Student Success in MLCS

The six major areas of focus of the MLCS course at Rock Valley College are numeracy, proportional reasoning, algebraic reasoning, functions, mathematical success, and student success.  Each unit addresses all of these facets.  Specifically, the course and accompanying lessons are designed to improve a student’s chance of success in a math class.  Here are some examples:

The approach of the course begins with real, relevant content and covers topics differently than they are in a traditional text.  That automatically increases motivation, an important component of student success.  Students have commented repeatedly that the course is interesting; they like what is taught as well as how it is taught.  For example, direct instruction and group work are balanced with each lesson beginning and ending with group work.  This improves attention, understanding, and engagement.  Students are shown respect for their prior knowledge by allowing them to tackle real mathematical problems instead of working from a premise that all the content is new.  Many of the specific skills of the course are not new to these students because in reality, most of them have had several years of algebra prior to the course.  What they lack is understanding, retention, and application.  To improve that, considerable time is spent on solving thought-provoking problems and seeing traditional topics from a unique perspective.  All problems are taught through a context and do not start with abstract ideas.  Instead, the development moves from concrete to abstract, which builds student confidence and understanding.  Further, students are treated like adults, most of whom work and have many varied experiences.  They learn how math is used in the workplace and see those ideas in practice in class.  For example, they learn how Excel is used.  They also learn how the concepts taught can be used to solve problems they will likely face in and out of college.

Next, specific student success activities are included in every unit.  Each student success lesson is different but all have mathematical ideas in them.  So beyond the traditional ideas of time management and test anxiety, issues that these students will face are covered.  For example, students learn how college math is different than high school math.  This is done in the context of determining what components are necessary to be successful in a college math class.  To visualize the various components, students hone their skills with graphs and percentages.  They study job statistics to compare STEM and non-STEM fields in terms of their earning potential and unemployment rates.  This approach brings in some statistics concepts.   The topic of grades is addressed often and deeply.  Components include how grades can be figured (points vs. weights), how GPA is figured and how it can be increased, and why it is difficult to pick up points at the end of the semester as opposed to the beginning of the semester.  Students learn about means, weighted means, what can and cannot be averaged, and how algebra can help solve problems that arise in this context.  Additionally, the first week has many activities to help students begin the semester on the right foot in terms of prerequisite skills, working in groups, and understanding course expectations. 

Another component of the course is helping students learn how to study.  Students think they should just “study more” but do not understand what that means in practice.  To remedy this problem, students are given very specific and explicit strategies that they can act upon.  Students receive a detailed list with actions they can do before class, during class, between classes, before tests, during tests and after tests.  Also, students tend to really like online homework but they can get dependent on help aids and sometimes can’t write out their work.  So every online assignment has an accompanying brief paper set of problems similar to the online ones, but they must write them out and have no online help aids.  With skill homework, they have conceptual homework on paper that is about quality over quantity.  That is, they have fewer problems that take more time so as to work deeply with the concepts at hand.  The test review has a detailed plan to teach students how to study for math tests, beyond just working problems.  Additionally, students are held accountable for all the work assigned so that they learn good study habits and personal responsibility.

Lastly, metacognition is emphasized regularly.  The developmental student often doesn’t fully understand how they think or learn.  Most problems are taught using 3-4 approaches to work at verbal, conceptual, graphical, and algebraic understanding.  For example, when students solve equations, they do so first with tables, then with algebra, and last with graphs.  Once they have learned all those techniques, they are asked to think about which makes most sense for them and keep that in mind going forward.  This approach of solving problems in multiple ways is used often in the course to broaden their mathematical skills, but also give them a deeper understanding of the topics.  This method has an additional benefit for students on test day.  They have several tools in their tool belt to use if one technique is not making sense or their anxiety is affecting their memory.

Together, these techniques support the developmental student in being successful in this course and future math courses.

Kathleen Almy     kathleenalmy@gmail.com
Heather Foes    heather.foes@gmail.comRock Valley College
Rockford, IL

For more information, please check out this blog: http://almydoesmath.blogspot.com

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FOIL in a Box (algebra!)

Some of us have a ‘thing’ about FOIL as a topic in an algebra class; there are concerns about emphasizing the FOIL process as it can submerge the real algebra going on.  Some (perhaps the majority) are not significantly handicapped by being “FOILed”.  This post is not about FOIL itself … it’s “FOIL in a Box”.

Okay, so this is what I am talking about.  The problem given to the student is to multiply two binomials, such as (2x – 3) and (3x +4).  Here is the “FOIL in a Box”:

Some students like this approach, and I think this is because the box lets them focus on one small part of the problem.  The overall process is submerged, and the format does all of the work.  Of course, this is exactly what many procedures in arithmetic do.  The FOIL in a Box method is much like column multiplication, where partial products are arranged in a mechanical way to produce the correct place value.  If correct answers to multiplying were the primary goal, there would be nothing wrong with either FOIL in a Box or partial products in arithmetic.

My observation has been that almost all students who use FOIL in a Box are handicapped in working with polynomials.  Students have trouble integrating the Box into longer problems.  And, though they may have some ‘right answers’ for factoring trinomials, the transition to other types is more difficult. 

What should we do instead?  My own conclusion is that we need to keep emphasizing the entire idea involved.  FOIL is used for “distributing when both factors have two terms”, and “distributing is used to multiply when one factor has two or more terms”.  We too often assume that students will keep information connected to the correct context … they don’t automatically know that distributing does not apply to 3 monomial factors [3(2y)4z ≠72yz], nor to a power of a binomial  [(x + 3)² ≠x² + 9]. 

The achievement of correct answers in the short-term should not come at the price of handicapping the student’s future learning.  All learning should be connected to good prior learning, and imbedded within the basic ideas of the discipline.  We need to be comfortable articulating the full name of what we are doing (multiplying two factors each with two terms), and not use a mnemonic such as FOIL as a container for knowledge of mathematics.

 
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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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Memorize This!

Slope formula?  Area of triangle? Quadratic formula? Basic number facts?

What is the proper role of ‘memorization’ in learning?  Specifically, is memorization needed in a math class?

Let me start with a short anecdote from today’s class.  A student needed to divide 16 by 8; he pulled out his graphing calculator.  Another student needed to know what the common factor was between 16 and 40; he also got out his calculator, and tried dividing each by 4 …

We have been stuck on a rejection of ‘rote’ learning, with a poor association of ‘memorizing’ with ‘rote’.  Now, there are actually times when rote learning is fine — though most of us (myself included) do not use this very often, in favor of more active learning.

This has gone so far as to result in students being told to NOT memorize; one of my colleagues tells students that they can always look it up (in the ‘real’ world).  On balance, this has harmed far more students than it has helped.  Let me explain why this might be true.

First of all, the human brain ‘wants’ to remember things (including formulas and facts) — you dial an arbitrary sequence of 10 digits more than once, and your brain is likely to work on remembering that phone number.  Telling somebody to ‘not memorize’ comes very close to telling them ‘turn your brain off!’.  We can’t condemn memorizing and condemn lack of learning; they go together.

Secondly, the progression from novice to more expert states involves a process called ‘chunking’.  We, as mathematicians, have a very large chunk size in domains where we have practiced and thought; while a student sees 15 steps in a series, we see 2 collections of steps.  When faced with a novel problem, we bring these chunks and our understanding of their connections to the problem.  Experts in any field have a large chunk size, often numbering 10 to 15 specifics in a chunk.  Telling a student to ‘not memorize’ is telling them “it is okay to have a chunk size of 1 (one)” — which means that they are likely to appear as a novice in that field, no matter how much they work.  [Memory, especially clusters of connected memories, seem to be a critical building material for our ‘chunks’; some writers call these ‘schema’ instead of ‘chunks’.]

Thirdly, and fortunately, it does not work to tell somebody to not memorize (see the first point).  The bad part is that some students actually listen to us, and they remember less because of it.  Basically, this is saying that our advice has damage that is limited by accident, not design.

In all my reading of learning theory, over a period of decades, I have yet to find a cognitive scientist say that ‘memorizing is bad’.  From a learning point of view, it is all just learning.  If a person memorizes a formula, without having practiced in varying contexts and without connecting it to other information, then they will be limited in how they can apply this formula; if a person does not memorize a formula, they have to organize their learning around other information — not connected to a formula.  We see students who have a vague notion that area is length times width, and connect all ‘area’ information to this; this incomplete learning creates unnecessary barriers.  If students know multiple formulas for area, as an example, they connect all of these to their understanding of ‘area’; they become better problem solvers … and transfer of learning is much more likely with this more complicated mental map of ‘area’.  The best situation is one where students have several area formulae available from memory, all connected to a concept of ‘product of two measurements, and perhaps a constant.

Mathematics is not the only domain with an interest in (not-)memorizing.  Language learning has also dealt with this, as well as others (see http://scottthornbury.wordpress.com/tag/cognitivist-learning-theory/, and you might also enjoy http://thankyoubrain.com/Files/What%20Good%20Is%20Learning.pdf).

The part that actually bothers me the most, however, is the attitude resulting from students not remembering basic information.  As long as he has to get a calculator for ’16 divided by 8′, he is going to feel dumb about math.  A sense of proficiency and competence goes a long way towards persevering.  Our students do not need a barrier added to their challenges, a barrier constructed out of our good intentions when we say DO NOT MEMORIZE!

Memorizing does not need to be ‘rote’, as we all know from personal experience.  Memorizing happens due to time on task, with a little reflection on the learning involved.  Memorizing is a natural process for a human brain; we need to take advantage of this capacity.

Memorizing alone will not be enough, and never has been.  However, without memorizing we limit the long term mathematical development of our students; we reinforce negative attitudes, and we create learners who have trouble transferring their learning.  Let’s keep a healthy balance — some memorizing, a lot of understanding, building connections, and enough practice to build competence. 

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