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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The Magic Solution to Learning in Developmental Mathematics

Contextualize … discovery learning … group work … experiments … homework systems … calculators … modules … learning communities … clickers … tutoring … and smiles.

What was that a list of?  To some extent, that was a list of ‘magic solutions’ offered by somebody to improve (often ‘dramatically improve’, according to that person) the learning in our developmental mathematics classrooms.  Every single advocate of these solutions has some ‘data’ (often labeled ‘research’) to support their answer to our problems; if they don’t have this data themselves, then they are a convert or follower — often a person at a foundation or policy group.

These are not solutions, let alone magic solutions.  Solutions deal with problems; solutions make sense.  Solutions fit within the surrounding systems to enable both long-term maintenance and ‘scalability’. 

Here is the magic solution to learning in developmental mathematics:

Offer sound mathematics with academic value, supported by skilled professional educators who can help every student learn by employing a diverse set of tools, focusing on cognitive growth in students.

We currently do not have sound mathematics in the majority of our courses; there are emerging models that provide some specific alternatives (New Life, Carnegie Pathways, Dana Center Mathways).   Many of our colleagues (perhaps the vast majority in some places) have limited skills further hampered by a limited conceptualization of their profession; organizations such as AMATYC and its state affiliates provide professional development to supplement the internal opportunities.  As a profession, we have not articulated a standard set of tools necessary for faculty to meet the needs of our students.  And, far too often, we look at surface outcomes of success (completion, passing) instead of looking at measures of meaningful growth in our students.

As you can see, there is nothing simple or quick about this magic solution.  I still call it ‘magic’, because this solution creates a qualitative shift in our profession — instead of ‘avoidance’, we have a positive target; instead of a discouraged and sometimes desperate people, we can be inspired and proud (both as mathematicians and as educators).  I admit that this magic solution is not quick, nor is it easy; however, it is a real solution, not a temporary distraction like the items listed at the start of this post.  [Those items are possible tools to use, not solutions.]

Many of us are currently involved with projects that are not really a solution, whether this consists of modules or mastery learning or a temporary redesign such as emporium.  Do not worry about this work; it is part of the process … not the end.  Whether it takes 2 years or 5 years, the incomplete solution will be identified as such, and the next stage will be started.  THAT (the next stage) is what you should be concerned about. 

Behind this basic change is a more developed and refined use of research.  Much of the ‘data’ used in our profession (internally or externally) is just a little better than the charts in USA Today — they are not statistically sound, and do not fit into a body of research for our profession.  Most of this data is better left ‘ignored’.  Our work should be informed by theory and research that develops over time; fads are a distraction from basic change.

I hope that you can focus on the larger picture, on what is a ‘magic solution’; perhaps you can look at the emerging models for inspiration or encouragement.  Our success in this endeavor called developmental mathematics depends more on our internal visions of solutions than on a temporary distraction or ‘data’.

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Research Trends in Developmental Mathematics

If you teach basic statistics in any form, you have probably dealt with the sharp contrast between ‘statistics’ and ‘statistical study’; in other words, there is a large difference between statistical data and the practice of statistics.  Having data does not mean there has been a statistical study.  In a similar way, having data does not mean that there has been research.

Research is an abstraction of this ‘data => statistical study’ to a higher level; research involves a prolonged effort to answer meaningful questions in a field of study, usually involving multiple researchers.  Research, in this meaning, is rare in developmental mathematics — we have lots of data, quite a few studies, but not that much research.  Research strives to provide richer and more subtle answers, and deals with a common core of issues.

One of my friends (thanks, Laura!) recently passed along a link to an article on research in developmental mathematics; this article is by Peter Bahr, whom I had read a few years ago (he’s been busy!).  The current article is called A Case for Deconstructive Research on Community College Students and Their Outcomes, and is available online at http://cepa.stanford.edu/sites/default/files/Bahr%203_26_12.pdf 

This article places research on developmental math within a larger framework of research in community colleges, focusing on student progression.  Which factors in a progression make a difference in the eventual outcome?  One of the conclusions Dr. Bahr reaches is that beginning algebra is a critical course; not passing this course on the first attempt raises the risk that a student will not complete — even if they persist to try the course again.  The article has several other points with practical implications for us, and for policy makers.

Instead of saying that remedial math is part of a ‘bridge to nowhere’ (the mistaken message of Complete College America), research into developmental mathematics takes a more intelligent (and difficult) approach of identifying specific features that have positive or negative impacts on student outcomes.   This research is too specialized for policy makers to understand, even if they understand research as opposed to statistics; part of our responsibility is to articulate what this research means in a manner that policy makers can understand.

I hope that you will use research like the Bahr article to suggest basic changes in your developmental math program.

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Algebra, the Quadratic Formula, and Life

Like most of us, I am still teaching a traditional developmental algebra course — two of them, in fact (beginning algebra, intermediate algebra).  This traditional content includes standard topics, and one of them has been bothering me for some time.  Can we drop the quadratic formula from our courses?

First, let’s come up with a list of good reasons to study the quadratic formula.  Well, did you think of any?  The most common rationale given is that “the quadratic formula solves ANY quadratic equation”.  That is partially true; the formula CAN solve any equation that is strictly quadratic (but not equations that are quadratic in form).  Let’s ignore that shortcoming, as perhaps it is not a significant gap.  The importance of the quadratic formula, then, depends on how important it is to solve a strictly quadratic equation.  Is it the only way?  Of course not — completing the square can also solve any strictly quadratic equation … often with less computational effort.  We also can use numeric and graphical methods to solve any quadratic equation.   Another common rationale for the ‘QF” (as it is sometimes known) as it facilitates the use of the discriminant; whether the discriminant is worth the bandwidth depends on how we use it, and how the study of the discriminant contributes to the mathematical reasoning of our students.  Some people use the QF to determine linear factors of quadratic expressions, which fits in to the ‘correct answer’ world view; I doubt if using the formula to factor expressions contributes to an understanding of equivalence.  [However, I have to admit that our normal instruction of factoring is not really designed to produce understanding of equivalence.]

How about good reasons to NOT study the quadratic formula?  Well, did you think of any?  Quite a few of my students dislike the formula because they realize how likely they are to make a minor arithmetic mistake which results in catastrophic failure to solve the equation.  Some of these students have a strong preference for completing the square — because it provides a logical sequence of steps that avoids many mistakes.  We also have a mythology among our students that says success in mathematics depends on the mastery of formulas to generate the correct answers required in a class.  Few of us concur with the importance of ‘correct answers’ in that myth, but many of us contribute to the myth by placing an emphasis on the quadratic formula.  I would say that the use of the quadratic formula to solve an equation detracts from the mathematical reasoning that I am trying to develop in my students.

Of course, the ‘elephant in the room’ with us is the role of quadratic equations and expressions in general.  Why are they important?  We could spend several blog posts on that topic, and we might go there someday.  For today, here is a brief summary:  the quadratic equations are included to foreshadow some authentic uses in STEM courses later, so we include some puzzle problems that result in quadratics in the developmental course (rectangles of a certain area, projectile under the influence of gravity; we also use quadratic equations as a field test of other algebraic skills (factoring, radicals, complex numbers, etc).  Very few processes (either in nature or in society) are essentially quadratic; the most common quadratic equations in valuable applications come from modeling data (such as fuel efficiency vs speed, profit vs production, etc). 

Very few of the applications leading to quadratic equations have a value in helping our students become more sophisticated in mathematical reasoning, nor in problem solving in general.  The solving of these problems is an exercise.  Therefore, this exercise should develop something of value in our students … and this does not mean ‘correct answers’.    Many applications are solvable by using square roots (like x² = 18), and that method can be connected to a series of related knowledge.  If the problems involve a full-quadratic, numeric & graphical methods provide solutions to most with connections we can make to other knowledge.  Resorting to the quadratic formula bypasses connections and understanding the process, and the QF stands isolated from other knowledge (for almost all of us).

How about a reality check:  How many of us “reach for the formula” to solve a quadratic equation arising from a situation or problem that is worth solving?  These problems often involve non-integer coefficients.  We are likely to reach for the QF primarily when the solutions are complex numbers, where numeric and graphical methods are less accessible.

Unless we teach the Quadratic Formula in a connected fashion, richly connected to basic concepts of mathematics, I think we do more harm than good.  Without those connections, the formula reinforces the myth of right answers.  Mathematics is important in life; the quadratic formula has few contributions to make.

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