Arithmetic before algebra?

In our college ‘developmental’ mathematics curriculum, we place arithmetic before algebra … and tell students that they need to know their arithmetic before they have a reasonable chance of passing an algebra course. In many ways, we have this backwards!

Think about this: Arithmetic, as normally taught, involves ideas relating to multiple sets of numbers (whole, fraction, decimal, integers) and procedures for multiple operations (binary operations in particular, and some unary operations), with concepts from geometry and dozens of cultural contexts for ‘word problems’.  Within these topics, few direct connections exist; students are faced with a problem learning (remembering, as they would call it) the diverse material.

Algebra brings a structure and connections between topics that makes the learning easier.  Of the arithmetic topics, only a limited number are directly prerequisites to some algebraic learning.  Fractions have little to do with the concepts of solving linear equations or combining like terms; we might force those issues to come up with some contrived problems, but the algebra itself is quite basic. 

Another point of view:  A 8 hour sequence of class time in an arithmetic course is likely to involve a wider variety of problems than an 8 hour sequence in a beginning algebra course.  Furthermore, the algebra course will provide a clearer logic for the work as well as connecting material to prior learning compared to the arithmetic course.

I’d also point out that arithmetic is not nearly as practical as it once was.  Current occupations have a greater need for quantitative sense and reasoning, and we could dump much of an arithmetic course to make room for these topics … and help students in the process (with no harm done to any student).

Take a step back, and really think about your developmental math curriculum.  Do you have the important stuff in the right order to help your students?

 
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Mile Wide … Mile Deep!

Actually, I wanted to say “kilometer wide … kilometer deep!” — but then some people would not get the reference. 

At the recent AMATYC conference, I attended a session by Xiaoyi Ji titled Investigation of Math Teaching in the U.S. and China which I found inspiring.  One of her main points to explain the large gap in ‘performance’ between Chinese and US students is the Chinese committment to depth AND breadth.   You can see her presentation at http://www.amatyc.org/Events/conferences/2011Austin/proceedings/xiaoyiS75.pdf , and you can see the entire list of proceedings at http://www.amatyc.org/Events/conferences/2011Austin/proceedings.html.

Our recent drive to avoid a ‘mile wide & inch deep’ is a false dichotomy.  The implication is that we can not have both depth and breadth.  This is one that I think the Chinese system has right — we truly need a kilometer wide and a kilometer deep; depth without breadth results in students who know a fair amount about isolated pockets of mathematics … and I suggest that this is a self-defeating goal.  We create more problems than we solve.

Breadth refers to two dimensions — one is the domains or categories of mathematics, the other the major areas in each domain.  Within polynomial algebra, for example, we have some areas which receive most of our attention (simplifying, solving) while other areas are neglected (conic sections come to mind).  We often see ‘functions’ and ‘modeling’ as alternatives, when both have a purpose.  We often omit other basic forms (exponential, trigonometric).  As a result, we create pockets of knowledge and chasms of ignorance … and wonder why our students have such fragile knowledge.

Depth refers to levels of knowledge, and we actually do not share a good understanding of what this means.  Too often, we look at surface features of the questions we ask (skill, application) rather than a more sophisticated analysis.  When better work is done, it is sometimes framed within Bloom’s Taxonomy which is not particularly well suited.  A better framework for the depth of knowledge is the ‘five strands of mathematical proficiency’; you can see an excellent presentation (in fact, the original) in an online book at http://www.nap.edu/openbook.php?isbn=0309069955.  This material was originally written for a school mathematics audience; however, I think you will find the concepts transfer to our level quite nicely.

Of course, we can not achieve ‘depth and breadth’ in one or two college mathematics classes.  On the other hand, we can ensure either an inch deep or an inch wide in one course  by the choices we make.  Let us all contribute to both depth and breadth at every opportunity.

 
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Math is Non-Linear

Within our current curricular structure, there is premise of linearity — that the topics need to be studied in a particular order (in general), AND that students will not be able to understand a later topic that depends on some earlier topic.

This is not true.

At the global level, we have “pre-algebra” → “beginning algebra” → “intermediate algebra” → “college algebra” → “calculus”.  We have an existence proof that this is not true … students (in larger numbers than we’d like to admit) can perform as well in the college math class without taking the prior course ‘required’ according to placement tests. 

At the micro level, each course is constructed (normally) by a list of topics and then outcomes within each topic.  If the topics were basically linear, however, we would always teach these topics as connected to other topics (prior and future); we do not do so, making the non-linear nature evident. 

You might consider both of these points ‘logic chopping’, and classify this as a worthless post.  So … let’s move on to a different analysis.

One of the imbedded linear conditions is ‘fractions’.  Let us assume, for the moment, that we have sufficient rationale to justify the inclusion of rational expressions in the algebra curriculum.  Back in pre-algebra, we cover various operations; we suggest to our students that they need to master simplifying arithmetic fractions prior to simplifying rational expressions (‘algebraic’ fractions).  In between these two topics, we covering factoring … and feel good about the parallels between prime factoring in arithmetic and polynomial factoring in algebra.

I see two reasons why the sequence of these topics is not linear.  First, arithmetic fractions deal with place-value numbers; students need to transform these additive forms into multiplicative forms to simplify by factoring … and this is a more advanced topic than algebraic fractions (which are often multiplicative in the first place).  [Just show a group of students the fraction “54/24” and see what fun they have with these hidden binomials… compared to ‘8x²/4x’ (obvious monomials) or ‘(6x+24)/(8x+32)’ (obvious binomials).]

Second, the fraction topics are not linear because of the extra rules often imposed for arithmetic — improper fractions and mixed numbers.  These rules do not exist for algebra.  Because arithmetic is a relatively advanced topic, we often cover these topics with a series of guidelines or procedures for each type of problem; few of these items transfer to algebraic fractions.

Personally, I would rather help my students understand a mathematical topic like ‘algebraic fractions’ without having to cope with layers of bad learning relative to arithmetic fractions.  In this area, I can not expect this to occur.

I suspect that you have noticed some of what I am talking about.  We cover linear forms before quadratic before exponential — and yet some students ‘get’ the more advanced topic while they still struggle with the earlier one.  You might have noticed, in particular, that some students just don’t understand these properties of real numbers (associative, commutative, etc) … and then they start simplifying expressions with terms and parentheses, and they get an insight into what the properties were saying.

Math is not linear.  We are not building machines that have a clear dependency in design; we are dealing with human beings working with ‘mathematics’ — a collection of scientific domains dealing with different types of objects.  Our job is to identify the most important mathematical concepts appropriate for each course, and allow the course to be non-linear; we can revisit concepts, bring in a new perspective, look at a different context.

Math is not linear.  Our curriculum tends to consist of a series of two to five courses, presumed to be linear … and we create many opportunities for students to leave math.  We tend to put a lot of “preparing for the future” procedures in each course, which tells students that they will not study the good stuff of mathematics until much later (if at all).  Our job is to show good mathematics to students in all courses; rather than seeing mathematics as a difficult set of hundreds of procedures, they might just see mathematics as interesting.

We can not expect to inspire students by using a linear curricular model which ensures that the early courses cover very little of interest … and even less of intellectual value.
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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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