Wednesday, April 25, 2012

MLCS: A lesson with algebra

I thought about titling this blog, "where's the math?" because that's a question I get often when instructors first look through our lessons.  There aren't x's up and down each page, but there is definitely math on every one.  Since today's lesson was such a fun one and it has so much algebra, I thought I'd share the lesson pages and explain the approach in more detail.

Our approach to algebra is different than a traditional text.  It's common practice in an algebra book to group a related set of skills together in a chapter with the goal of building that skill set, say graphing.  So a traditional chapter would have this outline:
  • The Cartesian coordinate system
  • Graphing lines
  • Slope
  • Equations of lines (y=mx+b, y=b, x=a)
  • Writing equations of lines
  • Extensions and applications

In MLCS, we hit all of these topics but use the Chicago philosophy:  do it early and often.  Day 2 of the course we learn about the Cartesian coordinate system and graph nonstop throughout the course, using graphs as visuals to understand a problem or idea better.  Slope comes up early too as we discuss rate of change conceptually, then numerically, then graphically, and finally as it relates to the equation of  line.  Students write linear models from the 2nd week of class on but we don't define them as y=mx+b yet.  When they see that form (y=mx+b), many students shut down.  But just ask them to look at a table and write a formula that will get to the y's from the x's, and they'll work just fine.  Not all students like the intuitive approach, thus necessitating today's lesson.  The lesson below can be enlarged for easier viewing.
Lesson 4.8, Chain, Chain, Chain starts by looking at alkanes which are a type of carbon chain.  We expect no understanding of chemistry for this lesson.  Any chemistry information is provided.  Students explore a few alkanes and then draw many more, listing the number of carbons and hydrogens using notation from chemistry.  They then generalize the number of hydrogens based on the number of carbons intuitively.

Next we talk about slope, trying to see it in several ways (from a table, using points, from the rate of change in the physical situation, and finally from the graph).  That leads into the idea of writing a formula, using the idea of y=mx+b.  The result confirms the intuitive result.

We graph the points we create, achieving a line.  But where does the line start?  In this case, the y-intercept (0, 2) means when there are 0 carbons, there should be 2 hydrogens.  That doesn't make sense in real life.  This relates to the concept of domain.  So we start the graph at the point (1,4) and extend it from there. 

Next we work on finding various numbers of carbons and hydrogens given one or the other.  Plus, the idea of what type of number the H's will be is discussed.  The number of hydrogrens is always even.  Students have to explain why.

Summarizing and generalizing the process, student practice writing the equation of a line given a point and slope or two points.  We also cover vertical and horizontal lines.  This is the theoretical part of the lesson.

Closing the lesson, we connect the new skills to the original context, but now with alkenes.  Students study the pictures given and list the number of carbons and hydrogens.  The roots of the words connect to geometry and help make sense of the pictures (cyclopentene is a ring of 5 carbons with one double bond).  They generalize a formula.  We discuss how they did that.  Most like using the intuitive approach but we want students to see that the skill learned today (writing equations of lines using y=mx+b) gives the same result.  Which makes more sense to a student is a personal choice.

I ran out of time today so I had students finish the last 3 problems of the lesson in homework as well as work on MyMathLab to practice both theoretical and applied cases of writing equations of lines.

Not only is the lesson interesting, it connects so many ideas learned to date.  Students use patterns and inductive reasoning, graphs, slope, rate of change, equations, numerical ideas, and science.  Plus, we did a lesson in the second unit on the number of tables given a number of chairs in different configurations.  About 2 minutes into today's lesson and a student calls out, "Hey, this is the tables and chairs!"  So they remember what we did and can see the connection two months later.

So, where's the math?  It's everywhere.  :)

Tuesday, April 24, 2012

Webinar documents

Thanks to all attendees for coming to the webinar today.  Below is the slideshow with all the graphics (some did not show during the webinar).  For the handout, click here.

MLCS webinar today is open to anyone

Webinar update:  You do not need to be registered to participate in the webinar today.  Just click the link below at the appropriate time.  There will be a packet available to you with a sample lesson and much more information about MLCS if you attend.

To enter the webinar, go to

Notes from the webinar organizer:

The webinar is today, Tuesday, April 24 at
3:00pm EDT / 2:00pm CDT / 1:00pm MDT / 12:00pm PDT
We will begin AT these times, so please try to show up a few minutes early to make sure your sound is working properly.
All you need to attend the webinar is a computer with Internet and speakers or headphones. You do not need to have a microphone. Questions during the webinar will be taken primarily by text.
If you are going to have a group at the webinar (which is fine), please make sure you also have a projector and speakers for the computer. Please ask one person to be the "spokesperson" for the group in the chat window and relay comments from all group members.
Ask to enter as a guest - you do not need a username or password. It may take up to 30 seconds for us to "clear" your entry, so please be patient.
Adobe Connect might suggest you download and install a small file to make the webinar software run more smoothly. Please don't be alarmed when/if this happens. 

Monday, April 23, 2012

MLCS Webinar tomorrow

I'll be giving a webinar tomorrow for AMATYC about the course.  If you are a member and want to learn more, please register.  A few weeks after the webinar, the recording (audio and slides) will be available.  I'll post that to the blog as well.

Webinar information:

Title: New Pathways for Developmental Math: A Look into Mathematical Literacy for College Students

Presenter: Kathleen Almy, Rock Valley College

Description: Mathematical Literacy for College Students (MLCS) is a new course that is part of an AMATYC initiative called New Life for Developmental Math as well as the Carnegie Quantway project. It is an innovative way to redesign the developmental curriculum, providing pathways for the non-STEM student. The course uses integrated, contextual lessons to develop conceptual understanding and technology to improve mastery of skills. In one semester, a student placing into beginning algebra will gain the mathematical maturity to be successful in statistics, liberal arts math, or intermediate algebra. Reading, writing, critical thinking, and problem solving are key components to reaching that goal. Webinar participants will learn much more about the course as well as receive ideas for course development including a sample course outline and a sample lesson.

Date: April 24, 2012

Time: 3pm EDT / 2pm CDT / 1pm MDT / 12pm PDT

Committee:  Developmental Math Committee

Friday, April 20, 2012

A Flowchart using MLCS

To help understand how MLCS can be incorporated into a developmental math program, I've included a picture of our flowchart below.  We are using the course to provide an accelerated way to statistics or liberal arts math for non-STEM majors.  Because of the requirements of Illinois, ours is a 6 credit course.  Many states are doing a 4 credit version because they may different requirements.  One of the many perks of this course is its flexibility.  Heather and I worked to create materials that would support a variety of implementations of this course.

If a student takes MLCS at our school and decides to head the STEM route or finds out his/her program requires something like college algebra, we have them take intermediate algebra next.  That is our current bridge to STEM.  In our version of MLCS, we have quite a few intermediate algebra topics but the development is conceptual, numeric, graphic, and applied.  The focus is not on procedures.  A traditional intermediate algebra class will give the student the procedural base they need to be successful in a college algebra or precalculus course.

Ultimately, I would like to have a bridge course that picks up where MLCS leaves off and continues the work using this integrated, interactive approach while developing the needed procedural fluency to be successful in STEM courses.  Likewise, I would like a replacement for prealgebra that develops strong number sense and computational fluency.

The benefit to the approach we've taken with our redesign is that it gives everyone options, both faculty and students.  We all learn differently and our goals are not the same.  Providing options respects the differences we have and allows students to be more successful faster since their needs are being met.

For more information on our redesign and the module courses listed in the flowchart, click here for a packet.  Our geometry course is not included in the flowchart below since that requirement is specific to Illinois.  It is mentioned and explained in the packet.

Clicking on the flowchart opens it in a window with a larger version of the graphic for easier reading.

Saturday, April 14, 2012

MLCS: Addressing the Common Core

This week ended the third unit of the MLCS course for this semester, concluding with a test and reflection on the open-ended problem for the unit.  Heather and I both noticed growth in the classes.  Our students are working, they are learning, they are progressing.  Connections are being made.  And that's very satisfying because our goals are two-fold:  prepare them for the next math class they will take and prepare them for college level work.  We call that mathematical maturity. 

The place where this growth was most evident this week was their open-ended projects.  Grading them was a very positive experience.  They made connections between algebra, graphs, and numbers while solving a non-routine problem.  We don't hold their hands on these problems.  We let them be problems, not exercises.  Raise the bar but provide the preparation necessary to clear it.  Heather remarked happily when we compared the project solutions, "This is what developmental students can do." 

New courses like MLCS, Statpath, Statway, and the like aren't about lowering standards.  They raise them.  It's an unspoken truth that many developmental students pass developmental algebra without ever doing much thinking.  They learn to decode, memorize, and mimic their way through skill lists and tests.  The idea that algebra is a treadmill for the brain is pervasive but not necessarily a reality.  Certainly algebra can be that, but many students find a way around thinking if given the chance.

In these new courses like MLCS, they don't get the chance.  Students have to read and reason through new problems constantly.  Application, connection, and retention are required to move forward. 

And those goals are the same as the Common Core. 

Nationally, mathematics educators are coming to the same conclusions.  We need to update our curriculum both in terms of objectives but also expectations.  Jobs that require mindless repetitive skill are being eliminated constantly.  Today's employers expect and require much more from their employees.  They expect new hires to learn a new set of skills, integrate them with the knowledge already gained, and apply all of that to new problems.  It's not, "here's a problem I need you to solve.  Here's how you do it and 10 to practice.  Now go."  It's, "here's a problem.  Go."  We do students no favors by shielding them from what's coming down the road.  Life isn't simple.  It doesn't come in bullet points or with View an Example. 

This isn't to say that online homework systems are bad.  They aren't.  Actually, they're a key cog in the machinery of this new course.  I need time to problem solve with my students, not do 10 iterations of a skill.  But the skills have to be learned.  Online systems like MyMathLab make that possible. 

My concern is the overdependence on online systems and the conclusion drawn by many that they are the way out of developmental math.  There is no shortcut to learning.  A student can relearn a skill they didn't get in high school, but that does not equate to understanding, retention, connection, or application.  It means they can now do that skill, whereas they couldn't, period.  And I disagree with the notion that the human element equals lectures and is therefore bad.  My presence in the classroom does not mean I'm a lecturer.  I'm a teacher.  I create lessons, tasks, and problems for my students to solve and facilitate that in many ways.  Sometimes that's direct instruction.  Sometimes it's circling the room to give feedback and keep students progressing.  It's varied and complex and not duplicated by a computer. 

A friend of mine recently pointed out that advocates of computer-based lab models decry lecture only to put students in front of computers to watch lectures.  Yes, students can repeat and rewind the lecture as much as they want.  But there's even less engagement in terms of mathematics because the student can't converse with the instructor.  I believe engagement is more than two students talking in a math class or lab.  It's what they're talking about and what the goals are that matter.  In MLCS, I've seen students really debate and discuss mathematics.  My eyes have been opened to what this student is capable of.  And it's a quite a terrific sight to see.  I'd always believed they were capable of more than skill manipulation but I hadn't honestly seen it to know it could happen.  But it can.

The Common Core seeks to do the same as we in higher education want to do:  raise the bar but make it appropriate as well.  However, training, materials, and support are not yet where they need to be for true change to occur.  Placing a new cover on a textbook and saying it addresses the Common Core doesn't make it so.  To accomplish the new goals of change and growth, we have to throw out some of the structures and mindsets of old and start fresh.  It's hard for everyone at first.  But the outcomes are worth it.  Now my students don't want a different kind of math class whereas when we began in January, they couldn't imagine how this class would work.  They've been pushed and they've risen to those challenges. 

It certainly makes me want to explore what more can be done with the instructional and materials design we've developed in terms of other courses.  I've told my colleagues that MLCS has ruined me on traditional courses.  I've seen more so now I want more in all my classes. 

I guess it's a good thing to have goals.  Keeps one on her toes.

Saturday, April 7, 2012

The Ostrich Syndrome: An End to Developmental Education

Connecticut is proposing legislation to end remediation education as it currently is: a separate entity.  Instead, the idea is to place developmental students into college level classes and provide support there.

I refer to approaches like this as the ostrich syndrome.  Basically, ignore it and it will go away.  My hypothesis is that this approach will not be successful and could actually create new issues that are more serious than the current ones.  Having taught developmental and college level math for years, my concerns come from both sides of the fence.

First, this idea is against all the research done in the past 30 years on mandatory testing and placement.  Years ago, most colleges allowed to students to take whatever courses they liked, regardless of a placement test score.  And the outcome?  Students failed.  Research and practice has shown consistently students will not be successful in courses above their placement.

This is not to say the placement measurements are perfect.  They're not by a long shot.  Could some students who place into intermediate algebra try a liberal arts math class and be successful? Absolutely.  And that's why so many of us are working are new pathways to college level courses.  We want to shorten the path but we want to do so appropriately.

There are many remediation programs that can help students brush up and improve placement, potentially out of developmental math.  And data shows those students who place up can often pass those classes.  But this isn't all of the students we teach in developmental math.

The issue are those huge numbers of students who place into beginning algebra and below, and those numbers are significant.  Those students lack so much more than a few algebra skills.  They often have cognitive issues, learning disabilities, and socioeconomic issues.  They are not ready for math specifically and college in general.  So the legislature suggests placing them up but just providing additional support there.

Can that work for some?  Yes, but it's that same body of students I wrote about earlier who could probably brush up and place up.  They aren't that far from college level and the leaps to be made are fewer.  That's why some initiatives include embedding students into college level and giving them help.  I won't be surprised that those ideas can work for some.  But for the majority who place low?  I'm not so optimistic.

The premise behind the proposal of embedding all remedial level students in college level courses is that anyone can learn anything given enough time and support.  Those who teach math and science know that is almost always not true.  I know that I could spend the next 5 years working on astrophysics and it's not necessarily going to make sense for me.  And one semester with intense help in addition to all my other obligations?  Even less likely to happen.  And for some students who are at the level of "what is 3 times 3?", the idea of college statistics or college algebra is like astrophysics to them.  It's intangible.  Pretending they really are college level and just need "some help" is unrealistic and I think, detrimental to this student.  They will feel more frustrated and lose more faith in their future in college. 

Yes, we absolutely need to get developmental education in better shape.  Less money, less time, and better outcomes are the goals.  There are many schools like mine that are getting their developmental programs into tremendous shape.  The pass rates are strong and the outcomes in college level course work is too.  Plus, we now have new accelerated options, like MLCS, that update the curriculum and tailor it to the non-STEM bound population that is not served by the traditional curriculum.

The problems with the current developmental math curriculum is that is fast-paced, tedious, skill-based, a repeat of high school, not engaging to most students, and outdated.  Plus, the student is not getting the "college knowledge" skills they also need to be successful.  Learning how to add rational expressions does not give students what they need in statistics or general education math.  And it takes a really long time to learn how to add rational expressions well.  Is that worth it?

This is what I believe we need to make developmental math work:

1.  Provide course options for all levels and goals (fast, slow, self-paced, online, face-to-face, hybrid, STEM-bound, non-STEM bound). 

This is a huge population of students who don't all learn the same way or at the same speed.  They have differing goals.  Give them options that make sense for their needs.  Then they will learn and pass, therefore, getting out of the program faster while getting the skills and knowledge they need.  We have seen this work in practice at my college.  It's also a trend taking hold at other colleges throughout the country.  In essence, one size does not fit all.  Let's stop pretending it does.

2.  Provide alternatives around traditional placement.

There are some students who just need a brush-up and who will place up and out of developmental math.  Institute sound policies with flexibility.  Data supports that students can place out of developmental math and be successful. 

3.  Improve the courses we offer.

Content needs to be relevant, engaging, and yet still challenging.  It's hard enough to be in a class for which you're not getting college credit.  If it also seems to have no bearing on your life or major, your motivation to learn can drop quickly. 

Instruction needs balance between active parts and direct instruction.  This student is not the same student from 1970.  Our world has changed and students expect more activity.  Their lives revolve around electronic screens with constant activity.  Give them active learning opportunities and they will learn.

The development of content needs to move away from linear approaches that often leave students with a list of skills instead of mathematical understanding.  We need to spiral and connect topics, and bring in new ones that we don't currently cover.  Bring in statistics, talk about data, expose students to modeling.  Connect all of those topics to algebra, geometry, and numbers. 

Keep expectations high.  Put students into courses with college level expectations with some material they haven't seen before so that students are pushed and prepared.  Developmental math is more than plugging holes of missed knowledge.  It's about building a strong base from which to build upon in college level course work.  That base not need be high school all over again.  We need to move away from mimicking and repetition since jobs that require those skills are being eliminated by the day.  Instead, create courses that force students to think and work in new ways (projects, open-ended problems, modeling).  Flexibility, adaptability, agility, and critical thinking are skills that will serve them well in all their college courses, not just math. 

Support these courses by providing strong support structures.  Integrate student success in new ways and talk about traits needed to attain mathematical success.  Give the teacher time to remediate as needed individually during class.  Incorporate online systems for skill development and mastery beyond the time that is provided in class.  Give the student the help they need, but in a course designed to push them up to college level.

4.  Train faculty to work with the developmental student.

Provide workshops, mentoring, ongoing support and collaboration for the faculty teaching this population.  The student has unique needs, necessitating faculty prepared to deal with them.  They are not college level students yet.  We can get them there but it takes knowledge of their challenges to get there.

I believe MLCS is a good first step in the direction of making significant change to developmental math in ways that embody the four points made above.  It's not the magic bullet, but a sign of progress and change.  We need more innovative options and practices, more pilots, more testing, and continual refinement.  I know from practice that this approach can get a program to a successful level and keep it there.

But throwing the baby out with the bath water doesn't solve the problem.  It simply creates new ones elsewhere.  Those who teach college level courses will be overwhelmed with students who are simply not ready to sit in those courses.  Those students tend to stop attending or make their frustrations loudly known during class, negatively affecting other students.  I've taught classes where the placement was wrong.  As a teacher, one of your options is to forge ahead and leave many behind.  Frustration, irritation, and complaints ensue from students who are not prepared.  The other option is to work with the student at their current level.  Before long, the teacher is not teaching the content of the course, but the course below it:  developmental math.  The complaints will also occur with this approach, but in this case from the student who signed up a college level course and is not getting one.  Ultimately, this approach can water down our college level courses so that they mean little.  They become the developmental courses students were getting before.

I wish developmental education was not necessary, but the fact remains that it is.  Ignoring it is like noticing giant cracks and crumbing rock in your house foundation and choosing to just build a bigger house on top of it so the cracks aren't seen.  Eventually, the cracks make themselves known, but the problem is even greater than originally.  That's not an outcome I want to see.

Wednesday, April 4, 2012

Upcoming MLCS Webinar

I'm giving a webinar for AMATYC on April 24 at 2 pm CST.  Here is the description and registration information.

Title: New Pathways for Developmental Math: A Look into Mathematical Literacy for College Students

Presenter: Kathleen Almy, Rock Valley College

Description: Mathematical Literacy for College Students (MLCS) is a new course that is part of an AMATYC initiative called New Life for Developmental Math as well as the Carnegie Quantway project. It is an innovative way to redesign the developmental curriculum, providing pathways for the non-STEM student. The course uses integrated, contextual lessons to develop conceptual understanding and technology to improve mastery of skills. In one semester, a student placing into beginning algebra will gain the mathematical maturity to be successful in statistics, liberal arts math, or intermediate algebra. Reading, writing, critical thinking, and problem solving are key components to reaching that goal. Webinar participants will learn much more about the course as well as receive ideas for course development including a sample course outline and a sample lesson.

Date: April 24, 2012

Time: 3pm EDT / 2pm CDT / 1pm MDT / 12pm PDT

Committee: Developmental Math Committee

Click here to register.