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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What good is algebra?

Developmental algebra is the most studied course in American colleges … well, at least the most enrolled!  Studying is another thing. 🙂

Why?  What value does this activity add?

I’ve noticed something about students who have passed our beginning algebra course, and I am not happy about this.  We have several math courses that can be used to meet the requirement for an associates degree, and one of these math courses is a mathematical literacy course.  This course involves a lot of problem solving, based on understanding relatively few concepts.

Consider this sequence of problems and typical student responses:
Item: A company has $38 million in sales this year, and expects it to rise by 10% for next year.  What will the sales be next year?
Student: Okay, 10% is 0.10 … we better multiply … 0.10 times 38 is 3.8.  That’s too small for the sales, so we add 38 + 3.8.  The sales next year are $41.8 million.
Item: A company has $38 million in sales this year, and expects it to rise by 10% per year for the next several years.  Write an expression for the sales based on the year n.
Student: What?  38 times 0.10.  Where does n go?  Is it 0.10n + 38?
Me: Okay, let’s look at a simpler problem.
Item: A company has $38 million in sales  this year, and expects it to rise by 10% per year for the next several years.  Estimate the sales for the next 4 years.
Student: Okay, the first year is like the one we did earlier … $41.8 million.  Do we do the same thing again?  [me: might be — would that make sense?]  Yes, I think so  … {calculates}. 
Me: That is looking good.  How about the expression … does your work here have anything to do with the expression we need?
Student: You got me!

Of course, our beginning algebra course has a lot of applications, and students see like terms and a lot of exponents.  We cover percent applications, including some where we know the value after the 10% increase and need to find the original.  In spite of the appearance of ‘mastery’, most students do not connect their knowledge with the concepts in a novel situation.  Quite a few students will actually deny the connection between the algebraic expression and the computations they do.

We often ‘sell’ our courses because of a belief that passing a math course indicates a better capacity to reason and to think logically. 

However, the traditional courses do not deliver on this promise (in my opinion).  Almost all textbooks have repetition of skills, and we cover too much material to work on applying anything to novel situations.  Sadly, almost all useful applications of mathematics (in life and in occupations) begin as novel situations.

I personally dislike (strongly!) the phrase “a mile wide and an inch deep” (for one thing, we are all adept at 90 degree rotations to get “an inch wide and a mile deep”).  Slogans like that do not help us.  What might help us is thinking about what we believe is valuable in mathematics … and delivering courses that build this value for our students. 

As long as we attempt to ‘remedy the deficiencies’ of our students, we will miss the benefits.  Their deficiencies are many; most adults have similar deficiencies (even those employed in occupations that our students are preparing for).  Our attention should be on “what mathematics is needed for community college students” or “what mathematics is needed for university students”.

I really believe that we can provide courses that students will see the value of, and that we can be proud of as mathematicians.  I think that the New Life model is a good starting point, and I hope you will consider becoming a supporter of this work … and consider offering these types of courses at your college!

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Mathematical Reasoning?

We, as mathematicians, really appreciate definitions — concise and consistent definitions.

What is ‘mathematical reasoning’?  How does it differ (if it does) from ‘quantitative literacy’?

This post focuses on ‘mathematical reasoning’ to clarify my own thinking.  Mathematics is the science of quantities, perhaps better stated in the plural — the sciences of quantities.  A science (singular) refers to a field in which there are shared concepts and theories.  In mathematics, we have several basic domains which have their own concepts and theories — geometry, statistics, arithmetic, calculus, algebra (a vague term), and more.  Within the context of general college mathematics, the first four listed are the most likely sciences involved.

If ‘mathematics’ is plural, what meaning does ‘mathematical’ have?  It might simply mean ‘related to one or more of the mathematics’.  Should ONE of them be sufficient?  What does ‘reasoning’ mean if there is more than one mathematics involved?

The more I ponder this problem, the more I am drawn to ‘literacy’ instead of ‘reasoning’.  My expertise is not that deep in all of the mathematics; however, it seems to me that the ‘reasoning’ involved is unique to each mathematics.  I can hear some of the readers saying “but, they are all LOGICAL!”, and that is true … but not sufficient.  Labeling something as ‘logical’ simply means that there is some systematic process involved in the reasoning, and I again suggest that there are many substantive differences in this reasoning between the mathematics involved.

For example, geometry involves both formal and informal logic; the reasoning often is based on identifying basic shapes and objects within different configurations and after different transformations.  We use phrases like “spatial sense” and “part-whole”, which also come up in calculus.  On the other hand, statistics involves descriptive work and inferential work; ‘hypothesis’ is used differently than we do in geometry, and nothing is ever proven … it’s all a matter of probability.

Could ONE mathematics be sufficient for ‘mathematical reasoning’, in the context of general mathematics at college?  I hope not.  There is little value in providing one science only in mathematics, just as there is little value in providing one science only in the ‘hard sciences’, for general education.  Specializing has value for advanced work.  General education needs to focus on a broader view, both to show the nature of the field of mathematics and to provide a set of ideas that students are likely to find useful.

I think I would rather use the name ‘mathematical reasonings’ (plural).  A course in ‘mathematical reasonings’ would likely be a more advanced general education course than we normally offer.  When I look at courses labeled ‘reasoning’, what they really focus on is ‘problem solving’; this is laudable, and I have such a course that I love to teach. 

My conclusion is that we should not use the label ‘mathematical reasoning’, both because the mathematics involved being plural and because we do not really focus on the reasoning.

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Quantitative literacy?

We, as mathematicians, really appreciate definitions — concise and consistent definitions.

What is ‘quantitative literacy’?  How does it differ (if it does) from ‘mathematical reasoning’?

This post focuses on ‘quantitative literacy’ to clarify my own thinking.  Since mathematics is the set of sciences of quantities, using ‘quantitative’ instead of ‘mathematical’ does not necessarily change the meaning.  However, the use of the word ‘quantitative’ implies that we might emphasize more the application of mathematics, rather than the structure of the sciences of mathematics.

To many people, ‘quantitative’ will tend to suggest the science of arithmetic (known quantities) rather than other mathematics.   When I look at courses that include quantitative in the title, I generally see applications of arithmetic … with perhaps a little basic geometry.  Only occasionally do I see statistics in such a course, and I have yet to see calculus included.  Since the science of calculus involves quantities under change, this seems ironic.  Are the concepts of calculus so advanced or obscure that students in a general education math class can not understand them?

I am concluding that I would prefer ‘mathematical’ to the ‘quantitative’ — not that I want to have the theory of mathematics exclude the application of the mathematics.  Rather, I want us to focus on multiple mathematics, not just arithmetic and some geometry.

How about the word ‘literacy’?  This word is problematic, since the synonyms include ‘knowledge’, ‘learning’ and ‘education’.   However, we can overcome this problem by being precise and consistent in our definition.  Perhaps we can define ‘literacy’ to mean ‘understands and can apply basic concepts’, as a parallel to the language literacy definition (‘can read and write’).  With that definition, I rather like the word ‘literacy’ appended to mathematical.

Of course, we have much work to do before we KNOW what ‘mathematical literacy’ means.  Which mathematics? What are the basic concepts of the ones we include?  Our professional community needs to deal with these questions, as many of our colleges have shifted away from a pre-calculus/calculus type of general education course … and towards a reasoning/literacy type of course.  Much valuable and creative work is being done; however, we need to develop some shared conceptions of this type of curriculum.  A lack of shared curricular concepts creates problems for articulation and transfer, and causes us to develop this part of the profession in more isolation than would be ideal.

 
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