The Power of Understanding Math

One of the most rewarding experiences we can have is when a student exceeds our expectations.

This is a story of a student who initially struggled with everything and is now being successful within an individualized course structure.  In this class, I never ‘lecture’ to a group of students.  Class time is used for studying, help, consultation, and testing; we call it the “Math Lab” though it’s not what most people mean by that phase.  The common meaning of “Math Lab” is a drop-in help center open to a variety of students in a set of math classes.  Our Math Lab is a way to take a few math courses … our math help for other classes is separate from the Math Lab.

This particular student (I’ll call him Philip) was clearly having trouble on the first day.  He did not want to use the online homework system, and that was not a problem for me.  However, he opened the book to the first page of the first section and had lots of questions about the names for types of numbers, about order of numbers on a number line, and (shortly after) about adding signed numbers.  The second day brought questions about the meaning of words in statements of properties, and about the meaning of variables.

It’s not that most students “get” these things, nor that they do not need to work on them.  What was unusual was the level of the struggle (basic) along with the sheer quantity of questions.  I never tell students what my prognosis is for them (I’m sometimes wrong) but I thought this student was going to spend weeks on every chapter.

Philip did, indeed, spend weeks on chapter 1 … a chapter about real numbers in a beginning algebra course.  Following those weeks, Philip then missed several classes due to medical problems related to his PTSD and physical injuries.  With over 6 weeks gone, Philip had only tried that first chapter test.  He was about to encounter the chapter on linear equations and applications, a classic “speed bump” for students struggling to learn algebra.

Somewhere in the month after that, however, Philip began making consistent progress.  In fact, he was getting through the third chapter faster than many students.  That progress has continued, and Philip is very likely to pass the course.

The main point is that something in the way Philip dealt with the struggle made a difference in how he succeeded in the entire course.  Philip works towards understanding everything, including ideas the are relatively minor.  He writes down lists of both vocabulary to learn and problems that he needs help with.  My guess is that his turn-around from struggle to success was caused by his hard work at understanding (and not just knowing what to do).

We all have students in this level of course who interact with the material at a low level; for them, it’s more about remembering what to do than it is about understanding.  I think Philip’s intense effort at understanding provided him with a cumulative positive improvement in the ability to learn new material.

Like most of us, I strive to have all students look for that understanding in learning mathematics regardless of the specific math course.  With other students, I end up trying to pull them someplace they have no intention of going (understanding) while Philip approached the material that way without any influence from me.

As a minor point in this post, I will point out that a struggling student such as Philip will be lost prior to getting any success.  Taking several weeks on one chapter is not an option within a fixed-pace class; instead of accumulating benefits, struggling students accumulate bad grades on assessments.  Our Math Lab, with its focus on individual learning, allows a struggling student to truly become a successful student.

A fixed-pace class has a limited capacity for helping struggling students; they need to be within a relatively small range of struggle in order to succeed.  Our Math Lab expands that range considerably (though there are still limits).

Understanding … a focus on understanding … enables students to obtain power in mathematics by raising their level of functioning to a higher point.

 
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What’s in that Fraction?

Sometimes students conceptualize math problems in ways that are mysterious to experts, but make sense to them.  On occasion, a bad conceptualization seems to be reinforced by features in the technology they are using.

I was helping a student work with rational expressions in our intermediate algebra course.  This particular student finds the material difficult, and often puts off dealing with the course.  Today, he was starting the first section which includes this problem:

If f(x) = 10/(x+1), find f(½)        [Presented in typical rational expression format.]

I think the student conceptualized fractions as two connected buckets (one for numerator, one for denominator) without seeing any particular meaning for the buckets together.

This student was doing most of their work on an older Casio graphing calculator, which shows fractions like this:

In other words, the calculator has a “a b/c” key used to enter fractions.  The student was trying to type in “10_½+1” so the calculator was showing ’21’ for an answer (which was a mystery answer for this student).  When I suggested using the division symbol instead of the fraction key, there was a resistance … until he discovered that it gave the correct answer for the online homework system.

I think it is pretty common to have students missing concepts in the meaning of fractions.  Frequently, they have trouble connecting a fraction with both one division AND with a combined product and quotient … where this last meaning allows for most of our algebraic work on rational expressions. Our instructional materials frequently emphasize the first concept (a single division), and never make explicit that a fraction also means multiplying and dividing … that “(3x)/(x²+2x)” means multiplying by 3x and dividing by (x²+2x).  Result: memorized rules for how we reduce a fraction.  It’s so much easier to focus on ‘multiplying and dividing by the same factor results in one’ as a concept … rather than ‘cancel common factors’ alone.

We might blame such misconceptions on an over-use of technology, or on a given calculator providing the ‘a b/c’ fraction key.  I think students have the misconception independent of the technology, and that the technology my student was using made it easier for me to identify the issue.

When a person looks for either research on learning fractions, or for suggested instructional sequences, there is agreement that a flexible and more complete set of concepts is critical for the diverse settings where fractions are used.  Our course materials (especially in developmental math, both in pre-requisite and co-requisite models) tend to focus so much on procedures that we never develop any further concepts about fractions.  That is really a shame, since students will forget the procedures; the concepts have a longer shelf life in the human brain.

We should always start with meanings and concepts … especially with fractions.

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Can We Save “Order of Operations”??

In one recent post, I looked at some basic flaws in the mnemonic “PEMDAS” (there are several fundamental flaws). In another recent post, I talked about how unimportant a ‘correct answer’ can be in a math class.  Let’s examine the intersection of those thoughts, and deal with saving the important topic of ‘order of operations’.

The two most common statements about why “order of operations” is important are:

• “The order of operations is just an agreement so we all get the same answer.”
• “You need to follow the order of operations so that you will get the correct answer.”

Both of these miss the point; their implication is that we can change the correct answer just by changing the ‘agreement’ about order of operations … that we could declare subtraction is always done before multiplying, for example.  The order of operations is not just some coincidence of the mathematical language which will evolve to be anything fundamentally different.

The reason the ‘order of operations’ is so important is that the meaning of a mathematical statement is based on understanding the order of operations.  In natural languages, the presence of multiple verbs in a statement is unusual … in mathematics, this is commonplace.  Multiple operations in a statement with nouns and adjectives provides an efficient method of communication, which is why scientific advances increased dramatically after the use of symbolic mathematics (as opposed to the original verbal forms).

Not only does “PEMDAS” have little to do with correct order of operations, the way ‘order of operations’ is typically taught has little to do with mathematics.

When we learn a computer programming language, we face the issue directly — what is the precedence order for operations?  Although there are some minor differences in the details, almost all precedence orders are based on a fundamental mathematical idea:

The more advanced operations are done prior to simpler operations.

We teach students that exponents are repeated multiplications; what we don’t divulge is that this means that exponents are more advanced operations … and therefore are done prior to multiplying.  We cover the procedures for multiplying and dividing fractions, but do not make sure that students know that these procedures are based on the fact that multiplying and dividing are at the same level of complexity, mathematically speaking.

The fundamental idea that “more advanced operations are done first” covers the majority of what we try to do with ‘order of operations’.  The difference is this:  order of operations is treated as a memorization issue, while ‘more advanced operations first’ is calling for understanding and communication.  How students get to a ‘correct answer’ is more important than the fact that they got a correct answer.

In those computer programming languages, operations are categorized into binary and unary types, just as mathematicians do.  The ‘more advanced first’ principle handles almost all cases in both types.  Even the type some of us complain about:

-5²

Even though this ‘ambiguity’ is not encountered very often in real-world problems, this is a core issue in communication.  How do we interpret:

-x²

We certainly don’t want people to apply the opposite operation prior to squaring, and we certainly don’t want the answer to change when given in variable notation.  In both of these problems, the “-” means opposite … which is less advanced than squaring; therefore, square first, then apply the opposite.

The few places where ‘more advanced first’ fails are also places where ‘order of operations’ fails, and these are often due to our failures to maintain integrity in our language.  Our notation for trig functions is sometimes bad, or even incorrect (when it creates an inconsistency with other operations or functions).  Even if we don’t change our behavior in trig functions, students will be better off with ‘more advanced first’ than they are currently.

I’d be happier if we never used the phrase ‘order of operations’; the entire implication of this phrase is ‘memorize the rules, or else’.  Our students would have a higher quality learning experience if we just focused on ‘more advanced operations first’.  The emphasis this involves on the meaning of expressions helps novices reach a deeper understanding of our mathematical language.

Which of these is a better answer to the question “why did you multiply before you subtracted”:

• I multiplied first because the order of operations says to multiply before subtracting.
• I multiplied first because multiplying has a higher precedence because multiplying is more advanced.

As we strive to help our students understand and reason in mathematics, an ‘order of operations’ has no place in the curriculum.  Knowing a structure for operations, including ‘more advanced’, is critical.

 
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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:

16÷(4)(2)

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):

16/4k

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 http://math.stackexchange.com/questions/33215/what-is-48%C3%B7293  )  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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