Basic Math or Pre Algebra or Nothing

What do students need before a ‘beginning algebra’ course?  Several of us (math faculty at my college) are working on this problem, with a goal of helping more students make a good transition to algebra while being aware of other expectations or demands.

My college does not have a basic math class, having eliminated that quite a few years ago.  There is still a prerequisite for the pre-algebra course (a placement test) though the cutoff is not very high, which means that one of the issues is students with extensive gaps in numeracy.  Our pre-algebra course has these components:

  • variables and expressions used from the first chapter
  • signed numbers start next
  • solving first degree equations (some with simplifying first)
  • fractions
  • geometry (formulas primarily)
  • units and conversions (the only math course doing this, for most students)
  • percents and applications (tends to be uncomplicated)

One of the issues I see us dealing with is our own views on “what students should know”.  In our course, we designate the first part ‘calculator free’ because students “should know” their basic facts about numbers; the remainder of the course allows a calculator.  We also expect students to use arithmetic procedures for fractions, though we do not check to see if they understand ‘why’.  We cover classic percent problems, because students “should know” these.

So, what essentials are needed to help students succeed in basic algebra?  In some ways, the answer has been “do some basic algebra”; the last course revision integrated algebra throughout.  We’ve looked at the data for the progression, and it is my opinion that the alumna of the newer course have similar struggles in basic algebra compared to the older course (with less algebra).  One observation is that the students struggle with the expressions and first degree equations that they ‘had’ in the pre-algebra course, whether the algebra was integrated or covered separately.

Here is the basic need I would identify for success in basic algebra:

Students need a core of understanding about numbers and properties, and need a sound beginning on procedural flexibility.

The traditional percent material focuses on correct answers, often using memorized procedures.  I would shift to questions about equivalence and multiple solution methods … because these are core issues in algebra.  My class work and assessments would focus on creating as well as identifying alternate correct methods.  The traditional geometry work in this course also tends to have a focus on correct answers (though we do not memorize formulas).  I would instead deal with how parts of shapes relate to the whole, and concepts of perimeter/area/volume; the same focus on multiple solutions would be appropriate.

The numerical demands of a basic algebra course are quite limited; we are not going to solve a lifetime of numeracy problems in 15 weeks of a basic math course.  A pre-algebra course gains little by making the attempt.   A reasonable goal is to develop a significant set of understandings about numbers and objects, along with the flexibility that this understanding supports.  Deliberate design, sophisticated pedagogy, and faculty expertise are required for this … just as is the case for most math courses that we should place in front of our students.

One of my colleagues used to say:

The student’s fragile understanding of mathematics begins in the pre-algebra course.

We need to shift our focus.  Without understanding, any math course becomes just a barrier to student success.  Without understanding, math is that subject that everybody says they are bad at.  With a focus on understanding, we offer an honest math course that can provide real benefits for students. With a focus on understanding, we demonstrate our commitment and respect towards all students … starting from the first day of our first math course.

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Would You Make This Trade?

Our department is beginning conversations about a new algebra course, with the immediate goal of making it easier to offer a ‘combo’ class for both beginning and intermediate algebra.  We might settle for that outcome, with a savings in credits for many students (from 8 credits down to 6 or 4).  However, the possibilities are not very limited … one advantage of developmental mathematics being on the hot-seat is that those in the approval process are more open to new ideas.

So, here is a possible trade.  We send two courses away (beginning algebra and intermediate algebra) and replace it with one course, for the same number of credits as one of those courses.  We can dream big like this by being willing to consider radical reformulations of developmental mathematics, going in to territory not yet explored by pathways or mathways.

Trade away:  Beginning Algebra and Intermediate Algebra (8 credits)

Receive: One developmental algebra course (4 credits?)

This might one way to get there … start with a set of outcomes from Algebraic Literacy, including the STEM-boosting outcomes, and incorporate a little just-in-time remediation work on basic algebra along with some increase in instructional time each week.  The new course could omit quite a bit of the procedural work that is not that important, and focus instead on goals that are more accessible to a broader section of our population: reasoning and applying.  These ‘higher level’ learning outcomes are more important for further mathematics as well as science.  We might be able to put 30% more students in to this new course than we can with the existing intermediate algebra class.

This type of new course offers great promise for our students; of course, there are challenges for us.  A core challenge: are we willing to give up existing content in this trade?  We get so accustomed to teaching certain skills, these procedures, and those types of puzzle problems; hidden (usually) within this are some good mathematics and valuable learning outcomes.  Getting a world-class course involves being willing to trade in old courses, being willing to let go, being willing to subtract content in a class.

In our situation, we want to expand our mathematical literacy course; this course would be appropriate for most of the students who did not place into the new algebra class, both in terms of prerequisites needed prior to the class as well as preparation for further mathematics.  The Math Lit class gets students ready for a college statistics class and a college quantitative reasoning class.

I do not know how far we will take our current opportunity.  I do know that my vision for a better mathematics program in college starts with Algebraic Literacy.  Whether we make a big change, or smaller, we will be taking another step on this journey.

Have you started the journey away from the old algebra courses?

Note: For those going to the AMATYC conference in Nashville, I am doing a session specifically on Algebraic Literacy; this is session S064 (Friday — November 14, 8am).

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Obi Wan and Mathematics Education

I’ve been thinking about my perceptions this semester.  You see, for the first time in about 4 years, I am not teaching a ‘reasoning’ course — neither our Quantitative Reasoning course (Math119) nor our Math Lit course (Math105).  Of course, I miss those classes.  However, I am actually not aware of missing them on a daily basis.  In fact, I am quite comfortable.

Which led me to the memory of a certain movie moment.  Jedi-to-be Luke is angry with the Jedi master Obi Wan, after learning that Obi Wan did not tell him the truth.

Luke, you’re going to find that many of the truths we cling to depend greatly on our own point of view.

Our point of view is primarily determined by our environment and attitude.  My environment is more traditional this semester; symbolic manipulation and correct answers are high on the list of outcomes.  Like most of us, my attitude when in this environment is impacted by the ‘comfort’ and ‘familiar’ feelings.  I know this … I have competence … this is good.

As a profession, most of us have not yet had the opportunity to take a different point of view about mathematics education.  The majority of math classes are traditional at this time; over the next 5 years, that will change.  So … what comes first: a point of view that supports a reform curriculum, OR experience teaching a reform curriculum?

Like most philosophical questions, there is not a good answer for this question.  However, I will suggest that some of us will need to support a reform curriculum before we have a point of view that is consistent with it.  Understanding comes from experience, and understanding something as complex as the mathematics curriculum in college is a long process.  Early in our New Life Project, some colleagues were suggesting that the best thing to do was to teach a lesson for instructors in the way a reform class would teach it; this would have been a waste of effort: those who do not yet understand why a class would be taught that way … would not understand what they are seeing.

Change just happens.  Progress occurs when some of us are willing to walk a path we do not yet understand.  In some ways, there is nothing more rewarding than beginning a journey without understanding and then finding both understanding and things of beauty along the way.

However you look at issues in developmental mathematics and college mathematics in general, do not let other people put you in a box that says ‘inferior’ or ‘will not change’.  I have faith in each of us, that we are able to become more than we have been.  Our environment determines much about our point of view, and it’s hard to move out of that causality loop.  It takes courage; it takes some inspiration.  I have been impressed by math faculty who have grown in this way.

Especially if you think that the traditional curriculum has much to offer, I hope you will join me on this journey to a better place … a place where we do more for our students, where students are enabled to reach their goals, a place where good mathematics shines in our classrooms.  You are needed; we can not reach our goal without you.

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The Forum on Community College Mathematics (CBMS; Oct 6-7, 2014)

The Conference Board of Mathematical Sciences (CBMS) is a collaborative effort of about a dozen professional organizations in mathematics (including AMATYC and MAA, as well as AMS).  Based on the CBMS view of current events, the group sponsors a Forum on issues related to those events; for example, Forum 4 was related to the Common Core.

This October, the CBMS is offering Forum 5 on “The First Two Years of College Math: Building Student Success”, to be held in Reston (Virginia) on October 6 and 7.  You can see the program at .  A unique feature of the Forum is that the breakout sessions are scheduled based on the wishes of those registering; during the registration process, you can select from the 18 offered sessions.  These are in addition to the plenary sessions.

Of course, I have a personal interest in this Forum … I will be doing one of the breakout sessions along with our friends Uri Treisman (Dana Center) and Bernadine Fong (Carnegie Foundation for the Advancement of Teaching).  Our session is #5, with a title “Increasing Student Success: New Math Pathways To and Through Gateway Mathematics Course“.  We are doing this session together because the work the 3 of us lead has had a long history of collaboration and mutual support; our projects are consistent with each other … more importantly, essential aspects of our goals are the same.

As is normal, travel funds are always a challenge.  In the case of the Forums, the CBMS has some funds to support those who wish to attend the forum.  During the registration process, you indicate your interest in these funds.  Priority is given to small teams (’2 or 3 participants’) from the same institution.

I am really looking forward to “Forum 5″ on the first two years of college mathematics.  Perhaps you can consider attending as well!

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