MOOCA MOOC is a Massive Open Online Course.  Many pundits and some policy makers are gaga (completely absorbed, excited or infatuated) by the vision of being able to deliver higher education coursework at incredibly low costs.  MOOCs are supposed to be taught by rock star professors who work at elite universities and can deliver a lecture better than anyone.  Only computers and one rock star are needed to deliver a course to thousands, world-wide (at least anywhere with a fast Internet connection and up to date software.)

There are three issues that most people who are gaga about MOOCS choose to ignore.

gagaThe first issue is that higher education involves more than absorbing facts and knowledge from a rock star.  Part of what makes a college degree valuable is the interaction that happens in a classroom.  Although discussion boards have their place, unless the student is going to live and work in a virtual universe, a discussion board is no substitute for face to face interaction.  Having professors who aren’t all extroverted rock stars is actually an important part of higher education.  Students learn to adapt and learn in different situations and from different people.


The second issue is motivation and persistence.  One of the major current challenges in higher education is the increasing number of students who quit when they are confronted with difficult situations.  These students floated through high school, choosing classes that they could pass by showing up.  There was always extra credit.  They cheated.  They know they can get into college if they graduate from high school, even with an extremely low grade point.  So, why do any more than the minimum?

The third issue is assessment.  MOOCS generally use non-proctored assessments, either delivered and graded by a computer or graded by another student in the class, a “peer.”   In either case, there is no guarantee that the student getting the grade is the person who actually did the work.  If the assessment is computer delivered, then the content is limited to what can be graded by a computer.  Showing the thinking involved in getting to the answer is not assessed.  Perhaps in other disciplines, peer assessment is valid.  In my experience, it doesn’t work very well in math.  A C student does not know enough to evaluate the work of another C student.

This is on my mind because of a conversation I had today after class with one of my students in my second semester Math for Elementary Education class.  She is not passing and is naturally discouraged.  She wants to improve her grade.  I did not have her in the first semester; she took it on-line.  It turns out that the on-line course was completely delivered using a publisher’s software package.  The tests were not proctored.  She figured out how to “game” the system.  She used the “try another problem” feature on the homework to get a step by step process for doing the exercise.  She printed it and then “did the steps” on the assigned problem until she got the answer.  Since she doesn’t believe that she is good at math, she didn’t really attempt to understand what was going on.  The tests were computer generated and included problems that were similar to the assigned problems within the chapter.  Since they were “open notes,” she used all of her copies of the worked out solutions for the exercises to figure out the answers for the test.  This strategy was sufficient to earn her a C but, as she readily admits, she didn’t learn much.  It is no wonder that she is struggling in my class.

Although on-line homework can provide valuable practice and immediate feedback to students, students also need to do work that is more complex than can be graded by a computer.  My feedback on their work can make a difference, if they choose to pay attention to it.

I heard a policy maker talking about education reform.  Her argument was that since college graduates make more money than high school graduates, we need to produce more college graduates.  Since not enough students who enter college persist to a degree, we need to change college so that they can be successful.  I wonder if we change college so that they can be more successful (reduce the number of credits required, allow MOOCs to be counted towards graduation, use more computer delivered instruction, even changing the content of the courses), will a college degree still be as valuable?

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Word Equations

How to Solve It

Dr. George Polya suggests that students and teachers organize their work using a framework of four steps:  Understand the Problem, Make a Plan, Carry out the Plan, and Look Back. A few years ago, I decided to ask my developmental math students to follow these steps as they worked on applications.  Because these are algebra courses, as part of Understanding the Problem, I required them to assign a variable to what they were trying to find out.  To Carry Out the Plan, I required them to write and solve an equation that included that variable.  I also required them to write their final answer in a complete meaningful sentence.

I saw some improvement.  At the very least, their work changed from being a random mess of expressions and numbers that I could not follow to having at least a modicum of structure.  I kept experimenting with ways to help students find a way to reflect on the reasonability of their answers and to help them express their understanding of the relationships in the problem.  The roadblock for many of the weaker students was the  “Make the Plan” step.  They still struggled with the very idea of looking for a relationship in the problem.  They would say things like “I can’t remember the formula for this one” or “If you would just give me the formula, I could do it.”   Textbooks can actually reinforce this attitude by sorting problem “types” into different sections, allowing students to learn what they see as a formula to do that problem type on their homework.

One of my colleagues, experimenting with a similar format, thought that the key was to have students talk through their understandings and then translate what they said into a word equation.  Then, they could learn how to replace their words with the assigned variable and the numerical information in the application.

For example, in the introductory chapter on solving linear equations, students are asked to solve applications like “The annual base salary for a job in sales is $40,000 plus a commission of 12% of the value of merchandise sold per year.  Find the amount of  merchandise that the salesperson must sell to earn a total of $90,000 per year.”   A word equation that is specific for this application is base salary + (percent)(base salary) = total.  Or: “The cost for standard shipping of an order of textbooks is $3 per shipment plus $0.99 per book.  Find the cost of shipping 12 textbooks.”  A specific word equation might be cost per shipment + (cost per book)(number of books) = total cost.  Or:  A car owner spends $7400 per year on car payments, insurance and standard maintenance.  His car averages 25 miles per gallon of gas.  He wants to limit his annual expenses to $11,000.  If gas costs $4 per gallon, find the number of miles he can drive per year.”  A specific word equation might be cost for payments and other stuff + (cost per mile)(number of miles) = total cost.  Or: “To encourage conservation of water, a city charges $50 per month per residence for 0 – 10,000 gallons of drinking water and $2.68 for each additional 1000 gallons.  If a family’s budget for water is $65 per month, find the number of additional gallons they can use per month.”  A specific word equation might be (base charge per month) + (number of additional gal per month/1000)(cost for additional 1000 gal) = total.  

I’ve discovered that when I model the use of specific word equations like these, my students are trying to memorize them for each problem type  just as if they were algebraic formulas.  They do not look for or see the similarities between the problem situations.  So, my newest strategy is to have them write more general word equations.  When we start doing an application, I say, “Okay, what big relationships do we know that we might be able to use to solve this problem?”  As I did applications similar to the problems above, I used the word equation fixed + non-fixed = total.

The other relationships we’ve talked about so far are the

percent (or fraction) = part/whole    and   part = (percent or fraction)(whole)             

original  + change = new    

part/whole = part/whole    (proportions)

For those of us who teach math, sometimes it is difficult to imagine NOT looking for a way to compare, contrast, and generalize.  However, this is new ground for many of our students.  We often call it by another name, “critical thinking.”  It is hard to teach, hard to assess, and hard to learn.

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Messy Papers and Rubrics

I’ve often used rubrics for grading applications in an elementary algebra class.  I think that the students have been somewhat oblivious to what I’m doing.  They look at the score, not the rubric.  If they get enough points to pass, they’re usually satisfied.

This semester  I am experimenting with whether emphasizing objectives and rubrics can result in better communication of the “clear and high standards” that I set for my students.   I’m including objectives and a rubric for each turn-in assignment.   To grade the first assignment, I printed out a half page sheet that included the assigned problems, the rubric, and room for comments.  Then, I graded every problem on every homework, made a comment about their presentation,  and showed how I used the rubric to get their final grade.  With 75 students, this was a big job.

The mathematics generally looks good.  I was pleased to see that most students were using = signs appropriately as they simplified numerical expressions.  I emphasized this during class, both in the examples I did and in the practice problems they did in pairs at their desks.  Most of them are doing well in following the order of operations.  Not surprisingly, there are a significant number of students who made mistakes in fraction arithmetic.

Aside from the mathematics, however, these students need to improve the visual quality of their homework.  Despite the directions, I had crumpled papers, spiral edges, loose sheets without staples, and messy work.  I used to not care about this kind of thing.  However, I now think that this lack of care in presentation is connected to and can continually reinforce negative attitudes about  learning math and the low expectations students often have for their work in math classes.

If I want neat work, I can’t just say “be neat” or “do your work in columns.”  I have to model and explain what that means.  I wrote comments on that first homework assignment.  And, first thing in class today, I showed examples of exemplary work on this assignment.  I showed examples of work that included the correct answers but just wasn’t presented very well.  I urged students to present their work with pride.  It is an accomplishment to do arithmetic and algebra well.

This was also my first opportunity to be very clear and blunt about academic dishonesty.  I showed some exercises where a student used a calculator when the directions explicitly said not to (academic dishonesty) and  exercises where students copied the answer from the back of the book even though their work did not support that answer (plagiarism).  Students are often shocked to learn that copying answers from the back of the book is academic dishonesty.  They have done it for so long that it seems like normal behavior.


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Balancing Act



First day of class in elementary algebra today.  We reviewed some essential arithmetic topics but the real task was to start the process of building a sense of community.

“Negative math experiences” are almost universal in this population.  Some of these experiences seem to be fallout from family breakups, illness, abrupt moves, loss of economic security, or just adolescent immaturity.  Others happen because of what a teacher or classmate did or said, in elementary school all the way to college.  I don’t know the whole story; I just hear student perceptions of what happened.

The challenge is to balance their need for encouragement with the reality that they will be held accountable for learning the course objectives.  Students need a safe space to learn, free of shame or ridicule, but they also need honest feedback and realistic evaluation of their progress.  They need to perceive me as someone who understands what it is like to be frustrated and struggle and make mistakes and yet they also need to trust in my expertise and knowledge.

Today was the day to emphasize that I’m not a Math God on a Throne.  As part of my introduction of myself, I mentioned that I own two goats, Spot and Petunia.  I said that of course I did well in math classes along the way but I also experienced what it is like to withdraw from a math class in graduate school because I  was struggling to find childcare, I couldn’t stay after class and get extra help, and I felt like everyone in the class was getting it while I was totally lost.  I told them how it took 8 years to write and publish two math textbooks and, at countless places along the way, the work was so overwhelming that I just wanted to run away and hide.



I told them about how the public embarrassment I’ve experienced trying to back up a trailer, how people keep telling me how to point the trailer where it needs to go but that just doesn’t help.  (Because of the emphasis on outdoor activities, the rural origins of many of my students and their smiles as I talked about this, I think that most of them can probably back up a trailer.)

I had them fill out a half-sheet of paper with “Tell me something about yourself,” “What is your major or career goal?”  “Tell me a positive math experience you’ve had,”  and “Tell me a negative math experience you’ve had.”  I had them talk in pairs for a few minutes about their answers to the questions and then asked them to volunteer a negative math experience to the group.   This pair and group discussion helps students realize that others in the class, despite their poker faces, are also worried about learning math.

Some college instructors think that this sort of thing (about 20 minutes today) is a waste of time.  Our students are adults and should be treated like adults; they will either act like adults and step up to the challenge or they will fail.  I look at it a little differently.  My elementary algebra students are adults but they have been damaged by failure or discouragement, whatever the cause.  They need extra support and explicit direction so that they can finally experience what it feels like and what it takes to be successful.

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Final Exams

Back in August, we were at the start line of the semester.  Our classes were full and our students attentive.  Now we’re at the finish.   As we grade common finals together as a department and see the many scores that are just barely passing or below, the conversation always seems to turn to “Why don’t they study?”  and “How can they not know this?”

Why don’t students do better on finals?  They almost all start out with the best of intentions and visions of success.  The ones who do not succeed often don’t seem to have a plan for studying for their courses during the semester or, if they have a plan, they don’t follow it.  Instead, they put out fires.  They write papers when they’re due in a week or the night before they are due.  They study for tests the night before.  They often don’t study for finals at all, except for completing the review assignments that I ask them to do.

I use a weighted average for grades.  I’ve talked about the way their grade is computed and given them tentative grades (based on a final exam grade that is equal to their current test average) more than 6 times during the semester.  Last week, once again,  I led all of my classes in computing their grades, up to the final.  On a scale of 100, the final is worth 20 points.  Then, I had them figure out what effect the final could have on their final grade.  Watching their faces, it was clear that they had not really paid attention to their progress.  For some, it was a relief to find out that they could pass the course even if they did not pass the final.  For others, it was a shock to find out that they could not pass the course, even if they earned 100% on the final.  They had not faced up to the reality of what they had done up to this point, for 17 weeks.

It used to be that students could survive failing a course or even two every other semester and still qualify for financial aid.  This year, the rules have changed.  They need to make “satisfactory academic progress” every semester.  They need to keep their GPAs above 2.0 and they need to complete a certain percentage of their classes with passing grades, every semester.  The rules have a purpose:  get serious.  The people (i.e. the federal government) are not going to give or loan you money unless you translate that money into progress towards a degree.  The old days of “I had a bad semester” are gone.

This is a new experience for students who slid by in high school, especially those from school districts where failure of students who show up and don’t cause trouble is not an option embraced by the patrons or parents.  The financial aid and advising people are trying to impress upon these students the consequences of not passing but it doesn’t seem to be sinking in.  Perhaps this is because there were so many second chances at their high school:  extra credit, accepting late work, mercy D’s.

Of course, I recognize that many of my students have  what we call “complicated” lives.  They almost all work, many of them working more than 20 hours a week.  Some of them have children.  Some are dealing with substance abuse issues or the criminal justice system or family members that need them.  Some of these are situations that they have control over; others are not.

It is hard to watch students fail, despite my best efforts and the plans and exhortations of many others who really want them to succeed.  Many of these students come from families where there are no resources to pay for college.  This is their chance. They aren’t ready for it. On the other hand, it is exhilarating to watch those students who do succeed, reaching these small goals on the way to their large goals. They and their families make incredible sacrifices in order to haul themselves up the ladder towards a degree.

I just wish there was a better way to help students understand the immense consequences of what they do, and don’t do, day to day.

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Absence of Consensus

TeamworkAt a faculty meeting of mathematicians this week, we got into a discussion of common finals, multiple choice questions, mastery skill quizzes, technology, and curriculum in general.  The one thing that was clear to me at the end of the meeting is that we don’t agree.  We started common finals about three years ago for all entry level (gen. ed core) and developmental math classes.  We needed to do something to show that we were doing departmental level assessment, for accreditation.  We also knew that students who were taking the exam later during finals week were finding out information from students who took it earlier in the week.  Some of the faculty thought it was a good idea, long time coming.  Others did not like it at all.  Our division chair essentially said thou shalt.  That is where we still are.  Although we are all dedicated to teaching and to the best interests of our students, we do not have a “common vision” of what that means or how to measure it.

The job of writing each test is traded among those who teach the class.  The tests are circulated for comment after Thanksgiving.  All of the students in a course take the exam at the same time, in the same room, in the late afternoon and evening of the first two days of finals week.  We all proctor.  The next two days, we all grade them together, one exam at a time, starting with prealgebra.  Each page of a particular exam is only graded by one person, to keep the grading consistent.  The idea is that we see what our colleagues expect from their students and how they grant partial credit.

I think this process is probably the best way to meet the accreditor’s expectations and to hold instructors somewhat accountable for teaching the generally expected content.  However, the process has not ended up changing many of our individual attitudes or behaviors very much.  We seem to be able to see our colleagues’ points of view but struggle to find a way to compromise on changing any of our individual practices.

I heard last week that now the accreditors will be requiring us to write detailed objectives for each course (not just a course description) that are published and available for students.  What process will we follow?  We could meet, discuss at length, and come to consensus on a list of objectives for each course.  Or, each faculty member could individually write the objectives for a course and ask for any comments by e-mail.  Many mathematicians are introverts and are used to thinking and working in isolation.  They often prefer to be given a job and be left alone to do it.

Personally, I would prefer the former method.  However, I recognize that my perspective is different than many of my colleagues because of my background.  I began my career as a high school teacher.  Starting with objectives, then designing instruction, then designing assessment, and then using the assessment to inform further instruction is how I was taught to teach.  I use the book as a resource to meet objectives rather than defining my course content by selecting sections from a textbook.  It does not seem strange to me that people who teach different sections of the same course would want to work together to produce a common set of objectives and assessments.

Can college faculty be effective when they are asked to teach “out of their comfort zone” or when they do not have the freedom to choose their own assessments?  Should they be required to use a particular pedagogy, to follow a strictly proscribed curriculum, to use others’ assessments?  In the past, the answer would have been a resounding no.  Today, I am not sure that accrediting bodies, state legislators, and state boards of education are going to take that no for a final answer.

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Turkeys and Algebra

Thanksgiving break is almost over.  I was fortunate to spend a few hours at a state park near here, climbing up to the top of Puffer Butte for a great view of the Wallowa Mountains.  As we hiked back to the parking lot, about 30 wild turkeys crossed the trail in front of us.  They followed the leader, heads bobbing.  About three of them ventured off to the side for a few steps but they quickly merged back into line and disappeared with the rest into the brush.  (Photo source:

When we teach developmental or entry level college mathematics, there is a lot of flock mentality going on.  We teach what everyone else teaches and we follow the leader.  It takes tremendous effort to even go off to the side for a few steps.  Why is it so hard to change curriculum?

One major barrier to innovation and change in my classroom is that it may not be accompanied by change in my students’ next classroom, especially if my students transfer.  In Intermediate Algebra, rationalizing the denominator is an obvious example.  By tradition, a simplified expression does not include any roots in the denominator.  This is a sensible practice if the expression is going to be approximated without a calculator.  (Approximating the square root of 2 divide by 2 is a much simpler arithmetic process than dividing 1 by the square root of 2.)

However, in algebra, we are generally not finding an approximation.  In Intermediate Algebra, we rationalize denominators so that the expression is considered to be simplified.  I tell my students that they need to learn the customs and traditions of mathematics.  Leaving a root in a denominator is like chewing with your mouth open: it is no more and no less than bad manners.  I can make a solid argument for taking rationalizing the denominator out of the Intermediate Algebra curriculum.  But, what happens when my students go to the next class, either at my school or at another school, and are expected to know how to do it?  Some might argue that the student can then learn this skill “just in time” and will be fine.  For many of my students, this is not true.  They struggle to learn new material at a typical pace and putting them in a situation where they need to learn material that the instructor assumes is prerequisite could be a tipping point towards discouragement and failure.

Because I have to teach rationalizing the denominator, I choose to use the topic as an opportunity to again emphasize that 1 is the multiplicative identity in the real numbers.  When I’m working on the board, I always write the fraction that is equivalent to 1 in the same color marker.  I ask over and over again, “why can we multiply this expression by this new expression?”

Another barrier to change can be the textbook.  I’m just about finished with an eight year effort to co-author two textbooks, one for intermediate algebra and one for elementary algebra.  When I started, I had a vision  for the content in these books.  That vision emphasized conceptual understanding and applications and deemphasized algebraic manipulations.   As the first drafts of both the table of contents and manuscript were reviewed, the publisher requested changes.  With all of the costs associated with technology and ancillaries for textbooks, they cannot afford to target a text at a limited market.  It has to have broad appeal.  The dominant struggle in this phase of writing was to preserve the emphasize on conceptual understanding and applications while also including the amount and rigor of algebra needed by reviewers.  The changes I wanted to make had to co-exist with tradition.

It is also important to recognize that past experiences of students are also a barrier to change.  Many of my students were accustomed to getting by by  memorizing “steps” long enough to take a test.  When I ask them to “explain why” or to solve multi-step applications that include extraneous information, they feel like they’ve jumped in the river without a life jacket.  As I’ve written before, I think it is critically important to establish a safe classroom in which it is safe to ask questions and struggle.

Finally, we do not work in isolation.  The attitudes of our colleagues have a great deal to do with how much change can happen.  Opportunities for professional development can help create an environment where changes, either minor or major, are expected.   When colleagues are not open to innovation or if workload is so onerous that change is a huge burden, then the alternative is to be creative at the classroom level.  If presented in a non-threatening way, successes at this level can lead to change in department attitudes and practices.  In some cases, the turkeys in the middle can start down a different path and convince the leader and the rest to follow.

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Best Practices and Pig Iron

I’m signed up for a daily e-mail from NCTM called NCTM SmartBriefs, which helps me keep informed about what is going on in the K-12 math community.  This week there was a link to an article in Education Week (behind a pay wall).  The authors teach 6th, 7th, and 8th grades at an independent school in the LA area.  They write about going to a conference in 2011 where they were told that “The only way to be sure that students are learning is through varied and frequent assessment.”  They contrasted that with a conference a year later in which the presenter said that “frequent assessment is unnecessary and actually hinders learning. According to the tenets of this conference, our main instructional objective should be to “get out of the way of student learning.””

The authors also commented on the tone of both presentations:  if the teachers were to be effective, they needed to choose.  There is a Right Way and a Wrong Way.   They also noted that many K-12 administrators have the power to label a teacher as ineffective if the teacher does not “adhere to the optimal form of teaching — in the administrator’s opinion.”

In the past, higher education faculty have been evaluated with a variety of metrics, depending on their position and responsibilities.  Standardized testing of student outcomes has largely been limited to broad general education exams and exit tests in the major.  Professors have had broad freedom to choose how  to teach their classes.  As states move down the path towards performance based funding for higher education and rely more on recommendations by outside consultants, this may change.  Like the two presenters at the conferences mentioned above, these consultants often make claims of best practices supported by the “research.”    I think that this situation poses significant challenges for higher education faculty.

Leaving the issues with making overarching recommendations based on research with limited generalizability for another day, I think the premise that there are specific “best practices”  is itself flawed.  The language and the conclusions may be appropriate in manufacturing, but we are not making steel out of pig iron in a classroom.  The raw material (our students) varies and we cannot reject or control it.  The strategies that work for one group of students may not work for another.  The strategies that I can implement effectively may not work for a colleague.

Although exposure to and analysis of new ideas and strategies is essential, I think imposing teaching strategies on faculty is often counterproductive.  Doing so assumes that faculty are not capable of reflecting about and choosing effective teaching strategies.  Instead of being professionals, we are trainees.  Instead of our chairs being mentors, they become enforcers.  Instead of being on a path to reflection and improvement, faculty become discouraged by top-down imposition.  In the culture of higher education, faculty know that top down “recommendations” often follow the hiring of a new dean or provost, awarding of a big dollar grant, or adoption of a new strategic plan.

One of the justifications for requiring a best practice is to improve student learning.  From a manufacturing standpoint, the qualities of the raw materials and the finished product are relatively easy to define and measure.  Although there are some outcomes of education that we can measure, there are many more that are extremely difficult to describe and assess.  Imposing an industrial model often means that we limit ourselves to outcomes that can be assessed by machine scoring, hardly the glorious improvement in higher order learning often claimed by the supporters of a new approach.  The research that confirms a best practice is frequently limited in scope and riddled with assumptions.

The other justification is to improve student learning by getting rid of the “dead wood,” the faculty who are not keeping up with content, have lost their desire to teach, or are not reflective about their teaching. The reality is that imposing a new pedagogy is not going to change this situation.  The rest of the faculty should not be cudgeled in order to deal with this small minority.

The authors conclude that “As teachers, we have to be educated consumers at the shopping counter of pedagogy, simultaneously remembering our own teaching voice while remaining open to new ideas.”   I hope that higher education faculty, increasingly on annual or part-time contracts without tenure, will continue to be allowed to “shop,”  exercising their professional judgment on how they should teach.

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Nothing to Do with Mathematics

I just had a vet in my office, taking a skill quiz.  He looks healthy and normal, he’s 24 years old.  But, under that normal exterior is a body that is not normal.  He served in the Army, qualified as a paratrooper, was injured by an IED.  He has back injuries, a damaged hip, arthritis, and constant pain.  He is struggling to get treatment other than pain killers, mostly because he “toughed it out” and didn’t get all of his injuries documented when he was deployed.  That is part of the culture but, long term, makes it difficult to prove that injuries occurred from military service.  He is working with a VA advocate and his congressmen to qualify for help.  All of this came pouring out when I asked “how are you doing?”

Another of my students is also a veteran.  He told  me last week he has PTSD.  He would probably benefit from accommodations such as a private testing room and extended time for tests but he is not ready to admit that he is one of “those guys that can’t handle it.”

In a single class, I have four students who I know are under medical care for anxiety or other mental health issues.  Their affect and behavior is well outside the normal range.  Although other students are decent to them, it is difficult to assign them to group work as they do not deal normally or well with frustration or “cognitive dissonance.”

Teaching developmental mathematics is not just about teaching mathematics.  Compassion, kindness, appropriate boundaries, and a variety of teaching strategies are essential but not always sufficient.


What is the Population of the United States?

I’m grading a test I gave in Intermediate Algebra yesterday.  These students have a minimum math ACT score of 19 or have completed an Elementary Algebra course with a grade of C or better.  This test assessed their knowledge of basic problem solving, basic polynomial arithmetic, basic factoring of polynomials, and solving absolute value equations.  I will focus here on student responses to the two application problems on the test.

Students spent a full day in class learning about problem solving.  I did some examples, they worked on some problems together.  The homework that night was all applications; each of the next 5 days they did additional problem solving.  The practice test included four problems.  I encouraged but did not require them to include “word equations.”  They were required to assign the variable, write and solve an equation, and report the answer in a complete sentence.

The test included two problems.  On the first problem, students knew the price of wheat in September and the percent increase in the price since May.  They had to find the price of the wheat in May.  The price of wheat has a significant impact on the economy of our region.  In the second problem, students knew that 15.7% of people in the U.S, 48.6 million people, did not have health coverage.  They had to find the population that this statistic was based on.  On to the exhibits.

This student solved the problem correctly.  Notice that he wrote down his approach to the problem: 46.8 million is 15.7% of what?  He then translated this into an equation and correctly solved it.  Most of the successful students followed this strategy.






This student did not solve the problem correctly.  Most significant is that he solved the problem using an equation that would be appropriate if he were trying to find the previous price of wheat.  Since students struggled with understanding that problem situation, we did several of those in class.  When faced with this problem (which he had done successfully in the homework), it looks like he put the numbers into a framework he knows.  The answer is not reasonable.  If he realized this at the time, he did not try another strategy.




This student found the right answer but did it using a method she learned previously.  I did not teach them to use proportions for percent problems (that’s the subject of a different post).  Notice that she documents her thinking by filling in the blanks of the proportion before writing the equation.   She recognized a problem that she is comfortable solving using something she has done in the past .  However, she also is flexible enough to use other strategies.  She solved the wheat problem without using a proportion.





This student used still another strategy to find the correct answer.  Rather than using a proportion, he seems to have used his understanding of a percent as being part/whole to write an equation.




After finishing grading most of the tests in the class (some have not yet come back from the disabilities services office), 76% of the students correctly solved the wheat problem and 68% correctly solved the population problem.  Five students scored less than 60%. Two did not show up but probably would have failed it if they had taken it.  Three students have recently transferred to an elementary algebra class as it became obvious that their ACT scores did not correlate with what they remember.



Here we see one of the fundamental issues in the current model of education:  if students can achieve a score of 70% or better, we judge that this is good enough to move on.  That means that they cannot reliably do 30% of the material on the test.  This could be due to a “hole” in their understanding of particular concepts, such as problem solving.  Or the errors might be classified as “careless” mistakes i.e. these students have not learned to proofread their work.  Although I will continue to assign applications in the next weeks, we are on to polynomial, exponential, logarithmic, rational and radical functions.

It bothers me a great deal that students can successfully complete this course and not be able to solve a basic percent problem, especially since this course is prerequisite for elementary education majors.  It bothers me still more that students faced with a word problem apparently do not try to reason out a credible solution.  Instead, they search their memories for what they typically call a “formula” but I think of as a template.  They try to force the problem to fit into that template and they do not try any other strategy, even when their answer is not reasonable.

Aside:  This is one thing I really like about having students do some of their homework using an on-line homework system.  Because the system immediately grades the response, the student always knows if the answer is wrong and cannot avoid that reality (e.g. by not checking the answers in the back of the book or looking at their graded homework when it is returned)

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