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I’m currently muddling my way through the interpretations of “algorithm” for Grades 2-5. Looking through the Standards, the Progressions, and previous blog posts, it seems that for addition and subtraction there is a sequence of moving from “written methods” (2.NBT.7) to “algorithms” (3.NBT.2) to “standard algorithms” (4.NBT.4). My reading of these terms is:
* written method: an informal recording of a process or observation, e.g. “24 + 13 is the same as 20 + 10 plus 4 + 3”
* algorithm: a set of steps that may lead eventually to a standard algorithm, e.g. recording addition of places on new rows as shown on the top of page 9 of the NBT Progressions:
456
+167
———
500
110
13
———
623* standard algorithm: a set of steps that is similar to that shown for addition in the middle diagram on page 9 of the NBT Progressions
Arising from this, I have a few questions:
a) Have I identified that sequence correctly?
b) Have I defined those terms correctly?
c) I understand that using standard algorithms are not forbidden in Grade 2, but for assessment purposes should the students only be expected to show a “written method”?
d) Is it the expectation that standard algorithms be used by ALL students in Grade 5 for decimal fractions? Or is the expectation that a “written method” (as defined above) is all that is required, while allowing some students to use the standard algorithms if they desire to?
On this last question the Standards ask students to “relate the strategy to a written method” in Grade 5 (5.NBT.7) and use “the standard algorithm” in Grade 6 (6.NS.3) so my first assumption is that standard algorithms are not required in Grade 5. But the NBT Progressions for Grade 5 (p.17) suggest that algorithms are to be used, as “the same methods of recording numerical work can be used with decimals as with whole numbers”. Examples in the top third of the page allude to vertical formats for addition and subtraction, such as aligning places and recording what are generally known as carry digits. Other comments are made about using “general methods” for decimal fractions as are used for whole numbers when dealing with multiplication and division (pp. 17-18). So it’s a little unclear exactly what is expected for ALL students in Grade 5.
Shedding any light on these issues would be appreciated. Thanks!
An extra point: reading through a response Bill made on the NBT blog on Oct 31, 2011, he notes in regards to addition and subtraction of whole numbers that “I think the standard algorithm would have to be introduced earlier than Grade 4, although fluency is not required until then. ”
How would this work with assessment? Are we able to assess the standard algorithms before Grade 4 in this case? Or do we only assess alternative algorithms?
Duane, I’ll answer both posts here. You are certainly on the right track with all of this; let me add a few comments. In addition to written methods and algorithms, the standards refer to strategies in a number of places. The distinction between a strategy and an algorithm is that an algorithm is general, it works in all possible cases, whereas a strategy might be specialized (e.g. shifting a 1 when one of the addends ends in 9). There are a number of places where the standards ask students to use strategies and then relate them to a written method (in fact I think all the occurrences of the phrase “written method” are side by side with strategies). So the idea here is really that students should be able to formulate the strategy in writing (“you break a 1 of the second number and add it to the first to make a number in the tens, and then you add the two numbers”). So I would modify your progression a bit and say it is strategies/written methods -> algorithms -> the standard algorithm.
As for your question about adding and subtracting decimals, the standards don’t require use of the standard algorithm for that until Grade 6; students can use other algorithms in Grade 5.
And, as for assessment, I don’t think you can assess standards before the grade level in which they occur. But, as you noted, a curriculum would probably be moving towards the standard algorithm before the grade in which it is mentioned in a standard. Actually, I’m not sure how you can tell in assessment which algorithm a student used, so the standards which require fluency with the standard algorithm might be classroom standards, not assessment standards.
Thanks for the confirmation of the sequence, Bill, and the clarification on strategies. With assessing algorithms I’ve found it pretty easy to identify with method students use on class tests – they always have plenty of room to record their methods. If we need students working with the standard algorithm before grade level expectations I guess I’ll just need to do a separate test to keep track of their progress and ensure it doesn’t get included in any reporting on the Standards.
Thanks once again!
I didn’t mean to say that kids couldn’t use the standard algorithm on a test before the grade level where it is explicitly expected. After all, the standard algorithm is a “strategy or algorithm based on place value and the properties of operations”, and so could occur as early as Grade 1. It’s just that it doesn’t have to occur that early.
Hmmmm… can I check I have this right please?
We can teach the standard algorithms earlier than where they appear in the Standards if students are ready for it. And we can assess the standard algorithms in those earlier grades and record the students’ outcomes against the Standards because they qualify as a written method or as an algorithm (inclusive of many different types). But the preference is that we follow the sequence laid out in the Standards.
Is that correct?
Duane, you are putting too much weight on the standards for determining decisions that should properly be made by curriculum and assessment writers. The standards describe certain achievements that students should have by certain grade levels. There is no “preference” that curricula wait until the grade level where a standard occurs before starting work towards that standard. Different curricula might have different approaches. In particular, I suspect that there are quite different opinions out there about when to start teaching the standard algorithm for addition and subtraction in order to reach the Grade 4 fluency standard. All the standards say is that you have to get there by Grade 4; they don’t say how.
Bill,
Do the “general methods” shown in the margin on pages 13 and 14 of the 7 April 2011 “K–5, Number and Operations in Base Ten” progression qualify as “the standard algorithm” for the purposes of 5.NBT.5?
Andy
As an elementary math coach and intervention teacher I question the wisdom in teaching the US standard algorithms for addition and subtraction to whole classes of 2nd or 3rd students. Many students are still formulating their understanding of the Base 10 system. To teach them the way we were all taught causes confusion in many students I work with. Unfortunately much math instruction is still teacher-centered as opposed to student-centered. I think it’s perfectly acceptable for teachers to meet with small groups of 2nd or 3rd graders that have STRONG base 10 understandings and demonstrate the US algorithms (chances are their parents already have) in that small group. What upsets me is students trying to use these procedures with no deep understanding of them. Subtraction, in particular, is a weakness for my 3rd, 4th and 5th graders! They make a royal mess of ungrouping, especially across zeroes. I appreciate the fluency for addition and subtraction being addressed in the 4th grade. Now if I could just convince 2nd and 3rd grade teachers to spend a lot more time with base ten manipulatives, drawings, expanded methods ,etc!, and let 4th grade teachers connect these models to the shortened US algorithm.
I don’t disagree with Kipra, but I think we need to articulate what is meant by “strong base 10 understandings.” Then, we need to articulate which part of that strong understanding is appropriate at what grades. Furthermore, I think we should realize that a part of the reason for teaching multi-digit addition/subtraction calculation is to help students deepen their understanding of the base-10 numeration system, as well understanding of the properties of operations – not just having an efficient calculation method. So, by learning how to add two 2-digit numbers, students should come to understand that in order to add two numbers, they must both refer to the same unit. That idea carries through addition (and subtraction) of not only whole numbers but also fractions and decimals. 2nd/3rd grade instruction of multi-digit addition/subtraction should keep that in mind. Another important rule of our base-10 numeration system is that we must use one and only one numeral in each place. That’s the reason we must re-group. So, when we teach addition/subtraction with re-grouping, we are not just teaching an efficient calculation method. We are trying to help them understand how our numeration system works.
From that perspective, I don’t necessarily see anything wrong with start teaching algorithms in Grade 2. Understanding of the base-10 numeration system and understanding of calculation algorithms are intertwined, and they should be taught with their connections in mind.
Whether and when to teach the standard algorithm was a hotly contested topic during the writing of the standards, and now some of that debate has transferred to the meaning of the term. Some think it is the algorithm exactly as notated by our forebears, some think it includes the expanded algorithm, where you write down all the partial products of the base ten components and then add them up. Ultimately this is a question that has to be settled by discussion, not fiat. My opinion is that the standard algorithm has two key features; like the expanded algorithm it relies on the distributive law applied to the decomposition of the number into base ten components, but in addition it relies on the fact that the order of computing the partial products allows you to keep track of the addition of the partial products while you are computing them, by storing the higher value digit of each product until the next product is calculated. I don’t think different ways of notating this constitute different algorithms. So, in particular, the algorithm that Scott was talking about, bottom of page 13 in the margin, would qualify in my opinion, but the partial product algorithm in the middle of that page would not.
Thanks, Bill. This is helpful — but I’m not sure what you mean by ” the algorithm that Scott was talking about, bottom of page 13 in the margin…” In the margin at the bottom of page 13 of the 7 April 2011 “K–5, Number and Operations in Base Ten” progressions document there are three ways shown for computing 549 * 8. Are you saying that all three of these qualify as “the” standard algorithm, in your opinion? Or only the one on the right?
Oops, I meant page 14, not page 13. But, also, we are talking about different versions. I am talking about the one here, which is a corrected version of the one you are looking at. It has the 3 ways for 549*8 at the top of page 14 in the margin. I didn’t give an opinion about those before, but I would say that only the one on the right is the standard algorithm. The one Scott was talking about is at the bottom in the margin on page 14 in this version. You can see the discussion here.
Looking at the division algorithm, the Grade level description for Grade 5 of the Standards (p. 33) says that students “finalize fluency with multi-digit addition, subtraction, multiplication, and division”. However, in Grade 6, students “fluently divide multi-digit numbers using the standard algorithm” (6.NS.2).
My interpretation is that in Grade 5 the students are meant to fluently divide using a variety of means, perhaps (but not necessarily) including the standard algorithm. In Grade 6 they must demonstrate fluency with the standard algorithm. Is this correct?
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