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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Does Math Exist?

Okay, you have to pick ONE answer for this question, and the only available answers are ‘yes’ and ‘no’.  What say you?

Some of us will say math exists because it provides practical answers to problems and describes the world around us.  Some of us will say that math does not exist because math itself is an abstraction about quantities — not the quantities themselves.

I got thinking about this question as I ponder again our basic label ‘math’ or ‘mathematics’ (though I like the UK version ‘maths’ better than ‘math’).  It’s ironic that there is so little difference between the english word ‘math’ and the english word ‘myth’, because very few people have an accurate picture about math.

How about we start a list of “FAQ” (frequently asked questions) about the nature of mathematics?  We would state the answers in non-technical terms to help the public, especially policy-makers, understand what maths are.  Too often, we have well-intentioned authorities or agencies decree that a specific math course (or set of outcomes) is good for all students … or they decree that all students must pass a certain math course — and place math in the position of gate-keeper.

Here is a start of the FAQ (feel free to add your own):

Question 1: Is, or should, mathematics be practical?
Answer 1: We noticed that the question was not qualified … if this was ‘always’ practical, or ‘all mathematics’, the answer would most definitely be NO.  Mathematics is a collection of sciences dealing with ideas about quantities, which means that most mathematics is practical to somebody.  However, a basic observation about the history of mathematics is that most of the ideas started out being considered impractical but most of them turn out to be practical.  People learning mathematics have to accept that some ideas may not seem practical at the time; a good mathematics course will combine practical uses with ideas that are not always applied.

What questions (and answers) would you add to this FAQ list?

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Good Teachers … Bad Math??

I was in a store today buying cat food and litter (really!!), and the person in front of me at the check-out did a double take … and he asked if I still taught math.  When I said yes he said “I pass math because of you!!”  (Warm fuzzies?)  At that point, the cashier said “Where was he when I was taking math?”  (Cold pricklies?)

Part of today’s educational climate is the push to evaluate teachers, especially in K-12 settings, partially based on student academic performance.  Those who produce higher levels of improvement, or absolute performance, are rewarded with good evaluations; those who do not produce run the risk of being dismissed.  This obsession with evaluation has not reached colleges (yet), though I am really looking forward to the evaluation system like this for politicians.  Somehow politicians can say “it’s the other guys fault” and get re-elected, while teachers saying “other factors negatively impact learning” gets ignored and then dismissed (if their evaluations are not good enough).

It is far too easy to feel smug when a student says “I passed math because of you”.  Why do people say this?  Is it because the majority of teachers are, well, ‘bad’?  Or, is it because the math involved is so distorted from any reasonable need for one person to know, that we are faced with a random function (input is teacher behavior, output is ‘success’)?    If we are facing this random function, we would observe almost all teachers having a student say “I passed because of you” … and I believe that this is, in fact, the case.

We need to push for good mathematics that people actually need to know.  At the college level, the New Life project is based on this goal.  In the school setting, there is the “Common Core” … however, I believe that the Common Core does NOT describe good mathematics that people need to know.  Instead, the Common Core seems to be a laundry-list of topics and skills that members of a group nominated, without sharing an underlying criteria the discriminate between good math and math that gets in the way.

We also need to work on ‘advancing the profession’.  Too much of our work is based on oral history and local traditions, without a common framework for building methods that support good learning.  The New Life project has a hope of facilitating this community-building, just like the Carnegie Foundations “Networked Improvement Communities” strives to build the profession.

I did, of course, thank the student for the comment about helping him pass the course.  If it wasn’t such a public venue, I normally also comment about their hard work being critical. 

I hope you will join me in building good mathematics and advancing the profession.
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Online Homework Systems?

I have been grading final exams this week, and having to resist the temptation to vocalize dramatically when I see what students do too commonly on basic problems.

This, of course, is nothing new; I suspect that most of us have this reaction at the end of a course, and that my students have not created anything that has not been seen thousands of times before.  During formative assessments, this ‘interesting’ mistakes are actually a great opportunity to explore the thinking and improve understanding.

My worry is that the rate of doing these ‘interesting’ mistakes might actually be increasing in my courses.  We adopted e-books and homework systems for our developmental courses this year — students pay a  course fee about $80 that covers the whole thing.  Since all students pay this as part of registration, all students have access to the ‘textbook’ from the first day of the semester.

Access has certainly been improved. Performance has not.  It’s possible that my subjective assessment is not valid; however, I am fairly sure that students are doing less well in this system.  One thought I have — does the online homework system create a false sense of mastery?  Students can get quite a few correct answers after looking at hints and doing some multiple choice questions.  Or, perhaps it’s the process of doing online homework, where writing problems might or might not happen … how does this impact memory?  [We can be pretty sure that writing out problems will improve memory and learning.]

I like my students having access to the book from day 1; I really like all of the resources that come with the e-book (like videos).  Informal conversations with some colleagues suggests that the impact on learning has not been that good.

Since online homework is becoming fairly standard … I wondered, and thought I would raise the question.  Feel free to comment!!

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