You may have noticed that I am back to publishing regular blog posts! My goal for now is a blog post every second Wednesday. I am now also trying to answer forum questions promptly. I want to thank the readers who took up the slack for the last year and a half in answering questions in the forums. In particular, I’d like to call out abieniek, Alexei Kassymov, and Lane Walker, whose answers were always spot on.

Now to the topic of this post. There has been a lot of talk since the standards came out about what they say about multiple methods for arithmetic operations, and I’d like to clear up a couple of points.

First, the standards do encourage that students have access to multiple methods as they learn to add, subtract, multiply and divide. But this does not mean that you have to solve every problem in multiple ways. Having different methods available is like having different means of transportation available to get to work; flexibility is good, but it doesn’t mean you have to go to school by car, then by bus, then walk, then bike—every single day! The point of having multiple methods available is to encourage students to think strategically about what might be the best method for a given problem, not force them to solve every problem four times.

Second, the different methods are not unrelated; they form a progression, with the ultimate goal being the standard algorithm. For example, when students are first learning to multiply two digit numbers, they might use a rectangle to represent a product such as $42 \times 71$.

This shows the fundamental role of the distributive property in multiplying multi-digit numbers. You have to multiply each base ten component into each other one. Indeed, the same rectangle representation provides a visual proof of the distributive property itself.

At some later point students might just start writing down all the partial products, without using the rectangle to derive them.

Note the correspondence between the rectangle method and the partial product method, indicated by the colors. The first row of the rectangle and shows all the products by the 2 in 42 (in red); the second row shows all the products by the 40 (in blue). The products in the partial product method are grouped in the same way. There are many ways you can order the partial products, but if you group them as I have here, going from right to left in each two-digit number, as in the standard algorithm, you make an amazing discovery: you can add up all the partial products in each group (blue group or red group) in your head as you go along. That’s because, in each case, adding the 2 to the 140 or the 40 to the 2800, there are enough zeroes in the second addend to accommodate the first, so it is easy to write down the sum right away, without writing the addends separately.

OK, so it’s not always quite this easy, because every now and then you will have to keep in mind a bundled unit from the previous step (aka carrying), but you will never have to remember that for more than one step at a time, because each bundled unit gets used up at the next step. So if you invent a notation for remembering the bundled unit (what we used to call “little 1 in the corner” when I was growing up) then you can still avoid writing down all the partial products, and just compute the sum within each group as you go along. You have just created the standard algorithm.

The different methods are not isolated different ways of doing the same thing; they are steps towards fluency with the standard algorithm, fluency that is not fragile because it is supported by understanding.

People seem to be having trouble posting comments. Here is one from Dev Sinha:

I’d like to split hairs a bit, as your phrase “with the ultimate goal being the standard algorithm” is likely to be misinterpreted. While the standard algorithm through its universality is an ultimate goal in a narrow sense, I’d say the ultimate goal is to have access to both for purposes of wider efficiency as well as the number sense which goes with it. For example, one shouldn’t have to “borrow” (ack) a couple times to subtract 6 from 204. Though when I talk to parents about CCSSM (which led to the notes you posted here: http://commoncoretools.me/2014/11/21/common-core-math-parent-handouts-by-tricia-bevans-and-dev-sinha/ ), they often comment that they’d have to use the standard algorithm. That’s fine, but I think having both is better, and highly numerate people such as professional mathematicians, scientists and engineers have often taught themselves these alternate strategies. I’m reminded here of a post by Jason where he outlines how one can implement a curricular sequence in which the standard algorithm, with understanding, comes early on and then the strategies are brought in in part of more efficiently deal with special cases.

Early after CCSSM came out, I heard some suggest that it was too much to aspire for all students to have access to both, but now I’ve seen a variety of implementations with a wide range of student populations which have achieved exactly that.

Good point; in context, “ultimate goal” refers to the progression of methods; all I intended to mean was that the standard algorithm is the endpoint of the progression of methods, nothing more.

Let’s see if I can comment 🙂

What do you do when the students perceive the multiple methods as isolated things to be memorized?

Ask students to find the connections between the methods? I see the 140 in this method, where is the 140 in this method? How are they similar? How are they different? “To find one’s way around the mathematical terrain, it is important to see how the various representations connect with each other, how they are similar, and how they are different. The degree of students’ conceptual understanding is related to the richness and extent of the connections they have made.” – National Research Council, 2001

It worked! My guess is that student perceptions arise from cues given by the teacher, so I would start by thinking about what those cues might be. If the teacher is always prompting the student who has solved a problem correctly to solve it again another way, that might be sending the wrong message. If a student is struggling with a method, or using it inefficiently, then it’s worth prompting for a different method. But correct answers arrived at by a correctly executed method should be highly valued. Particularly on written work it should not be a requirement that more than one solution be given. It should be an option, of course, and comparing different solutions by different students can be valuable.

I’m a little out of my league here, but I wonder if there are some general heuristics kids can develop about when to use which method.

For example, in upper grades, there are choices to make when you see something you know is true but want to prove—this one feels like induction will work, this looks like a “suppose not” indirect method, or maybe a generic calculation will work. Or in algebra, the CCSSM advice about “different forms for different purposes” encourages students to transform an expression according to what they want to see. Similar heuristics live in geometry

Is there anything like that in these multiplication strategies, or are they all focused on getting to understand what’s behind the efficient methods?

Al

The different methods that Bill described are essentially ask students to engage in the same kinds of reasoning and thinking. The scaffolding for the notation varies. I do think the partial products and rectangle model allow for more flexibility in how student compose and decompose the factors though, which helps students develop other mental math strategies, not described in this post.

The heuristic you are looking for really comes into play as students develop fluency with the operation. Fluency is students ability to solve a problem flexibly, accurately, efficeintly, and choose a strategy appropriately. The context (e.g., situation or numbers) drive appropriateness. Do I need to need to calculate or estimate? How accurately? Does it make sense to use an algorithm or a counting strategy? For example, what is efficient for 204 – 199? or Do I have enough money to by xxxxx?

In the end these methods are all based on the distributive property, and there are certainly situations where it would be more efficient just to use that (e.g., multiplying any two-digit number by 101). I wouldn’t say they are all focused on getting to understand what’s behind the efficient methods. Rather I would say that they are all focused on understanding how the distributive property applies in multiplying multi-digit numbers, and they are all ways of organizing and adding up the partial products that result from that application. So yes, understanding what’s behind the efficient methods is part of that.

I agree tht alternate strategies help students to understand the end goal of algorithm.

I like that this article stated the fact that all strategies don’t have to be used. So many parents think they have to do it “the new way” Parents need to be aware that they don’t have to use all strategies. It is simply a way to help more students understand

I would like to bring up a point that seems to be being de-emphasized in the discussion. It seemed to me Bill that you started wrote this blog about the use, in general, of multiple representations to whit you included a specific example. For fear of losing the forest we have discussed the tree of multiplication methods.

Multiple representations of mathematical concepts exist for pretty much every concept taught within the CCSS. This is part of the beauty of mathematical thinking. To be able to see, for another instance, the Pythagorean Relationship can be viewed within a Number Theoretical, 2-D Geometric, Volumetric, or Trigonometric etc. perspectives each of these sheds light on and improves the depth of understanding of the basic principle. Fostering these multiple perspectives and representations of ideas or concepts is what good teaching should do.