Finding Statway(tm) materials (and Quantway!)

Are you looking for the Statway™ materials … so you can become familiar with them, and consider that kind of approach?  Are you intrigued by the Quantway™ ideas, and want to check it out?

You can email pathways@carnegiefoundation.org with the request; the Carnegie Foundation will send you a collection of lessons.  [Currently, a sample of Statway™ lessons are available; a set of Quantway™ lessons will be available soon.] 

Next year, all lessons will be available under a Creative Commons license.

Radicals, the Game

We have been working on simplifying radicals this week in my intermediate algebra classes, and I can’t help but think … why?  Where does anybody need to (1) simplify square roots and cube roots that represent irrational numbers  and (2) perform operations on radical expressions?

I am not opposed to including some radical concepts, especially connecting them to fractional exponents (so we can include exponential functions), and knowing that some radicals represent irrational or imaginary numbers is basic enough — and useful enough — to include.  The issue is WHO CARES if we can write √(80)  as 4√5  ?

Within our algebra courses, we use ‘simplifying’ radicals when we solve quadratic equations with irrational solutions (or complex).  That is not what I am talking about; rather, in situations leading to a quadratic equation with either irrational or complex solutions, do we really need to express those solutions in ‘simplified radical form’?

Factoring is often listed as a, well, useless topic in algebra.  Until we replace our algebra courses with something better, I actually do not mind covering some factoring — even trinomial factoring.  The value with factoring is that it really deals with basic concepts of terms and equivalent expressions; much of our mathematical capacity is based on our being able to envision alternative but equivalent expressions. 

I do not see the same payoff with radicals, either simplifying or operations (which also involves simplifying).  When I explore with my students why they make mistakes with radicals, it’s not normally some basic issue that is a barrier; more often than not, it is the strange radical notation involved in the work.  It’s not that they do not understand that an index of 3 means ‘cube root’, it’s the standard moves of the radical game that cause the problems.

Sure, a radical is equivalent to a product of radicals involving factors of the original.  Students get that a number can be factored in different ways.  It’s the particular partitioning of factors.  Sure, we can teach prime factoring and circling groups according to the index, or we can teach a calculator trick to find the magic partitioning of factors.  This partitioning is conceptually similar to partial fractions; with radicals, I think students feel like we are working backwards, as we are with partial fractions.

Seems to me that we’ve made ‘radicals, the game’.  The focus ends up on the legal moves allowed, and the format the answers can be.  The basics (meaning of radicals or fractional exponents, domain) tend to be de-emphasized.  And, honestly, I could use these two weeks for other topics that would help my students more than this game.

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Dear Aunt Sally … Please be Excused (from math)

In developmental mathematics classes, as in school mathematics, Aunt Sally seems to be everybody’s friend.  As in “Please Excuse My Dear Aunt Sally” as a memory aid for order of operations (aka “PEMDAS”).  I would, indeed, like to excuse Aunt Sally from ever being in my math class. 

In another post, I talked about the “Sum of all Shortcuts”; in this post, the issue is mnemonic aids.  You can improve your students’ “learning” if you minimize the use of these ‘easy to remember’ tricks. 

This may sound counter-intuitive … isn’t it a good thing if students can remember something?  Well, it CAN be a good thing; the issue is what exactly do they remember?  In the case of PEMDAS, they remember ‘do inside parentheses first’ or ‘do parentheses first’.  Fine, to a point — students can evaluate  “8 + 10 ÷ 2” and “(8 + 10) ÷ 2”.

The student then sees these ideas, which do not follow PEMDAS:

• 8x + 2x   (can add before multiply)
• 8x + 2y    (can not add)
• 8(x + 2y)   (can multiply ‘first’)
• (2x²y)³     (can ‘power’ first)
• f(x)=8x + 2    (what does that x mean?)
• f(-3) for that function    (what do we do with the -3 on the left side)

The “P” in PEMDAS is especially worrisome.  Parentheses have multiple purposes in mathematics, and only some of them relate to the order of operations.  We also use other symbols of grouping, some of which are another operation (radicals, fractions, absolute value, etc).

Now, we actually make students do too much with expressions of extra complexity just to see if they can follow the order of operations.  We create our own need for an easy-to-remember tool (PEMDAS) which then results in students having to unlearn later when we do other work in ‘simplifying’.  This is a bit like designing a tool to require disposable parts, in order to keep a business active; I would suggest that our artificial level of difficulty with numeric expressions serves no purpose, not even our own.

It’s important, however, for our students to be literate and comfortable with the basic meanings of expressions and forms.  As I talk with my students, I am impressed by how many of them remember ‘PEMDAS’ years later and by continuing difficulty in doing work that does not involve applying ‘PEMDAS’.  We are not doing our students any favor by giving them an easy thing to remember which does not transfer to future work.

Some readers are likely upset by my suggestion; yes, I know PEMDAS has helped millions of students in their math classes; yes, I am aware of research showing mnemonics help learning disabled students in particular.  However, the benefits for most students do not seem that great to me; the long-term result may be more negative than positive.  [Many of my most struggling students have learning problems, and survived by using tools like PEMDAS; they have difficulty in the situations listed above that do not follow the PEMDAS priorities.]

We know that PEMDAS does not cover most expressions involving variables.  I am suggesting that PEMDAS directly interferes with the algebraic literacy of our students; quite a few students suffer needless discouragement when their algebraic difficulties increase as they painfully discover the real limits of PEMDAS.

Let’s send Dear Aunt Sally on a much needed vacation; she has been used for many years, and perhaps is ready to retire.  Instead, let us focus on basic literacy dealing with reasonable objects of valuable mathematics.

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When Are we Going to USE this?

I think all students receive the same email when they register for a math class; this email says “The BEST way to irritate your instructor is to ask ‘When are WE going to USE this?’ at the first sign of confusion.  Not only does this irritate your instructor, but this will distract the instructor from your lack of understanding.”

Of course, my introduction above is a weak attempt at humor (a dangerous choice on my part!!).

Seriously, the issue is this:  Do students NEED to see the application of everything that we tell them to learn?  Are some concepts okay to learn, even if there are no applications that the student values?

Underneath this issue is the curricular issue:  What is the purpose of this course?  For developmental mathematics, we are in a “preparing for” business.  Our students need to take further mathematics (of various kinds), science classes, and technology classes; we also prepare students for college in general.  These future situations are a justification for the mathematics they learn … is it reasonable for us to expect students to appreciate the applications while they are learning the mathematics?

 In a recent post (https://www.devmathrevival.net/?p=282 ) I emphasized mathematics as a practical science.  For a recent conference in Michigan, I gave a talk on general education (http://jackrotman.devmathrevival.net/General%20Education%20Mathematics%20in%20Michigan%20May%202011.pdf) in which I highlighted the ‘usefulness’ of mathematics we require students to take.  It’s pretty common for people to conclude that I think developmental mathematics should be applied and in-context, in almost all topics we teach.

However, that is not my judgment about what is appropriate.  The traditional developmental mathematics courses are predominantly procedures; we do ‘applications’, but 90% of these applications are puzzles (you have to know the answer in order to write the problem).  This is not enough practicality to show students that mathematics is powerful and practical.  We need more applications meaningful to students, and we need content which will benefit our students.

At the same time, it is obvious to me that we would make a mistake to limit the mathematics we teach to those ideas for which we have applications that students would understand at that time.  A math course is not employment preparation (only), nor is it solving problems (only) … just like it is not beautiful mathematics (only).  We need a balance.  Students will need to learn material with no clear applicability to them in other classes, and learning in the face of ‘not useful right NOW’ is a critical survival skill.

We can also use learning research to provide guidance.  My own reading of theory and research indicates that context, and applications that students value, plays a positive role for motivation when done in moderation; an emphasis on context complicates the learning process, and may make the important seem invisible.  [For some references, see http://jackrotman.devmathrevival.net/sabbatical2006/9%20Situated%20Learning.pdf.]

I hope you will think about the purposes of our math courses, and reach your own conclusion about the appropriate role of applications that students will understand to support the learning of mathematics.  I am sure that we can design our courses so that students can learn powerful mathematics that WILL be useful to them, and that we can incorporate applications along the way.

 
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