New Life vs Emporium Models

I am currently at the AMATYC conference in Austin — very good conference.

Earlier today, I had a session entitled “New Life Takes on the Emporium Model for Redesign”.  My intention was to provide a viewpoint on these alternatives, both of which are currently popular. 

Here is the file

Here is the handout from the session (1 page summary): https://www.devmathrevival.net/wp-content/uploads/New-Life-takes-on-the-Emporium-Model-for-Redesign-HANDOUT-final.pdf

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Evidence Based Decisions

How do we prevent ‘evidence based decisions’ from becoming ‘evidence constrained decisions’?

First, let’s get clear on what ‘evidence based decision making’ is about.  Primarily, the idea is to apply evidence from the scientific method to decision making.  This is the definition given at http://en.wikipedia.org/wiki/Evidence-based_medicine ; much of the current push to use this method in education comes from improvements in medicine due to using scientific evidence as a basic methodology.

The idea is to base decisions on the evidence, when appropriate evidence is available.  Remember that we are talking about scientific evidence — which is a stronger standard than ‘data’.  The scientific evidence provides a connection between a practice or treatment with the outcomes (usually stated as a probability or odds).  Sounds good, doesn’t it?

Well, in education, there are difficulties in getting scientific evidence.  We have tons of data, which are raw measurements organized in some manner; however, this has little to do with scientific evidence.  Most commonly, we have either before and after data relative to some change; sometimes, we have data from two groups under different treatments … data on the outcomes, without data on other variables that we suspect have an impact on the outcomes. 

Scientific evidence does not come from one set of data.  After one set of data suggests, scientifically, that we have reason to believe that this treatment results in a change in the outcomes, this hypotheses gets tested by replication — done by different practitioners.  The idea of scientific evidence is that we achieve something close to an empirical proof that we have a cause and effect relationship — not just a one-time correlation.

I can not resist bringing up one of my favorite oxymorons — “data based decision making”.  Data is simply organized measurements; no decisions can be made based on data, because data is not evidence of anything.  I use brand X gasoline one week, and the next week I use brand Y — and get 10% better mileage … which means nothing; this data just means that I get slightly different outcomes, nothing else.  I normally find the phrase ‘data based decisions’ to be used as a cover for a hidden agenda.

Back to evidence based decisions … as mathematicians, we are all scientists; we understand the power of research — and it’s limitations.  The presence of evidence (in the scientific sense) suggests better courses of action (decisions) to the extent that the probable outcomes are ‘likely’.  The presence of evidence does not determine the best decision … wise people still need to evaluate the current situation and apply their understanding of the evidence.

What do we do when there is no scientific evidence relevent to our decision?  Are we constrained by the evidence available?  Even in medicine, with its superior collection of evidence, decisions are not constrained by evidence.  We should be guided by the evidence we have, and use our wisdom combined with our understanding of the outcomes desired to determine the best available decision.

Relative to mathematics education in colleges, I would present these observations:  We have large bodies of evidence about learning which can (and are) being applied to our courses.  We often mistake data for being evidence, and mistake reporting data for research, and this has led to some dramatic failures (and some less dramatic).   When we do remember the distinction between data and research, we tend to skip the step of ‘replication’ before announcing a conclusion; this has led to cynical colleagues and a skeptical public.

If we do not understand what the word ‘evidence’ means, who will?  Certainly not external forces such as politicians.  We need to be much better at articulating what we are basing a decision on, and clearer at describing results.  We need to focus on our shared values, and use them to describe the desired outcomes.  We need to focus on our wisdom, to provide guidance in the absence of evidence.

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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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