Creativity vs Standardized Tests

Realizing that this is a little outside of the stuff usually discussed here, I wanted to share somethat that one of my students shared with me.

You might already know of his work (I did not), this is a short presentation by Sir Ken Robinson: http://www.youtube.com/watch?v=zDZFcDGpL4U&feature=youtu.be 

One of the points he makes is that the standardized tests that are so much emphasized these days are part of the industrial model of education, which served students well in the early 20th century and probably well enough in the mid-20th century.  However, in the world of today, the industrial model misses the real needs dealing with creativity and collaboration. 

As usual, there are points where I disagree with what he says; however, it’s an enjoyable 11 minutes … complete with a good sense of humor and some nice ‘animation’ (a little bit like a “Prezi”, if you know what that is).

So … enjoy!
Join Dev Math Revival on Facebook:

Prerequisites

Prerequisites are placed on courses for various reasons, from convenience to supporting student success.  Few prerequisites are placed on courses based on validation studies … some prerequisites are used based on professional validity, while others have even less of a scientific basis.

Let’s say that you are working for the department of education in a state such as mine (Michigan), and you have been getting more concerned about the possibility that academic standards are not consistent across regions or levels of education.  You look at mathematics, and notice that college courses use a different organizing system than high schools … and you do not want students to get credit for a college math course that is really a high-school level course.  An easy, and somewhat logical, approach is to enact a rule (or law) that says any college level math course needs to have at least an intermediate algebra prerequisite.

What’s so bad about that arrangement?  If intermediate algebra is at the level of 1oth to 11th grade high school, this seems like a pretty low standard for ‘college’; when challenged, you might add that the really logical rule would be that a college level math course needs to have at least a college-algebra prerequisite … and you’d like to do this, if WE would just make up our minds about what ‘college algebra’ really is.

However, the problems with this approach are too basic to be resolved by this framework.  First, it assumes that all mathematics builds on the stuff in an intermediate algebra course; several basic areas, most notably statistics, do not have any relationship to the concepts and skills of intermediate algebra.  By requiring intermediate algebra as the minimal prerequisite, we mislead students and cause them to take unneccessary courses … both problems are non-trivial.

Second, this approach assumes that a subtle concept like ‘rigor’ can be measured by the prerequisite.  This is not one of the valid uses of prerequisites; rigor is measured by properties of the course in question (the content, concepts, assessment and practices) … which do not necessarily change just because we list “IA” (intermediate algebra) as a prerequisite.  If we want sufficient rigor in college level mathematics classes (and I hope we do), we need to measure those courses — not a prerequisite to those courses.

Third, prerequisites tend to disproportionately affect underrepresented groups.  At my institution, it is not unusual to have 30% of a pre-algebra class be minority; the courses which immediately follow intermediate algebra are often 90% majority.  Sadly, our curriculum is still not a pump … more filtering happens, so any unvalidated prerequisite can lead to wasteful reductions in minority completion.

I’m pleased that my own institution has 5 college-credit math classes that do not have an intermediate algebra prerequisite.  Two of these 5 courses transfer to several institutions in the state.  However, students are still advised to take intermediate algebra “just in case” … they don’t really need it, but they might change their mind later.

If this topic is of interest to you, you will want to follow a position statement being worked on in the Developmental Mathematics Committee (DMC) of AMATYC.  The DMC web site is http://groups.google.com/group/amatyc-dmc?hl=en   The motivation for this position statement is to help institutions and states use appropriate prerequisites, based on validation — not prerequisites to enforce an abstract policy on ‘rigor’.

In the long term, we will replace “IA” with a more reasonable course like the New Life Transitions course — at least for the majority of students who do not need a pre-calculus type course later.  We will also replace beginning algebra with something like Mathematical Literacy for College Students, and this kind of course could really serve all students.  Until this change happens, we can work on better prerequisites relative to IA … and for all courses (including Transitions).

 
Join Dev Math Revival on Facebook:

Ban Intermediate Algebra!?

Sometimes, there is a fine line between ‘reasonable interpretation of reality’ and ‘bad idea’.  Should we ban intermediate algebra in colleges?  Would it hurt anybody … help anybody … would anybody notice?

My current ‘reality’ includes teaching an intermediate algebra course that is quite traditional, except for us using an ebook (to save students money, and provide equal access).  This course has the usual combination of topics — functions, absolute value statements, polynomials, factoring (lots), rational expressions, rational equations, rational exponents, radicals, radical equations, quadratic equation methods, and quadratic functions (along with a variety of word problems, which are mostly puzzles).

In case you did not know, I have been teaching for quite a while (something like 39 years).  Originally, intermediate algebra was taken primarily by those who needed pre-calculus … and most of them needed calculus.  For a variety of reasons, the vast majority of my current students are not in this category; for them, intermediate algebra is part of their general education process.  [At my college, intermediate algebra is the MOST commonly used course to meet a gen ed requirement.]

Outside of the small minority of my students who actually need calculus (a group which should be larger), most students are not well served by an intermediate algebra course.  The traditional course does little to enhance their mathematical literacy or reasoning, with its focus on symbolic procedures; the traditional course does not contribute to the GENERAL education of students, since it is fairly specialized (polynomial arithmetic and related symbolic procedures).

For many of my students, intermediate algebra is where their dreams and aspirations wither and die under the negative influence of a curriculum which does not serve their needs.  Even for those who need pre-calculus, the traditional intermediate algebra course does not signficantly increase their mathematical proficiencies.  [The procedures learned are soon forgotten, and not much else was learned in the first place.]

Let’s ban intermediate algebra.  In its place, we should offer a version of the New Life “Transitions” course.  The Transitions course learning outcomes focus on providing mathematical preparation as part of a general education, especially if the student will take science courses (biology, chemistry, etc).

If you do not know about the Transitions course, take a look at the learning outcomes listed at https://dm-live.wikispaces.com/TransitionsCourse.   This course focuses on concepts and connections between concepts, so that students gain more than procedures.  The particular outcomes were chosen to be part of the general education of students needing science courses; some ‘STEM enabling’ outcomes are listed as an option for a course preparing them for pre-calculus.  The “Instant Presentations” page here has a presentation on the Transitions course; see https://www.devmathrevival.net/?page_id=116 

Of the two New Life courses, the first course “Mathematical Literacy for College Students” (MLCS) has generated more interest as an alternative to a traditional beginning algebra course.  I find this interesting, since we could argue that intermediate algebra is a worse match to student needs.  Curiously, the Transitions course is somewhat similar to some materials that are already on the market … which means that implementing Transitions avoids some of the challenges faced by those working on MLCS.

Some of you have been thinking “hey, we are required to use intermediate algebra as the prerequisite for all college-credit math courses”.  Well, I know … our profession needs to work on that problem.  Presently, the Developmental Mathematics Committee (DMC) in AMATYC is working on a position statement related to this problem; see http://groups.google.com/group/amatyc-dmc 

Obviously, I do not really expect us to ‘ban intermediate algebra’ (though I can dream!!).  Perhaps some of us can help our students by using the Transitions course as an alternative for those students who to not need pre-calculus.

Join Dev Math Revival on Facebook:

The Calculator Issue

I was talking with an editor from one of the larger publishing companies earlier this week, and one of the issues the editor saw as critical was “the calculator issue”, which this editor saw as both basic and divisive in the profession.

I would like to start by asking “When do YOU reach for your calculator?”

For my own work, and perhaps yours, I use calculating devices for several categories of work.  First, if the quantities are ‘complex’ and a precise answer is needed.  Second, if a procedure will need to be repeated more than twice (like finding a table of values for a function).  Third, if the calculation deals with a high-stakes question (like grades for my students).  Fourth, if the situation involves the exploration of ideas which are still in the ‘learning process’ (like a new mathematical concept or a review of long-lost treasures).  There might be a few other situations.

You might wonder why I start with our use of calculators.  So often, the comments we make are about our students’ “over-use” or “dependency” on calculators; we see calculator use as creating a risk for learning mathematics.  Many of us do allow calculators, and even embed their use in the learning process.  Some of us forbid their use, and some of us have a blended approach.  Most of us, however, believe that there is an issue with calculator dependency.

My conclusion is that the problem is not with calculators being used.  The problems occur when student attitudes about mathematics and their own efficacy create motivation to use calculators when the human brain is a better device.  If a student is doing a problem in the homework, or an example in class, and reaches for their calculator to add two one-digit numbers, this is part of the problem — the human brain is a better device, and the use of the calculator in this situation provides a clear ‘bad at math’ message about the student (and the student is the one sending the message).

Of course, we can not ignore the impact of culture on the use of tools, even calculators.  Some students temporarily feel ‘smarter’ when they use a technological tool frequently; I suppose this is not so much ‘smarter’ as ‘good’.  In the case of mathematics, the cultural bias towards technology combines with a norm that “it is okay to be bad at math” to encourage over-use of calculators.  These comments about culture are not universal for our students, some of whom come from cultures where very little technology is available … some even come from cultures with a positive attitude towards mathematics.

If this analysis is correct, then the issue is not whether we allow calculators or not.  The first basic issue is our outlook on learning — maximizing understanding, connections, and abilities to work with quantities … these are traits of the emerging models of developmental mathematics.  Students should develop their strategies for good uses of calculating technology, and they can do this as long as we do not focus so obsessively on ‘correct answers’; if we assess representation and communication, the use of calculators will not create problems.  As you probably know, technology does not really “solve” problems either; problems are solved by people and what we do.

I encourage you to ponder your approach to calculators in your classes.  Are you creating a calculator policy to make a personal statement, or are you creating a calculator policy which reflects your understanding of how technology affects learning of important mathematics?  Does your calculator policy encourage, or does it discourage, the development of mathematical proficiency in your students?

These issues are complex, and will not be solved by a simple yes/no policy on calculators.

 
Join Dev Math Revival on Facebook: