Wednesday, May 23, 2012

The Story of an MLCS Textbook

Heather and I have been working for over two years writing a text that completely embodies the goals of MLCS.  Those goals are as follows:

1.  Prepare students for non-STEM college level courses like statistics and general education math
2.  Develop the maturity necessary to be successful in college level courses
3.  Develop an algebraic base to give students the option of intermediate algebra upon completion

Ultimately, we began with the goal of appropriate preparation for non-STEM math, which is not intermediate algebra.  Having a student who is able to add rational expressions enough to pass a test but with no real understanding of them has no use to me as a statistics teacher.  Intermediate algebra has been used a prerequisite because it gives us a student who has a certain level of cognitive ability and maturity.  In other words, it's been used a hoop for statistics and general education math.  I believe all developmental courses should prepare students for where they're headed and that passing them means the student is college ready.  But each course in the developmental sequence should be supportive of the courses that come after it.  If your path includes college algebra or precalculus, then intermediate algebra is an excellent preparatory vehicle.  For other college courses, it is insufficient.

So that mindset of where these students will actually go has been at the forefront of our book development.  It drives every decision and omission.  Nothing is accidental.  Absolute value equations serve no purpose in this course so we don't do them.  If a school wants to add that topic to their developmental math sequence, it would make more sense to add them where they are a benefit (intermediate algebra) instead of where they feel like an odd add-on in MLCS.  All the topics in MLCS have to serve a greater point than "we've always taught this topic" or "I had to learn it."  We're not letting history guide our decisions other than the fact that history has taught us the current developmental math sequence is not working.

But we want schools to have options when using this book.  Our version of MLCS is large and won't always match what other schools want to do verbatim.  The same goes with most developmental courses in that variability is large.  So we have a large number of topics to accommodate a variety of needs.  One of those needs is that a student can go straight into intermediate algebra after our course and not have to go back to beginning algebra if they change their mind and head towards STEM courses.  And students do change their minds.  We have plenty of algebra in the book and students can go onto intermediate algebra and pass it.  But MLCS is not a beginning algebra course even though there is a large amount of overlap between the two.  We have incorporated many intermediate algebra topics but do so differently.  Our students who go onto intermediate algebra will see many familiar topics, but in that course will get the procedural aspect to them.  We do some procedures with quadratics, rational functions, radical functions, and exponentials but that's not our goal.  We spend our time modeling, graphing, exploring, and understanding them.  We build the base that intermediate algebra can add onto if further skills are needed there. 

The key thing we did was throw out old conventions and expectations of the type and order of topics often seen in developmental math.  In other words, this is not an algebra book.  And it's not an algebra book repurposed either.  The beauty of not having written an algebra book is that we don't have that to work from.  We started from scratch, which was incredibly daunting.  But ultimately that served us and our students well because we didn't feel tied to a model that isn't working.  Like in home building, it's often easier to start from nothing than to remodel.

To address the second goal mentioned that I skipped over with all this talk of intermediate algebra, we were intentionally building a cohesive product.  It's not a collection of neat lessons.  Its order is intentional and thoughtful and when used, works.  Students slowly but surely grow mathematically and become stronger problem solvers.  And along the way they get algebraic skills.  If reordered to make an algebra book, our lessons would certainly support the development of algebraic skills and they would show students why the skills matter.  They show the relevance of algebra.  But the mathematical maturity would not necessarily come.  The whole is greater than the sum of its parts, as Aristotle said.  In this case, order matters.

It was important that we establish and maintain rigor.  The bar is so high after this course, college level math, that we have to make sure students can handle that.  I speak often on this course and I always say, "I teach the college level courses on the other side.  Why would we want to pass students to put them in a course they will fail?  That is a hollow victory that ultimately hurts the student."  And the rigor is definitely there, even without adding rational expressions.  It's a tough course when done the way we do it in Illinois because it's a very large course.  It would still be a challenging course with fewer topics and more time on them, but it would have a less rushed feel.  This is something we're dealing with.  I guess it's better to have more content and cut than to find our students are underprepared and have to add.  But I always err on the side that more is better.

We had other goals too, like the book being successful in any developmental classroom, not just ones with very experienced full time instructors.  So it has a wealth of support and tools to help any instructor.  Plus, it's built with the instructor in mind.  We do unusual lessons that involve things like chemistry and nursing applications and physics.  I say unusual, not because they've never occurred in a math classroom, but because they are unheard of in a developmental math classroom.  Developmental math classrooms are primarily about algebraic manipulation and a small set of contrived applications that involve numbers, investments, trains, coins, and mixtures.  Occasionally a projectile will pop up (and then down, ha) but that's about it.  Our book has one new context after the other.  New is exciting but can make some instructors uneasy at first.  I was certainly nervous the first time I taught order of magnitude.  Now it's one of my favorite topics and lessons; I can't imagine not teaching it.  So we were thoughtful to make sure the background is sufficient for the instructor to feel confident, and that the focus is always mathematical and not the context itself.  The whole point is to build mathematical problem solvers, not movers of the letter x or chemists, for that matter.

Our perspective is one of the teacher.  We worked on this book because we believe in this particular pathways movement.  That's why we haven't worked on a Statway book.  It's a great philosophy but not one that we can implement at our school or perhaps even in our state.  It was important that we piloted every lesson and really work and finesse the lessons until they are class tested, instructor approved.  If you build this course, you have to get it through your curriculum committees and state level rules.  You will likely have adjuncts teaching the course.  We love MyMathLab but feel there's more to mathematics than algorithmically generated problems.  We want students who do math on paper and on the computer.  Being in the classroom, we have empathy for these issues and have worked to find solutions.

I've been asked why we are getting a book published and not just using the materials ourselves.  Well, there are two reasons.  First, we know the hurdle of finding good materials is huge and daunting and therefore can stall out the best laid plans.  Getting this initiative off the ground matters deeply so we wanted to eliminate the worry about locating good materials.  We want instructors to be able to focus on logistics and getting the course approved, which is plenty enough to think about.  But second, we've had very positive responses to our written materials used in the classroom for years for other courses like developmental algebra, statistics, general education math, precalculus, college algebra, and math for elementary teachers.  Heather and I have long written workbooks that support the text they're based on so well that students usually don't buy the text.  They use our workbooks and MyMathLab.  Using that approach as a stepping-off point, we started researching and reading about interesting contexts and situations where math pops up.  Then we crafted the lessons, organized them to achieve the goal of mathematical maturity, and tied sets of them together to form cohesive units.  We then worked, reworked, and reworked again the book formatting so that the content functions on the page and in the classroom.  It's been a very organic process.

What is exciting now is that the book has been through our field tests for a year now.  And it branches out to other schools starting this fall.  If you are interested in class testing some or all of the book, please contact me.  That feedback from instructors and students only makes the product better.  It's been a time-consuming process writing a book, possibly only matched by all the years I worked on our redesign.  But the publishing process vets the work in a thorough way, helping create a product that works on a large scale.  The book has already been reviewed multiple times.  Add to that editors, a team who is producing it, and all the class testing so far and to come, and the work keeps improving.  It's been a labor of love, but ultimately worth it.  Our goal was not to think outside the box, but instead to throw the box away.  That's a scary thing to do because it's so unfamiliar.  But algebra books are not achieving all the goals we have in developmental classes, so we figured we had nothing to lose.  If you're searching for an algebra book, please look for one of the excellent ones that already exist.  But if you want something different that works and supports this course, we may have just the book for you.

Thursday, May 17, 2012

MLCS: Lessons learned after one year

Our spring semester and second semester of piloting MLCS is complete.  And I'm declaring it a success.  I'll share about our statistics later in this post, but let's start with lessons learned.  There have been many.

1.  We have figured out what we are doing and can communicate that clearly.  When Heather and I started teaching the course in August, it was still a fuzzy idea.  Now we know that are working to develop mathematical maturity in a very specific way.  The goal is college readiness in one semester.  The order of lessons is very intentional so that the development progresses continually.  And it works.  For example, we started working with linearity and exponential change in the first unit.  Just numerically to start, as we do with every concept, but we built to graphs, algebraic models, and problem solving with the models.  This progression has happened with all the topics be it writing and solving an equation, working with units, using proportional reasoning, or working with graphs.  Our approach to content seems like a lot of interesting activities.  But underneath is a very specific design to ensure students have the skills and vocabulary they need and are putting ideas together continuously.

2.  We've learned how valuable open-ended problem solving is.  It's so valuable and rich that we intend to count it for more in the course points and may progress to an individual open-ended problem solving assignment at the end of the semester.  Students do these types of problems in groups because they are very hard.  But we believe some of the greatest learning of the semester happens through them.  The problems force students to connect a myriad of ideas, do research, work together, write a coherent solution, and work over an extended period of time.  The mathematical lessons are great, but it's the student success skills that they come away with are also so valuable.

3.  The two most important elements for success in MLCS are work ethic and attendance.  Simply put, students must attend, period.  If they miss, they can't just watch a skill-based video or phone a friend.  The interaction that happens in the class as well as the questioning and discussion can't be easily duplicated.  Heather and I still have a goal that the course can be done in a hybrid or possibly fully online format.  We're not there yet, but it's a goal.  We want the course to function well face to face first before branching out to other modalities.  But the face to face experience requires excellent attendance.  Just like missing work at a job, there are consequences.  A student recently said that when he misses a class, it feels like a week was a missed.  And if you miss a week, it feels like a month.  I don't see that as a bad thing.  If anything, it's bothersome to me when students can miss multiple class periods of a class and there's no negative outcome.  The classroom experience should be of benefit to students.

The question comes up, "what about student's lives and the natural things that happen?"  That is a reality of working with developmental students, but the fact is you can't get out of developmental math in one semester and it be easy.  If it was easy, we wouldn't have the nationwide crisis we do right now.  If life happens and a student has to miss a lot, it will be incredibly difficult to catch up.  He or she may have to drop and try again another time.  That is a reality to any course.  Most students cannot miss long periods of time and pick up again without issue.  If they had that level of knowledge, they probably wouldn't be in the course in the first place.

4.  At our school, this course is not for everyone.  Due to the strict requirements of my state, we have a lot more content than many states will with their versions of MLCS.  So we have a giant, 6 credit course.  That's a tough thing to take on for a developmental student.  That's why at our school we have so many options like 8 week courses that move slow and steady as well as online and hybrid offerings for our traditional courses.  We offer an accelerated combined algebra course too.  It's also very challenging.  But again, you don't get out of developmental math in one semester without doing some major work.

5.  Accountability is everything.  This is something Heather and I are still finessing.  We need more measures in place to check on each student's paper conceptual homework that don't include collecting homework every class period.  We're reflecting on the semester, like we ask students to reflect on each lesson and unit, to determine what worked and what needs improvement. 

6.  The idea of getting the content eventually is a helpful approach to grading in this course.  Students don't have to ace every test to pass.  They have to work really hard and "get" the content eventually.  We used a version of gamified grading courtesy of my friend George Woodbury.  Students could earn up to 10 points on each of the 4 units.  The final exam was 100 points, giving the class total 140 points.  The 10 unit points were based on how well students did on their open ended project (2 points), MyMathLab (2 points), in-class quizzes (2 points), and the test (4 points).  For example, an A on a test is worth 4 points.  B's are worth 3 and so forth.  We will adjust the breakdown of points and what's needed to attain a 2 in a category because that's not quite right yet.  But it was still successful this semester.  It worked more in terms of mastery and a level of success instead of students constantly worrying about how many points something was worth.  The idea has really good bones but with more discussion and research, I think we'll get it to an even better place.

7.  Our placement tool is not sufficient.  We use Accuplacer and it's not enough.  It only tests algebra skills and this isn't an algebra course.  However, students can do algebra (more on that later).  We're looking at additional measures that students have to complete prior to registering, to ensure we get students who are willing to really work and/or a critical thinking test instrument as an additional placement measure.  Those ideas will be worked on this fall.  I don't need a student with incredible algebra skills.  I need instead someone who has a sufficient cognitive level for this much reading and critical thinking and who is willing to work.  We've had several students who would not consider themselves a traditionally successful student in a math class, but they've shined in this course.  I've had students tell me that they love that the course is not about pushing symbols around but that every skill has a point and every lesson is grounded in something real.  As I said earlier, work ethic and attitude matter far more than prerequisite skills.

8.  Everyone who comes in contact with these students must understand the approach and methodology of the course.  It's not enough for the instructors to understand the approach.  Anyone who tutors these students must be on board too.  I've recently worked more with faculty tutors in our Math Lab to talk about the goals of the course and how to help these students.  The most common problem is wanting to throw techniques at the student that we aren't using yet or ever.  For example, we build exponential models as a comparison to other models.  We'll ask students to solve an exponential equation, but with the graph, a table, or guess and check.  We don't take the log of each side of the equation to solve it.  Sometimes a tutor will jump to a technique like logs instead of going with the approach taught.  There's no malice intended, but it's still an issue all the same.  The main goal we have is understanding and knowing when and why a technique makes sense, not just getting every algebraic tool possible to say we can.  Pushing symbols with no real understanding of them is not our goal. 

9.  We are seeing success in a traditional numerical sense like pass rates, placement scores, and the like.  I had 55% pass and Heather had 62% pass.  Keep in mind our sample is small, but I'm still encouraged.  We also have students take the placement test at the end of the course just to see how that measure is affected.  All but 2 students in a each class went way up on their placement score.  The most common outcome was a placement score at the very upper end of our intermediate algebra range (almost college level) and several placed into college level.  There are many students who are near the college level cutoff who could be successful in a statistics or general education math course, so I'm buoyed by those results.  Our college level cutoff is really meant for college algebra and statistics and is too high for our non-STEM college level courses.  It's also very comforting to see that all the algebra we do is enough.  No, we don't do every algebra topic under the sun.  But we do cover a large amount and certainly everything they need in the outcome courses.

Beyond numbers, I see and hear so many encouraging things in this class.  Students really talk about math and mathematical ideas.  And I see progression and growth in the students as the semester progresses.  Heather and I will sometimes get frustrated if attendance is down or students are getting lazy about homework.  But then they'll say something in class that we have never seen or heard in a beginning or intermediate algebra class.  And then we know something new and something good is happening.  That makes all the work and time worth it.

Tuesday, May 8, 2012

Pathways Primer: MLCS, Statway, Quantway, Statpath

The idea of new pathways through developmental math is a popular approach these days, especially for schools interested in redesign in ways outside of restructuring the traditional curriculum.  However, there are many choices that are often used interchangeably.  In so doing, it's easy to confuse the various pathways. 

First, all pathways courses use technology in some way to augment the course and develop skills.  Second, all pathways courses are based in contextual, integrated content instead of linear, skill-based units that dominate developmental math.  Word problems and problem solving are the norm.  Skills are developed but only as a means to continue with problem solving.  Let's explore the big 4 pathways in development currently.

1.  Mathematical Literacy for College Students (MLCS)

MLCS is a one-semester developmental course designed to take students who are at the beginning algebra level and give them the mathematical maturity to be successful in liberal arts math or statistics.  It varies from 3 to 6 credit hours based on state and school curricular needs and requirements.

It works under the assumption that the student is at the developmental level, and that the goal is college level by the end of the semester. 

There is a strong algebra component to the course, especially the development we are using in our book and course at my college.  Our goal is to give students the option of taking intermediate algebra upon completion if they need to bridge to the STEM path.

2.  Quantway

Related but different than MLCS, Quantway is a one-semester developmental course that takes the student who is at the beginning algebra level and gets them ready for liberal arts math or statistics in one semester.  It is usually a 3 or 4 credit hour course that students take in place of beginning algebra.

It has the feel of a quantitative literacy course with college level problems and work.  Any prerequisite knowledge is addressed just-in-time. 

Algebra is in limited quantities, just enough to complete any problems at hand.  Students who change their major to a STEM field will need to take beginning algebra after Quantway.

3.  Statway

Statway is a two-semester integrated statistics course with developmental content addressed just-in-time.  Students who place at the beginning algebra level qualify for the course.  At the end of one year, students will have met their statistics requirement if their program of study has one.

It is a statistics course done over one year, allowing the content to progress slower.  Content is at the college level with developmental content woven in as needed.

Algebra is in limited quantities, just enough to complete any problems at hand. Students who change their major to a STEM field will need to take beginning algebra after Statway.

4.  Statpath

Statpath is a one-semester developmental course covering descriptive statistics.  It has no prerequisite.  Content is at the college level with developmental topics covered just-in-time to continue progression through the material.  Typically, it is a 6 credit hour course.

Algebra is in limited quantities, just enough to complete any problems at hand. Students who change their major to a STEM field will need to take beginning algebra after Statpath.


MLCS is not better than the other three, but it is different.  All pathways support students moving through the developmental sequence differently and more quickly.  However, students need to know their program of study and required college courses if they are to use Statway, Quantway, or Statpath to accelerate their sequence.  Otherwise, they will need to take beginning algebra after the course if they have a STEM math requirement like college algebra.  In developing MLCS at our school, we had the goal of giving students more options and a broader base in case they do change their mind.

Another difference is philosophy.  MLCS operates under the assumption that students are at the developmental level and need help, both mathematical and student-success oriented, to get to the college level by the end of the semester.  The other pathways operate under the assumption that the student really can do college level content with the right support structures, be they more time or tailored instruction.  Personally, I believe there is a small set of developmental students who could do college level content with support, but the majority of developmental students are not ready for the level of rigor and abstraction present with college level material.  I chose to work on MLCS because I believe in the philosophy of helping a developmental student move from where they are to where they want to be in one semester.

In terms of materials, Carnegie will have open source materials for Quantway and Statway within the next year.  Schools who are under their grant structure are piloting those materials now.  Myra Snell has written materials for Statpath.  I am writing a text with a colleague, Heather Foes, for MLCS that will be published with Pearson.  If you are interested in seeing materials and/or class testing them, please contact me.  Other publishers have MLCS books in development as well. 

As for technology, Carnegie is developing their own programs, MyStatway and MyQuantway, to support their courses.  We use MyMathLab with our course and it will accompany the text we are writing.


There are many choices and ways of incorporating pathways.  Your choice depends on your needs and goals.  If you are interested in developing a version of MLCS and need assistance, workshops, or professional development, please contact me.

Tuesday, May 1, 2012

21st Century goals mean tough choices about algebra

The role of algebra is a hot-button issue when considering the idea of new developmental math pathways like MLCS.  A common reaction I get is one of wanting all the content we do now in developmental math but done with more relevance and engagement while adding in new topics and skills to address 21st century needs.

The problem with this approach is that it's almost impossible.  Doing everything already in the curriculum but with a new approach and adding content results in large credit hour courses and more time, not less.

New goals mean tough choices.  So what's holding us back? 

Usually it's familiarity and history.  Sometimes it's state guidelines.  But often it's our preconceived notions of what developmental math must look like.

We've always covered certain topics in certain orders in developmental math.  Doing topics in a different order or omitting something can be uncomfortable to instructors.  But what's interesting is what happens when you make these changes.  The classroom is different, the level of engagement is different, and the learning can be greater.  Do we have to graph linear inequalities in a developmental algebra class?  What about absolute value inequalities?  Actually, no.  Look forward to where these topics will be used.  Graphing linear inequalities is used in finite math and business calculus with linear programming.  Ask any teacher of these classes and they will tell you that they have to reteach this type of graphing because students have virtually no recollection of it.  Likewise, absolute value inequalities are use in delta-epsilon proofs in calculus 1.  This is a topic that some calculus teachers omit.  Those who cover it always have to reteach this kind of equation solving because students have no memory of it.  If absolute value functions are used in precalculus, solving is rarely needed.  We graph them and analyze them but solving doesn't usually play into the problems.

We spend a great deal of time on certain topics in algebra, like these, that doesn't pay off in the long run.  Most students don't end up needing the topic and those who do relearn it in the later course through just-in-time instruction. 

But what about what they do need right now?

Right now, the students in developmental math are trying to get the skills necessary to be successful in college and their upcoming math class.  For most, that is not calculus.  Yet, we prepare them all as if that is the goal. 

Across the country, math educators are reaching a common sentiment that calculus is not the end goal for most students, nor should it be for all students.  NCTM and MAA have released a joint position statement acknowledging a change in perspective on this issue:

MAA/NCTM Position

Although calculus can play an important role in secondary school, the ultimate goal of the K–12 mathematics curriculum should not be to get students into and through a course in calculus by twelfth grade but to have established the mathematical foundation that will enable students to pursue whatever course of study interests them when they get to college. The college curriculum should offer students an experience that is new and engaging, broadening their understanding of the world of mathematics while strengthening their mastery of tools that they will need if they choose to pursue a mathematically intensive discipline.

They advocate an engaging college curriculum that is different than K-12 AND one that prepares students for the direction they're headed.  If students are STEM-bound, the traditional algebra route makes a lot of sense.  For most students, their next course is statistics or liberal arts math.  Algebra is not the predominant goal for these outcome courses.  Hence the design of new courses like MLCS that support these needs and goals. 

But MLCS does not feel like an algebra class.  It's very different.  That difference is a good thing.  Just as students have to adjust at first (and they do), instructors do too.  They have to give up old notions that developmental math must include certain topics.  Instead, the goals of this course include critical thinking, reading, problem solving, discussion, and mathematics.  I sometimes wonder where the math is in an algebra class.  Students can mimic their way through without understanding much of anything they're doing.  They're not solving problems, but exercises.  We don't push deeper to challenge understanding but instead work at a skill level.  That isn't good enough anymore.  It's comfortable for us as math teachers, but it doesn't work.  Students aren't prepared for their next course unless it's college algebra.  And many aren't really ready for college level expectations regardless.  The concept of college readiness isn't really addressed in developmental math because our time is spent looking back at high school and plugging knowledge gaps.  Heather often says we're preparing students for something they'll never be asked to do and not giving them they skills they will need.  I'm glad that MLCS is helping change the developmental math landscape, but if anything, it shows me there is more work to be done.  We need to look before and after MLCS and examine our offerings and approaches.  We cannot let history guide all future decisions.

As we wind down on another semester of MLCS, I'm more encouraged by the development of this course.  We are growing our offerings at my college and I expect that will continue each year.  Students like what happens in the classroom and they're getting what they need for their next course (math and otherwise).  Will we still tweak and improve?  Absolutely.  The work is not done.  But I'm encouraged by the progress made in this academic year alone.  3 years ago, 15 math faculty met in Seattle and dreamt of new options and new experiences for students.  Yes, that meant a lot of work and difficult choices, including the role of algebra.  But the outcomes are already proving to be worth it.