Sunday, December 11, 2011

Pilot Recap, Week 16: Lessons Learned (Part 2)

Last week, I blogged about the lessons we learned on a large scale items (pedagogy, role of technology, metacognition, etc.).  Today I'll add to our lessons learned but this time in terms of the course and structure.

We surveyed our students continually throughout the semester and also had informal check-ins with them to find out what was working and what wasn't.  That helped us tweak as we went and also maker bigger improvements for next semester.  During those surveys and chats, we started to inquire about what they could have done to make the experience better.  Yes, we realize that we had a huge impact on the success of the course since it's brand new.  But students are a part of this learning equation too.  What they said was interesting.

1.  MyMathLab homework matters and helps to develop skills but the paper homework matters more.  Many of our students wanted to gloss over it because we didn't collect and grade each problem for correctness.  This was the place where went "old school" and used the approach used by many countries who are successful with math eduction.  That is, give students fewer but more involved and more meaningful problems and assign all of them.  Some paper homework assignments may have only had 5 problems but they were 5 tough problems that would take some time to make sense of.  And that sense-making process is where the learning happens.

2.  Never, ever skip class.  We had an attendance policy but it needs changing.  Still, it did help to get students to class.  But they found that they had to be there regardless of a policy or not.  In a course like this, you can't just "get notes and do the homework."  Much of the learning experience involved the students.  In a typical lecture, the instruction happens through the instructor.  But much of our instruction happened when students were working hard problems and talking to us and each other about them.  Yes, we had notes and reference content on paper but nothing was boiled down to "here's a skill, here's 6 examples of it, and now there's 30 problems with multiple variations."  What happened in class just couldn't be recreated outside of it.  I don't think it necessarily can in a traditional course either.  But that fact was glaring in this course.

3.  Apathy will kill your grade.  If they didn't care enough to show up, get help when things got tough, or even try the homework, they were sunk and quickly at that.  Again, this is not all that different from any other math class.  But where I see that most often is in my college level classes because the content is new and harder.  In developmental classes, they've seen the content before.  So they can ride on their memories from high school, help from their friends, MML, etc.  We did topics that their friends had never seen either so they really needed to work on the content and get help ASAP. 

4.  Work more hours than you think you need to.  We told them the college rule of 2 hours of work per week outside of class for every one hour in class.  But they ignored us and tried to get by with less.  And when we asked them if the time spent studying mattered, it was an unequivocal yes.  They wished they had spent much more time each week.

Basically, the premise of this course is to take care of all their developmental math in one semester.  Well, there is no free lunch in life.  If you're going to do a year's worth of content in one semester, you're going to have to work at it.  And no amount of us nagging or reminders about the obvious rewards (much less time and money in developmental math) mattered.  In the end, it comes down to them.  We had a lot of students who were successful in the course.  And we had a fair number who withdrew or failed. 

5.  Points, even extra credit, were not strong motivators.  This was such a disappointment to us.  All our 'tricks' learned from years of teaching just did not work.  We offered bonus points for various things and opportunities for extra credit with some special assignments.  We had a points structure with lots of points for all the various things we did beyond tests.  And in the end, few got any bonus points.  They just didn't care enough to do what was needed to get them.  I hated to see that but that again shows where they're just not quite college ready yet.  We wanted them to learn the value of attending, working hard, and aiming higher.  And some just wouldn't do those things.  You can lead a horse to water...

Still, better learned now than in a college level class.  That's the point of developmental courses:  develop the content and student success skills to be successful in college.  It's not just about math.  So we're trying a new points structure in the spring based on a friend's approach with course contracts and a gaming principle.  George Woodbury has been trying and refining his use of these approaches for a while.  With his ideas and knowledge of our students and this course, we've developed a new structure that encourages work.  Because honestly, that's what we needed them to do more than anything.  It also is built on the concept of learning the material eventually.  Yes, students need to pass each test but they also need to eventually make sense of the material by the end of the semester.  We tried to encourage that this semester by letting them do test corrections on their exams and offering bonus points towards their final exam.  Again, not that many actually did this.  Frustrating but a recurring theme to this semester.

All of these lessons were hard but necessary if these students are going on to a college level course next semester.  I don't hold my students' hands in my college level courses and neither do my colleagues.  I hate to see a student not get these lessons until it's too late, but I can't make the learning happen for them.  We learned a lot this semester, good and bad, and so did they.

We've got finals this week and then it's on to planning for next semester.  I'm encouraged about it.  One, because I'm an eternal optimist and two, because of what our students said on Friday of last week.  "This is a great course."  "Definitely keep offering it.  We learned so much."  "We really learned how to apply math."  I know we're on to something good.

My blogs will continue with updates about materials and the course so please keep checking back.  The first pilot is complete but the process to make this course come to life is not.

Saturday, December 3, 2011

Pilot Recap, Week 15: Lessons Learned (Part 1)

This week was the last week of new content.  We decided to spend next week (the last week of regular semester) to work on getting ready for the final exam.  It's not a typical thing for either Heather or me to take a week to do that in any class but we thought it could help them do better on the final exam.  They will work on MML and paper problems that recap the course and cover all the key content.  And in the spirit of full disclosure, we're exhausted after this pilot so this idea for the last week was a welcome relief.

So what did we learn this semester?

There's not enough space to list it all so I'll be brief but still use two blog posts to cover it.  In short, we learned a lot.  Here's the first half:

1.  As I've said numerous times on this blog, you can make a really hard, rigorous course without all the algebraic skills of a typical beginning and intermediate algebra course.  We did that and then some.  The course was a real challenge for students.  We found ways to make it more accessible but the fact remains, it's quite hard to read and solve word problems all day, every day throughout the course.  But it's a worthy challenge.

2.  It can be really fun to do real mathematics with developmental students.  Not canned problems, not rote skills, but real, rich mathematics.  Problems that don't have a quick, pat answer.  Situations that are fuzzy and ill formed.  The discussions we had and the engagement of the students was really enjoyable.  They were thinking and talking and learning about the power of mathematics.  That's valuable regardless of passing or failing or what happens next semester. 

3.  Balance is everything.  We finally got there but wow, was that a rough road.  Finding just the right amount of MML vs. paper homework, lecture vs. group work, open-ended vs. single solution problems, skills vs. applications, the list never seems to end.  Basically, this course turned our regular course approach upside down.  So all the things we knew how to do well inside and out didn't work so well anymore.  Finding ways to take students with us on this journey and make it work for all involved was hard but now I feel like we know what works.  Or at least we're a whole lot closer to that point.

4.  It's frightening to see how little numeracy and literacy skills our developmental students have.  But I tend to not to focus on that.  We get what we get and there's no point in getting upset over it.  Instead, we consistently challenged them to read, research, and think in ways they hadn't.  We stretched them.  For that, I'm proud.

5.  Developmental students really lack metacognitive skills.  They don't know how they think or learn.  We constantly asked them to take a problem that started out in words and translate it 3 more ways:  numerically, graphically, and algebraically.  But when we were done, another goal was to find out which way made sense to them.  While we worked and worked on that, I think more work needs to occur there in an overt way.  We focused so much on other student success skills, not knowing that metacognition would be a key one throughout.  Usually student success is about attitude, test strategies, and time management.  Those are included in our course but so are ideas about majors such as STEM fields, how to calculate and understand grading practices, and how to understand themselves and their learning processes.

6.  Technology need not be that lightening rod of controversy it often is in mathematics education.  Our approach was simple:  use technology the people do in the workplace.  That is, they use whatever they have or makes the most sense given a situation.  So that's usually a calculator (scientific, could be graphing or not) and a computer but it could just be a pencil.  We used Excel nonstop, and students really liked that.  The graphing calculator was zero advantage to the students who had it.  We're not doing a lot of "graph:  y=3x + 2" so it's no leg up to have one.  And the numeracy required is so consistent and involved that having a calculator didn't mean they had all the answers.  Plus, our tests pushed them to explain calculations and delve deeper, beyond simple calculations.  The calculator was just a tool, not a crutch.

7.  My last lesson in this part is for all those instructors out there who are feeling disenfranchised by some of the approaches of redesign which seem to be replacing us with technology.  Computers are just tools.  The human element matters in learning.  It always has and it always will.  The role we had in the classroom was essential.  We lectured some, not a lot, but were constantly questioning them, challenging them, and keeping them moving forward when they got stuck.  We weren't guides on the side or sages on the stage.  We were their teachers, in every sense of that word.  Heather and I both walked around every inch of our classroom every class period, moving to where we needed to be and doing what needed to be done.  Sometimes that's whole group, sometimes it's small group, sometimes it's with individuals.  Whatever was necessary, we adapted. 

I truly enjoyed it and feel re-energized towards teaching.  I've also joked that I'm ruined on traditional classes.  Now that I've had the opportunity to teach a course that every aspect is real and relevant, that's interesting and not contrived, and that students really enjoy, I can't go back to teaching traditional algebra face to face.  I may continue teaching it online and will definitely continue teaching statistics, general education math, math for elementary teachers, and calculus face to face.  I need to get college algebra using this approach but that will happen down the road.  My goal is to take this approach to as many classes as I have the time and energy to do so.  It's just too rewarding for them and me not to.

That's all for now.  Next week, I'll recap the lessons we learned about the course structure like grading, advising, etc.

Friday, November 18, 2011

Pilot Recap, Week 13: Winding down

We're making our way toward the end of the course now.  Next week we have our last unit test.  After Thanksgiving, we plan to test a few additional lessons that hadn't made their way into the current units yet but will in the future.  The last week, we have a big project planned to pull the entire course together.  It will be an in-class, open-ended problem that students solve together as a group.  They will also have an individual component to turn in to ensure accountability.

I was out of town for AMATYC last week and sick at the beginning of this week, so there's less to report.  Still, it was a good week.  We explored viral marketing with students, building exponential growth models and exploring social networking.  The statistics are amazing in terms of how much time is spent worldwide on social sites.  A few years ago, it was a fun past time.  Now, it's a major means of marketing and sales for companies.  Analyzing that was interesting mathematically and context-wise.  Students definitely liked discussing it.

Today we talked about order of magnitude, pH, and the Richter scale.  It was really interesting showing students how some contexts have really large and really small numbers in them simultaneously.  Making sense of them and graphing them can be very difficult.  Going to new scales based on order of magnitude makes the process accessible.  It's interesting to show students the human side of mathematics, how we take numeric truths and use techniques to make them easier to deal with.

Heather and I can see the light at the end of the tunnel.  A great deal of learning occurred this semester for our students and us.  We've got lots of ideas for the spring to make things even better in terms of grading, testing, and helping them adapt to the level of work.  The content and lessons have really worked well and those that haven't are in the process of revision.  I believe we'll get this course to an even better place than it already is, which is pretty good so far. 

Tuesday, November 15, 2011

AMATYC Workshop Handouts

Below is the packet we used during our workshop on MLCS.  It contains the course description as it is being used in Illinois, topics in each unit, implementation options, and a sample lesson.  Based on your state or needs, some topics and lessons can be omitted without a negative impact.  We are using a big version of this course (6 credit hours) but it is easily scaled down to a fewer objectives and credit hours.

One of the things we talked about during the workshop was the trouble Heather and I had when choosing a sample lesson.  We can't pick just one lesson that encompasses all the great facets of this course.  So this lesson does not contain theory that is being developed.  It comes after a lesson with theory.  But it does showcase the relevance, difficulty, and multiple approach process we use when developing the content.  Also, the homework with the lesson looks brief.  However, it isn't if the student works all the problems as intended.  Also, there are other components to the homework not shown here such as online homework.

Keep in mind that the lessons are still being edited and are not in their final forms.  But they still give you a taste of our approach.  Looking at the presentation below will also give you more snapshots of how we approach the content.

MLCS AMATYC 2011 Workshop

Saturday, November 12, 2011

AMATYC Presentation

Below is a link to the slides used in the presentation.  They're not the zooming variety since that takes a special app (pptPlex) in PowerPoint to see.  But all content shown today is included below. 

If you are interested in seeing more lessons and/or class testing, please email me

AMATYC 2011 Presentation

For more information on the MLCS pilot, including video and weekly updates, please keep reading below.

Friday, November 4, 2011

Pilot Recap, Week 11: Intentional development

I feel like we got back on track this week after the test.  Basically, Heather and I sat them down said, "it's time to work."  We gave them specific strategies for learning the material each class, after class, before tests, and after tests.  Not all that advice was new but it was repeated and elaborated on so students could use it to their advantage.  We built in some small incentives but the main incentive is passing gets them into a college level class next semester.  We reminded them that had they taken beginning algebra this semester, they would have had 3 semesters before college math (beg. algebra, geometry, intermediate algebra).  Illinois requires geometry but this course satisfies that so it really shortens the sequence if they will work.  Otherwise, it doesn't.  Right away, I saw a huge improvement in terms of work, questions in class, and questions between classes.  I think it being this point in the semester wakes them up too.  Time is running out to raise their grade.

On another up note, the lessons for this week worked excellently.  Debate was lively, students were engaged, and progress is being made mathematically and in terms of student success.  This week we saw a recurring theme that is good to see, especially in the midst of the constant mistakes lessons learned this semester.  Our development of content has worked incredibly well.  Our approach has been go slow in terms of how long it takes to accomplish all facets of a topic.  Also, start with numbers and learn them well before attempting algebra.  For example, with equations and proportions, we started solving numerically for a while and then eventually solved equations with algebra.  But we stayed with proportions using numerical techniques for quite a while.  This week we got a proportion that couldn't be solved with numerical scaling, so we used algebra.  Eventually, cross products being equal came out of it.  And they were easy.  But what was great was that students had no difficulty setting up proportions or using units by this point.  And cross products are easy to find and solve.  Building a good foundation proved incredibly helpful.  By the time we got to something that required more theory, that was easy.  Also, they saw value in both techniques (using proportional reasoning and using algebra).

Same with dimensional analysis.  That topic is usually forced on students and they resist.  They just want to multiply or divide.  Plus, we do it once and rarely do it again.  Our entire semester has had problems with conversions.  So we started the semester learning how to do them just in terms of multiplying and dividing as well as judging which operation is appropriate and if our result makes sense.  This week, we got some conversions that were very involved but asked them to use that technique (multiplying or dividing) to answer the question.  And it was very frustrating.  So we offered dimensional analysis and it was very welcomed.  Not only did they use it, they understood it, and have continued to go to it since. 

Again, as a math teacher, that is music to my ears.

We closed today with a lesson that integrates lots of content.  That's something we do constantly as well.  We wanted to know if the reduction in plastic made by water bottle manufacturers amounts to much and if so, how much?  We couldn't find the volume of a water bottle cap so we had to use its density and weight.  We weren't given the original amount of plastic, just the new amount and the fact that it's 30% less.  Once they found the original amount, they found the amount saved for one bottle and then all the bottles in the U.S. for a year.  That number is large so we converted it from cubic centimeters to cubic yards (still too big) to Olympic swimming pools (9, by the way).  Not trivial, interesting, and integrated.  I'll take that every day.

Heather and I will be AMATYC next week presenting a workshop on the course.  We hope to see you there!

Sunday, October 30, 2011

Going to AMATYC? Experience an MLCS lesson

If you will be attending the 2011 AMATYC conference in Austin next month, consider our workshop on Saturday November 12 at 10:45 am to 12:45 pm.  Anyone can attend; there is no advanced registration or special fee.

Below is the description from the program guide.  We will explain where the MLCS course (in the Quantway pathway) came from and what it includes, showing examples as we go.  We'll take a large portion of the workshop to do a lesson from the course so that attendees can experience how the classroom feels.  We'll debrief that lesson, talk about the pilot, and ways the cousre can be implemented in a school's sequence.  There are many options including some very simple ones for emporium redesigns.

We hope to see you there!

Mathematical Literacy for College Students: Bringing New Life to Life

New Life is an AMATYC initiative to create a new experience in developmental mathematics. This workshop will explain the project history as well as allow attendees to experience the new course, MLCS, by participating in a class activity. Attendees will receive ideas for course development and a sample course outline.

Friday, October 28, 2011

Pilot Recap, Week 10: Tough, but necessary

Great class periods with exciting content?  Check.

Engaged students appreciating math?  Check.

Improved performance? 

Nope, no checkmark there yet.  That's where Heather and I sit now, dealing with the frustration of students not accomplishing what we want for them.  The open-ended problems we graded for this unit were excellent.  I saw real progress made with them in terms of students building models and using graphs, algebra, and numbers to solve a big problem at hand.  They enjoyed the problem too.  But today was test day and that's altogether different.  Showing what they know individually without being able to work with someone else or look something up is a real challenge, but a necessary one.  I enjoy students talking and working on math as much as anyone but I also want to see what they can do on their own.  Whether it's a test or quiz or in-class assignment, as a teacher I've got to have that kind of information to assess students.

After two hours of Heather and I hashing out the issues at hand and solutions to them, we know that we can't solve all of them ourselves.  Sure, we'll do more quizzes in MML and more time in class discussing problems they've tried for homework.  We'll work more on metacognition, something many incoming freshmen lack.  But a big part of the problem is lack of motivation on the students' parts.  Yes, they enjoy class.  But going home and doing work on their own is work, hence the name.  They have MML assignments and conceptual work that is really challenging, but again, necessary.  That's where learning happens, not when I'm speaking but when they start doing.  And some are just not doing what they should.  You can lead a horse to water...

So what's the take-away?  The course is solid and has phenomenal potential.  We see that and so do students.  We've gotten comments like, "this is first time I could see why we do the things we do in math."  "This is the first time math has really made sense."  But we are dealing with a student who is not really ready for college and not just mathematically.  They still have skills to learn and habits to form.  That process is tough but necessary.  We are actively cultivating the skills, thought processes, and behaviors necessary to succeed in this course, college, and work.  Still, a lot rides on them.  I have the same challenges as a parent, that you prepare them but ultimately they have to sink or swim on their own.  And it's a painful process to watch when it doesn't go well.  But, like I do with my kids, I will be there on the sidelines when they come to me for help.  I will do my part and eventually, I hope, so will they.

Saturday, October 22, 2011

Pilot Recap, Week 9: Attractive Algebra

We reached a new milestone in the MLCS pilot this week.  Students actually wanted to use algebra to solve a problem when they weren't required to.

Upon hearing that, I immediately thought:  I can retire now.

But seriously, I have never had an experience teaching developmental math where students wanted to use algebra.  In a traditional curriculum, we force it on them promising a world of brain calisthenics unmet by any other means.  They balk because they don't like it and find it useless.  Whether it's useful for everyone is a matter of debate, but there are many folks like myself who will use algebra to solve problems in real life because we know how to.  I see it as a tool in my tool belt along with pictures, graphs, numerical approaches, and technology.  And now, so do my students.

This week began with us solving all kinds of equations physically with manipulatives and then in written form.  Students did not like using manipulatives at first but we forged ahead, telling them, "this is a broccoli moment.  You may not like it but you need it."  And it did help.  They understood why we subtract something from both sides or add it.  They could see why we don't divide off 2x on both sides when finishing a problem that ends with 2x = -16.  They saw that adding 3x to both sides is not the same adding 3.  But our goal in the course is not algebraic manipulation for the sake of it.  We want to solve big, interesting problems and let the algebra come along for the ride.  And it does.

So once we had established how and why we solve equations like we do, we got back to the task at hand:  solving a problem and determining if we should write an equation or use numbers or a table or graph.  And that's when we heard something very satisfying for a math teacher:  "Can we use algebra to solve this?  It would be so much easier."

Why yes, you can.

We started with a fun problem setup that was completely real:  you only have so much money (a $20 bill) and are heading to a restaurant for dinner with your friends.  How much of something can you get for your money?  We started simple with just the food items and then built all the way up to incorporating tax and tip.  Not calculating tax and tip, but solving a problem like this:  you have $20, you want to figure out how much of an item you can buy knowing its price and that after you purchase it, you'll add on 7% tax and a 20% tip.  That's not a trivial problem.  And we told them:  solve it with just numbers and then with algebra.  And along the way, we asked them to determine which method they preferred with each problem and why.  When we finally got to this scenario (with tax and tip), students could not see their way out of it numerically.  But they could with algebra.  This is the equation they built and then solved:

1.20[1.07(1.50 + 5 + 0.25w)]=20

And without question, the solving was the easy part.  It was creating that monster that was hard.  By the time we got to the solving, they were breathing a sigh of relief. 

But the best part was that they were completely engaged and really thinking.  One student said, "my brain hurts but I like this."  Heather said it's like they're sweating on the inside, meaning really working their brain.  I used to say to my algebra classes that algebra will do that but often it doesn't.  It just frustrates students and reduces their motivation to enjoy and succeed at math.  When you tell them they can solve a problem however it makes sense and not force a method on them, they'll gravitate towards the things (like algebra) that we know can be quicker and more useful.  It's just human nature:  we like choice, we don't like mandates.

We ended the week with more problem solving that resulted in equation building, graphs, looking at linearity vs. other types of graphs.  And then out of nowhere today, we saw a connection and had them build a rational function.  We explored fitting data with the best type of function and how the graph doesn't tell you the whole story about data.  But a table and an equation together with it can tell you everything you need to know.  Again, algebra came out of nowhere (seemingly to them but we had planned as such).  We had a situation that we wanted to understand more about and doing the calculations to be able to make the graph was getting time consuming.  So we went to Excel.  They've started to see the power of Excel.  We talked them through programming the columns to do the calculations we wanted, filling in cells, and then graphing.  To do that programming, they have to understand the calculations at hand to generalize them. 

It's amazing how motivating something can be when it has an immediate point.  I love math just for its own sake, for the beauty of it.  But most of my students don't and need more to keep them interested and willing to work.  Just having a real reason to do something that isn't contrived can go a long way to keeping them in the game and moving forward.

Altogether, a good week.

Friday, October 14, 2011

Pilot Recap, Week 8: They get it

That title sums up where Heather and I have gotten to:  they get it.  We have finally gotten to the point where students have completely bought in and more than that, they like it.  Meaning, they like the course, the approach, the goals, and the execution.  Sure, there are about 1000 things we can do to improve and we will.  That's the nature of a pilot with a brand new course.  But in the meantime, they like coming to math class.  They like seeing what's going to happen next.  And here's what I care about:  they are learning. 

A sample of what we did this week:

Topic: Study census data on median weekly earnings vs. unemployment rates as well as educational attainment vs. unemployment rates.
Goal: Develop scatterplots, idea of association, best fit lines, predictions. Discuss how level of education affects potential earnings and likelihood of unemployment.

Topic:  Origami done in a roundtable group relay
Goal:  Analyzing the role of steps, their order, precision, and teamwork

Topic:  Looking at proportions within recipes and chemical equations, balancing chemical equations
Goal:  Work on developing proportional reasoning; see similarities between chemical equations and algebraic equations (terms, coefficients, checking work, strategy for starting a problem)

Topic:  Students measured an ingredient by volume, then weighed results.  Found standard deviation.
Goal:  Looked at role of variability and how it can measured, that measures of centers are not enough to tell the full story about data.  Used numeracy and algebra to develop the standard deviation formula, order of operations to utilize it.

Topic:  Analyze problems and write 1-step equations to solve them.  Solve them numerically and compare/contrast methods.
Goal:  Understand basic priniciples behind solving equations to set foundation for more involved problems next week.  Analyze the role of numerical and algebraic methods, when each has an advantage.

This is just a taste but I think it shows the variety.  We've found a way to make the topics connect within a unit and have some relationship to each other.  We're always moving forward, building their numerical and algebraic skill set.  But with that, we work on proportional reasoning constantly, functions, and bring in geometry and stats whenever possible. 

If you're wondering how origami could end up in a unit that has solving equations, the idea is this:  show, don't tell.  We want students to understand that you can't "wing" everything in real life.  Sometimes order, accuracy, and precision really matter.  I can say that till the cows come home; it will mean little.  But put students in a situation where they experience that firsthand and all of sudden, the message is very clear.  And beyond that, it's just fun to see 20 year olds fold paper boats.  :)

Tuesday, October 11, 2011

Emporium Models & Quantway/Statway: Living in Harmony

It may seem counter intuitive that these two initiatives could work together, but I believe there is a way to make that happen.  Most faculty are in one camp or the other.  That seems logical since emporium models are about lab-based learning, working on an individualized program of study at one's own pace.  The Pathways initiatives are about students working together in a classroom at the same pace.

So how could they marry and live happily ever after?

Well, consider this:  both need the same things but offer them in different percentages.  Both want to offer the student a new experience that serves their needs more than the traditional lecture-based approach.  How they achieve that is different, for sure.  But both want students to get the skills and connections.

Emporium models focus heavily on skills and at many schools, solely so.  But ask instructors and students and they will say that they do want interaction, they just don't want lecture.  Students want to gather together and instructors want to talk with them as a group.  But problem solving together would be the ideal, not just answering questions as a group to students who are at different places content-wise.

The Pathways models focus on group interaction and problem solving, but skills have to be included.  Skills exist in the classroom as does lecture, but in lesser proportion to problem solving and connections.  They're often reserved for time outside of the classroom so that students can spend as little or as much time as they need.

Both models use interaction for connections and online homework systems for skills. 

In our course, we've already seen students need more skill work than we can do in the classroom, so we're beefing up the online component to allow for that.  Emporium schools often complain that they need more than just skills so they try to beef that up with a once weekly problem solving session.

How about the best of both worlds?

Emporium schools could use Quantway or Statway lessons during their once-a-week problem solving sessions to make connections and solidify understanding.  Students who haven't seen the topic will get an introduction in a way different than the static, online lecture.  Students who have learned the topic will make connections.  Everyone gets something out of it even though they're on different skills they rest of the week.

Pathways schools could have once-a-week lab time just for skills.  That would give students who need more time for skill development just what they need along with the instructor there to support their individual needs.

What I like about this idea is that it gets the best of both worlds and would not be that involved (or costly) for either camp to implement.  What do you think?

Saturday, October 8, 2011

Pilot Recap Week 7: What's STEM got to do with it?

One of the first reasons for implementing the MLCS course was the desire to tailor the traditional developmental math curriculum and move away from the one-size-fits-all approach.  There is a large sector of the community college population who are not STEM (science, technology, engineering, mathematics) bound.  Many students want an Associates in Arts degree of which typically has a statistics or general education math requirement.  I believe, as do many other faculty members, that intermediate algebra is overkill on certain topics for these students but that it is light on topics they will need.  So MLCS has been a great option (so far) for this population. 

However, an interesting discovery was made this week.  Heather has a strong background in science with a degree in chemistry in addition to mathematics as well as a specialty in physcial chemistry.  Watch her teach and you'll see it's clearly a love.  I respect and value science but favored pure mathematics when I was a graduate student.  We wanted to blend these loves into the MLCS course by inserting mathematical theory when it made sense but in an accessible way.  So far, that's been fun.  We've shown students why we need negative numbers, zero, fraction, percents, decimals, irrational numbers, and even complex numbers.  We've gotten at ideas of proof, again in unexpected ways.  That's just a sample of the math side.  For the science side, we are embedding lessons and problems throughout that have a scientific focus.  This week we worked on ions and atoms, finding the numbers of protons and electrons based on the charge.  Next week, we'll be balancing chemical equations and in the next unit, the majority of lessons have a science flair.

The discovery was for us more than our students.  We realized how much science we are bringing in which is so valuable to students regardless of their future paths.  Those who will major in non-STEM areas need to know more about STEM fields and have scientific literacy as well as math literacy.  For our students who will head towards STEM fields, which is a fair number, they are getting their feet wet in situations that aren't contrived but are very real.  They're also getting a preview of topics to come in their science courses with time to understand the math behind those topics.  By the end of this course, students will have seen a wide variety of topics from biology, chemistry, and physics. 

Additionally, we are doing some STEM recruitment as well. We have a lesson that will allow students to study STEM fields in terms of their earning potential, unemployment rates, and skills necessary (both math/science related and not). These fields are so valuable to the U.S., so we want students to have a chance to see what they entail, that they are interesting, and that they have strong salaries to boot.

Just something unexpected in this process. I'm not sure why I'm still surprised by things like this but I am. This process has been so organic and non-linear. The act of discovery and problem solving, like we use in mathematics, is a very satisfying thing. Not easy, but incredibly fun.

RVC Developmental Math Redesign 2011

I've updated a PDF packet I made a while back that gives you a brief summary of what our redesign looked like in terms of broad ideas and many specific ideas, our flowchart, and our data.  I've added information to this packet about the new MLCS course and how it is being incorporated.  Check out the link below to download the packet:

RVC Developmental Math Model 2011

Saturday, October 1, 2011

Pilot Recap Week 6: Hitting our stride

This was a great week in the pilot because for the first time, every lesson had the right mix of instruction vs. small group work.  We had challenging problems that drew their interest, enough theory to explain how to complete the problems that arose, and time to solidify skills with additional problems.  The biggest challenge of this whole project is determining its structure and order.  There isn't any course like it anywhere or any book to model off of so we just have to try, fail, correct, and try again.  Some days that's tough because solutions are so obvious in the classroom whereas they're not so apparent on the page. 

One really fun thing students encountered this week was the difference in finding a measurement in real life compared to finding it on paper.  Students are often put off by traditional math approaches that try to keep everything on paper.  They want to experience the situation at hand, not just talk about it.  But when we did that (find measurements physically), they could see there are all kinds of additional issues that come up.  Solving them makes for good discussion and an interesting class period.  They could also see the advantages of paper:  it's often more accurate and it can be faster.  That's not to say that finding a real life measurement isn't valuable.  It is.  But it's worthwhile to talk about the advantages and disadvantages of each approach as well as when each is the most useful or appropriate.  Like calculators and computers, the less we are completely dependent on one means of solving a problem, the more skillful we become as mathematical problem solvers.

We also realized that we are flipping the classroom in our own way.  MyMathLab is used throughout the course but for skills only.  We don't have 10 skill examples written out on paper in any lesson, like a traditional text would.  We do a few key skill examples, which Heather refers to as naked problems:  problems stripped of all context.  But practicing more of those can be done outside of class in MyMathLab own their own time.  If they need 10 minutes or 2 hours, they can get that.  What the classroom provides is a completely different experience, one that is built on talking about and doing mathematics, not skills.  There is a movement nationally to put students in a computer lab and have them complete skill problem after skill problem, all the while calling that mathematics.  Certainly skills matter but they are not enough.  Being able to recognize when to do a skill in a context is one of the key uses of mathematics.  I constantly tell my students, no one is going to come to you in real life and say:

Simplify:  3 - 2(x - 1) + 8.

It's not happening outside of a math class or math test.  But will they be faced with complex problems where having mathematical skills could serve useful?  Yes.  That's why I like this course so much.  They are talking, arguing, and explaining mathematics to each other, not skills.  We had problems yesterday that were pretty meaty and within them they had to simplify an expression like the one I listed above.  What was interesting to see was that virtually no one had trouble with the algebra; that was the easy part.  The hard part was solving the original problem at hand that required generalizing a situation into an expression.  But they figured it out and it was pretty satisfying for them and us.

Next week wraps up another unit and we test.  I'm not as nervous as the first time around.  They are getting the goal of the course and how it operates.  But I still keep my fingers crossed that they'll do well.

Wednesday, September 21, 2011

Pilot Recap Week 5: Striking a balance

I'm writing early this week because I'll be leaving soon to present at Pearson's Course Redesign conference in Orlando, Florida.  Instead of giving our students the day off on Friday, we've provided them with the classroom and time to complete their second open-ended problem.  I can't take credit for that idea; it's all Heather's.  We're not at the point where we can have subs yet, so that solution makes use of the time and teaches students a valuable lesson in time management.  Beyond content, one of our primary goals in the course is to help students understand what college level expectations and work look like.  The better we do at that, the better prepared they are in the spring when they head to a college level math class.

Now back to this week.

There has been a prevailing theme for a few lessons now:  our balance is off.  We see how much students enjoy being given challenging but accessible problems to solve with their groups.  But we have to have some mini-lectures to teach new skills.  We started the course with more of a block approach:  they work together for a period of time, then we work as a class for a little while, then they go back to their group to apply knowledge.  Now we're doing more back and forth (whole class/small group) after every few problems and it's just not as effective.  They get antsy to keep working together and I feel like we're drawing things out too much, interrupting the flow.  Luckily, my co-author is the queen of logistics and efficiency.  She can easily see how to regroup certain problems and streamline others to make the class time better.  It's funny that neither of us noticed this when we were writing but that's what the classroom does:  it shines a spotlight on all the weaknesses of any print materials.  We joked that if this were a traditional class with someone else's book, we'd be saying, "What were they thinking?  Good bones but needs some reorganization."  Unfortunately, those people are us.  Hindsight's 20/20 for sure.

Are they learning?  Absolutely.  I'm continually amazed and pleased with what they can do with the challenges we present.  The rigor is there but the packaging is different.  We just feel that the pace of the class is dragging some which causes another problem:  overconfidence.  When students feel that everything is simple and/or they've seen any of the content in previous classes, they will jump to the conclusion of knowing it all.  When we challenge them with extensions, connections, and explanations, they sometimes swing to other extreme of frustration.  That comes from being overconfident and not listening as closely to the somewhat familiar content (even with a new twist of going deeper and focusing on understanding).  We've been working on that for a little while and already see improvement.  Again, it's an issue of balance.  We have to have enough challenging problems to lure them into wanting to know more, but not get so challenging that they give up.  We're diligently working on productive struggle and persistence.

I'm pleased that both sets of students are real troopers.  They do what we ask and stay positive in the process.  They realize that the course is a work in process and for that I'm grateful.

Really, this course is like writing a proof in graduate school.  You're posed with a problem that looks insurmountable at the beginning.  You try this and that and slowly see you might be getting somewhere.  Add some time, sweat, perseverance, lots of erasing and rewriting, and before long, you can see the finish line and a path to reach it.  I'm seeing a way to the finish line and I like what I see.

On to Florida.  Let's redesign!

Sunday, September 18, 2011

Pilot Recap, Week 4: Busy, busy

In week 4, we tested, debriefed the first open-ended problem, conducted a unit 1 survey, and debriefed the whole unit.  Both Heather and I were interested in finding out what students liked and didn't.  A brief summary:

What they liked:

Active learning
Content is interesting and real-life
They're not bored
Getting to use MML
Working in groups to talk about problems

What they disliked:

All the word problems
Long class period
Not being able to make up assignments

Heather and I are unapologetically strict on attendance and class participation.  The course necessitates us to be so.  We meet for a long class period and go through a lot.  That experience cannot be duplicated at home since it's usually very active, has some kind of exploration, always has discussion, and relies on working as a whole group and small group to go through the content.  Students like that dynamic because it's interesting.  But missing class has consequences.  Since developmental students can be their own worst enemies sometimes, we want them to learn the importance of doing work on time and attending every class.  We're not unreasonable about either but our expectations are high and there are consequences to missing.

The first tests were not great nor terrible.  They are learning to read the problems closer and study more.  We spent time working on them afterwards and I believe there will be a better outcome on the next unit. 

Since our class periods are 100 minutes each, that was just one day.  The rest of week we began a new unit with a new focus.  One of my favorite parts of this second unit is the incorporation of manipulatives.  We're constantly noting things that could be improved and one is more hands-on learning in the first unit.  They are eager to dig in and participate so we've got to integrate more of that into the course earlier on.

We investigated the idea of center using shapes, physical situations, and numbers.  It was interesting to look at the concept of the mean in so many different perspectives.  As with almost every activity we do, it starts out looking deceptively simplistic but always goes deeper than students anticipate, challenging them. 

We closed the week on a somewhat frustrating note.  It's always aggravating to test a lesson and find it's just not there yet.  But that's the pilot process for you.  My notes from that day are covered in annotations and post-its about all the ways to make it work better.  It wasn't a complete loss as students got to explore something they can do usually (multiply fractions) but have little understanding of the procedures involved.  This is not a course in arithmetic; that's the prerequisite.  But we know this student is usually very weak on fraction and percent understanding.  So we've incorporated those concepts within other lessons to strengthen their abilities whenever possible.

I almost forgot:  we closed with a very short lesson that incorporated logistics by looking at an academic department's flowchart.  Again, seemingly simple, it wasn't.  There are lots of little details to read and understand to answer the questions.  I always like the activities that incorporate a student success element into the math.  Student success courses that are disconnected from content don't always have much punch.  It means more to discuss ideas related to student success when they directly relate to the content or point in the course.  We're trying to capitalize on that approach and timing whenever possible.

Next up:  integers.

Sunday, September 11, 2011

Quantway and the Complete College initiative

Complete College America is challenging and supporting states in efforts to improve completion rates.  Recently, they awarded ten $1 million grants to ten states.  The grants will help fund initiatives that should allow students to move forward through college towards degrees and ultimately, careers.  One common theme amongst the states who received grants:  initiatives to improve and accelerate the developmental math sequence.  We know that it is often a bog for students and can prevent students from ever getting to college math and therefore, a degree.

Most colleges looking at ways to accelerate the process through developmental math are considering modular models based on the emporium concept of self paced, lab based learning.

But another option, Quantway, could potentially be faster for students and simpler for colleges to implement since it is only one course. 

One of the main reasons we wanted to develop the Quantway course Mathematical Literacy for College Students at our college was the overwhelming number of Associate of Arts degrees awarded each year.  Students need a statistics or general education math class to satisfy the majority of AA degrees.  We saw time and again that when students had a high enough ACT or placement score, they could start in one of these courses and usually complete them.  Both courses have a pass rate of around 70-75%, regardless of instructor, day pattern, or time of day.  If we can get students into these classes, they usually pass and can complete the math requirement for their degrees.

The problem was so many students are just shy of meeting the requirement for entry.  Enter a lengthy developmental math sequence emulating high school math with courses like intermediate algebra that can be more rigorous than the college level courses it feeds into.  There lies the bog.  At our school, we worked very hard to get them over that hump only to get into a class like statistics or general education math that is more accessible and does not depend on the prerequisite skills they worked so hard to get.

Yes, emporium models can accelerate the process.  But they are costly initially (and potentially long term if they are not successful) in terms of both time and human investment.  For the student headed to statistics, they still don't address the skills that student needs to be successful in the course or in their career.

Quantway's MLCS course is not proven yet to work since the pilots are still just beginning.  But as we've already seen, something good can come of this approach.  If it is successful, and that is a big if, schools can have a viable option to redesign that only involves the investment of teachers learning about the course and its pedagogy.  Part of our pilot is to develop all the resources an instructor would need to be successful, not just the materials for a student.  As someone who looked at option after option to try in a large scope redesign that would not be grant funded, it was always encouraging to find ideas that weren't costly and had real potential.  We've already seen that with our combined algebra course.  In one semester, a student can take beginning and intermediate algebra (if they place high enough) and go into college level math the following semester.  We hope that MLCS will do the same for our program, but this time serving a different population with different goals.

Saturday, September 10, 2011

Pilot Recap, Week 3: Real, encouraging signs

Throughout this first unit of the Quantway MLCS course, Heather and I have both had lots of moments of questioning whether or not students were benefiting from the approach.  They like most of the lessons and are engaged but still have that somewhat skeptical reaction to the course approach.  It's just so different from what they are used to that the adjustment is real.

But yesterday, something wonderful happened.  We graded the first set of open-ended problems they were asked to solve as a group.

Part of our idea in approaching this course is that we want to develop real problem solving skills as well as critical thinking.  As anyone who's held a job knows, problems don't come boiled down, well organized, or well defined.  Part of the problem with problems is just figuring out where to start.  Plus, you have to figure out what you know, what you don't, what's relevant, and what's extraneous. 

Traditional developmental math word problems are very nicely presented to students.  They're short usually, 1-3 sentences, and full of key words.  Often, they follow a "type" that students can recognize.  Once they know how to deal with that type, they can solve the problem.

This sounds like the problems are easy.  But anyone who has taught word problems knows that even with all these nice facets, students often seize up when presented with one and have little success with them.

One perk to this course is that like a statistics course, every problem is presented in a word problem format.  The catch is they don't look like it.  So students get used to reading the scenario presented and are willing to attempt them.  They're not always successful but at least the willingness level is higher.

Every lesson is filled with problems to solve based on a scenario.  But nearly all the problems have a unique solution.  This is so we're not so out of the box that students resist entirely.  They like solving a problem and checking their answers.  The problems are realistic and difficult but they can find out if they're right or not.

That's all well and good but we do want them to see what real problems look like.  Enter our once per unit open-ended problem.

On day 1 of the unit, students are presented with a large paragraph that explains the scenario and problem.  Students have 3 weeks to work with a group to solve it.  We intentionally help them through this by having regular revisits to the problem, each time with specific tasks for the group to complete.  The first time through, we have them identify what they know, what they don't, what terminology they need help with, etc.  The second time around, we have them develop a list of tasks for the group members as well as a timeline for completing them.  They're also challenged with starting their rough draft.  The third time has them looking at the rough draft with the group, refining it and making a plan for the final solution.  They submit, we grade, and we debrief the whole thing the fourth time in class.  We show them a strong solution, talk about the problem, and make notes for the next one so that they can do better when facing another problem at this level.

It's an interesting approach and we had our fingers crossed the whole time hoping they could do it.  Every time they worked together, they went into the typical mode students do when they're confused.  "We're lost.  We don't get it.  Can you help us?"  Usually we swoop in as teachers and hold their hand.  Heather and I both made an agreement:  we will not do that or else they will never be able to do these problems.  Instead, we ask questions and answer questions with questions in an attempt to get them talking.  This helps them from getting too frustrated, but also keeps them in charge of figuring out their solution.

When they submitted the solutions, more than one student said they had done some research on their own, talking to people who might have been in the situation in the problem.  They were talking and thinking about math outside of class and not just in the "what's the answer?" way they usually do.  We were very pleasantly surprised.

Grading them, we were even more surprised and pleased.  In short, they rose to the occasion.  The grades were much better than we ever expected, some C's but mostly A's and B's.  No perfect papers but that was to be expected.  They followed the expectations, and we could see that they finally got to the other side with the problem.  In class, so often they were struggling to get started but at some point, they dug in and solved it.  We had them do an individual assessment of their work and the group's after submitting their solution.  Repeatedly, we read that students liked working with the group and enjoyed the process.  We didn't ask them "do you like your group?  Do you like this approach?"  We just asked them to give a percentage to the level of work each person did and comment on anything if they wanted to.  If someone wasn't pulling their weight, this was a private chance to tell us.  What they volunteered was unexpected yet great to read.

All in all, it was a very positive way to close out the unit.  Monday is the first test.  My fingers are going to cramp from all the crossing but I do hope they can do well and rise to our expectations again.  We've prepared them; the ball's in their court now.

Monday, September 5, 2011

Pilot Recap, Week 2: Does a Quantway approach dumb down standards?

I get many comments and questions about the new course Mathematical Literacy for College Students (MLCS), which is in the Quantway model.  One that I don't get directly but I hear of often is "won't you be dumbing down the content and standards by using this approach?"

It's a valid concern because we are changing the focus off traditional algebra.  Whenever that happens, educators worry that we'll be expecting less of students.  I cannot say strongly enough that this concern is not an issue whatsoever.  Actually, quite the opposite is the reality.

Students in a beginning or intermediate algebra class see content that's very prescriptive.  And while they may not understand it, they can often get by with mimicking and memorization.  As good intentioned as we all are to bring in conceptual understanding, applications, and understanding of the processes going on, the bulk of the content is rote use of skills that usually can be classified by a "type":  one step equations, "work" problems, mixture problems, etc.  A student who doesn't really know what they're doing can find ways to pass tests despite their lack of understanding.

In this class, the skills are not the focus.  They exist and are plentiful but they're just one cog in a much bigger machine.  The approach is something like this:
  • Rich situation initially explored
  • Skills identified that need development
  • Development of skills
  • Application of the new skill in the original context
  • Further exploration of the situation in a deeper way while making connections
Our lessons often have a MyMathLab skill assignment and a conceptual paper homework.  Students love the MML assignments; they struggle with the paper ones.  We don't want them to be surprised on test day and constantly emphasize that the paper homework problems are at test level. 

In the U.S., we have a well-defined view of what a math classroom should look like.  It may or may not be the best view for employers or jobs but it's one we know and understand.  Many instructors and from what we've seen, many students, believe that to do math, it must look something like this:

Solve:  -3(2 - x) = 4- 2x

Certainly that is in the spectrum of mathematics, albeit in the skill band.  But this is mathematics too:

There are two pay structures, a 5% raise or a 3% raise with a $1000 bonus.  For whom is each option best?

Because these students are so new to a large problem like this, we're walking them through it and taking the opportunity to show graphs, tables, patterns, the role of variables, and develop numeracy whenever possible.  But at the end of the day, solving that problem requires solving this equation (that they wrote):  1.05S = 1.03S + 1000.  So we still get to equations of all sorts just as you would in an algebra class.  This is simpler equation that they'll solve in the course.  The large ones are seen as well.

The reality is when you do the skills and the concepts and make connections and apply every skill developed in multiple contexts, you get a hard course.  It's an interesting course, but it's hard nonetheless.  We're watching closely to see how students fare in the pilot and studying the methods we use such as sequencing and pacing of topics to ensure they'll understand them.  But regardless, this is a tough approach for students.  They can't fake it or mimic anything and that can be frustrating, because that's a technique they may have come to trust in times of struggle.  The ones who are successful will do well in a college level course, that we're confident of.  At this point, we're just hoping that most can be successful at this course and make it to the college level course. 

We're not going to walk away with a 90% pass rate so worries of this being an easy course to make students feel good that puts them in college level math for which they're unprepared are unfounded.  The ones who succeed will be excellently prepared.  

I hope that we will work our way through this journey this semester, over the hills and valleys, and come out on the other side with the great majority of the students with us and better for having the experience.  My Pollyanna ways just expect that to happen.  But we have to wait and see.

Enough with the metaphors and referencing of old movies.  It's time to get back to work.

Monday, August 29, 2011

The Nitty Gritty: A Quantway-type lesson in action

One question I have received often about this course is, "what does a class look like?"  So today's blog has the sole purpose of showing you the class atmosphere and a typical large scale lesson.  I say large scale because lessons vary in length.  All lessons are 25, 50, 75, or 100 minutes long.  The one described below was 100 minutes.  I've shown bits and pieces (not all 100 minutes) to give the flavor of the class period.  We've already seen that having lessons of various lengths where the activities change regularly goes a long way to maintain focus and productivity.

The clips I show here show more of Heather so that you can see what the instructor does.  However, most of the period involved students working and the instructor circulating.  It's a back and forth motion, whole group - small group - whole group and so forth.

Warning:  Video quality is moderate and the videographer needs to work on her technique.  I tried to avoid a Blair Witch effect but there are a few shaky moments.

The lesson filmed was called "Higher or Lower."  Its premise problem is that a bargaining unit is negotiating a contract with two pay structure possibilities.  One is a 5% raise; the other is a 3% raise with an additional $1000 added on.  We begin the lesson by describing the scenario.  Students have a few questions to work on related to this such as "which would you pick?", "if you made $20,000, what would your new salary be under each option?", "does the order matter when adding the $1000 and applying the 3%?"  Below, Heather is doing the initial problem description and putting groups to work on these beginning questions.

In the next clip, you get a sense of what the class atmosphere is like.  Small groups, lots of discussion, chairs grouped together.  Heather circulates and helps groups progress without answering everything for them.  Our motto is "answer questions with questions."

Part of this lesson involves taking the percent of a number.  The goal of the lesson is larger than this skill but it's still a vital part.  In the next clip, Heather conducts a mini-lecture explaining how to take the percent of a number.  Many students know a rule but don't understand what they're doing.  Heather works off of a picture, moves to a scaling technique, and then generalizes the process with a rule.  Our goal is to develop agility with skills.  This is hard for students but worthwhile.  It's not just can they do something.  It's do they know when to do the skill and can they perform it in various contexts.  Heather integrates multiple representations, estimation, and reasonableness to work on one of the course goals:  numeracy. 

Not all lessons have a specific skill taught but many do.  MyMathLab is outstanding for skill development.  However, students abuse help aids and often forget to write down legible work.  One goal we have for the materials we're writing is to help students develop good study and work habits so that they're successful in a college level math class.  So we have made a designated page for every MML skill homework.  In the next clip, Heather explains this sheet and what we expect them to do to make the most of the program.

An important note:  the MML assignments that we're building embed the skill in a context to get as much practice as possible with realistic situations.  We do not want students to be in "mimic mode."  The goal is learn the skill and transfer it.

After the mini-lecture, students were asked, "if you wanted to know who each salary structure is best for, what would you do?"  We want them to think about solving large problems before diving in and showing them everything.  They brainstormed ways and we did have students who said, "pick a bunch of salaries and find the outcomes under each."  So students worked on completing a table doing just that and then answering questions with their group about the table.  Questions like, "who benefits the most under each option?" and "can you generalize the calculations?"  Note:  questions are more explicitly defined on paper than my brief descriptions here.  Generalizing a calculation is challenging but begins the process of bringing in variables.  Our method is intentional.  We want them to see how generalizing the calculations allows the use of spreadsheets like Excel to accelerate calculations.  Below, Heather debriefs the class after they've worked on these tasks.

Lastly, Heather shows students Excel and how this problem can take advantage of spreadsheets to gain more insight.  Because there are so many clips, I cut off the end of class.  At the end, we asked students if there was a salary for which the pay structures would give the same amount.  They wanted to try guess and check using Excel (as did we) so we did that until we found the exact amount, $50,000.  Then we helped students write that problem mathematically which led to this equation:  1.05D = 1.03D + 1000 where D is the salary in dollars.  We can't solve this yet but it was interesting for students to see variables used and for equations to come into play.  They're used to being given an equation to solve so it was surprising to them to see one develop organically.  We're building to the point of being able to solve one.  That comes in a later unit.

An additional homework assignment on paper accompanies the lesson and MML assignment.  Paper homework is conceptual and applied, looking similar to test type questions.  They're not long assignments but they're deep and challenge students to explore concepts further.

It's a lengthy blog, for sure, but I wanted to give a real feel for what we're doing in the classroom.  More blogs and videos to come.

Saturday, August 27, 2011

Pilot Recap, Week 1: They don't know what they want

This week began the pilot of the MLCS course in the Quantway path at my school.  My colleague and co-author, Heather Foes, and I each teach a section and sit in on each other's class.  We're writing the materials for the course and using MyMathLab for skill homework only. The bulk of homework and all of the assessments are applications or conceptual questions and done on paper.  We are using objectives from Carnegie's Quantway course Mathematical Literacy for College Students but are not funded by the foundation. 

It's been an exhilarating and exhausting week.  The course is something out of everyone's comfort zone for sure.  Heather noted on Monday that it's definitely quicker and easier to lecture the whole time.  You talk, they write, you move through the material.  And based on our student's reactions, that's what they expected and sometimes seem disappointed by.  Where's the lecture?  Funny, when we had some mini-lectures, they didn't look all that elated.

It's been surprising to see so many students wanting the traditional class format.  It makes sense on one level because it's familiar.  But when given that format, as math teachers know, students complain that there's not enough time to work with the concepts, that they're bored, and that there's no point to the content.  So color us surprised to give them relevant and applicable content with time to delve into the concepts and make connections and they're not much happier.  They really don't know what they want.

We had an interesting problem this week where we showed a concept being taught two ways and then studied the approaches.  Nearly every student liked the straightforward, "here's the rule approach" over the conceptual one that ended up with a rule.  But when we did some applications afterwards and asked them how they got the answer, no one said they used the rule.  When pressed to verbalize their thought processes, they would start to explain how they thought through the problem and then arrived at the answer.  When we pointed out that the rules approach is efficient but can be lacking if you don't remember it or know how to apply it, there was definitely an "aha" moment.  Again, what they think they want and what they actually use and need don't mesh.

Reading this you may think it has not be a successful week.  On the contrary, it's been a positive experience so far and I have real hope for something significant to occur this semester.  Students were engaged, talking in groups, and working through the problems well.  They've done a prerequisite skill module in MyMathLab to address any holes in their arithmetic knowledge and they're doing well with all the class expectations.  Heather and I have been pleasantly surprised with that.  They've done what we asked and then some but there has been the occasional, "this isn't what I expected with a math class" comment.  It's just an adjustment on everyone's parts that will ease as we progress in the course.  On the up side, Heather and I have said for over a year that we want to do something truly different in developmental math.  Based on everyone's reactions to date, I believe we are succeeding.

On another positive note, there's a lot to be said for an integrated approach.  By the time we get to theoretical rules and skills, they've had enough time to process what we're doing since the development of content is gradual.  This week alone, they've seen fractions, percents, ratios, rates, spreadsheets, generalizing calculations, and models.  We haven't done every facet of each of those topics but we do some in each unit and continue that approach doing more and going deeper over time.  I'm cautiously optimistic that the comprehension will be better as well as application of concepts and skills since every skill that is developed is applied immediately and repeatedly in different contexts.

I videoed the class yesterday and will post it soon with explanation of a typical class period.  Look for that in the days to come.

Back to planning and writing...

Monday, August 15, 2011

Pilot Plans

Well, only a week to go before our semester starts and the pilot of the Mathematical Literacy for College Students (a Quantway type course) begins.  We're gearing up, getting things copied, working on data collection, and more. 

As we pilot, I'll be blogging regularly about the experience and what we're learning from it.  I really enjoy the excitement of pilots.  The unknown and "flying by the seat of your pants" experience keeps an educator on their toes.  My approach to the classroom has always been to plan to the hilt but be able to go with the flow.  I work to maintain pace and plans but I also allow for the natural spontaneity that comes with a classroom.

Look for updates starting next week.  We'll be videoing lessons too, so I hope to be able to post some video during the semester as well.

Thursday, August 4, 2011

Emporium Models or Quantway & Statway?

In today's current educational landscape, "redesign" is the key buzzword in developmental math.  Several states are embarking on across-the-board projects to change their programs.  The common approach involves the emporium model with modularization.  The premise is that the prealgebra/beginning algebra/intermediate algebra curriculum is thought of as one large sequence instead of individual courses.  Schools divide the content in various ways (units, weeks, chapters, etc.) and often call those pieces "modules."  They then develop a way for students to start either at the beginning and move quickly out of the content they know or place in the sequence based on a test.  Either way, a self paced approach to completing the remaining content is usually taken.  Students watch videos, read the book or use workbooks, and work homework problems in an online homework system.  Tests are taken at the end of each module.  In theory, students can move through content quickly.  Lectures are often not included but some schools incorporate weekly one-hour sessions for problem solving purposes.

Having used self paced approaches (unsuccessfully and then successfully) for 10 years and helping schools who are using them, I'm familiar with the emporium model.  It has a new name but the approach has existed for decades.  It's improved, certainly, with current online homework systems. 

Some students will thrive and love this approach to math education.  The ones that are close to being at college level and are motivated can do especially well.  Students who have seen the content before, need a brush up on missing items, and who are not that far from the finish line are best suited for the approach.

Likewise, some instructors love moving around a lab full of students, answering questions that vary wildly.  The challenge in being prepared for hundreds of topics interests them and they enjoy not preparing lectures.  It can be a refreshing change of pace from the typical classroom environment.

The issue is that it is not the right approach for everyone, especially students who have large gaps in understanding and/or place very low in the developmental math sequence.  It is also not attractive to many faculty.  Since faculty support and buy-in are critical to a redesign's success, that matters.  Schools and states can impose mandates but if the faculty don't believe in them, they will wait out the administration until the movement passes and revert to their prior techniques.

These statements will incur disagreement.  The emporium model is controversial so heated discussions come along for the ride.  I've been told by some of its advocates that everyone responds well, some take longer than others, but it works for all.  And I disagree.  First, because I've seen it firsthand and heard from many around the country.  But for another perspective, consider this:

This summer my children have 3 full months off from school, longer than they ever have had due to school construction.  We always do math, reading, and writing throughout the summer but this year I feel it especially important to have a deliberate approach since they're away from school so long.  We have workbooks and library time but we've also added the Khan Academy to our repertoire.  I thought, "they will love this!  It's interactive.  There's videos when you need help.  The practice sets are like a video game.  They earn points.  They can move as fast or as slow as they want.  This is awesome!" 

Except it didn't turn out to be that way at all.

They liked it initially but the novelty wore off.  Also, they didn't want to watch a video about a new topic.  They wanted me to explain a new topic and us talk back and forth as they had questions.

It was a nice addition, something different, and something they liked on occasion.  But they did not want it to be their bread and butter.  I asked them both (aged 7 and 11) if they could imagine only learning math by working in a program like the Khan Academy with a teacher answering questions.  They both looked at me like I had grown a third eye.  "But what about the teacher?  Don't they get to teach?"  They don't equate teaching with answering questions from someone else's explanation.  And maybe it is teaching to some but many think otherwise.

Reading that, you  may think, "but they're children.  These are adults who are in a completely different place in their mathematics education."  Are they really in that different of a place?

I see students constantly who know so little about math that their understanding is like that of a child.  Years of rules and varied approaches has left them insecure and confused with little natural curiosity or confidence.  I often think how great it would be to start them over and let them explore math, ask the questions they have, work with more manipulatives, and build their confidence with their understanding.  Sure, use worksheets and computers and books but that's not enough.  The human element is needed to create it, direct it, and solidify it.  I don't mean lecture needs to be there 100% but a person needs to be a part of the equation, working with students to build their understanding.  People, tools, computers, and paper need to exist but with balance.

Ask students when they remember loving math (and most will) and they will nearly always say elementary school.  Ask them when they stopped liking it and many will say 4th or 5th grade.  The change came when the subject stopped being physical and meaningful and started becoming procedures and rules.  Both are necessary but when we emphasize only one aspect to the subject, we often lose students.

One of my main problems with an emporium approach across the board is that it works under the assumption that our subject is just a stack of skills to be learned and that our curriculum is fine.  The problem just lies with the speed or delivery of it.  Also, when used across the board, it works on the assumption that one solution is right for everyone.  In the age we live in of choice and options, it can be infuriating to students to lose that entirely.

So where do Statway and Quantway come into this?

Both pathways strive to look at the curriculum and how we teach it, changing that experience from primarily lecture on skills to some direct instruction but more time problem solving with students.  There is deliberate design of activities to create engagement of students with one another.  Can you get engagement with emporium models?  Yes, if you accept the idea that engagement is two students discussing how to work through an exercise.  That's fine but I want to see more than that.  I want to see them also discussing problems (not just exercises) and mathematics in a larger sense. 

The pathways can be a cost effective addition to a department.  They also accelerate the timeline for students.  So they offer solutions to some key, common problems in developmental math programs.

Lest you think I believe the pathways to be the solution to all of our ails, I don't.  I don't think there's any singular approach that will fix the problem for everyone and even if it there is one, I don't profess to have it.  The problem is just too big and too complex for that to be the case.  A myriad of approaches is necessary to solve the myriad of issues. 

The pathways are an excellent addition to a department to serve students with specific goals, specifically the ones who are not headed toward a STEM direction.  Emporium models are excellent for those students who are headed to STEM areas but are close to college level, motivated, and just need a little time.  For the students who aren't in either category, a well designed curriculum using instructional principles from research on developmental students can work to serve them as well.  We've seen that in our school's redesign.  We get 65-70% of students to pass nearly all our developmental math classes.  The sections are not self paced; most live in a classroom.  A few are computer assisted with about 70% of the class time on direct instruction the remainder spent with students on computers and the teacher helping.  We have many options, offerings, and solutions to help solve the problems we have.  Most work well, some are still being massaged.  If you want to read and hear more about our redesign, click here.

My point is that I don't believe there is a quick fix to this issue of developmental math education and my antennae go up when I hear someone professing there is one.  The latest ideas for innovation (emporium, pathways) can live together in harmony as just a few pieces of a much larger redesign puzzle.  Schools have multiple options for making change based on their funding and their culture.  One size need not fit all.

Monday, July 25, 2011

Hot Topics: But what about factoring?

One of the common questions I get about the MLCS course has to do with the big guns of algebra:  factoring, quadratics, and rational expressions.  Instructors want to know if those topics are included.  Here's the short answer:  yes and no. 

First, the yes.

Each of these topics is seen in the course and materials I'm working on. 

Now, the no.

They are not developed in the depth comparable to a traditional algebra text.

Here's why:

These topics are lightening rods in the debate over curricular reform at the developmental level.  Scrapping them entirely alienates some faculty which is never the goal.  However, we are limited on the time we have in the classroom.  There simply isn't enough time to develop all the finesse with these topics normally seen in a traditional algebra course and accomplish the goal of this course.  That goal is specific:  in one semester give students the mathematical maturity necessary to be successful in a general education math class or an elementary statistics class.  The reality is that procedural fluency in each of these three topics is not necessary to succeed in either of the college level courses listed.  So each topic is seen but not developed deeply.  That is intentional.

These are good topics and do naturally appear.  Our goal is to expose students to them on a conceptual and applied level.  Let them see what factoring does first and why it could be used along with its inherent limitations (that is, most things do not factor).  Let them see that not all functions are visualized by a straight line.  We spend more time on exponential functions than quadratic but quadratic functions do make their presence known.  Likewise, there are some functions that appear that naturally include rational expressions.  It has been quite interesting and exciting to see functions arise through another problem or question and see the form they take on. 

For example...

Recently I was writing a problem where students need to make a conversion using pi and average several numbers.  NOTE:  This was not an exercise in calculation but the calculation that arose from a realistic problem being solved.  As I was writing the solution, I thought about how students would approach the calculation.  They could convert all the numbers and then average them but that was cumbersome.  It would be faster to average, then convert.  But are the results the same and why?  When I started proving this to myself, I saw rational expressions and their arithmetic including division and addition.  It was fascinating to see this occur naturally and look absolutely nothing like any application I've seen in the rational expressions chapter of an intermediate algebra book. 

Because this question is real, involves proof, and includes algebra, we will ask students to work through this question.  It won't be simplistic but it's worthy of time and attention.  It also does not take 3 weeks on rational expressions to answer it.  But it does expose students to something they haven't seen that is definitely worth the time.  This is not the only place rational expressions appear but I've listed it for illustration purposes. 

The key thing to remember about this course is that it takes students on a journey where skills are developed but are not the end goal.  So some skills won't be seen and that's ok.  That's why we have intermediate algebra for the students who will eventually need those skills in a precalculus or college algebra course.  We are not trying to be all to all but instead give to a certain population what is more in line with their needs and goals.

Friday, June 24, 2011

Two online resources

In February of this year, Pearson hosted a day of webinars on course redesign.  I gave a session on our school's redesign.  Many documents are listed at the left but if you'd like a recording of the presentation (audio and video of PowerPoints included), click here.

The Carnegie Foundation has started a new blog on its Pathways initiatives, Quantway and Statway.  The most recent post describes quantitative literacy and how it differs from mathematicals and statistics.  It's a great read with lots of reference material.

Wednesday, June 15, 2011

Doomed to repeat history?

Those who do not remember the past are condemned to repeat it.

George Santayana

I have received questions about Quantway/Statway and the new MLCS course.  Some want to know, "how is this different than the integrated high school curricula of the 90s?"

Good question.

In the late 80s and early 90s, NCTM released a new set of standards encouraging a different way of teaching and learning mathematics K-12.  Following those standards were years of grant projects, pilots, materials, and eventually textbooks with a new approach:  an integrated approach.  The idea was not to do algebra 1, then geometry, then algebra 2 but instead combine content and integrate it.  A worthy concept but definitely not how we typically teach mathematics.  It has strong supporters and opponents, with good cause.  Because while a curriculum like this can work, it takes a tremendous amount to do so.  It requires good materials, a change in mindset, time to learn how teach in a new way, and abundant professional development.  For more on this subject, please read this article from NCTM's website.

Because these necessary elements do not always occur, integrated models are not always successful.  For a different take on this subject, please read this article about Georgia schools and their experiences with integrated courses.  It is very forthright in both the pros and cons of the approach including why they are dropping this instructional approach.

So what can we learn from this?

Integrated courses have a lot to offer students but execution is key.  This is a new approach to teaching and vastly different from the stack of skills approach most instructors are used to.  It forces us as teachers to integrate our teaching and move between multiple contexts and concepts in one lesson.  Truthfully, not all teachers are comfortable with that nor want to teach that way.  And in my opinion, that's ok. 

Just as students learn differently, instructors teach differently.  It is very difficult to find an approach that suits everyone.  So part of the issue with integrated curricula at high schools was that everyone had to do them.  And they're just not right for everyone.  One person's brain may move in several directions at once, circling constantly to connect ideas.  Another's brain work may work very linearly and sequentially, moving steadily from one idea to the next.  Since we think differently about things, it stands to reason that we should have the opportunity to teach and learn to our strengths. 

Adding to this issue is the concern of what people are using to teach from and how they are teaching it.  Good materials are a must.  But teachers have to know how to use them.  Training and support are just as important.

Integrating these issues (pun intended), here are some reasons courses like Quantway's MLCS can work:

1.  MLCS is one course and does not have to replace the traditional beginning or combined algebra courses.  It augments a program and offers an option for the non-STEM student or the school who wants to offer beginning algebra differently.  But beginning algebra can still live and exist for faculty and students who want it.  It's an option, not a mandate.  Thus, it's more likely to be taught by teachers who like an integrated approach and taken by students who want something different than a traditional algebra course.

2.  The initial scale is small.  Meaning, we're just developing the one course, MLCS, right now.  If it's successful and there is demand, I hope to work on its precursor, a course about numeracy using an integrated approach.  Also, New Life has sketched out a Transitions course to follow MLCS for the student bridging back to college algebra.  There has also been talk of a college level version of MLCS that would be different than the general education math courses that are surveys of areas of math.  Ultimately, all these courses would be options, not replacements, to courses that exist.  And they would be built using a model that would be proven to work.  This "start small and build" approach was not always used by high schools.  Many scrapped their traditional algebra 1-geometry-algebra 2 sequence and instituted Math 1-Math 2- Math 3 sequences.  That can work but if it doesn't, it's hard to change horses midstream.

3.  Materials and a training program are being developed with the goal of addressing issues that will always exist.  I'm a big believer in the idea that "how" is more imporant than "what."  We have components and whole courses in my department's developmental sequence that work extremely well but were flops at other schools.  Why?  Because the execution was different and we tried to learn from mistakes of others.  And I know others have used our mistakes and done better on something than we did.  We're all a part of a large assessment chain.  So, one flaw of the implementation of integrated high school curricula was professional development.  Knowing that, we're designing a training program that will support the materials and course development.  But beyond that, we're developing the materials with the known issues in mind.  One of those issues is that a wide range of instructors will likely teach the course, some who are not familiar with a new type of pedagogy. 

For example, notes to the intructor on teaching the ideas are interspersed throughout the lesson plan and materials.  Instructions on pacing, assessment, grouping, rubrics, and examples of each will be provided.  Additionally, the layout of a lesson attempts to address a common concern:  what's the point of this lesson?  Students will often like integrated lessons but get caught in the detail of the context and miss the forest for the trees.  So we close every lesson by asking students to step back and determine what was learned (mathematical and otherwise) and how it can be used again.  To wrap a unit, we are building specific active tasks that students will do to assess where they're at with all the skills, concepts, and techniques of the unit.  Again, we want students to see the bigger picture and connect what they have learned to date so that they don't get lost in the content.

This is all well and good but ultimately we need to know if it will work.  The Pollyanna in me says yes because I've been through the nonstop assessment process of redesign.  Redesign is not about saying, "we're doing X, now we're done."  It's about implementing something, scrapping what doesn't work, and working until a solution is found.  The pilots this fall around the country will tell us a lot.  Having piloted numerous new projects, I know how exciting and educational the pilot process can be. 

When we redesigned our traditional developmental curriculum, we did so on a large scale right out of the gate.  Meaning, all the pre, beginning, and intermediate algebra classes used the new modular approach with MyMathLab, standard policies and new pacing.  That approach was fine because the changes were largely administrative and structural; teachers could still teach how they liked.  When we redesigned our geometry course, it was just me and two sections of students.  That smaller approach to piloting helped when the content and pedagogy were so vastly different.  Each day, I made modifications and by the end of the semester, we had materials and a method that worked.  And it's continued to work for many more instructors and students.  Partly because of lessons learned during the pilot but also because of the training and instructor resources that were developed after the pilot.  That smaller system approach to piloting will be used as we implement the MLCS course as well.  I'm writing materials and instructor notes now that will be revised as they're piloted, but I'm also keeping notes and doing research for the training program to follow.  One can't work without the other.

So yes, we're in wait and see mode.  But I have a strong suspicion we will see something we like.