This is the first week of our summer classes, so I have been working with my introductory algebra class on basic concepts.  We actually do very little with operations on signed numbers (traditionally the start of an algebra course); instead, we spend 3 class days mostly working on the language of algebra.

My interest is in having each student understand the objects we are using.  When we see ’3x’, I want them to have multiple and correct ways of expression this verbally.  When we see ‘the square of a number’, I want them to have at least one correct symbolic expression they can write.  I deliberately do all of this work without any context for each problem; in other words, the problems are not framed in terms of a situation with physical objects or meaning.

In our Math Lit course, we also do some of this same work.  The difference there is that we introduce algebraic reasoning by talking about some contexts where algebra might be helpful, and then deal with understanding the objects when there is no context.  Does it help to have the context first?  Not really.  It’s fun to have a context, and it motivates some students (though not most).

What seems to happen with context is that ‘understanding the context’ takes quite a bit of energy; I think the brain tends to then organize related information as being connected to that context.  Making the ‘math visible and general’ is not easy, when students begin in a context.  In some ways, beginning in a context comes across as just being a more complicated puzzle word problem (“two trains left at the same time …”).  Students seem to feel like the context was just there to give them another word problem.

One of the myths seems to be that “we need to make it relevant”.  In some cases, we have gone so extreme that we refuse to cover a topic if we can not show students a context that they can see the math within.  I think we have confused math education with something else — having a context for everything is a basic property of occupational training.  Unless we are teaching an occupational math class, context is a tool to use when it helps; context should not be a cage that prevents good mathematics from being learned.

Whatever we might call a course (introductory algebra, mathematical literacy, whatever), a core understanding of basic ideas is critical.  Think about this problem:

2x+4x=??

Without further learning, something like 30% of students will give either 6x² or 8x² as an answer.  [Even among those who generally give the correct answer, their confidence may not withstand a little questioning about 'why'.]  I’m not talking here about understanding operations on rational expressions, or factoring trinomials with a leading coefficient greater than 1, nor about simplifying radicals with an index of 3 and a radicand containing constants and variables.  The issues here deal with the initial constructs of an algebraic language system.

A related issue is ‘transfer of learning’ — context generally creates barriers to transfer.  Context is a concrete approach, and serves an instructional purpose when used appropriately.  However, an initial learning (in context or not) does not enable transfer to situations where the knowledge is needed.

In reforming the math curriculum, we need to keep aspects of the prior design that have benefits for students.  Think about (1) Transfer of learning and (2) Student confidence.  Known factors support transfer of learning — ease of recall, connections, and flexibility.  Student confidence seems to be impacted by feedback and repetition.  The presence of repetition can support both transfer and confidence — it’s not the presence of any repetition; rather, it’s purposeful repetition (including the use of mixed repetition) that provides the benefits.

When people say that algebra is not needed in occupations, this is often based on people in those occupations looking at a list of typical topics in an algebra course.  I think different results would be obtained if we asked about a different list — variables, algebraic reasoning, functions and models, graphical interpretation, etc.

I’d encourage us, as we re-build our curriculum, to incorporate more context — but not be limited by context.  I’d encourage us to help students learn deeply by providing sufficient repetition (with mixed practice especially).

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Most of the data I have seen suggests that the traditional developmental math curriculum tends to reinforce existing achievement gaps.  Students who had done well overall, but not in math, pass our courses.  Students who have struggled pass at a much lower rate.  Access is not the same as equity.  In particular, minority students tend to have much lower pass rates than majority Caucasian students.

In Uri’s talk, he tells the story of how Boeing became successful at building airplanes … by designing ‘fault-proof’ planes, where one failure would not cause a catastrophic event.  Uri calls us to design fault-proof educational systems to avoid catastrophic events for our students.

A Math Lit class is one attempt at a fault-proof course.   In the traditional curriculum, there is a tendency for students to be defeated by mathematical ideas that they did not understand.  The Math Lit approach for this problem is to avoid catastrophic failure; within each class, we identify students who did not understand enough to succeed and provide an opportunity to learn.  We focus on the more important mathematics and cover a few less topics; however, the course provides more hope that all students can succeed regardless of their prior mathematics.

A central part of this fault-proof system is the instructor ‘assessing’ every student’s understanding in every class.  Work shown and dialogue reveal a much richer map of knowledge than can ever be achieved by technology such as homework systems.  Online platforms such as My Lab, Connect Math, and Web Assign play a role for students; however, they are not fault-proof — I believe that they tend to be even more ‘reinforcing existing achievement gaps’ than the basic traditional curriculum.

In general, the New Life model looks at the problem of equity by designing a curriculum that provides powerful opportunities to learn.  Our goal is to create a system where hard work will result in progress for every student.  Because equity is so important, we in the New Life project base our work on the value of instructors working with students on important mathematics in prolonged and intense ways.  No student should be blocked from success by the accidents of their prior learning experiences; no student should be blocked from considering STEM fields by faults in their mathematical knowledge.

If you’d like to see the talk by Uri Treisman, it is available on the Dana Center web site at http://www.utdanacenter.org/its-50-of-the-best-minutes-you-can-spend-to-get-a-detailed-examination-of-educational-inequality-in-america-uri-treismans-equity-address-at-the-nctm-annual-conference/

I hope that you will work with us to build a mathematics curriculum that avoids catastrophic failure, where every student can succeed in learning important mathematics.

 http://www.utdanacenter.org/its-50-of-the-best-minutes-you-can-spend-to-get-a-detailed-examination-of-educational-inequality-in-america-uri-treismans-equity-address-at-the-nctm-annual-conference/

If you’d like to download this as a PDF file, click here: Summary of Three Emerging Models for Developmental Mathematics

The New Life “MLCS” course has updated information in this chart.  Specifically:  New textbooks are being published later this year (Pearson) and early next year (McGraw Hill).  The chart also now uses the Algebraic Literacy title for the second course.  In the Dana Center New Mathways information, some updates are also included.

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The probability for a sequence of (relatively independent) events is the product of the probabilities for each event.

The probability concepts provide a more subtle way of looking at the problem.  Let’s take the simplest possible sequence — two courses.  There are three events involved:

1. Course A
2. Transition to Course B
3. Course B

Clearly, there is an event (or multiple events) prior to Course A.  However, those factors deal with systematic factors generally outside of the mathematics curriculum.  Event 2 is a retention or continuation measure, subject to impacts from within the mathematics curriculum.  However, this transition is an event with a probability less than 1.

For a two-course sequence at my college, the approximate values for the probabilities are:  .68, .75, and .55.  The product of these probabilities is about .28; approximately 28% of students starting in course A will pass course B .  In this case, the conditional probability in course B hurts; however, even if the probability in course B is equal to the pass rate of that course, the result is only a little higher — 33% in our case.

For students placed one level lower, they have a 3-course sequence with 5 probabilities:

For a three-course sequence at my college, the approximate values for the probabilities are: .65, .80, .58, .70, and .64, which have a product of about .15 — approximately 15% of students starting in this course A will pass course C.  [The 'course A' in this sequence is not the same as 'course A' in the prior sequence.]

When our department did a 3-year study following students in a 3-course sequence, we came up with a net rate of 18% (compared to the theoretical value of 15%).  The difference was caused by some additional students who repeated and passed one or more of the 3 courses.

Clearly, the primary method to reduce this net probability — the negative impact of exponential attrition — is to eliminate events in the sequence.  Some acceleration models seek to eliminate transition events — two classes combined into one semester; in some designs, this truly does produce a unitary value for the transition event (100% move from course A to course B).  However, the majority of students probably can not manage a doubling-up like this where they have 6 or 8 (or even 10) credits of math in one semester; this combination model also creates challenges for math departments — small and large.

Another approach is to eliminate the need for a given student to take course A.  In some cases, this is done by state mandate.  More professionally valid solutions involve early testing and intervention programs like El Paso Community College (see http://achievingthedream.org/college_profile/el_paso_community_college ) or boot camps.  Some of these models eliminate both course A and the transition event; most eliminate course A and still have the transition event to course B.  Some other models are described at the California Acceleration Project (see http://cap.3csn.org/ )

The New Life model seeks to eliminate courses from the general sequence and from a given student’s sequence.  A ‘typical’ student faces a 3-course sequence such as beginning algebra, intermediate algebra and then a college-credit math class.  In the New Life model, this 3-course sequence would often be a 2-course sequence (saving 2 events in the probabilities).

For more information on the New Life model, take a look at the Instant Presentations page (http://www.devmathrevival.net/?page_id=116)

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New presentations include:

1. Reform — the Big Picture
2. Reform — the New Life Model
3. The Mathematical Literacy Course overview
4. The Algebraic Literacy Course overview
5. New Life at your Institution

Instead of a redesign, or just flipping a classroom, look at ways to provide better mathematics to your students.  We can create shorter paths through math and enable students to learn sound mathematics that means something.

If you have ideas for other quick presentations, let me know!

 
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Well, a similar thing happens with the content.  We had an ‘algebra II for everybody’ flurry, and we are starting to see the ‘algebra II for a lot less’ movement.  One recent report is being used to say that the math that students need in life is generally taught in 8th grade — fractions, rates, proportions, and simple equations http://chronicle.com/article/High-Schools-Set-Up/139105/ ).  Of course, if we read the original source for this article (see http://www.ncee.org/wp-content/uploads/2013/05/NCEE_MathReport_May20131.pdf ) the conclusions are much more subtle; the source actually says that students need conceptually understanding in general, and list ‘functions’ as a needed topic — and mentions complex measurement ideas and geometric visualizations.

In many ways, the actual source (at the National Center on Education and the Economy) is very consistent with what we found in the New Life project.  The ingredients of Math Lit (MLCS) are based on a very similar list of quantitative needs in occupations and client disciplines at the basic level.  I encourage you to read their math report (link given above at NCEE).

The worry, however, is that people will remember the articles talking about the study; that people will see that story line repeated enough that we begin to believe that it has to be true.

We need to keep our voices in the public conversation so that policy makers hear a more informed point of view, one based on professional expertise and information about what students really need in college for mathematics.

No, Virginia, students do not just need 8th grade math in life.  Many college programs will involve courses which depend on other quantitative abilities, and many occupations involve more than just 8th grade math.

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For both students, they have not seen exponential models in their college (developmental) courses; none of the current Applications for Living students had the Math Lit course previously.  (That will change as some Math Lit students take Applications for Living.)  In both cases, we explore models from numeric and symbolic forms; the Applications for Living course includes more variety, and also requires active graphing of exponential models.

In both courses, students have a difficult time leaving the linear world of adding and subtracting.  There is confusion about the role of slope in an adding world; during the exploring process, we take the time to show repeated adding as a multiplying, and identify the number as the slope.  When we work in exponential situations, the linear view seems to dominate.  During the exploring process, we show repeated multiplying as an exponent and learn about the role of the multiplier.  The performance learning outcomes are not what we would want; there are some differences between numeric and symbolic problems.

For example, the final exam in the Math Lit course had a doubling problem for which students needed to write the model.  Something like:

At the start, 25 people knew about the latest i-product; this number is going to double every day.  Write the exponential model for N (the number  who know) based on t (days since the start).

Another problem for the Math Lit final was a growth pattern from a numeric standpoint:

The cost of a machine is $400, and this is expected to grow by 10% per year.  Complete the following table of values.  [The table shows years 1 to 5, where the value for each year needs to be completed.]

In Applications for Living, the corresponding problems were this symbolic one:

The value of an investment is expected to grow by 6% per year.  Write the exponential model for the value in terms of the number of years.

And, this numeric one:

At 3pm, 20 mg of a drug were in the body.  At 4pm, 15 mg were in the body.  Complete the following table of values.  [The table shows hours 1 to 5, where the amount of drug needs to be completed.]

Almost half of the Applications for Living students treated the last problem as a linear one: They showed values of 10, 5, 0 and 0 (sometimes with a puzzled comment about having zero as the amount).  In class, we had done drugs in both half-life and percent decrease models; we had calculated the multiplier as well.  They did a little better on the symbolic form; part of this is the fact that this course also does work with finance formula, and one of those formulae is basically the answer for this problem.

The Math Lit students did well on the numeric problem; part of that success was the remediation we did earlier when most students had difficulty on all things exponential.  Few of the Math Lit students wrote a correct exponential model, which is noteworthy since the problem is a slight variation of a situation we used to introduce exponential models.  Most of the incorrect answers were variations on y =mx + b.

Clearly, this assessment feedback is indicating a need for an adjustment to the instructional cycles.

However, I also think that the results reflect a math curriculum that tends to treat topics in isolation.  How often do students need to deal with both linear and exponential models in one assessment?  Also, do we use the word “always” with students?  As in: “Compare the y-values; the difference always tells you what the slope is.”  Or, “If you can see how to get the next value in a table, you can always use this to complete a table.”  Or, “In a function, you can always get the next function value by adding or subtracting.”

During the instructional cycles in both courses, I can see the resistance to leaving the linear model.  It’s a bit like distributing, where students become fixated on one process.  I want students to see the power of understanding exponential models; students want the comfort of one model for all situations.

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I am started an updated series of presentations on the “Instant Presentations” page http://www.devmathrevival.net/?page_id=116

Two of the new videos in this series are now available — “Reform, the Big Picture” and “Reform: The New Life Model”.  I hope you and your colleagues find these videos helpful!  If you have suggestions for other presentations, pass them along.  The planned videos at this point include: the Math Lit course, the Algebraic Lit course, and Implementing New Life at your institution.

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As you would expect, the first time through presents some challenges; we already know of several things to change for next time.  In general, the high use of small groups did what we wanted and got students directly involved with the material.  Naturally, this process uses quite a bit of class time.  My major change for next time will be an adjustment to the balance between group work and whole-class work.  When we are developing new concepts, I will keep the focus on group work; when we are more into rules and procedures, I will blend more whole-class work.  This is mostly an issue of practicality, as we ran out of time on most class days.

The Math Lit course is more about understanding than a traditional course, and this is a good thing for our students.  However, students have a harder time judging ‘did I get that’ when the focus is more on understanding.  To help them, I will be doing more daily assessments.  Obviously, this takes class time — which was a problem already.

One specific observation that I did not expect to see — students needed a symbolic rule for slope.  They generally understood that we were looking at a rate of change, but the concepts (rise and run) did not communicate what comparison to make; the ‘(y2-y1)/(x2-x1)’ statement cleared up problems they were having with two things — which values go on top, and keeping the order consistent.

Related to that is another problem — since we saw both linear and exponential models (which both involve two parameters), there remains a bit of confusion about how to write each model.  We still emphasize linear more than exponential, but I can see students mis-match the parameters between and within models.

The single biggest problem?  Getting students to do online homework!  When I could see that students were not doing any homework in the first two weeks, I started checking homework everyday.  The assignment in the book was generally done (though not always including a comparison with the answer key).  The online homework had a rate less than 10% for the class of being done — and this is with one of the better online systems.  In talking to students, internet access was the single biggest problem (though this perhaps was a polite excuse, better than ‘I did not want to do it’); they reported that they could not get online at home, and the usual work/school schedule made it difficult to get to a library or computer lab.    I can not solve the access problem; however, I will apply some additional motivation for them to keep up with the homework.

One of the pleasant surprises is how well ‘dimensional analysis’ went.  The first two times students ran into this, they really did not get it (in small groups, and whole class discussion the same day).  After a quiz and another whole class discussion, most students understood enough to do 3 to 5 step conversions in this style, with work that looks reasonably good.

Overall, the Math Lit class is off to a good start.  The focus on understanding and the use of small groups resulted in a good attitude about learning for most students.  With some adjustments to class procedures and more assessments, the class will work well next time.  [Yes, the outcomes this time were not that good; partially, this is due to a system error in registration which allowed some non-qualified students to be enrolled.]  I am making some changes to the daily schedule, along with the group/whole class shifts and more assessments.

Math Lit is a productive approach for students learning mathematics with understanding.  I am hoping that you will look into developing such a course at your college.

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You may have seen some articles (opinion, more than articles) about the use of Pell grants for developmental courses.  Michael Petrilli wrote one recently, basically saying: If students knew that Pell grants would not cover developmental courses, they would study harder in high school to avoid that later economic problem.  Fortunately, most articles like his are not well crafted, so that only those who already agree before they read it will agree with it when they are done reading it.  (See http://www.bloomberg.com/news/2013-04-30/pell-grants-shouldn-t-pay-for-remedial-college.html)

We seem to live in an era of simple ‘solutions’ to complex problems.

In talking to my students, quite a few worked as hard as they could in high school.  Sometimes, there is a learning challenge involved that remains undiagnosed.  Other students describe situations in which the daily decisions are more about survival in the now than thoughtful consideration of the future; education is often the first casualty of both poverty and family disruptions.  In more urban regions like mine, the high school environment is ‘challenging’; crime and safety compete with academics, and students often attend schools with a long history of problems.

We need to keep our voices raised for those who may not have a voice in the discussion at all — our students.  Those attacking the use of financial aid for developmental courses often lack an understanding of the factors that result in students ending up in our courses.

It is true that we place too many students in developmental courses.  It is also true that our curriculum needs some work.  Attacks on access will not solve these problems.

Here is my simple solution for a complex problem: All politicians and policy makers must begin every speech with an articulate statement on the value of learning in their own lives; further, all politicians and policy makers must complete another degree (at least at the bachelor’s level) every 10 years at risk of losing their job.

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