AMATYC 2014 — Accelerate and Improve Dev Math with “New Life”

Here is the presentation file, as well as the handouts, from the November 15 session.

Presentation: New Life Accelerate and Improve Dev Math 2014 AMATYC

Main handout: References_NewLifeSession_AMATYC2014

Math Literacy Outcomes and Goals: MLCS Goals and Outcomes Oct2013 cross referenced 2 by 2

Algebraic Literacy Outcomes and Goals: Algebraic Literacy Goals and Outcomes Oct2013 cross referenced 2 by 2

Summary of 3 Emerging Models for Dev Math: Summary of Three Emerging Models for Developmental Mathematics Updated 2014

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Technology and Transfer Credit for Math Courses … the Value in Mathematics

At a certain university in my state, there is a policy which states that they will not grant transfer credit from an institution if that institution offers the course in an online format; this is applied even if they know that only 1 section is offered online and 100 are face-to-face.  The policy is applied regardless of the course’s policy on proctored tests for online courses.

At a certain university in a different state, there is a policy which states that they will not grant transfer credit from an institution if that institution allows the use of any calculator in the course; the policy is applied even if students can only use the calculator for trivial purposes (computation).  The policy is applied regardless of the course’s assessments of outcomes and regardless of the overall quality of the course.

These issues are coming up in conversations here at the AMATYC conference in Nashville.  Both policies are implemented out of negative motivation on the part of the universities … whether a lack of trust for their colleagues or a lack of understanding concerning the uses of technology to support the learning of mathematics.  Certainly, universities need to stop their use of arbitrary policies concerning technology, which amounts to a conceited attempt to impose a narrow view of what a ‘good’ math course must be like.

In other conversations, some of my colleagues suggest that we need to present arithmetic and basic skills without the use of a calculator.  One person presented a good point in this regard:  Some students confuse the input/output from a machine for the mathematics.  I agree that students need to have a personal understanding of mathematics.  However, we too often present arithmetic as the initial barrier in front of students, a barrier with little redeeming value and almost no long term benefits to students.

At the same time, I routinely see us in a general consensus of what good mathematics is … and what value it has for students.  Concepts, properties, choices … reasoning, communication, problem solving.  We generally support a ‘common core’ of properties that describe good mathematics.  How, then, can we let minor details about technology determine the transfer of credits and the nature of a student’s first “mathematics” course in college?  Are we so easily fooled by a surface feature (technology) that we do not see the value of the work going on?

This is not to say that all uses of online learning and calculators is good or valuable.  Not at all.  If we use that criteria — sometimes not used wisely — we would not grant transfer credit for any course taught in a face-to-face format because research shows that a significant portion of such classes provide no significant learning of mathematics.  No technology, no pedagogy is beneficial without regard to the quality and wisdom of usage.  Every tool can be used poorly.

It’s time for all of us to make decisions based on an evaluation of all components of a course — the outcomes, faculty, instruction, assessment, and integrity.  There is no room for prejudice in dealing with people … or with courses.  If a person feels that they are unable to evaluate the quality of a course due to the presence of a particular technology, then their professional responsibility to allow others to make the determination.  I would prefer, however, that a person with such a prejudice to seek a better understanding so that their prejudice does not exist anymore.

This is not a problem about ‘us’ and ‘them'; this is a problem about ‘we’.  A professional community, committed to providing good mathematics in service to our students and their success. This is not easy work; rich communication is required, and levels of trust. The path forward is always walked by all.

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AMATYC Presentation: Missing Link — Replace Intermediate Algebra

Here are the files and documents for the November 14 presentation called “The Missing Link: Algebraic Literacy to Replace Intermediate Algebra”.

main presentation: The Missing Link presentation AMATYC2014

References and Curricular Model: References_NewLifeSession_AMATYC2014

Algebraic Literacy Goals and Outcomes: Algebraic Literacy Goals and Outcomes Oct2013 cross referenced 2 by 2

Sample Lesson on rates of change: Algebraic Literacy Sample Lesson Rate of Change Exponential 2 by 2

Here is that sample lesson as full-page: Algebraic Literacy Sample Lesson Rate of Change Exponential

Extra handout: MLCS Goals and Outcomes Oct2013 cross referenced 2 by 2

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A Natural Approach to Negative Exponents

For today’s class in our beginning algebra course, we took a different approach to negative exponents.  The decision to do something different is partially rooted in my conviction that most of our textbooks are wrong about what negative exponents mean.

To set the stage, the first thing we did was a little activity on basic properties of exponents.   The activity is based on this document Class 22 Group Activity Exponents

This activity uses the type of approach many of us use for a more active learning classroom.  I suspect mine is not as polished as many; several students found the ‘long way’ a bit confusing.  As usual, I did not present any of the ideas before students got the activity and worked in their small groups.

One of the problems on this activity Example for negative exponents Nov2014is the problem shown here.  In the ‘long way’ method, students easily wrote out the factors and found the answer.  Quite a few of them used the subtraction method to create a negative exponent.  In a natural way, we noticed that m^-4 is the same as having m^4 in the denominator.

 

Negative exponents indicate division!

We did not create negative exponents in order to write reciprocals.  We started using negative exponents in order to report that we divided by some factors.  I find it troubling that we have focused on a secondary use for the notation, when the primary use makes more sense to students.

If you want to see what is so important about this, give a problem like this to your students.

Negative Exponent Divide not Reciprocal example Nov2014

Direction: Write without negative exponents.

 

Almost every student focusing on the reciprocal meaning will invert the fraction — making the 4 a multiplier instead of the divisor it really is.  Most students focusing on the division meaning will see that the m cubed needs to be in the denominator.

In part of this activity, students also dealt with a zero power.  In doing the long way (write it all out), quite a few students wrote that variable in their work; it made sense, though, to omit that factor because it said “zero factors” … and then we can talk about what value that ‘zero factor’ has in a product (one).

As we shift towards more work with exponential functions, it becomes critical that students understand the meaning of all kinds of powers.  A core understanding of negative exponents is part of this; fractional exponents are important too (though we tend not to cover these in either our Math Lit course or beginning algebra).

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