More on the Evils of PEMDAS!

The most common course for me to teach is ‘intermediate algebra’, and I’ve been thinking of the many issues with that course as part of the college curriculum.  However, my interest today is in poking at PEMDAS … and the poor way we often teach the order of operations.  As you know, understanding the order of operations concept is one key part of understanding basic algebraic notation.

An easy poke at PEMDAS is the “P” (parentheses for us, ‘B’ bracket in some other countries).  The problem below is actually from our beginning algebra curriculum:


Operator precedence usually places products and quotients at the same level, with the normal parsing from left to right (answer: 8).  Of course this ‘tie breaking’ rule is arbitrary; however, a convention about this is necessary for all machine calculation … and our students interact with these machines.

I’ve seen people say that this is a silly point, without merit … and they suggest including sufficient grouping to avoid any “ambiguity” from the expression.  I’ve also seen people say that there is no such thing as implicit multiplication (as in the problem above, or as in an algebraic term like -3x).  What they mean is that implicit multiplication has the same priority as explicit multiplication; some programming environments do not allow implicit products in order to avoid issues with that precedence.

If we state the problem algebraically, it might be:

16÷4k, where k=2

We, of course, prefer fraction notation for quotients due to the ‘confusion’ created by the divided by symbol (which our students write as a slash):


One discussion site has a comment that we should use those grouping symbols to be clear, and concludes with a comment that the answer changes when we use algebraic notation for the same quotient & product expression.  (see  )  This ‘changing answer’ feature should bother all of us!

In the original problem above, the product involves parentheses … so our PEMDAS-based students always calculate that product first.  They have no idea that there is an issue with implied products when variables are involved; I’m okay with that at the time (we get to it later).  In all of my years of reviewing missed problems like that one, I’ve never heard a student justify their answer by ‘implied products have a higher priority’.  They always say “parentheses first”.

If we could say “GEMDAS” (for “grouping”) we would be more honest.  I’m not sure what “G” means for my poor aunt Sally … but, then, having a sentence for an mnemonic with no connected meaning is likely to be a bad thing.  When we continually talk about ‘remember my dear aunt Sally’, we encourage students to process information at the lowest possible level — instead of a beginning understanding, all they get is a memorized rule which is fundamentally flawed.

The role of mnemonics in ‘remembering’ has been studied.  The book Cognitive Psychology and Instruction, 4th edition Bruning et al has a review of research on this on pages 72-73 (it’s also in their 5th edition though I don’t have that page reference).  The basic conclusion was that mnemonics help students remember when mnemonics help students remember … and can interfere with remembering when the student does not find them helpful.  That means the some students can use them to remember, some students get confused … and (in my view) all students have negative consequences for using poor aunt sally.

I think the emphasis on PEMDAS also creates a mental ‘twist’ in our students’ minds.  They take expressions which do not have stated grouping and insert parentheses so that the basic meaning is changed:

5x²  is mistakenly processed as (5x)²

In the intermediate algebra course, some strange things happen relative to parentheses.

(3x² – 5) + (4x + 3) is treated as a product

A good portion of my class time is spent on un-learning PEMDAS and building some understanding of notation with order of operations.  The biggest problem … grouping that is done with other symbols besides parentheses (fraction bars, radical symbols, absolute value, etc).

Because I’ve been teaching so long, I’m occasionally asked about any changes I notice.  Folks expect me to report that students are less prepared now compared to 30 or 40 years ago.  Actually, there have been improvements in the mathematics preparation of our students.  However, these improvements are not uniformly distributed both in terms of students and in terms of mathematics.  In particular, students struggle more now with order of operations; some of that degradation seems to be due to the over-use of PEMDAS.

We should avoid books that build in PEMDAS, and we should avoid the mnemonic in our classes.  Understanding something is much better than memorizing an erroneous rule.

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Understanding the Data on Co-Requisite Remediation

We need to change how we handle remediation at the college level, because the traditional system is based on weak premises … and the most common implementations are designed to fail for most students.  Where we have had 3 and even four remedial courses, we need to look at one for most students.

Because of that baseline, the fanatical supporters of “co-requisite remediation” are having a very easy time selling their concepts to policy makers and institutional leaders.  The Complete College America (CCA) website has an interactive report on this (   ) where you can see “22%” as the national norm for the rate of students starting in remediation who complete a college math course.  With that is a list of 4 states who have done co-requisite models … all of whom show 61% to 64%.

One obvious problem with the communication at the CCA site is that the original data sources are well hidden.   Where does the ‘22%’ value come from?  Is this all remediation relative to all college math?  The co-requisite structures almost always focus on non-algebraic math courses (statistics, quantitative reasoning).  One could argue that this issue is relatively trivial in the discussion; more on this later.

What is non-trivial is the source of the “61% to 64%”.

One of the community colleges from a co-requisite remediation state came to our campus and shared their detailed data … which makes it possible to explore what the data actually means.  Here are their actual success rates in the co-requisite model they are using:

Math for Liberal Arts: 52%

Statistics: 41%

These are pass rates for students in both the college math course and the remediation course in the same semester.  Another point in this data is that ‘success’ is considered to be a D or better.

For comparison, here are similar results from a college using prerequisite remediation, showing the rate of completing the college math course for those placing at the beginning algebra level.

Quantitative Reasoning: 53%

Statistics:  52%

In other words, if 100 students placed at the beginning algebra level in the fall … there were about 52 who passed their college math course in the spring.  Furthermore, this college considers ‘success’ to be a 2.0 or better.  The prerequisite model here has higher standards and equal (or higher) results.

The problem with the data on co-requisite remediation is that only high-level summaries (aggregations) are shared. Maybe the state average for the visiting college really is “61%” when they have about 45% (they have more in statistics than Liberal Arts).  Or, perhaps the data is being summarized for all students in the college course without separating those in the co-requisite course.  One hopes that the supporters are being honest and ethical in their communication.

I suspect that the skewing of the data comes more from the “22%”.  The source for this number usually includes all levels of remediation followed to any college math course (including pre-calculus).  The co-requisite data is a different measurement because the college course is limited (statistics, quantitative reasoning).

Another interesting thing about the data that was shared from the co-requisite remediation college is this statement:

Only about 20 students out of 1500 in co-requisite remediation had an ACT Math score at 15 or below.

At my institution, about 20% of our students have ACT Math scores at 15 or below.  Nationally, the ACT Math score of 15 is at the 15th percentile.  Why does one institution have about 1% in this range?  Is co-requisite remediation being used to create selective admission community colleges?  [Not by policy, obviously … but due to coincidental impacts of the co-requisite system.]

Sometimes I hear the phrase “a more nuanced understanding” relative to current issues in mathematics education.  I suppose that would be nice.  First, though, we need to start with a shared basic understanding.  We can not have that basic understanding as long as the data being thrown at us is ill-defined aggregate results lacking basic statistical validity.

Perhaps the co-requisite remediation data has statistical validity.  I tend to doubt that, as we use a peer review process to judge statistical validity … we we know that has not been the case for the co-requisite remediation data we are generally seeing (especially from the CCA).  The quality of their data is so bad that there would be a failing grade in most introductory statistics courses for a student doing that quality of work.  It’s discouraging to see policy leaders and administrators become profoundly swayed by statistics of such low quality.

Reducing ‘remediation’ to one measure is an obviously bad idea.

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MichMATYC 2016 conference schedule (Saturday, October 15)

The MichMATYC conference planners at Delta College are doing the final tuning of the program for October 15 (2016).  Here is a almost-final schedule of sessions:



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The Right Answer is Not the Thing

This is not another post on assessment, though the content is related.  The central theme in this post is faculty being wise about the process of helping students navigate through mathematics in an efficient manner (something we might call “learning mathematics”) 🙂 .

As context, I want to share part of a lesson from our math literacy course.  Like many such courses, we both use accessible situations and recognizing patterns in the learning process.  This particular lesson uses interest (simple and compound) with these basic steps in the process.

  1. We deposit $500 in an account that pays 6% interest each year, on that $500.  Find the interest earned in the first 4 years by competing the table.  [The table shows a row for each year.]
    Find the total money by adding the interest and the original $500.
  2. We deposit $500 in an account that pays 6% interest each year, on the current balance (including prior interest).  Find the interest and current balance for the first 4 years by completing the table.  What is the total money for this account?
  3. Which account results in “more money” for us?
  4. We found the current balance by calculating “0.06 × 500 + 500”.  Is there a way to simplify this calculation so there is only one multiplication and no addition?

Of course, much time is used in the first two steps.  Students often have misunderstandings about percents, but these are motivational questions … as is #3.  However, the learning in the problem is all about the fourth step, which is looking for “1.06 × 500”.

Many teachers will present the 4th question in a manner that defeats the purpose of the question … “we added 6% to 100%; what do we get?”  This approach ‘works’ in that many students will see how we got the 1.06, and we feel good that they got the right answer.  Unfortunately, we just avoided all of the meaningful learning in this context.

First of all, students need to really know that percents do not have any meaning by themselves.  When we say “added 6% to 100%”, we have reinforced the basic misunderstanding that percents work like decimals in all situations.  It’s easy to determine if students have this misunderstanding by asking a variation of the classic question:

We had a 10% decrease in pay last year, and this year we got a 15% increase in pay.  Our current pay is what percent increase or decrease compared to the pay before the decrease?

This problem is tough for students because it does not explicitly state the core situation … that the base for each percent is the current pay … and we might think that this is the main reason we get the wrong answer “5% increase”. However, even when this fact about the base is pointed out, students continue to add the percents.

Secondly, the “we added 6% to 100%; what do we get?” question divorces the situation from the algebraic reasoning.  We’ve done adding of fractions, where a common base is required.  Somehow, with percents, we are comfortable leaving the base out of the problem when this produces more ‘right answers’.  Each of those percents has a base, which happens to be the same number in this ‘interest’ situation.  A more appropriate instructional move is to provide a little scaffolding:

Let’s write 0.06 × 500 + 500 this way:  (0.06 × 500) + (1.00 × 500)

Remember how we added 4x + 2x?  We got 6x.

Does that suggest how we might do the adding first?

Now, this instructional move will not make the problem easy.  The goal with this move is to connect the new problem to something fundamental in mathematics:  “like” things can sometimes be added.  Having the right answer without applying this concept is not learning any mathematics.

In our Math Lit course, this lesson introduces the concept of ‘growth factor’ which is then used as we identify sequences that are linear versus exponential.  That discrimination in sequences can get quite sophisticated, though we generally keep the level reasonable for the needs of the course and students.  The phrase ‘growth factor’ is used temporarily until we consider declining situations … however, this “adding to get one multiplication” is behind all exponential models.

Unrelated to the main point of this post, it’s interesting that many of us think of the number ‘e’ when exponential models are being discussed.  There are, of course, very good reasons why that is the most commonly used base in mathematics; unfortunately for the learning process, using base ‘e’ presents a disguise of the direct process involved in the situation … a multiplicative factor based on a percent increase or decrease.  I don’t see using ‘e’ prior to a pre-calculus course, in terms of helping students.

Back to the main point … whether you are teaching Math Literacy, Algebraic Literacy, or even the old-fashioned courses, “right answers” are a poor measure of the quality of learning.  The learning process itself needs to be richer and more valid than using a measure known to have limited validity.

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