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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Functions, Models, and Dev Math

First of all, if you are not ‘doing exponentials’ in developmental mathematics then you are missing many great opportunities.  From finances to environmental studies to biology, repeated multiplying is a very common process — and is at least as practical as repeated adding (linear).

This post is about two issues … first, what functions are relevant for which developmental math course, and second, how to present the distinction between functions and models.

A pre-algebra level course should include practical experience with linear and exponential situations.  Linear relationships can be used as part of working on proportional reasoning, where the rate of change (like ’12 in/1 ft’) can be written in two forms depending on which value is the input.  Various representations are accessible to students, so an understanding of graphs of data can be included — even without dealing with concepts like slope and intercepts.  Exponential relationships can be used starting from a practical context such as compound interest or indices such as the “CPI”, where the multiplier can be written as a percent added to 1 … including negative changes.  Representations can be included.

A first algebra course should formalize the practical work with these functions to include the symbolic forms normally seen, and concepts related to graphing — slope, intercepts, base, initial value.    The first algebra course can introduce quadratic relationships based on geometry, but it is more important that students understand function terminology and some notation. 

A second algebra course should ease away from practical contexts to deal with topics from a more scientific point of view; half-life and doubling-time would be appropriate.  The second algebra course could include work on conic sections, especially if the course serves to prepare students for pre-calculus.

The distinction between functions and models should be included in both algebra courses.  Even a pre-algebra course should have measurement concepts such as precision and accuracy.  The first algebra course can use this to describe the distinction — functions represent data where the only variation is due to measurement error and show a known relationship between inputs and outputs, while models represent data where other sources of error cause variation and reflect an educated guess about the possible relationship between inputs and outputs. In the second algebra course, students should have experience in judging the distinction between functions and models for themselves.

Functions and models should form a significant part of any algebra course, with more attention than symbolic manipulations.  This emphasis should be especially strong in the first algebra course; the second algebra course can reasonably incorporate relatively more symbolic work.  A pre-algebra course should be mostly about developing a quantitative sense concerning numbers in relationships.

The “New Life” model courses have learning outcomes that reflect this point of view, at least partially.  Whether you can do New Life courses, or can just make changes to existing courses, I encourage you to strengthen the work done in your developmental math courses on linear and exponential relationships.

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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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