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.

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.

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...

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.

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.

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.

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.