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.

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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