Sunday, October 19, 2014

Teaching what can be discovered instead of what can't be

Barry Garelick's recent piece Education News, Undoing the ‘Rote Understanding’ Approach to Common Core Math Standards, got me thinking about one of the things that Reform Math has backwards.

Barry talks about the emphasis by today's math educators on ad hoc methods like "making tens" at the expense of traditional algorithms like borrowing and carrying. And so we see more and more worksheets like this one:

And fewer and fewer like this one:

And, in promotional, Common Core-inspired videos like this one, we see just how painfully slow the "making tens" method can be--as well as how it, by itself, does not give students a general method for solving more complex addition problems.

As I wrote in a comment on Barry's piece, I don't remember ever learning officially how to make tens. I remember it instead as something I discovered on my own--in the course of computing all those long columns of sums that students used to be assigned (sometimes upwards of six addends!) and eagerly looking for shortcuts.

The standard algorithms, on the other hand, I most certainly did *not* discover on my own, and am quite glad to have had teachers that were willing and able to teach it to me.

It's ironic how "discovery-based" Reform Math spends more time showing students how to do stuff they might discover on their own than it spends showing them how to do stuff they almost certainly won't learn on their learn own.


Anonymous said...

You continually rave about how wonderful Singapore math is. I wish you would actually crack open a Singapore Math book. Decomposing numbers and making tens comes straight out of Singapore Math. I teach 3rd grade and we begin the year reviewing making tens and then go on to expand it to making hundreds.

Barry Garelick said...

Actually, Singapore Math introduces making tens in first grade. The difference is that students are also expected to memorize math facts, and for two digit and more numbers, Singapore also introduces the standard algorithm for multi-digit addition and subtraction in 2nd grade. That algorithm is not delayed until 4th grade as many schools are doing in the name of Common Core. Because the standard algorith appears in the 4th grade CC standard, the popular interpretation of CC is to delay teaching it until 4th grade. In fact, there is nothing prohibiting teaching it earlier such as in 1st or 2nd grade--something that William McCallum and Jason Zimba (lead writers of the CC math standards) have said, though not in wide public venues.

The use of number bonds to solve elementary addition and subtraction equations is not limited only to Singapore’s math textbooks. It has also been used extensively in U.S. textbooks, such as my third grade arithmetic book (Arithmetic we Need by Busell, Brownell and Sauble; 1955).

See also here.

Katharine Beals said...

Hi, Anonymous,

By "continually rave about how wonderful Singapore math is" I assume you mean "periodically post comparisons between Singapore Math and other math programs"?

"I wish you would actually crack open a Singapore Math book."

It would be hard to post those Singapore Math comparison problems without cracking open a Singapore Math book.

"Decomposing numbers and making tens comes straight out of Singapore Math."

As Barry points out above, and in his piece on Ed News, decomposing numbers predates by many decades U.S. imports of Singapore Math. The issue here isn't that it shouldn't be done, but that it shouldn't eclipse standard algorithms-- or delay more advanced math.

"I teach 3rd grade and we begin the year reviewing making tens and then go on to expand it to making hundreds."

Singapore grade 3 workbooks (Primary Mathematics 3, Standards Edition) contain no making tens or making hundreds problems. Arithmetic in these books begins at a much higher level than that.

Anonymous said...

Third grade Singapore Math workbook pages 28-29, making tens. Third grade Singapore Math workbook page 30, making hundreds.

Katharine Beals said...

On page 27 of my edition (2008), there are a total 8 problems in which making tens is presented as an option. The directions are simply "add." On p. 29 there are two subtraction problems that have the same graphical "look" as "make 100s", but that don't involve making hundreds. Here the directions are simply "subtract."

As Barry points out above and in his Ed News piece, Singapore Math introduces making 10s in 1st grade. By 3rd grade it is old hat. The point Barry and I are both making is that that in Singapore Math, unlike in U.S. Reform Math, the making tens methods (and other ad hoc methods) do not eclipse (or take substantial instructional time away from) the standard algorithms, which, by 3rd grade, are solidly in place in the curriculum.

lgm said...

There have always been third graders who didn't figure out how to use the 'make a ten' number bonds before third grade started. In the past, they were put into a small group and given direct instruction at their level of need. Now, everyone in the class is held hostage and not allowed to continue their math instruction until their classmates 'catch up'. That is the real issue. Low expectations for those who are relegated to thumb twiddling while they wait. Of course, if the parents pull them and send them to private school they are called 'elitist'.

Anonymous said...

On pages 28 and 29 of the Primary Mathematics 3A Workbook Common Core edition, there are some problems for making 10. These involve 2-digit plus 1 digit, e.g. 55 + 7, or 2-digit + 2-digit, e.g. 58 + 34, by making a ten with a number close to a ten (the 58). These two pages are almost identical to pages 26 and 27 of the 2008 Standards edition. They go with a chapter in the textbook specifically titled Mental Calculation, which does do a brief review of making tens or adding tens first and then ones when adding 1 digit to 2 digits, or 2 digit numbers.

In the original third edition, to which the US edition is closest, this review was in 3B, not 3A, was even briefer, and was immediately followed by some mental math strategies for multiplication and division of tens and hundreds.

Note that students have already learned to add and subtract 2-digit numbers both mentally and with the standard algorithm way before 3A, plus they have added and subtracted 3-digit numbers in 2A, extensively, using the standard algorithm, in all three editions.

The original third edition had no review of this at all. It had a section on sum and difference, an introduction to bar model, and problems involving addition and subtraction of 3-digit numbers, without any strategies, even standard algorithm, as its review of second grade. Then it jumped into adding and subtracting 4-digit numbers using the standard algorithm. No mental math with 1- and 2-digit numbers in 3A originally. They already knew this, presumably.

Which is why, when US schools adopted Primary Mathematics way back when, they would have had to go back to 2A.

So, now in the Standards edition and the Common Core edition, for the US, review of mental math strategies is in 3A. And the section on sum and difference and introduction to bar models no longer includes 3-digit numbers. Only 2-digit numbers. They can focus on the concepts in that chapter rather than having to add and subtract 3-digit numbers which they might not know how to do yet, if they came from a US math second grade. Then there is a review of adding and subtracting 3-digit numbers using the standard algorithm, under the guise of estimation in the Standards edition (something not in the US edition at all) and called Looking Back in the Common Core edition.

It is interesting that in the Primary Mathematics this is called mental math. Which to me means doing it in your head, without paper and pencil. And the strategy is introduced in 1A concretely, then pictorially, but then they should be able to do it abstractly and can use that strategy in their heads, as they learn the facts. That is, to easily think 10 + 5 when seeing 8 + 7. Mentally.

Katharine Beals said...

Thanks for this explanation, Anonymous@1:18 PM. It's notable that making tens is at this point simply a mental math strategy (not a "deep understanding" strategy), and that the closer you get to the original Singapore math, the less review there is of it in 3rd grade (down to zero review in 3rd grade in the original).

Anonymous said...

There is a brief review in 3B for the third and US edition, But mostly to look at what to do when a number is close to a ten, like 48, somewhat of a new strategy. So it is not accurate to say no mental math. Plus, mental math strategies are taught in other contexts (money and measurement. But in grades 2 and 3, mental math chapters comes after the standard algorithm. But making a ten or subtracting from a ten is important in grade 1, to lead up to that algorithm. It is just that in the US, they don't have that understanding by third grade. Hence the more extensive review.

And, in these grades, the standard algorithm is taught with place-value discs, in a way to foster "deep understanding" of it. Mental math is good for certain situations, numbers close to ten or hundred, adding and subtracting money that ends in 0 and 5 (in 2B they are allowed decimal notation for money, in Common core not until grade 4). Adding and subtracting measurement in the metric system. Nice little strategies that help speed computation.

Catherine Johnson said...

"By "continually rave about how wonderful Singapore math is" I assume you mean "periodically post comparisons between Singapore Math and other math programs"?"


though I must say, comparing Singapore Math to other math programs is tantamount to raving about how wonderful Singapore Math is ----- !

(from me, Catherine)

ChemProf said...

One issue is that increasingly I find teachers mean "Math in Focus" when they say "Singapore Math," not what the Primary Mathematics, which is what homeschoolers usually mean. Math in Focus is based on a second Singapore curriculum, intended for weaker students, called My Friends are Here, and is slower and more like typical US textbooks.

The Common Core-aligned Math in Focus books actually say Singapore Math on the cover, which is part of the confusion.

Anonymous said...

So, in the first example, how do the kids know 3+1=4 if they did no "rote" memorizing? I guess they could count on their fingers. But then they'd need to recognize somehow that the 3 adds with 7 to make the sought-after 10. Without memorizing. I guess they can use their toes for that, since their fingers are busy finding all the addends for 4.

Katharine Beals said...

Even knowing the meanings of "1," "3," and "4,"--and that "3" is one more than "4"--involves rote memorization.