FOIL in a Box (algebra!)

Some of us have a ‘thing’ about FOIL as a topic in an algebra class; there are concerns about emphasizing the FOIL process as it can submerge the real algebra going on.  Some (perhaps the majority) are not significantly handicapped by being “FOILed”.  This post is not about FOIL itself … it’s “FOIL in a Box”.

Okay, so this is what I am talking about.  The problem given to the student is to multiply two binomials, such as (2x – 3) and (3x +4).  Here is the “FOIL in a Box”:

Some students like this approach, and I think this is because the box lets them focus on one small part of the problem.  The overall process is submerged, and the format does all of the work.  Of course, this is exactly what many procedures in arithmetic do.  The FOIL in a Box method is much like column multiplication, where partial products are arranged in a mechanical way to produce the correct place value.  If correct answers to multiplying were the primary goal, there would be nothing wrong with either FOIL in a Box or partial products in arithmetic.

My observation has been that almost all students who use FOIL in a Box are handicapped in working with polynomials.  Students have trouble integrating the Box into longer problems.  And, though they may have some ‘right answers’ for factoring trinomials, the transition to other types is more difficult. 

What should we do instead?  My own conclusion is that we need to keep emphasizing the entire idea involved.  FOIL is used for “distributing when both factors have two terms”, and “distributing is used to multiply when one factor has two or more terms”.  We too often assume that students will keep information connected to the correct context … they don’t automatically know that distributing does not apply to 3 monomial factors [3(2y)4z ≠72yz], nor to a power of a binomial  [(x + 3)² ≠x² + 9]. 

The achievement of correct answers in the short-term should not come at the price of handicapping the student’s future learning.  All learning should be connected to good prior learning, and imbedded within the basic ideas of the discipline.  We need to be comfortable articulating the full name of what we are doing (multiplying two factors each with two terms), and not use a mnemonic such as FOIL as a container for knowledge of mathematics.

 
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Remediation as a Remedy

Over the past 40 years (roughly the period of my work in developmental mathematics), our approaches have shared one focus — we tend to ‘fix’ (provide what is needed) in our students by focusing on small (or even smallest) parts of the problem.  Algebra, in this approach, means that we study some properties of numbers … simplify expressions … solve equations … translate word problems … graph equations … and so on.  Does this approach result in any significant improvements for our students?

Earlier today, I was searching for some information and ideas on retention — especially of minority students.  In the process, I encountered a blog by Mike Rose at  http://mikerosebooks.blogspot.com/   … in a  post, Mike says:

Such problems of status and institutional structure interact with flawed beliefs about cognition and motivation that run throughout basic-skills instruction. One of those flawed beliefs is that the way to remedy a problem is to focus on the smallest units of the problem—in the case of writing, it would be rules of grammar, often treated out of context in a workbook or in an entire course focused only on the sentence. In such settings, students don’t get to work with language in a way consonant with the intellectual and rhetorical demands of the writing they will have to do in college. Another false belief is that underprepared students’ motivation and self-esteem will be hurt by a more-challenging curriculum. That is a one-dimensional, not to mention patronizing, understanding of motivation. There’s no scientific basis for such beliefs, but they persist.  (see http://mikerosebooks.blogspot.com/2010/08/smithsonian-of-basic-skills.html)

 As you might guess by the content, Mike Rose is talking specifically about developmental writing classes.  However, I am intrigued by this analysis — he is suggesting that this ‘small unit’ approach to remediation actually conflicts with meeting the goals of remediation (as in being ready).  In mathematics, we tend to focus even more on small units of learning (iin recent years); when done to the extreme, we see modular learning.  However, I encourage you to not make the mistake of thinking that non-modular courses avoid this ‘small unit’ problem — the small unit problem is dealing with fairly trivial details to the extent of missing the basic value; we can solve equations … but we are unable to communicate and reason algebraically.

This ‘small unit’ problem is part of what has given “algebra” a bad reputation.  If we fail to correct our approach, we will find other domains taking over the field — like statistis.  Statistics is fine, and has a purpose.  However, I believe that algebra has much to offer as a domain to help students become ‘ready’ for success in college and in life.  The concepts of algebra, and our modes of expression, would serve students well — if we focus on those fundamental goals, and not the smallest units.

I hope that you will share my journey of making remediation into a remedy for our students, so our work provides what students need to be ready.
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Getting Into Statway or Quantway

In case you missed the webinar on January 24 (by the Carnegie Foundation for the Advancement of Teaching) … institutions can begin a process that may lead to being included in either Statway or Quantway.

Here is the slide from the webinar dealing with ‘getting into Statway or Quantway’:

Just to review … Statway is a two-semester sequence designed for students who place into beginning algebra, which takes them ‘to and thru’ a college-credit statistics course; statistical topics are the focus of the course, with developmental math topics integrated in both semesters.  
Quantway is a one-semester course designed for students who would take a ‘non-STEM’ course the next semester; it is also designed for students who place into beginning algebra, with a focus on numerical reasoning, proportional reasoning, algebraic reasoning, and functions & models.  After Quantway, students would take a college-credit math course such as quantitative reasoning, statistics, or math for liberal arts.

“Getting into Statway or Quantway” means more than just offering the course; joining a Pathway means being part of the Networked Improvement Community coordinated by the Carnegie Foundation … that is why institutions need to meet the criteria listed above.

If you are ready for this type of change, send your letter of interest to the Carnegie Foundation (email address on the image above).

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Free Online Materials from NROC?

The National Repository of Online Courses (NROC) has developed extensive materials for developmental mathematics, which are available to preview and pilot test and will be free for individuals to use this fall at HippoCampus.org. Currently, a good beta version of the material is being piloted around the country – with the full release of the material due later this year that will include assessment to individualize a learners’ path through the lessons. Right now, the beta materials for arithmetic and basic algebra are available for previewing – I spent some time looking, and am impressed. Even though the topics look old fashioned, the presentation is very different; the material is engaging and worth learning.

If you would like to get a preview, go NROCmath.org and look under the Higher Ed banner (right side); the materials are called NROC Developmental Math—An Open Program. On that NROCmath.org page, there are links to learn more and to view course now ‘learn more’ provides an overview of the course, and ‘view course now’ provides you access to preview (one unit) the materials. You can register for a password to preview all beta units. (Warning: This will also put you on the mail list to get news of the final release.)

To fully integrate the material into an institution’s curriculum, in-class, blended or online, as a supplement or full course, institutions, systems or agencies may become NROC Network Members and receive technical support, professional development resources and the rights to adapt and import all resources into their own learning management system to adapt and make available to all teachers and students.

Isn’t it great when people share good materials like this?
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