Category: math reasoning and applications

Meaningful Mathematics … Learning Mathematics

Among the pushes from policy makers is ‘meaningful math’ … make it applicable to student interests.  Of course, this is not a new idea for us in mathematics; we’ve been using ‘real world applications’ to guide some reform efforts for decades. #realmath #collegemath #learningmath

Some current work in reforming mathematics education in college is based on a heavy use of context, where every new idea is introduced with a situation that students can understand.  We know that appropriate contexts with meaning to students helps their motivation; does it help their learning?

Before I share what I know of the theory and research on these approaches, I would like you to envision two types of textbooks or instructional materials (print, online, or whatever).

  1. Opening up the start of a lesson or section, repeatedly and randomly, generates a short verbal introduction followed by formal mathematical symbolism related to the new idea(s) in almost all cases.
  2. Opening up the start of a lesson or section, repeatedly and randomly, generates a variety of contextual situations and mention of a mathematical idea or tool that might be used.

Many traditional textbooks are type (1), while a lot of current reform textbooks are type (2).  Much of the change to (2) is based on instructor preferences, and I am guessing that much of the resistance to change is based on instructor preferences for (1).

Let’s take as a given that contexts that are accessible to students improve their motivation, and that we have a goal to improve student motivation; further, let us assume that we share a goal of having students learn mathematics (though that phrase means different things to different people).  There seem to be a number of questions to answer dealing with how a context-intensive course impacts student learning of mathematics.

  • How uniform is the impact of context on different learners?  (Is there an “ADA-type” issue?)
  • Do students learn both the context and the mathematics?
  • Is learning from a context more or less likely to be used and transferred to new situations?

I know the answer to the first question, based on research and experience: The impact is not uniform.  You probably understand that there are language issues for quite a few students, perhaps based on a class taught in English when the primary language is something else.  However, most of the contexts have a strong cultural factor.  For example, a common context for mathematics work is “the car”; there are local cultures where cars are not a personal possession, as well as cultures outside the USA where cars are either generally absent or relatively new (and, therefore, people know little).  The cultural problems can be overcome with sufficient scaffolding; is that how we want to spend time … does it limit the mathematical learning?  There are also ADA concerns with context: a sizable group of students have difficulties processing elements of ‘a story’ … leading to problems unpacking the context into the quantitative components we think are ‘obvious’.

The second question deals with how the human brain processes different types of information.  A context is a type of narrative, a story; stories activate isolated memories and create isolated memories.  To understand that, think about this context:

You are standing on a corner, and notice a car approaching the intersection.  When the light turns red, the car applies the brakes so that it stops in about 2 seconds.  You estimate that the distance during the stopping process is about 100 feet, and the speed limit is 30 miles per hour.  Let’s look at the rate of change in speed, assuming that this rate is constant through the 2 second interval.

The technical name for a story in memory is “episodic memory”.  This particular story might not activate any episodic memories for a given student; that depends on the episodes they have stored and the sensory activators that trigger recall.  Some students will respond strongly and negatively to a particular story, and this does not have to depend upon a prior trauma.  More of a concern are students who have some level of survival struggle (food, shelter, etc); many contexts will activate a survival mode, thereby severely limiting the learning.  Take a look at a report I wrote on ‘stories’ http://jackrotman.devmathrevival.net/sabbatical2006/2%20Here%27s%20a%20story,%20Ignore%20the%20Story.pdf

What happens to the mathematics accompanying the story?  If we never go past the episodic memory stage, the mathematics learned is not connected to other mathematics; it’s still a story.  In the ‘car stopping at an intersection’ story, the human brain might store the rate of change concepts with the rest of this specific story, instead of disassociating the knowledge so it can be used either in general or in new ‘stories’ (context).  Disassociating knowledge is another learning step; many context-based materials ignore this process, and that results in my biggest concern about ‘problem based learning’.

Using knowledge (Transfer … question 3) depends upon the brain receiving sensory input that activates the knowledge.  This is the fundamental problem with learning mathematics … our students do not see the same signals we see, ones that activate the appropriate information.  For this purpose, traditional symbolic forms and contextual forms have the same magnitude of difficulty: the building up of appropriate triggers to use information, as well as creating chunks of information that work together.  We need to be willing to “teach less mathematics” so that we can focus more on “becoming more like an expert with what we know”.  For more information on learning mathematics based on theoretical (and research-based) points of view, see http://jackrotman.devmathrevival.net/sabbatical2006/9%20Situated%20Learning.pdf and http://jackrotman.devmathrevival.net/sabbatical2006/6%20Learning%20Theories%20Overview.pdf 

I can’t leave this post without mentioning a companion issue:  Contextual learning is often done by ‘discovery’.  Some reform materials have an extreme aversion to ‘telling’, while traditional materials have an extreme aversion to ‘playing around’.  From what I know of learning (theory and research), I think it is safer to take the traditional approach … telling does not provide the best learning, but relying on discovery often results in even more incomplete and/or erroneous learning.  Just for fun, take  a look at http://jackrotman.devmathrevival.net/sabbatical2006/8%20Telling,%20Explaining,%20and%20Learning.pdf

Looking for a brief summary of all of this stuff?

Contextualized learning comes with significant risks; use it with caution and a plan to overcome those risks.
Mathematics is, by its nature, a practical field; all math courses need to have significant context used in the process of learning mathematics.

These ideas about context are related to the efforts of foundations and policy influencing agents (like CCA).  We need to keep the responsibility for appropriate instruction … including context.

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QR Courses: Resources to Build Good Quantitative Reasoning Courses

We had a workshop this winter on Quantitative Reasoning courses (QR) in Michigan.  The information shared at that workshop is now available on the MichMATYC web site.  Here:  http://michmatyc.org/QRCourses.html

[This workshop was sponsored by MichMATYC with operational support from the Michigan Center for Student Success.]

Take a look … information from several colleges is included, and some math path maps are available on the page.

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Even Our Puzzles Are Outdated … Mathematics for 2025 (and today)

Earlier this month, the Conference Board of Mathematical Sciences (CBMS) held a forum on mathematics in the first two years; many of the presentations are available on the web site (http://cbmsweb.org/Forum5/)

As part of one of the first plenary sessions, Eric Friedlander commented …  Students in the Biological Sciences now outnumber those in the Physical Sciences in the standard calculus 1 course.  (David Bressoud shared some specific data on those enrollment patterns.)

Historically, the developmental mathematics curriculum was all about getting students ready for pre-calculus.  Our “applications” tended to be puzzles created with physical sciences in mind — bridges, satellites, pendulums, and the like.   Few problems in our developmental courses draw the attention of those in biologically-oriented fields (including nursing).

We could include:

  • Surge functions to model drug levels
  • Functions to estimate the proportion of a population needed to be immunized to prevent epidemics (P_sub_c = 1 – R_sub_0)
  • Models for spread of cancer … and for treatments
  • Pollution prediction (simplified for closed systems)

This list is a ‘bad list’ because there is no common property (except being related to biology) … and because I do not know enough to provide a better list.  Take a look at books in applied calculus for the biological sciences; you will see applications that are perhaps better than those above.

There is a trend in the new models for developmental mathematics (AMATYC New Life, Dana Center New Mathways, and Carnegie Foundation Pathways) to include a balance of applications — including more from biology.  We need to bring in more of these applications throughout our curriculum (from the first developmental course up to calculus).

Most of us realize that the ‘applications’ in our courses and textbooks are puzzles created by somebody who knew the answer; generally, these problems do not represent the use of mathematics to solve problems and answer questions in the world around us.  Sometimes, we are not able to provide enough non-mathematical information to provide representative problems … in those cases, some reduction to the ‘puzzle state’ is acceptable.

Our puzzles should represent the diversity in the uses of mathematics, with a significant portion of applications being realistic in nature.

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Is THAT the Best You’ve Got??

A student comes to college, and needs to meet their general education requirement.  One of those is in mathematics, and this student actually has some options:

  • College Algebra (called pre-calculus at their college)
  • Introductory Statistics
  • Quantitative Reasoning

Being a typical student, this student wants to avoid the college algebra course; they thought about being an engineer but are too frightened of mathematics.  The next choice would be statistics, because everybody seems to think it is the best choice.

In looking in to the course, the student discovers that the statistics course has some nice features.  Most of the material is taught by first looking at data from the world around us, and the description says that the quantitative work is somewhat limited.  The student becomes worried when they look at the content in the text materials used — it’s got words used in a weird way (normal, deviation, inference, significance); it’s like statistics is a foreign language without any visible culture, so the student feels like much of it is arbitrary.

So, the student tries to find out what “Quantitative Reasoning” means.  The course description talks about voting, networks & paths, logic, and ‘proportionality’ (whatever that is).  Like the statistics course, it looks like the material often involves data from the world around us; however, it’s not clear how much quantitative work is actually involved.  The student is not too worried about any particular topic or phrase in the content descriptions; however, the course does not seem to have any pattern to the topics … it looks like an author’s 15 favorite lessons.

The student thinks about the basic question:

Will any of these courses help me in college courses, in my work, or in my life in general?

Basically, this student will reach the conclusion that none of these three courses will be that helpful.  As a mathematician, I would summarize the basic problem this way:

  • The college algebra course and the statistics course focus on a narrow range of mathematics.
  • This quantitative reasoning course does not focus on any particular mathematics.

There is a mythology, a story repeated so often that we believe it, that statistics is a better pathway for most students.  The rationale is something like “our world is dense with data and decision making” or “making decisions in a world of uncertainty”.  I see a basic problem, that remains in spite of what has been written: statistics is an occupational science, with few broad properties or theories.  Statistics is about getting helpful results, and for statisticians, this is great.  How does it help students when we use “n”, “n – 1″, and “n + 4″ for calculations involving sample sizes; the ‘plus 4 rule’ is a typical statistical method for producing the results we want — even when there is no mathematical property to justify the practice.  [In a field like topology, we don’t let inconsistent procedures survive.]  I think we also over-estimate the value of statistics in occupations; there are limited uses in  other college courses, and some nice uses for life in general (for those motivated).

The quantitative reasoning (QR) course has a different problem — we don’t have a shared idea of what this course should accomplish.  For some, it’s an update to a liberal arts course (like the example above).  For others, QR means applying proportionality and some statistics to life.  Still other examples exist.

Is that the best we’ve got?  We are giving students options now (a nice thing), but the options are really not that good for the student.  For the student above, they really should take the college algebra course — perhaps they will find that mathematics is not their enemy after all; they might become an engineer, an outcome not likely at all with the other two choices described.

As mathematicians, we need to claim the problem and be part of the solution.  That college algebra course?  Modernize the content and methods so that it actually helps students prepare for further mathematics without becoming a filter that stops students.  That QR course?  We need professional conversations around this course; MAA and AMATYC should jointly develop a curricular model of some kind.  In my view, the QR course is the ideal general education math course; we should include significant mathematics from multiple domains, done in a way that students can discover that they could consider further mathematics.  The statistics course?  Let’s keep a realistic view of the value of this course; it’s not for everybody, and we tend to think of statistics as the option for people who never need anything else.

No, THAT is NOT the best we have.  We have some basic curricular work to do; together we can create better ideas, and help our profession as well as millions of students.

 
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