Integrated Math vs. Non-Integrated Math
In high school, most of us took the following sequence of classes: Algebra, Geometry, Algebra 2, Trigonometry/Pre-Calc.  Then many went on to either a Statistics of Calculus course if there was time.

Unfortunately, rich mathematical problems are not often contained within a single subject.  Instead, a good problem worth solving might contain elements of geometry, algebra, and other branches of mathematics and thus it has become less relevant to teach these subjects separately.  Integrated just means that the four traditional subjects (Alg, Geo, Alg2, Trig), along with some other branches (Discrete, Probability, Statistics, Pre-Calc), have been mixed together and spread out over four years.  It's the same content, the same mathematics, but the order has been rearranged and the courses titled Integrated Math 1-4.  This has been around for some time now and is nothing new to colleges. 

One unfortunate thing that has happened over the years, is that schools have often run dual tracks (one traditional, one integrated) and have used the integrated track as their remedial one.  This is unfortunate because it incorrectly stereotypes one as inferior to the other.  At Castle View we offer only the integrated path.  There is no low level track, no high level track.  We believe very strongly in all students getting the best mathematics available and believe that an integrated approach is much more relevant to the 21st century learner for the reasons listed above.

Traditional vs. Reform
Since the early 1990's, there has been a tremendous amount of talk about reform mathematics.  Put (very, very) simply:

• Traditional mathematics emphasizes a teacher-centered classrooms where rules and procedures are passed down from teacher to student and practiced until committed to memory. 


• Reform mathematics emphasizes student-centered classrooms where students interact with the teacher and their peers to expand their current understandings into new ideas which are then formalized.

Castle View believes that this reformed approach is best for students to be successful in a 21st century society.  Again, we have moved past needing our members of society to merely play their assigned role or be obedient, compliant citizens or employees.  Instead, the world is demanding leaders, thinkers, innovators, and people that are able to self-direct and create.  Research has given tremendous amounts of evidence that the majority of students who come through a traditional mathematics program (or school for that matter) tend to be very passive about their approach to learning.  They simply want to be given the formula or procedure, they want to be shown an example, and they have a hard time making any progress in unfamiliar situations.  The argument is that in a reform classroom, students learn to think for themselves, and learn how to apply their current body of knowledge to an unfamiliar situation.  This is a powerful result that leads to students being unafraid to attack problems, unafraid of new situations, and unafraid to take the risks necessary to be leaders and innovators.

Traditionally, emphasis has been on the "correct" procedures to finding solutions in mathematics.  Thus, a teacher traditionally has been in charge of demonstrating these step-by-step procedures and the students have been in charge of routinely practicing problem after problem until they have "learned" this procedure.  There are several problems with this approach to learning, a few are listed here:

• When procedures are learned in this manner, they are not often retained for long periods of time.  Thus we find students (I was one of them) that are able to store them in short-term memory, pass their exams and classes, and 3 months later be scratching their head wondering "how do I do that again?"


• This emphasizes another person's way of solving a problem.  A student being 21st century ready needs to be able to think for them self and to be innovative and creative.  If the only way they've ever learned mathematics is to mimic the procedures of other people, then they've not likely learned to create and invent their own solution paths. (note: there ARE times to learn formal procedures and understand the work of mathematicians over the centuries, but it is important that this comes after a student has had time to explore, create, discover, discuss, and make personal sense of how it all might come together.)


• The very definition "learning mathematics" has evolved since many of us were in school.  It used to be as simple as getting the right answers and knowing the correct formulas.  Often most of this "learning" was prescribed to us in discrete situations and there was not a lot of mystery about what was expected of us (learn linear equations, practice linear equations, take a test on linear equations...now learn quadratics, practice quadratics, test quadratics...).  Instead, our students need to be able to apply their mathematics to unfamiliar situations.  This involves a much more complex process than simply being shown how to do something and practice it over and over again.  Instead, they need to learn how to access their current knowledge (stuff locked away deep in their brains), figure out how it might apply to this new situation, test some conjectures out, and make sense of their solution (whether it makes sense or not).  This is now what it means to "learn mathematics" - it is a very interactive and meaningful thing.


• This approach unfortunately rewards compliance and memorization.  Don't get me wrong, we are not promoting non-compliance or disobedience of any sort. We just don't want to call your student "good at mathematics" simply because they've followed our rules and our procedures and arrived at our answers.  Contrary to popular belief, there are often many correct solution paths and many correct solutions to a rich problem.  I am not talking about the simple arithmetic or algebra problems often thought of in math classrooms such as 23x16 or 4x +2x = 80...what I am talking about are the problems that involve many factors that students have to consider and come to some conclusions about before they solve the problem.  These are the richer, much more relevant types of task that students will face beyond high school and we need to foster this ability to think and discuss as much as possible.  Memorizing a procedure that a teacher tells you may help solve some simple algebra and calculus problems here and there, but it does not necessarily lead to a disposition of creative problem solving in unfamiliar situations.

The common link to everything above is preparing a student to face unfamiliar situations in college and beyond.  It would be nice if everything had a step-by-step manual attached to it.  It would be nice if everything had an example to follow and refer back to.  It would be nice if everything that a student will someday face had a teacher or an expert on hand to pass down the secrets and tricks of that particular situation.  And while many jobs in this word do, in fact, have these procedures, examples, and experts, the reality is that many of them don't.  The leaders and innovators of the 21st century are expected to author their own ideas and create new paths, ones that have not even been thought of yet.

Interactive Mathematics Program (IMP)
Put briefly, our program is an integrated and reform-based program meant to be a different approach to learning mathematics than what many of us had when we grew up.  This is because it starts with problems first, before learning formal procedures.  It is the thinking and exploring of these problems that help guide students to the formal, traditional mathematics.  The idea here is that any student can access these problems and begin thinking about what might go into their solutions without being dependent on a teacher's direction and authority; after all, the goal is that your student is learning how to be independent and capable in true problem solving.  This is where the teacher's important role comes in as facilitator to help students bridge the gap between what the students are concluding from the problem and the formal mathematics that we often think of and are used to.  By building this exploration and allowing them to make sense of what is actually happening, the mathematics they learn is considerably more meaningful and thus retained throughout their years.  It also becomes something they can continue to access and build upon rather than just another discrete skill to check off and likely forget.

To find out more about the philosophy and research behind IMP, I encourage you to look at the IMP website here.  In addition, I encourage you to see my address to parents for helping your student be successful in IMP and my frequently asked questions area.

Vince Parenti,
Mathematics Lead