This piece is a mix of content analysis and opinion. The former is part of my work at the University of Oregon, which is why I am posting here on my University pages, while the latter is solely my own and does not necessarily reflect any official position of my University.

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
 • • • • • • • • • • • • • • •
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)
 30 5 20 600 100 7 210 35
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.