Three Myths About Math Education: Part 1-Inch Deep and a Mile Wide

No school subject experiences more public controversy than does mathematics. Whether it’s the start of the school year, the fallout from a letter to the editor of a local paper or, perhaps, the release of an international measurement such as PISA, numerous occasions present themselves throughout the year that stirs up the intense negative feelings the general public seems to harbour toward mathematics.

These criticisms tend fall into one of two broad categories:

  1. Issues with what is taught.
  2. Issues with how math is taught.

Let’s start with Item 1, the what.

There seems to be an unchallenged assumption that math education is trying to do too much. In fact it’s become popular to characterize the math curriculum used in my province (Newfoundland Labrador, Canada) as “an inch deep and a mile wide,” a sentiment typically voiced by those apparently longing for the so called “good old days,” times when all students allegedly sat quietly in rows and assiduously practiced nothing but their math facts; sum after sum after sum, inching closer and closer to perfection one supposes.

Kind of reminds one of a pre-Christmas Carol Bob Cratchit, does it not? That and everything that goes along with it, you know, thrifty middle class that doesn’t ask too many questions, stays out of the way of the privileged ones, doesn’t rock the boat too much, especially with that pesky liberal minded thinking.

But, that’s a digression…

Let’s come back to reality and take a closer look at that apparently too-unfocused curriculum.

A few weeks back I took the time to dig out, from the education library stacks, the student textbooks and teacher resources currently in use in my province for grades 3 to 6. Why those grades? Simply put, there wasn’t the time to go through all of the grades. A quick glance through grades k-2 showed no great surprises anyway. As I was already intimately familiar with what’s there from grades 7-12 I decided that no personal gain would come from a review those so I settled for the middle ground of grades 3-6.

I then scanned through the materials noting the major topics and organized them under general headings such as Measurement, Geometry and such. The list, once completed, did not seem all that diverse.

I then took it to the next level and decided to do the same for the curriculum that had been experienced by those typically levelling the criticisms: the adults. This posed a bit of a challenge as the math curriculum has been changed several times in recent history. The current one dates from around 2009, the previous from around 10 years previous to that, then the late 1980’s, the early 1980’s and, finally the early 1970’s. Since I was quite familiar with all but the last one, having been a teacher / program development specialist during the years they were in use, and knowing that they were not substantially different, I decided to go for broke and take on the earliest one I could find in the archive.

After all, the so-called good old days that the loudest opponents seem to be calling for did seem to be set in those times.

The series was called “Investigating school Mathematics,” published by Nelson and was in use in this province (NL, CA) during most of the 1970’s. I dug the teacher and student materials out of the archives and repeated the exercise.

The lists were then matched and tabulated side by side using the best match between terms. The results are below.

Here’s Grade 3:

Grade 3
1973-74 2009-10
Patterning (increasing & decreasing patterns)
Counting & Measurement (length, rulers, half-units, area & volume with squares & cubes, liquid measure) Measurement (time, calendar, ruler, perimeter, mass)
Number Theory (odd / even, multiples, factors, primes, place value) Numbers to 1000 (representing, ordering & estimating)
Addition & Subtraction (relation between the two, large numbers, regrouping, money) Addition & Subtraction (relation between the two, 2-digit addition / subtraction with regrouping, estimating)
Fractions (parts, naming, comparing)
Geometry (edges, faces, vertices, segments, angles, triangles)

Geometry (parallel & intersecting, angles, quadrilaterals, parallelograms, polygons, symmetry)

Geometry (point pictures, symmetry, translations, functions, bar graphs, negative numbers)

Geometry (polygons, prisms, pyramids, cylinders, cones, spheres, sorting)

 

Multiplication & Division (number line, sets, repeated addition, commutative, multiplication-addition, multiplication algorithm, estimation, division by rectangular arrays, repeated subtraction, number line, quotients as factors, relation to multiplication., finding quotients, remainders, checking) Multiplication & Division (repeated addition, grouping, division as sharing, repeated subtraction, arrays
Data Analysis (collecting, bar graphs)

There’s fairly similar, aren’t they. For the one in the 1970s you may have been expecting something that was heavy on the basic computations and memorizing of math facts, perhaps. While, for the earlier one, there are more words next to the Multiplication & Division section you will notice that the focus is on various strategies by which the process is understood and used instead of the “drill & kill” you were probably expecting or, for that matter, think you remember. It’s essentially the same for everything else. The one notable thing is the inclusion of a Data Analysis (focusing on reading bar graphs) in the newer curriculum.

Here’s the table for grade 4.

Grade 4
1973-74 2009-10
Patterns (in tables, in charts, solving problems and equations with patterns)
Numbers & Measurement (estimating, perimeter, distance on maps, area from counting squares, volume from cubes, liquid measure) Measurement (time, dates,  area & volume, cm2 & m2)

 

Numbers & numerals (place value, inequalities, large numbers)

Number Theory (even/odd, factors & products, primes, function machine)

Numeration (Place Value, number words, large numbers, number lines, inequalities, ordering)
Addition & Subtraction (function machine, regrouping, magic squares, money, estimating ) Addition & Subtraction (L-R, and R-L, regrouping, renaming, estimating)
Multiplication & Division (relation between, mult.-add. principle, Multiplication facts, function machine, multiples of 10, Multiplying 2 and 3 digit factors, problem solving Dividing through subtraction, estimating, 3-digit quotients, 2-digit divisors) Multiplication & Division (relation between, Multiplication facts review to 7×7, New to 9×9. skip counting, doubling & halving, multiples of 10, patterns in a multiplication Table, 3-digit multiplication, dividing through subtraction, estimating quotients)
Fractions (number pairs, parts of a whole, representing length, equivalent fractions, adding, mixed numbers) Fractions & Decimals (fractions of whole and group, comparing & ordering fractions, decimal tenths & hundredths, estimating, making change)
2D Geometry (points, segments, rays, angles, Parallel, pyramids, symmetry)

3D Geometry (cylinders, cones, circles, tangents, congruent and symmetric figures)

Motion Geometry (ordered pairs, stretching & shrinking, translations, negative numbers)

2D Geometry (symmetry, tangrams)

3D Geometry (prisms)

Data Relationships (pictographs, bar graphs, Venn and Carroll diagrams, solving problems)

Once again, the two versions are remarkably similar. Once again look to the multiplication and division section. In both cases the students are expected to know the multiplication facts up to 7×7 at the start of the year and up to 9×9 at the end of the year but in neither case does it seem that s single minded focus on drill is evident. Rather, it seems that the focus is on building in an overall understanding.

Grade 5 follows.

Grade 5  
1973-74 2009-10
Sets, Logic and Patterns (exploring sets and patterns) Patterns (modelling patterns, increasing & decreasing patterns, solving problems and equations)
Numbers (place value, different bases, old numerals)

 

Numeration (representing numbers, expanded form, rounding, large numbers, decimal place value, rounding decimals, comparing decimals, estimating with decimals, adding & subtracting decimals with regrouping)
Decimals (and fractions, adding and subtracting, with metric units)  
Equations & Operations (function machine, function rules, solving equations, multiplication-addition principle)  
Estimation (note appears throughout; not done separately)
Computing (area, perimeter, average) (note:  area & perimeter done in measurement unit)
Dividing (estimation, 2-digit, larger)  
Number Theory (factor trees, primes, Union, Intersection, GCF, LCM  
  Multiplication & Division (strategies, relating facts, by tens hundreds etc., halving & doubling, estimating, multiplying 2-digit numbers, multiplying using base 10 blocks & arrays, estimating quotients, using subtraction to divide, dividing by sharing, remainders as decimals, interpreting remainders)
Fractions (length, naming, comparing fractions, ratio, LCM, mixed numbers and addition, subtracting, measurement)

Multiplying and Dividing Fractions (whole numbers and fractions, dividing by faction, function machine, equivalent fractions, lowest terms)

Fractions (and area, equivalent fractions, on a number line, comparing fractions, decimals)
2D Geometry & Measurement (points, lines, segments, congruence, parallel, measuring angles, symmetry, congruent Triangles)

3D Geometry (space figures, volume, surface area, POV)

Graphing and Geometry (Symmetry, Rotation, Enlargement, Tessellations)

2D & 3D Geometry (lines & faces, parallel, intersecting & perpendicular, sorting quadrilaterals)

Motion Geometry (translations, reflections, rotations)

A New Math System (other kinds of arithmetic e.g. clock arithmetic)  
  Probability (lines, experiments, comparing probabilities, solving problems)
  Data relationships (gathering data, double bar graphs, solving problems)

Measurement (length, perimeter, area, volume in cm^3 and m^3, litres & ml)

Again, the two are more alike than they are different.

And, finally, here’s grade 6

Grade 6  
1973-74 2009-10
Sets, Logic and patterns (classifying figures, exploring similarities and differences, dot matrix printers) Patterns (identifying patterns in numbers, in tables, creating tables using expressions, equivalent equations)
Numbers (exponents, bases, scientific-notation) Numeration (millions, billions, large numbers, millionths, billionths)
Number Theory (primes, factor trees, sets, GCF, LCM) Number Theory (factors, multiples, primes, integers, order of operations)
Whole Numbers (function machine, distributive principle, equations  
Ratio and Percent (ratio, equivalent rations, scale drawings) Ratio & Percent (ratio, equivalent ratios, percent, scale diagrams)
Fractions (equivalents, lowest terms, Adding & Subtracting Fractions, Multiplying & Dividing fractions) Fractions (modelling fractions, improper fractions & mixed numbers, comparing fractions)
Decimals (multiplying & dividing, rounding) Decimals (money, multiplying and dividing by 1-digit numbers)
Geometry (perimeter, circumference, area, Pythagorean, volume

Geometry (congruence, construction, symmetry, angles, tessellations, cross sections

Geometry & Graphing (ordered pairs, reflections, rotations, translations)

Geometry (angles, angles in triangles & quadrilaterals, area, perimeter volume of rectangular prism, classifying triangles, sorting polygons, congruent polygons

Motion Geometry (reflections, translations, combining transformations)

Probability (experiments, outcomes) Probability (experimental vs. theoretical probability
Positive & Negative Numbers (meaning, adding & subtracting)  
  Data relationships (questionnaires, databases, experiments, line graphs)

Regardless of how closely you read the above tables it’s obvious that the curriculum of today is not so very different from the one experienced those who’d now be in their 40s and 50s. In particular, there’s no evidence that that today’s covers significantly more ground.

So where does the notion that “today’s curriculum tries to do too much and we should move back to the way it was,“ come from? Frankly I am at a loss and can only speculate that it is an erroneous thought, the product of selective memory, and one that that has never been challenged. As you can see, though, there’s scant evidence that supports that it’s much different from the one experienced by those fronting the accusation.

But, perhaps you’re thinking, “No, it’s not the what I was worried about. It is, rather, the how. Today’s math education is focused too much on “discovery learning” and it’s silly to expect that students, by themselves, can be expected to rediscover what took genius mathematicians many lifetimes to figure out.”

Let’s deal with that in the next post.

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4 thoughts on “Three Myths About Math Education: Part 1-Inch Deep and a Mile Wide

  1. I shall look forward to your ‘how’ post. Teaching methods are definitely different now to how I was taught. I never ‘got’ maths and still find the subject hugely challenging (apart from geometry) but when I’m in a classroom supporting students with their learning, I’m able to make some sense of it. Whether this is a maturity thing, who knows, but my understanding of a subject that used to terrify me is definitely improved. One of the maths teachers whose class I support regularly is appreciative if I let him know that I’m not ‘getting’ whatever it is he is delivering as he reckons there will be students there too nervous to own up that they too are struggling.

    1. You and I see it alike. Too often the approach is “it’s the student’s individual responsibility to get this and my class will just keep on the way it always has. As I see it that’s a cop out. Whenever students are having difficulty job one is to find out why. Perhaps it’s because the student just plain does not want to well in anything but more likely it’s because, for whatever reason, the student is having difficulty learning in the environment they are currently in and something needs to change if they are to have success. perhaps it’s that issues at home need to be resolved or perhaps they need basic remediation. Perhaps, though, it’s the case that the current teaching approach needs to change or at least be broadened.

  2. Very interesting! I am looking forward to the next posts in this series!
    I remember that my cohort in elementary school was subject to a so-called progressive experiment: Set theory was taught in grade 1, before anything number-related. Those parents who frantically tried to re-learn school math to supervise and support their children (something I personally never understood, fortunately I only know from 2nd hand experience) were totally at a loss 🙂

    1. It turns out I did the same thing. Union, intersection and such were things I learned right upon entering school. It’s served me well. Back then everyone was freaking out about the so-called new math. Everyone’s still freaking out but everyone seems to have forgotten the sweeping changes that accompanies the Sputnik era 🙂

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