every minute. Three minutes later, at 11:03, Lupe leaves a serving of yogurt on a heater. The number of bacteria in Lupe’s yogurt at that time is 1, but due to the increased temperature, the bacteria increases by a factor of 8 every minute.
At what time will the number of bacteria in each yogurt be the same?

At noon, a computer virus infects 1 kilobyte of a computer’s memory. Every minute thereafter, the number of kilobytes infected triples (is multiplied by three). At twelve minutes after noon, a much worse computer virus infects 1 kilobyte of another computer’s memory. Each minute thereafter, the number of kilobytes infected in that computer is multiplied by nine. When will the number of kilobytes infected be the same in both computers?

You bought two loaves of bread from the grocery store, one rye and one sourdough. When you got home, you realized the sourdough bread had one sq mm of mold on it. You measured to find the mass of mold tripled each day. The rye bread began to grow mold on the fifth day. You measured to find the mass of mold grew by a factor of nine each day. When will the masses of mold be the same on each loaf of bread?
Create a table to display your solution. Solve the word problem using equations to see if you get the same answer.[/quote][/quote]

Yeah, sorry for the long examples.  I chose them to illustrate not just the clunky writing but also how much is being swept under the rug in the instruction.

Before starting the math, I think the writing is amateurish and could use an editor but also the scenarios are contrived.  The chocolate bar problem reads like a laundry list and that's really all it is.  The answer is (1)(1/2)(1/2)(1/2)(1/2).  The same thing happens over and over yet there's no compression in the writing or the math.  There's no natural way to introduce a variable to answer the question or to explain the work.  But, whatever, it's review, still, I think any text book would do better.

The other three problems are about exponential growth.  Bacteria questions are standard--since bacteria reproduces by cell division, there are twice as many cells with each generation.  The first question is a little clunky, since no one talks about integer numbers of cells IRL but it's pretty much the model the other questions are riffing on.  The number of bacterial cells in Sally's yogurt is 4^n where n is minutes after 11:00.  Lupe's is 8^(n-3).  Since 4 = 2^2 and 8 =2^3, they have the same number of cells when 2n=3(n-3), n=9, 11:09.

My gripe with the second question, is a virus couldn't spread on a hard drive at an exponential rate.  A computer virus maybe infects a network at an exponential rate, but the damage to an individual computer would be linear because a computer processor, no matter how fast, is not exponential, it's linear and the virus uses the processor to corrupt the hard drive.  This may sound petty, but why have the students make sense of a scenerio that might be completely new to them, just to muddle the facts?

The third problem neglects to give the initial condition for the rye bread so it's not a fully formed problem (presumably the rye also begins with a 1 sq mm growth on day five).  The colony size is reported as an area, but then the question talks about "masses of mold".  Is this a different measurement or a colloquial meaning of mass, as in a tumor.  Again, all these are minor points but the sort of detail that a text book editor does attend to.

OK, the math, solving problems about exponential growth is appropriate in Alg 1.  But there's something else that's going on.  The function log isn't introduced in MCPS alg 1, instead students learn: 
[quote]There is a Property of [b]Equality for Exponential Equations[/b] that states:
     if a^x = a^y, then x = y.
In other words, if the bases of an exponential equation are the same, then the exponents must be equivalent as well.[/quote]

Which is fine except, what if one bacterial colony doubles in a minute and another triples?  There's no way to solve 2^x = 3^y?  Does this not happen in nature because cell division is binary?  Nope, that's not it, really exponential growth is exponential growth, the base is just a method of getting at the rate.  Lupe's yogurt bacteria grows more quickly because the bacteria split more often, not because they split into more pieces.  So really the question makes just as much sense, it's just not possible to solve without the log function.  And the Property of Equality for Exponential Equations" is an immediate consequence of the definition of log.

Log isn't introduced until two years later in Algebra 2, unit 1 and even then it's mostly looking at the graph of log, the curriculum guide says, [i]Note: Students are not expected to utilize the properties of logarithms to evaluate expressions or solve equations in this course.[/i]  It's not until Pre-calc that students learn to manipulate log symbolically.  But then, not only do they need to learn these properties, they need to use them frequently.  Getting a something out of an exponent needs to be second nature, but they've had no practice.

Does waiting to introduce log promote deeper understanding?  This leaves students with gaps in their understanding and questions they can't answer so I don't see how it could.  They'd be better served practicing with the concept year after year instead of being spoon fed at first.  Log isn't a difficult concept it's as fundamental as exponentiation, it just looks a little different because it's written out instead of having it's own symbol.

And this is the thing I'm seeing over and over in the curriculum, symbolic manipulation is deemphasized even though, by pre-calc it's very clear this is a fundamental and unavoidable skill for solving problems.  There's really no "understanding" that can substitute for experience and it's unfair to throw students into this later class without experience.  But given the design of the classes, the time before pre-calc is largely a waste, so the best approach is rushing through them (which is one of the selling points of the HS magnet, a selling point of the MS magnet is that these early courses are taught in more depth).

And, again sorry to go on, trying to be clear not to beat a dead horse, there are plenty of simple examples of mistakes, too, my own gripe is more the emphasis and organization of the class sequence.[/quote]

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