I find the Common Core State Standards in Mathematics to be more rigorous,
in both senses of the word, than traditional frameworks such as the one in which I
was taught. Consider for example the following question: why is 35 x 27 = 27 x 35?
Of course I mean more generally "why is multiplication commutative?"
If like many you approach this through using the standard algorithm then either you
automatically put the larger number on top, in which case the question is obscured,
or if you perform the algorithm in two orders you come to an additional mystery: why in this case
does 245 added to 70 shifted over (really, 700) equal 135 added to 81 shifted over?
At this point it is likely that your background doesn't provide you with
the requisite tools to
solve this mystery, which we will solve below.
The Common Core prescribes an age-appropriate way for third-grade students to answer
the original question.
Namely, in second grade students are to "Work with groups of equal objects
to gain foundations for multiplication" and in particular rectangular arrays (2.OA.5).
In doing so, teachers can call attention to the following array
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and call attention to the fact that if we look at columns we see five groups of three,
and if we look at rows we see three groups of five.
In third grade, students should understand the definition of multiplication as total
numbers of quantities which come in equal
groups (3.OA.1), which is equivalent to repeated addition.
They should have plenty of opportunity to see the same products coming in different order,
for example possibly filling out multiplication tables or playing games to achieve
fluency (3.OA.7). At some points, their teachers can then draw from their second
grade experience and explain using the array above why 3 x 5 = 5 x 3.
Similarly, if we consider an array of dots with 35 columns
and 27 rows, we can group by columns to see 35 x 27 and group by rows to see 27 x 35.
In fact for any numbers of columns and rows, we see the total number of dots as
the product of the number of columns with the number of rows, in both orders so to speak.
Alternately, one could think geometrically and "rotate by 90 degrees". This all
serves to help students "Understand properties of multiplication" (cluster heading
above 3.OA.5).
So third graders will see why 35 x 27 = 27 x 35, probably without knowing the most
efficient way to compute the relevant products. And
by the time students are fluent in multiplication of multi-digit numbers (4.NBT.5, 5.NBT.5)
they will be well-equipped to understand the additional mystery we ran into,
which presents a good
opportunity to better understand the standard algorithm. Consider the following table
(with apologies, since it would be better as a graphic of an appropriately labeled and
scaled rectangle)
If one uses the meaning of place value to see that
35 x 27 = (30 + 5) x (20 + 7),
then by distributivity one sees the product as the sum of the
four terms in the table. First adding the rows yields 700 + 245, as in
the way the standard algorithm with 35 on top comes to the answer, while adding
columns yields 810 + 135 as when 27 was on top. Having fifth-graders delve into this
deeply would provide good experience to build on
for multiplying polynomials in high-school.
In the end, we see that the Common Core approach is more rigorous in the mathematical
sense of providing reasoning in the mathematical development. The authors of the Standards
usually distinguish between this notion of rigor and a notion of being more
cognitively demanding, which I refer to as "juiciness."
As a mathematician I am devoted to rigorous argument,
and as an instructor I appreciate
students' abilities to retain and apply material which students have engaged with reasoning.
When mathematical facts are generally left unexplained, we are emphasizing
reliance on memory and a belief in authority (the teacher/ textbook) over reasoning
skills.
I have heard it said that the standard algorithms are the only "theorems" in
elementary mathematics. In some sense I agree. But mathematics does not only consist
of theorems, it consists of all of the following, which we saw in the above discussion:
- Foundational material (definitions and models),
- Elaborational material (properties, lemmas and key examples), and
- Refined material (efficient algorithms, and the Theorems mathematicians state at the
beginning of their papers).
Just as all of these deserve attention in exposition of advanced mathematics,
taking time to build and interrelate them, they all deserve attention
in curricular development as well. In particular,
the Common Core's choice to focus more narrowly in early grades in order to serve the
more rigorous work of relating these different kinds of mathematical material
- rather than, say, on the one hand obtaining
some kind of fluency with standard
algorithms as quickly as possible or on the other hand engaging in
problem solving not
grounded in mathematics of central importance
- strikes me as mathematically sound and
pedagogically wise.
Additional Notes
- Yes, I have seen videos and student work (e.g. third graders using diagrams
to reason about distributivity) which indicates that the reasoning above can be
approached in grade-appropriate ways, in particular avoiding unnecessary
terminology and especially formalism. For many, this is unfamiliar, and our
natural bias is to assume more difficult than what we did in school. Indeed, this is a common
issue to address in teacher professional learning around the Common Core.
But our children have
no such biases, and seem to take it as natural to ask questions such as to why equalities
such as 35 x 27 = 27 x 35 hold.
- For further discussion of commutativity of multiplication in the Common Core
see page 24 of the relevant
Progressions Document.
For a task which demands understanding of commutativity in an interesting way, see
this task from
Illustrative Mathematics.
To see how this plays out in a curriculum see a related
EngageNY Curriculum Module.
- Referring to the kinds of mathematical objects discussed above, a
brief caricature of the Math Wars is "only caring about the punchline (refined material)"
vs. "only caring about the set-up (foundational material)".
To elaborate a bit more, one can see "Emphasis on refined material, mathematical precision,
a narrow definition of what mathematics constitutes, and a faster pace" vs.
"Emphasis on foundational and elaborational material, process, a broader definition of
what mathematics constitutes, and a slower pace." Discussions about these choices will
continue, but I think the Common Core made some sensible choices from which we can work.
- In this discussion there was
a need to attend to cluster headings in understanding the demands of the
Standards.
See
the discussion of the Grecian Urn for an elaboration.
Because of this construction and choices of brevity etc., some find the writing
of the Standards difficult to penetrate. It certainly helps to get training
in order to read them as intended. But with the
Progressions Documents mostly
out there is a narrative form available (though still fairly dense, to most eyes).
And tasks from Illustrative Mathematics
can help clarify matters as well.
- While I've focussed on mathematical reasoning here,
more authentic modeling/ application is another primary sources of "juiciness" which
I would like to see in curricular materials.
Despite my own preference for theoretical mathematics,
I would prefer to see modeling play a greater role in high-school curricula.
In particular, I think there could be some strong curricula which serve Career and
Technical Education.
- In talking with K-5 teachers, it seems that students can feel well-grounded through
much of whole-number
arithmetic, but then have significant difficulties with fractions. They do
not understand, for example, why multiplying numerator and denominator of a fraction by
the same number should yield an equivalent fraction. The Common Core addresses this in two
ways. It gives a careful choice of definitions and reasoning, one which even some
of the most vocal critics of the Standards have lauded. Secondly, it incorporates
reasoning throughout K-12 mathematics, as we saw for multiplication above, so that at the
moments where it is essential for understanding it is not a foreign tool.
- The amount of exploration and discussion vs. direct instruction employed in
engaging these kinds of arguments is a matter of curricular and instructional choice.
My experience as a university instructor and limited experience as classroom observer
(mostly through videos) speaks to the value of both, with more
structured work as one solidifies mastery.
- Mathematicians may be reminded of
Lockhart's Lament,
in particular that "By concentrating on what, and leaving out
why, mathematics is reduced to an empty shell." The Common Core provides
a mathematical framework for answering why, with choices based on
pedagogical considerations.
As far as I can tell, full arguments of everything in K-12 mathematics except for
notions which require limits and continuity (such as extending standard functions from
rationals to reals) are embedded in the Common Core.
The Common Core does not prescribe more open-ended
classroom experiences, as Lockhart advocates for,
and indeed by placing demands on mastery and fluency could limit the time spent on them.
Also, most visions of the Common Core value connection with the "real world"
much than Lockhart does; carpenters, for example, do use fractions and
proportional reasoning.
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