What good is algebra?

Developmental algebra is the most studied course in American colleges … well, at least the most enrolled!  Studying is another thing. 🙂

Why?  What value does this activity add?

I’ve noticed something about students who have passed our beginning algebra course, and I am not happy about this.  We have several math courses that can be used to meet the requirement for an associates degree, and one of these math courses is a mathematical literacy course.  This course involves a lot of problem solving, based on understanding relatively few concepts.

Consider this sequence of problems and typical student responses:
Item: A company has $38 million in sales this year, and expects it to rise by 10% for next year.  What will the sales be next year?
Student: Okay, 10% is 0.10 … we better multiply … 0.10 times 38 is 3.8.  That’s too small for the sales, so we add 38 + 3.8.  The sales next year are $41.8 million.
Item: A company has $38 million in sales this year, and expects it to rise by 10% per year for the next several years.  Write an expression for the sales based on the year n.
Student: What?  38 times 0.10.  Where does n go?  Is it 0.10n + 38?
Me: Okay, let’s look at a simpler problem.
Item: A company has $38 million in sales  this year, and expects it to rise by 10% per year for the next several years.  Estimate the sales for the next 4 years.
Student: Okay, the first year is like the one we did earlier … $41.8 million.  Do we do the same thing again?  [me: might be — would that make sense?]  Yes, I think so  … {calculates}. 
Me: That is looking good.  How about the expression … does your work here have anything to do with the expression we need?
Student: You got me!

Of course, our beginning algebra course has a lot of applications, and students see like terms and a lot of exponents.  We cover percent applications, including some where we know the value after the 10% increase and need to find the original.  In spite of the appearance of ‘mastery’, most students do not connect their knowledge with the concepts in a novel situation.  Quite a few students will actually deny the connection between the algebraic expression and the computations they do.

We often ‘sell’ our courses because of a belief that passing a math course indicates a better capacity to reason and to think logically. 

However, the traditional courses do not deliver on this promise (in my opinion).  Almost all textbooks have repetition of skills, and we cover too much material to work on applying anything to novel situations.  Sadly, almost all useful applications of mathematics (in life and in occupations) begin as novel situations.

I personally dislike (strongly!) the phrase “a mile wide and an inch deep” (for one thing, we are all adept at 90 degree rotations to get “an inch wide and a mile deep”).  Slogans like that do not help us.  What might help us is thinking about what we believe is valuable in mathematics … and delivering courses that build this value for our students. 

As long as we attempt to ‘remedy the deficiencies’ of our students, we will miss the benefits.  Their deficiencies are many; most adults have similar deficiencies (even those employed in occupations that our students are preparing for).  Our attention should be on “what mathematics is needed for community college students” or “what mathematics is needed for university students”.

I really believe that we can provide courses that students will see the value of, and that we can be proud of as mathematicians.  I think that the New Life model is a good starting point, and I hope you will consider becoming a supporter of this work … and consider offering these types of courses at your college!

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The Math Dance

A step forward, step to side, bring feet together … a step backward, step to side, bring feet together.  Dance the waltz enough, and a person can do this sequence without any difficulty.  Many can become experts at the dance, and all can be included. 

Lost is why these are used as the dance steps. 

Of course, the ‘why’ does not really matter — it’s just a dance!!

I have taught a lot of students to dance.  The majority have been able to do dances like the waltz and two-step.  Sadly, the math dance has no particular value if a person does not know why the steps are done like this.  To understand means that a person can improvise; a little understanding allows helpful flexibility, and much understanding allows an artist’s rainbow of insight, logic, and problem solving.

Mathematics has become a dance, one that can be taught as remembered moves to particular musical themes.  There are some experts who assert that this the only possible outcome when society decrees that ALL persons must complete a subject, that mandatory always translates into a lowering of the value of this learning.  The evidence for this view seems abundant, and it is easy to accept this result (especially with the bright and blinding light of accountability shining on education).

We must not give up on mathematics so easily.

Mathematics has much to offer every student, our society, and the future.  Not the math dance — the real mathematics, science of relationships between quantities. 

We can create sound mathematics appropriate for all learners.  All students can learn, given the proper resources and conditions.  I might grant that the more extreme learning disabilities might present obstacles too large for a very small minority; this group is at least 2 standard deviations below the mean.

I encourage you to avoid the current rush of methods that might be more efficient at teaching the math dance.  We have seen these types of improvements before, which provide change but not progress.

I invite you to work with me to imagine a better mathematics program for all of our students, a program that shows the practicality and beauty of mathematics.  We do not need to make mathematicians of all students, just like we do not need to create math dancers … however, I believe that we can create a program that inspires more students to seek out more mathematics.

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Contextualized Mathematics

Should mathematics be learned within the context of a situation that creates (or at least, shows) the need for the mathematics?  Do students learn better when we do?  What do we want our students to gain?

Some of my recent posts might leave a person with a belief that I think mathematics should be highly contextualized.  However, I definitely do not think this is true.  My reasons deal with two discrete issues that connect within our classrooms — the impact of high context on learning, and identifying the goals (what we want students to gain).

The research on contextualized learning is not particularly strong at this point.  Certainly, the general researchers in learning & cognition conclude that context can actually interfere with learning; this is simply a corollary to the principle that learning is improved by making the target (the thing to be learned) as visible as possible.  Context can hide the big ideas.  For a good summary, see this article about cognitive psychology in mathematics http://act-r.psy.cmu.edu/papers/misapplied.html  — one of the best single sources I’ve ever seen.

A second component of the learning dimension is language and culture.  As soon as we present a context, we make demands of our students about other knowledge … sometimes fairly unrealistic.  One example I saw recently involves ‘shooting free throws’, rebounds, and ‘dunks’; another dealt with a baseball infield.  Sports are not uniformly followed by our students.  The same difficulty arises when we talk about projects around the house (whether it is sewing or woodworking).  These language and cultural factors affect both native speakers and those who learn English as a second language, and the problems cut across economic standing as well.  

The other dimension of my concern deals with our goals … what do we want our students to gain?  Some people bring in contextualized learning so that students can experience ‘doing mathematics’ like a mathematician does.  Other people use high-context because it motivates students.  To me, both of these goals are important … however, they are not the whole story.  A major goal of any math class should be to provide general tools that can be used, especially in future classes that the student needs to take, and this suggests a need to be able to transfer learning to other situations.  This transfer is inhibited when a learner has not practiced a skill or process repeatedly; meeting this threshold is very difficult if we contextualize most problems.  See http://jackrotman.devmathrevival.net/sabbatical2006/3%20Life%20in%20The%20Grey%20Zone.pdf, which deals with ‘how much practice is enough’.

I am especially concerned about preparing students to cope with the quantitative needs in their science and technology classes.  These needs vary from the very specific context to quite broad conceptualizations, and we seldom know which mixture of needs a particular student will have.  Developmental mathematics needs to deal primarily with broad sets of needs.  I do not think we can limit the mathematics to that which there are contexts that the student will understand.

It would be simple to say that we need ‘balance’ in our curriculum, and this would be true.  However, we should talk about what students should gain.  Some of their future classes are actually after some very specific skills (such as equivalence of different forms, or dimensional analysis); others are general … almost theoretical (such as behavior of types of functions for biology).  For a particular college, the needs might shift the ‘balance’ more strongly in one direction or the other.

The New Life reference material (at http://dm-live.wikispaces.com/) was developed to summarize many of these needs.  I encourage you to look at the sound mathematics described, most of which can be well served by combining learning about the process along with dealing with various contexts.  Needs exist for which contexts may not exist; some needs deal with theory, where context is a temporary step off the trail.

Yes, there are “what works” studies that conclude a high-context approach improves math classes.  These are not proofs of a result.  Like other ‘what works’ studies, there are many factors involved and only a few measured for inclusion in analysis.  From a learning standpoint, the best to be expected is “no significant difference” with high context … and this would reflect a great deal of effort to avoid the known difficulties with high context.

We are preparing students for success, and their success involves a multitude of needs.  Our math classes should focus primarily on general concepts, with a limited role for contextualized learning.

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Instant Feedback Lowers Learning

Online homework systems are “cool”.  We like them as faculty (in spite of our frustrations), students generally like them, and we believe that instant feedback is a good thing.

Learning is a different process than connecting a stimulus with the proper response (“conditioning”).  The effect of instant feedback might help conditioning, but can definitely interfere with learning in humans.  Schooler and Anderson published an article entitled “The disruptive potential of immediate feedback” (see http://act-r.psy.cmu.edu/publications/pubinfo.php?id=313 ).  The logic for being disruptive is that the instant feedback competes with the learning content for resources in the working memory.  Paying attention to feedback means that there is less attention available for the concepts and procedures.

 Related to ‘instant feedback’ is the general property of being FAST!   When learners complete activities quickly, research shows that the entire process tends to stay in working memory … never making the transition to long-term memory.  See O’Reilly (page 153), Leron and Hazzon “The Rationality Debate: Application of Cognitive Psychology to Mathematics Education”  (see http://edu.technion.ac.il/Faculty/uril/Papers/Leron&Hazzan_Rationality_ESM_24.3.05.pdf#search=%22co and Kahneman “Maps Of Bounded Rationality: A Perspective On Intuitive Judgment And Choice” (see  http://nobelprize.org/nobel_prizes/economics/laureates/2002/kahnemannlecture.pdf#search=%22Maps%20of%20bounded%20rationality%3A%20A%20perspective%20on%20int) and O’Reilly’s chapter “The Division of Labor Between the Neocortex and Hippocampus” in Connectionist Models in Cognitive Psychology (edited by Houghton, George).

There is a point of view that advocates learning within a gaming environment, which might seem to contradict these statements.  One distinction that might help understand the contrast is that of ‘awareness of learning’ — in many games, the learning takes placed without direct attention to the learning, meaning that the learner has less ability to explain (and transfer) that learning.  We would hope that mathematical learning needs to be transferable, and we like to have learners who can explain what they have learned.  I do believe that ‘instant feedback’ and ‘quick learning’ lowers the overall learning.

Why do I think this is important?  Much of the current ‘movement’ in developmental mathematics involves intensive uses of online homework systems for their instant feedback and quickness.  From a learning theory perspective, this is not a good thing.  My prediction would be that students using these systems have even more surface processing and lack of transfer (of knowledge) than our old-fashion textbooks. 

How should we design instruction for better learning?  Just because feedback can be ‘instant’ does not mean that it’s best; learning support systems (homework) should design the speed of feedback based on parameters from research studies to facilitate deeper processing in the brain.  These systems should also consider breaking up sets of problems to include other activities; a student who quickly completes 30 homework problems without a break might be processing only at the surface level … other learning processes within these sets can give the brain an opportunity to reconcile the new material with prior knowledge (a key step).    As instructors, we can monitor the time on homework to encourage students to slow down, to even take short breaks in the middle.

Given that students may tend to be less patient than in prior periods, we need to pay deliberate attention to slowing things down.  Part of this would be direct and honest statements to our students about how they can improve their learning and success in mathematics.

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