Well, I’ve been somewhat discouraged by the latest test in our ‘math – applications for living’ course. This test is all about understanding linear (additive) and exponential (multiplicative) change. In spite of approaching these models in different ways, followed up by repeated work, students had considerable difficulty translating verbal descriptions to anything quantitative or symbolic.
One basic problem seems to be that students did not start with much understanding of slope for linear functions. Every student in the class had completed beginning algebra (usually recently), where we do the usual work with slope — calculating it, using slope to graph, and finding the equation of a line given information about slope. When faced with non-standard problems with information on slope, the transition to quantitative or symbolic statements was not easy.
One part of this difficulty is the connection between input & output units and units in slope. Like most traditional algebra books, ours does not make an issue out of units for slope in ‘applications’; as you know, slope must involve two units. Because of this difficulty, students would see a percent change as a linear change.
Mostly, this post is a “note to self”: Learning slope is not really a fast thing. Repeated use, alternate wordings, and non-standard problems need to be used to build a more complete understanding. We too often spend a week or two on all linear function topics in a beginning algebra course; to get that ‘sufficiently good’ understanding, we need to allow more time so we can really dig in to what slope and the linear form are doing.
Join Dev Math Revival on Facebook:
Uri Treisman and I have been involved with efforts to systematically reform developmental mathematics, such as New Life, Carnegie Pathways, and the Dana Center New Mathways. Uri has been very supportive of our AMATYC work, including the New Life project.
On June 6 (4pm Eastern), we will be doing a joint Webinar on Issues in Implementing Reform in Developmental and Gateway Mathematics as part of the AMATYC webinar series. The goals of this webinar are to present some general concepts to guide our work in reform, and to share some practical means to implement those concepts.
Here is the way the AMATYC webinars work — AMATYC members can register for a webinar (at http://www.amatyc.org/publications/webinars/index.html). Registration usually begins about two weeks before the event (so you won’t see this one listed in April!). AMATYC members who register will receive an email with directions (the day before the webinar).
One thing to point out — people can watch the webinar as a group! One person needs to be an AMATYC member and register; you can include non-members in the viewing process. (The directions you receive will even tell you how to make the group process work better.)
I hope you can join us for this webinar.
Join Dev Math Revival on Facebook:
A recent post here dealt with the metaphor of developmental mathematics as a bridge, designed to help students reach the other side. The ‘other side’ is not just mathematics (would we really want that?), with a diverse collection of courses … some of which are called ‘gateway courses’, while others are ‘just’ college courses. So, the question today is “What about the math on the other side”?
Is the math ‘on the other side’ the good stuff (important mathematics)? Do courses ‘on the other side’ place a high priority on student success? If we reform developmental mathematics in to a program which makes a difference in the mathematical learning of students, will their ‘college math courses’ have the same vitality?
These are questions which I can not answer; I am not immersed in the world of college-credit math classes (just parts of it). However, I do know that our profession is rather silent on this component of our curriculum … we are talking a great deal about developmental mathematics, and I hear quite a bit about STEM and calculus. Not so much about college algebra … pre-calculus … liberal arts math … or math for elementary education majors.
The easy target in this list is college algebra. Pre-calculus … at least we know what the goal is (calculus), and students taking pre-calculus can be assumed to have that goal (even if incorrectly assumed). However, we have absolutely no agreement on what ‘college algebra’ is. For some of us, college algebra is what we happen to call our pre-calculus course; for this group, I would say “Hey, be honest … call if pre-calculus!” For others, college algebra is actually a prerequisite to pre-calculus; on this … “how much time is needed getting ready for calculus?” [Perhaps we place additional steps in between to make sure that only the best survive; I hope not.] For still others, college algebra is a course outside of the pre-calculus sequence, perhaps used as a preparation for symbolic-based science courses; this is a good reason to have a course … though I question whether ‘algebra’ is the majority of what the students need. Some use ‘college algebra’ as a general education course; I suggest to you that a course could be either college algebra OR general education … but not both. One of the problems with the ‘college algebra’ label is that the traditional developmental math courses generally have ‘algebra’ in the titles; is ‘college algebra’ more of that developmental stuff?
Perhaps my worries here are just due to my extensive ignorance of some aspects of our curriculum. Perhaps, outside of the college algebra mess … perhaps we have generally sound mathematics and important ideas in our curriculum. Perhaps my problem is that I look at textbooks. If most of my colleagues who specialize in these courses tell me that ‘things are okay’ on the other side, I would certainly be relieved. However, with all of the current focus on developmental mathematics, it is possible that we are ignoring something equally important.
In our bridge metaphor, are we working on improving the bridge … just so that students can be delivered to a great wasteland of college mathematics on ‘the other side’?
Join Dev Math Revival on Facebook:
In the world of web design, there is a concept called ‘sticky web pages’ or ‘sticky content’ … the concept being that a design can encourage people to click on links and/or return to the page. [A brief explanation at http://en.wikipedia.org/wiki/Sticky_content, and some tips at http://techtips.salon.com/sticky-pages-10404.html.]
If you are changing your developmental math program … are you creating ‘sticky math’? Are students motivated by the design to spend more time than required? Are students inspired to take more math than is required?
I can hear the cynics among us thinking ‘That is just not reasonable — students just will not do more math than required’. Well, this is not a question of past evidence … this is a question of the over-arching goals of a math curriculum. Are we providing the absolute shortest (and presumed negative or neutral) experience with mathematics … or do we seek to provide appropriate mathematics in an attractive manner that inspires students to be more mathematical?
I have been thinking about this concept for quite a while. Historically, developmental mathematics has been an overly long series of courses to prepare students for the ‘good stuff’ (calculus, in that paradigm). Some of the current redesign efforts have a deliberate goal of getting students out of mathematics as quickly as possible — often via a set of modules, of which most students need a proper subset. This “quick out” approach is an understandable reaction to the old courses, and has appeal to people outside of mathematics (like administrators and policy makers). Most “modularized developmental mathematics redesigns” are based on a quick out for students.
We can do better than a “quick out” methodology. A common theme of the emerging models for developmental mathematics — New Life, Carnegie Pathways, and Dana Center Mathways — is students are capable of learning sound mathematical concepts presenting in an engaging fashion, which will result in some students being inspired. Some students will be inspired to work harder on one course or just parts of it; other students will be inspired to consider taking additional mathematics.
Reasoning about quantities, core ideas about proportionality, key ideas of algebraic reasoning, and concepts of functions are components of ‘sticky math’. Even some traditional polynomial algebra can be ‘sticky’, though not when presented as a series of procedural skills disconnected from broad ideas. However … the most fundamental ingredient for ‘sticky math’ is the faculty students work with. Technology has strengths and a role to play; by itself, technology is not enough.
However you redesign or reform your developmental mathematics courses, I encourage you to create sticky math experiences for all of your students. Provide the ‘good stuff” (important mathematics) with faculty deeply engaged with the learning environment. Inspire your students!
Join Dev Math Revival on Facebook:
Content of developmental math courses, Learning math | Jack Rotman | 
April 25, 2012 9:17 am | 
Comments (0) 
Subscribe
Jack Rotman
NOTE: This blog will become 'inactive' on January 1, 2020.