Calculators as Problem … Calculators as Resource

Earlier this year, we had a post here on teachers as a problem or a resource (see https://www.devmathrevival.net/?p=1021).  Technology — calculators in particular — presents another problem/resource discussion.  Is the use of calculators a good thing, or an evil contribution to an ignorant population of math students?

For example, an article in USA Today mentions calculators as part of a discussion on math illiteracy related to pushing too much math too soon (see http://www.usatoday.com/news/opinion/forum/story/2012-07-09/math-education-remedial-algebra/56118128/1).  I don’t usually cite a USA Today item, as the publication presents so many examples of bad statistics and mathematics.  One line in this article did resonate: Nobody in a high school math class could tell the teacher what the answer is for 8×4 was — without using a calculator.

To some extent, we are still in the “back to basics” movement (basic skills). People who complain about calculators usually mention basic skills or facts as a goal of mathematics education.  We also have colleagues who see nothing wrong with intense use of calculators in math classes; and, we have entire colleges who ban calculators from math classes.  The question, then, is why use calculators?  Why not use calculators?

We need to answer this question within our framework for education in general, and math education in particular.

Education is about a process that creates a qualitative and quantitative change in the capacities of the student.

If a student leaves a class, or a college, with the same capacities with some added skills, we have not educated the student — we have provided some training.  Training is all about skills; education is about capacities.  This is the reason why college graduates do better in jobs and quality of life measures. 

Mathematics education is about a process that creates qualitative and quantitative change in the mathematical capacities of the student.

Knowing the answer to a problem like 8×4 is not an issue of capacity.   However, needing to use a calculator to find the answer to simple problems often means a lack of mathematical capacity.   Capacities are based on understandings and connections; a specific missing fact is not a matter of capacity.  Having a grasp (call it an intuitive grasp) of number relationships begins the network of quantitative structures that make up mathematical capacities.

At some point in reading this, it is likely that you thought of the word ‘memorization’.  When calculators are not allowed in classes such as developmental mathematics, we often justify it by saying that students need to memorize basic facts.  My guess is that students in such classes store number facts in special locations in their brain with an index like “stuff I have to remember verbatim in order to pass”; I would like to see good research on this learning issue.  I want the number facts stored in a more complex way related to indices such as “factors”, “multiples”, “sums”, “differences”, “divisors”, and “properties of numbers”.

In my own classes, I require a calculator for all students.  This happens to be a department policy, though I would do the same thing if it were my choice only.  The issue is not ‘memorization’ — the issue is ‘understandings’ (as part of capacities).  Allowing the calculator implies that I need to observe students and provide feedback about the goals of a math course (understanding).  This is admittedly tricky, and I know that I do not provide enough feedback to enough students. 

A professional use of calculators is to focus on the contributions to learning.  The presence of the calculator provides learning opportunities that I value — such as understanding the difference between (-5)² and -5².  As you probably know, the confusion between these forms is common and problematic; I have students (this week, in fact) who have learned to state the correct words (memorized) but enter it incorrectly on the calculator.

Another example:  One of the most common relationships in the world (natural and societal) is repeated multiplying.  These exponential relationships require sophisticated methods to solve symbolically.  However, a numeric and graphical exploration is within reach — IF we use a good calculator.  Exponential relationships, in fact, are behind many of the general education goals in colleges (science, economics, and politics as examples).  Without a calculator, we are saying that a student needs to complete the advanced symbolic work of a strong pre-calculus course in order to be generally educated.  This is exactly the approach of many universities, including a large institution located a few miles from my college.  Pre-calculus is not general education; it is STEM education, and using that course for general education is part of the larger problem in college mathematics.

One final thought on learning opportunities with calculators — with calculators, we can present reasonable approximations for ‘real world problems’.  The world is messy; few calculations out there deal with integers only, and many involve very large numbers … or very small numbers.  It might not actually help students transfer what they are learning, but it feels better in class.

Can calculators be a problem in a math class?  Obviously yes — depending on many factors.  NOT using calculators is also a problem; knowing how to use technology is an employment skill, and also can support learning mathematics.  Not using calculators puts mathematics in a make-believe world that has no connection to a student’s life; after all, almost all students have cell phones that they use as a calculator … some have a smart phone with a ‘math app’.  We might argue that a spreadsheet is a better mathematical tool than a calculator; as a learning tool, a spreadsheet has a learning curve and some limitations that make it more difficult.

Calculators, then, are both a good thing (resource) and a bad thing (problem).  The important decision is not ‘calculator’; rather, the important decision is ‘learning as building capacity’.

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Mastery Learning in Developmental Mathematics

I have never met somebody who used Mastery Learning in their classroom, nor have I heard of anybody using this method.  Like all absolute statements, this statement has a ‘if … then’ clause within it.  More properly stated, I would say that “Nobody understands what mastery learning is”.

Mastery Learning is not about a minimum performance level on one assignment; that is unrelated to the theory and conception.  Mastery learning is about the entire learning environment, and is based on two goals and a belief.  The goals are to create a learning environment with flexible and adaptive learning resources so that essentially all students (>95%) are able to learn at a high level (‘masters’), and to reduce the learning variation observed among groups of students.    The belief is that almost all (>95%) of students are capable of learning at the master level if provided the appropriate learning resources and instruction.  For some information on Mastery Learning, see http://www.nathanstrenge.com/page5/files/formative-classroom-assessment-0026-bloom.pdf — it is a good summary.  In the original model, ‘enrichment’ was part of the design for students who achieved the master level quickly — they would have time to explore and investigate, create and design.
Mastery Learning is hard work for the faculty and administration involved.  No excuses … if a student does not understand, you find another way to learn and reinforce.  Mastery Learning is likely to be ‘more expensive’ than other models for that reason.  Of course, ‘more expensive’ depends on your point of view — are we talking about the cost per enrolled student, or are we talking about the cost per passing student?

People often confuse Competency Standards with Mastery Learning.  With online homework systems, people set ‘mastery’ levels — but this has nothing to do with Mastery Learning.  If the instructional system is limited to software and tutoring, this can not be Mastery Learning.  Mastery Learning involves the entire learning process, not just ‘homework’.

What would a course look like in a Mastery Learning model?  Here is a brief sketch, based on a program at my College (long since evolved and eventually closed).

• Students begin with a standard assignment
• Students take an assessment (skills, applications, concepts — including novel situations)
• Those who perform at a master level proceed to the next unit (we were not able to design ‘enrichment’ in to the course).
• Students who performed below master level had a diagnostic interview with an instructor.  Options included:
  1. Media (video and/or audio) help
  2. Tutoring
  3. Computer tools (either custom written programs, or packaged, or both)
  4. Hands-on activities
  5. Small groups
• Students initially below master level then took the assessment again.  If not at master level, prior step repeated (with longer diagnostic interview)
• All units completed at master level
• Cumulative final exam, also done in the same master manner.

People using the ‘mastery’ label are generally only referring to the first step (assignments); however, this is not even an assessment — and Mastery Learning is all about assessments (formative and summative).  In Mastery Learning, we aim for the goal of 95% of the students achieving a master level on the assessments; how students do on the original assignments is not usually considered.

The other comment to make about Mastery Learning is that the model is not just about skills and procedures, even though most uses of ‘mastery’ refer to only that.  Mastery Learning is an approach for any content.  The financial resources make both Mastery Learning and ‘more than skills’ a challenge — they both cost more.  Does your college have the commitment to the goals of Mastery Learning?

In my view, the main disadvantage of Mastery Learning is that it tends to deal mostly with individual situations, not social; group learning processes tend to be very difficult.  I like to create a sense of community in my classes, and that would be very difficult in a Mastery Learning model.  Of course, that is also true of the ‘lesser cousins’ that we see much of today — online homework, modules, emporium, etc; these are all ‘lesser cousins’ of Mastery Learning, as they share disadvantages without any of the benefits.

The best thing about Mastery Learning is the demand it places on us — we seek to have 95% of all students become masters; ethnicity would not be a correlation with success, nor would economic status.  In Mastery Learning, all predictors of success are eliminated because all students (at least 95%) succeed at the master level.  As you know from your test analysis work, the highest discrimination is possible when the difficulty is 40% to 60% — which is exactly where our courses are today.  If we could do Mastery Learning in developmental mathematics, that outcome would be worth the disadvantages and costs.  We tend to reinforce inequities, not overcome them … Mastery Learning can be a part of a solution.

Most of us will not have the institutional commitment to make Mastery Learning work for the 95% of our students.  Look at the emerging models for developmental mathematics (New Life, Pathways, Mathways) for other ways to get our course difficulty out of the high discrimination zone (pass rates).  In those models, pass rates in the 70% to 80% range are possible … and that would be a big step to eliminate the high discrimination we currently see.

Whatever your model, never take the Mastery Learning label lightly.  It’s way more than a setting on a homework system.

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Variables Less Understood

In a traditional beginning algebra course (like one I am teaching), we spend much of the time working with variables in expressions and equations … and functions.  The course first shows variables within simple expressions with a progression towards complexity and equations.  One problem (and the student errors on it) really caught my attention this semester; it’s actually a simple problem, not requiring any procedural steps.

Here it is:

Solve   -6k + 3 = 3 – 6k

I have to admit that I did not emphasize ‘reading the equation to see what general statement it is making’, though we did actually talk about equations where the variable term was equal on both sides.  One of the common errors is shown below:

Every student making this error could some an equation like  ‘2y – 5 = 4 – y’.  What was causing the error?

Sadly, the problem was that many students are learning the ‘algebra dance’: Duplicate these steps, record the result.  Part of the dance is to write the opposite of the variable ‘thing’ on the other side to get one variable in the problem.  Students used this dance to solve a number of equations to produce quite a few correct answers.  For this problem, part of the dance was the ‘get one variable’ — the student knew that -6 + 6 was zero, so we just have the letter.  The variable was less understood than we thought, based on the consistent correct answers to other problems.

It’s very likely that you can list some mistakes that are similar in showing a less understood variable concept.  One of the errors I am seeing is “5 + 2x” becoming “7x”; the numbers and letters become the whole story … the operations are not even being read.

If you are curious, there is a wide body of research on learning variable concepts; for one summary, see http://www.nctm.org/news/content.aspx?id=12332 (an NCTM item).  Some particular research items:  http://www.emis.de/proceedings/PME29/PME29RRPapers/PME29Vol3KoiralaEtAl.pdf and http://www.merga.net.au/documents/Steinle_RP09.pdf  and http://elib.mi.sanu.ac.rs/files/journals/tm/16/tm915.pdf. 

What I am focused on, however, is not the research nor the particular misunderstandings — rather, I am thinking about WHY this happens.  It seems the problem is most likely when students have a higher motivation for ‘correct answers’ compared to their valuing of understanding (which is a combination of desire to understand and confidence in being able to).  In my classes, I often say that I am not that interested in the answer they get; I am more interested in the knowledge you have about that type of problem (the understanding).  Obviously, this statement from me does not change the drivers of student motivation (answers or understanding); I need to create instructional spaces where the understanding is the result being assessed directly.  I suspect that I will be using some type of writing for this purpose; this will be a challenge, given the range of writing abilities in the class.  For one reference on writing in math, see http://www2.ups.edu/community/tofu/lev2/journaling/writemath.htm. 

However, I can count on a basic human trait:  We (meaning our brains) naturally prefer to understand the world around us.  Knowledge organized by understandings is easier to maintain and use, compared to knowledge that is random memories.  The problem with ‘variables less understood’ is that this natural desire to understand has been subverted, perhaps caused by messages about ability … perhaps reinforced by social messages that math is about formulas and answers.

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Modules in Developemtnal Mathematics — pro and con

I am hearing about colleges either adopting or considering modules in their developmental mathematics program.  Sometimes, this is done as part of an ’emporium model’; however, other designs make use of modules.  Perhaps it would help to have a brief exploration of the pros and cons of modules.

The word ‘modules’ does not have a uniform meaning for us.  In general, a ‘module’ could be another name for a ‘chapter’ — each being a sub-unit within a larger organization of material.  However, most uses of the word ‘module’ refer to one of two approaches to content — uniform sequence of modules or customized sequence.

The difference between the two uses can be subtle, such as a case where the customized exit point is the end of a ‘course’ — some modular programs designate ‘modules 5 to 8’ as a course, and that is where the exit point is.  Customizing is done by either changing the ending module within a course or changing the entry point (starting module) within the course.  Conceptually the contrast for the two designs is important due to the fact that a customized program prevents a summative assessment common for all students.

Over the past several years, I have had discussions with faculty involved in a type of modular program.  Via this obviously non-scientific method, I have developed some pros and cons for modularization.  Most of these apply to either type (uniform or customized).

A modularized approach is usually based on an assumption that the delivery mode is a major source of problems, sometimes stated “we can’t teach this to them the same way they saw it the first time”.   I have not seen any evidence of this being true; it’s not that I want to teach them “the same way” (whatever that means) … it’s that this assumption about the delivery mode often precludes examination of larger issues about the curriculum.  Modularized tends to reinforce notions that ‘mathematics’ is about knowing the procedures to obtain correct answers to problems (often contrived and overly complex).  Our professional standards (such as the AMATYC Beyond Crossroads  … see http://beyondcrossroads.amatyc.org/) begin the discussion about mathematics by describing quantitative literacy.  This aspect — of modularization tending to limit the mathematics considered — is the largest factor in seeing this approach as being weak and temporary.

The other major area of concern, suggested somewhat in the pros and cons, is the professional status of faculty in developmental mathematics.  Administrators and policy makers often do not understand the professional demands of being developmental mathematics faculty; in the modularized approaches, faculty tend to look a lot like tutors.  This similarity then suggests to some that faculty are not necessary, and we can provide a larger pool of tutors.  Our professional standards call for us to see the work in math classrooms as being rigorous in both mathematics and education.  This aspect — the professionalism of faculty — is the most common concern reported by faculty engaged in a modularized program.

Summary:
The attractiveness of modular approaches is easy to understand.  However, the typical implementation of modular approaches will reinforce a traditional content with a weaker assessment system combined with a generally lower faculty professionalism.  When implemented, modular programs will tend to be temporary solutions.  The emerging models — New Life, Carnegie Pathways, Dana Center Mathways — provide a clear alternative to address the problems based on professional standards to create long-term solutions.

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