# Number sense or nonsense?

Twitchy features Common Core math problems that try to teach number sense.

3rd grade common core math. See image. I have a math minor and it doesn’t make sense to me. pic.twitter.com/gTuJmLUN6e

Saxon’s Core-aligned seventh-grade math book is riddled with errors, writes Michelle Malkin.

1. Sean says:

I’ll see the math minor and raise with a physics major with grad work in math. Doesn’t make sense still. I guess they’re trying to “repackage” something into 26 to fill that 10 group (30) then just adding the balance to get the total. (Little bit like filling orbitals in Chem). So it’s what? the associative property? 26 + 17 = 26 + (4+13) = (26+4) + 13 = 30 +13 = 43.

Imagine this:

I couldn’t figure out why my algebra 2 students couldnt manage multiplication and were so slow – here’s a culprit? Easier? I don’t know, but it’s slow as heck. Not entirely sure how it’d work for bigger problems – don’t want to know.

• It sounds like the nomenclature has been deliberately chosen to be obscure and difficult to understand, especially for people brought up with “old-school” methods (so parents can’t help?).

Perhaps it’s about “checking White privilege” or something.

• allen says:

No, the purpose in education of obscure and difficult to understand nomenclature is to provide the appearance of technically sophisticated, and thus unquestionable by the technically unsophisticated, professionals.

• Anna says:

No, it’s just one of those things that is easy to show in the classroom but hard to explain in words. Give a teacher some base ten blocks and 15 minutes and she could have a class of kids doing this no problem. It is hard to transfer to homework, since parents do need some kind of explanation and the explanation is hard to write succinctly and clearly. And most parents will just look at it and think its a stupid method anyway, I guess,based on this thread. Which is discouraging.

• Which makes it even more remarkable that time tested terms like “place value” and “carry the one” have been replaced by this gobbledygook.  Or is that unremarkable?  I forget whether the purpose of edu-jargon is to promote learning, or the parochial interests of the establishment.

• allen says:

Discouraging? Hardly. Teaching something as simple as arithmetic is established as being so complex that only highly-skilled professionals can do it.

How is that a bad thing?

• I tutored a middle schooler a few years ago who was WAY behind in math. One thing he could do, though, was apply one traditional algorithm, I think it was multiplying multi-digit numbers. He was embarrassed to admit that his little sister’s friend (his sister’s age) had taught it to him. But she did a good job and taught it effectively. (His schools to that point had used Everyday Math. I did a lot of basic fact drills with him, and his science professor father to this day says that he learned more in a few months working with me than with anything else they’ve tried to help their son.) The standard algorithms can be taught by children.

• Ann in L.A. says:

Does any curriculum other than Everyday Math emphasize the lattice method? Our EM kids spent half their multiplication time drawing stupid lattices.

One came home crowing about how lattice is soooo much faster. His teacher had done a side-by-side comparison with the traditional method; but, the teacher had cheated–which I pointed out to our boy, he had already drawn the lattice on the board before the speed test began. When we re-ran the test at home with a blank sheet of paper, the traditional method was clearly faster.

We also chewed his 4th grade teacher out at back-to-school night. The parents were in revolt over lattice, and one asked how you do it with larger numbers–the teacher hemmed and hawed. Then we asked about how you do it with decimals–the teacher didn’t even know how. Another parent, with an older child, began to explain, but was stopped. Another parent with an older kid said: yes, *we* know how to do it, the point is that *she* doesn’t know how to do it.

I find all the emphasis in EM classrooms on lattice ironic, because the Everyday Math Teacher’s Reference Manual actually states that the lattice method was originally added for its “recreational value and historical interest,” not its mathematics value, and that, “It is not easy to understand exactly why lattice multiplication works.” (K-3 Edition, 2001, pg 107.)

Which goes to show how many teachers actually read the manuals. If they read that line, would they waste half of their multiplication efforts on it?

• Educationally Incorrect says:

The idea that the lattice method is “better” in any sense goes back to what I’ve said about ed people thinking that people only need to learn to operate on small numbers.

I’ve heard of professional mathematicians being unable to make the lattice method work with decimals while the traditional algorithm works fine.

The traditional algorithm scales far better for bigger numbers. The side effect here is that in a given amount of time a student can perform more traditional multiplications (using less paper) than lattice multiplications. More practice for the same investment of time.

Learning to multiply two polynomials is less not that confusing when you’ve learned the traditional algorithm. Instead of lining up by place value you line up by like term. I can’t imagine how someone raised on the lattice method would even begin the process of trying to multiply to polynomials for the first time.

Makes sense to me. That’s how I add.

Of course, I did calculus in 8th grade and was doing graduate courses in math before I finished high school. But I don’t think my math background, Joanne’s math minor, Sean’s physics major or graduate work have anything to do with this.

Yeah, this is number sense. That exactly what it is.

We have historically done a horrible job of teaching number sense, and of teaching those who don’t already get math how to get math. I’m sure you both know that.

What’s really appalling, and common today, is adults making fun of stuff they didn’t grow up with and don’t understand today. Things taken out of context. As though they are the appropriate arbiters of what struggling students might or might not understand.

This one makes sense to me. But I am no better positioned to critique whether first or second graders will understand this than Sean or Joanne. That’s not where my training and experience is.

• Joanne says:

Rich Hoffman was a math minor. I majored in English and creative writing.

• My apologies, Joanne. I saw another story on this and saw that it was someone else who claimed to a math minor.

So, I apologize to you. I was so annoyed with him, and not you. I’ve read enough of your stuff that I should have realized that that was not quite your voice.

Again, my apologies.

• Making fun of? Unless you already do this method of addition yourself, the explanation looks like gibberish initially. I have a major in math and worked in a university math lab helping people of most levels for three years. I have never done my addition this way. EVER. It looks like a waste of mental effort. I refuse to accept that this method is at all necessary to help develop “number sense”. Rather, people without much number sense will find it extraordinarily difficult.
This method may come naturally to some people, but for a lot of people, it looks like magic and reinforces their conception that “math is hard”. There’s a straightforward way to do two-digit addition–the standard algorithm that is guaranteed to give us the right answer in just three steps (add the ones, rename a ten, and add the tens) as opposed to the (at least) four poorly-explained steps above. Unless a person just loves to play with numbers and go on to study real algebra or something, the main number sense he/she needs is the ability to promptly and accurately calculate or all his/her “sensed”-estimates are guesses that are going to be wrong much of the time.

• Hmm, I take back the “never”. Now that I think about, I do this method almost unconsciously with numbers that lend themselves to it. No one had to teach it to me, and I believe it is simply a result of doing so many addition problems in my life. I’ve memorized certain outcomes, and I apply that knowledge. I guess I support Michael E. Lopez’s query below. I learned very traditional math, and despite not liking to play with numbers, I ended up with a BS in math, a subject I showed no extraordinary aptitude for as a younger student. So I stand by my assertion that teaching methods such as the above is completely unnecessary for developing number sense. Someone with number sense will likely start doing the method on their own when it’s more efficient or they just don’t want to pull out a paper and pencil.

• Anna says:

This is where the original algorithm comes from. Make ten and carry it; place the ones digit in the ones place. Repeat. I know we learned it so long ago that it seems obvious and instinctual, but there’s a step where little kids don’t know the algorithm yet and you have to teach it. Making tens is an important prerequisite step to being able to actually DO the algorithm and understand it,

You are right that this isn’t how math has traditionally been taught in the US. But it is how it’s taught in those high performing Asian countries we are trying to catch up to… Maybe it’s worth giving their methods a try? Or are we so arrogant that, just because we don’t understand something that works for the rest of the world, we assume it must be a bunch of junk?

• C T says:

How does that work for three, four, five numbers being added together? I don’t see the value. It seems like a limited use method and a waste of instructional time if someone with number sense is just going to come up with it on their own.
I don’t “assume” it’s junk. The sample above really is unclear to people who don’t already do it consciously. And people can become very good at math without ever learning the method above. High-performing Asian countries also spend a lot more time studying and using explicit instruction. Simply trying to use parts of their curriculum won’t give us Singapore’s results.

• lulu says:

When you have more numbers, you mentally add them into groups of 10…so 24 + 44 + 36 + 35 would be 6+4=10 and 4+5=9, so 19 + 20 + 40 +30 + 30 = 100 + 39 = 139. I’ve always done math in my head this way, although my husband never has. My son uses it sometimes, and other times doesn’t. Singapore math encourages you to find groups that add up to 10, then in the 10s column, groups that add up to 100. There are lots of tricks that people use to do mental math, and the Singapore curriculum teaches them but doesn’t require that you use them. My son loves the idea of not having to write down steps, so he usually likes these techniques.

• Lulu, you’re appear to be doing a different algorithm than the sample one above. You’re not intentionally breaking apart the digits in the ones place with the goal of getting tens but instead adding them in the easiest order to keep track of.

• lulu says:

CT, if you have more than 2 numbers, you don’t need to do that technique, because you just look for groups of 10 within the ones column. The Singapore curriculum does have students add sets of 4-6 numbers, and teaches them to look for groups of 10 within the column. Although it doesn’t really use the terms ‘borrow’ and ‘carry’, it teaches that technique, saying that if the column adds up to 19, you put the 9 in the ones column and the 1 in the tens, but for each individual column you add looking for groups that add easily.

3. Kiana says:

This is not slow at all if you do it habitually and instinctively. It does take practice to get it to that level but I have found it faster than the traditional algorithm for mental arithmetic, especially when adding multiple numbers.

If I recall, this is one of the things that singapore math teaches rather early on.

4. Michael E. Lopez says:

I agree that this makes perfect mathematical sense. It’s sort of how I add in my head. Not quite, but close. (I would add 26+7=33, +10=43.)

It’s explained like \$#!+, though.

I also wonder if it’s not a case of, “People who have become good at X do Y, so we’ll *teach* Y as a way of doing X.” You see a lot of this with reading exercises: people who are really good at reading anticipate what is going to be said next, so we’ll bog down students who can barely read with drills in anticipation and prediction so that we can say that they’ve got advanced skills….

5. It’s explained really badly, but the method is sound. Here is the method, in English:

“Round one of the numbers, add the rounded number, then subtract the amount you rounded by.”

Eg. 26+17

Round 26 up to 30 (rounded by 4)
Add 30 to 17 = 47
Subtracted the amount rounded: 47-4 = 43

The advantage of this method is that it keeps you in the ballpark even if you make a minor error. There’s no chance of getting a result like 26+17 = 277. That’s the ‘math sense’ aspect of it. It keeps you thinking of proportions while you’re calculating.

6. Sean says:

I’ve always told students to make the numbers “work” for you. Certainly by the time you get to Algebra I there can be multiple ways to attack a problem (maybe I’ll multiply by the reciprocal of this thing in front of parentheses rather than actually opening it up with the distributive property). What you choose depends on the tools and aptitudes you have. Math is a game, just don’t break the rules and you’ll be fine – but which rule you play is a matter of style.

Just because some people who are good at math do it this way, and just because it makes sense; is this the way we want to teach it to all the kids? If you’re headed into a real mathy field, you’ll find this stuff or pick up on it. But is it reasonable for the 95% or the 98% or whatever it is?

Stuff like this also makes is tough for parents to help kids, even if they want to. Leaving them more dependent on the schools or who all else. That’s a whole other can of worms.

7. lulu says:

Singapore math teaches this method in its mental math lessons. It mostly encourages students to use it for numbers ending in 8 & 9′ so that 67+99 is either 67+100-1 or 66 + 100. It also teaches the traditional algorithm and students can use the method that they choose.

8. lulu says:

I just asked my Singapore math kid how he would do that problem, and he said that he would add 7+6, then add 13 + 20 + 10. I had forgotten that it also teaches a lot about grouping ones, tens, etc.

9. EB says:

The technique is useful, and many if not most people who have become accomplished at addition and subtraction (especially in their heads) use it instinctively. But I think that introducing the two-digit version too early, when children are “not there yet” in their ability to simply envision the single-digit number that, when added to the one at hand, makes ten, could be very confusing. As a third grade teacher (which I’m not; I taught first grade) I would be looking for signs in each child that they had or were near that level of number sense before doing much with this technique.

10. Lee says:

I have a book (“How to Calculate Quickly”, but there are several other books on “rapid math” and mental arithmetic) that has tips & techniques similar to this. Another way (as suggested in the book) would be to think “26, 36, 43,” thus splitting the 17 into 10 + 7 and adding that way.

My own kids are learning mental calculation through their use of the abacus – they’re currently working on mental sums of +/- 5 three-digit numbers. It takes practice, but like anything else, the more you do it, the better you get.

11. Richard Aubrey says:

Learned numbers so long ago that I can hardly recall. But I believe I would have added the right-hand numbers, recalled the sum, carried the excess to the next column to the left, no problem. Seems the easiest way to keep track of operations.
Since then, rounding one for ease of adding and then subtracting the rounding has occurred to me when the figures seem to be amenable to the process. Do that, too.
And, after a bit, you know that, when the sum of the right-hand column is more than ten, the sum of the next column to the left is going to be one–if you have only two figures to add–more than its ostensible sum.
Automatic.
IMO, teach the old way and, as the kids go through life, the other ways will occur to them, as they did to me who has a hate for math. IOW, if I can do it without effort, how about the smart kids? Which is probably almost everybody else.
Any possibiltiy of deliberately ramping up the confusion to impress the rubes?
This, of course, is mox nix compared to the assertion that there are numerous errors elsewhere.

12. momof4 says:

I recently visited my son’s family and did some homework with his second-grade twins, who turned 7 last week. The male twin has a good math mind and has enjoyed doing mental math at home for several years. He would add 20 + 10 + 13= 43. His school uses Singapore Math, but I’m not sure if his class has covered that kind of problem yet – I was there for the first two weeks of school and they were reviewing, with two and three digit addition and subtraction, on paper. Again at home, he was asked to multiply 196 x 3, he did 200 x 3 and subtracted 12 from it, to get 588. Some people, even at young ages, do “get” a number sense easily. However, for 4th-grade homework, this is well below ideal. I know Singpore will be doing this in second grade and they did simple multiplication in first grade.

13. Crimson Wife says:

This is one of the methods that Singapore Math teaches. My kids always found it easier to use one of the Right Start math approaches, which is 26 + 10 = 36, then 36 + 4 = 40, then 40 + 3 = 43.

14. Michael E. Lopez says:

Most logical & mathematical systems can be expressed as inferences from a core set of axioms. What is surprising to some, though, is that you don’t necessarily have to start with any one set of principles as your axioms. Usually, the entire system can be expressed as extensions of several different axiomatic beginnings. You don’t have to start with vertical angles being congruent… that can be inferred from other facts if you’d like to start elsewhere.

I suspect that how you go about solving arithmetic problems depends on what you have internalized as your axioms. Many of the ways people have talked about solving math problems in this thread rely on a much smaller set of axioms than, say, the way I would solve them. The axioms are addition and subtraction facts that result in single digit numbers and 10: 3+4=7, 6+4=10, 8-3=5, and so on. These facts are VERY easy to learn because they are simple and few, but the “back end” is that you need slightly more complex algorithms to put them into practice.

On the other hand, if your “axiom” facts includes double-digit results like 6+7=13 and 9+8=17, then you have more “front end” learning to do, but your algorithms can be done much more efficiently. (See my comment above for an example.)

My guess is that this translates into all areas of math: multiplication, logs, etc. The more stuff you have as basic, the more stuff you have to put in the effort to hard memorize, the simpler your operations can be. While the more restricted your axioms, the more work you will have to do to use them to prove things.

15. Bill says:

I’d just do the math in my head and say the answer when I figured it out (this got me in a LOT of trouble in many math classes, since I didn’t LIKE to show my work, despite the fact of getting the answer correct most of the time).

Sounds like a very bad way of explaining rounding and estimation, in my opinion.

• “Teachers” habitually marked my math work down for “not showing my work”.

I was willing to show them that I did it in my head.  They didn’t care.  I refused to come down to their level.

16. Roger Sweeny says:

As various commenters have commented (?), this method is perfectly logical and will get the correct answer. Many people who do a lot of addition use both it and the more familiar add and carry method.

The question is whether it makes sense to make it part of third grade math. Will students understand and be better able to do several digit addition? Alas, as in so much of education, we just don’t know. And as far as I know, no one does anything like randomized clinical trials (RCTs) to find out. Instead, we just charge into something that people with power say is the next big thing. Which often leads to disappointment.

• lulu says:

We know that the Singapore math curriculum teaches this in 2nd or 3rd grade. I don’t know that anybody has actually compared similar groups of students using Singapore or other techniques. Among the homeschoolers that I know, Singapore tends to be used by folks with STEM backgrounds, which may not be representative of the population for many reasons.

I’ve found that that curriculum teaches everything slightly differently from the way that I learned it. Then when they use a technique like this, it seems like a logical ‘next step’ based on how they had taught the earlier addition concepts. I’m not sure that sticking this approach into a standard curriculum would have the same effect, since the students have different ‘prerequisite’ understandings about how addition works.

• Florida resident says:

Hello, dear Roger Sweeney. Around September 6, 2013 You wrote this about the book by Robert Weissberg “Bad Students, _not_ Bad Schools”: “It sounds intriguing. My local library network (about 30 libraries) has only one copy, which suggest to me that it is either a bad book or politically incorrect, or both. Using Joanne’s “Search this website …” box indicates she hasn’t posted anything about it. Anyway, I’ll take a look.” – See more at: http://www.joannejacobs.com/2013/09/from-superman-to-teach/#sthash.dgB425r2.dpuf Did you have a chance to read at least some part of the book ? With invariable respect of Ms. Joanne Jacobs, your F.r.

• Roger Sweeny says:

I’m reading it right now but finding it slow going. It’s very different, and more than a little discouraging.

17. SteveH says:

What is number sense? What is an understanding of place value? Do kids have to know how to convert from decimal to octal? It’s hard to judge this out of context, but I would guess that it’s just another attempt to avoid mastering the standard algorithms. I would love to see schools work more on number sense, but I don’t think I would agree with their definition or approach. Why not take the time to extend something like partial sums to the full standard algorithm? I’ll call it ABTSA – Anything But The Standard Algorithm. Everyday Math wants students to select their own approach, as long as it’s not the standard algorithm. Apparently moving from partial sums to the standard algorithm doesn’t add any more insight, whereas the thing above does.

They could take a more formal approach, but go left to right:

26 + 17 = (20 + 10) + (6 +7) = 30 + 13 = 43

It’s so easy to see the carry in there. I do this all of the time.

They should talk about how efficient the standard right-to-left algorithm is for addition. But they can also talk about estimation as a left-to-right process. With two digit numbers, students should be able to do it accurately in their heads. With practice, it’s not too difficult to do three digits exactly. I don’t like shortcuts that depend on the numbers. I guess that this is an assignment that focuses on “understanding”, not mastery. I doubt they expect the kids to do this quickly for any two digit numbers.

I like the book “Arithmetricks”, but prefer shortcuts that work no matter what the numbers happen to be. For me, many good estimation techniques work left-to-right. The further you go, the more accurate the number. Rounding is also good sometimes, but you don’t want kids to have to learn all sorts of different rules.

What about 327 + 476? How does the left-right technique extrapolate to these numbers?

You get (300 + 400) + (20+70) = 790 for the first two numbers. That’s pretty easy. How would the example in the link be adapted to these numbers?

Most people can do that in their heads and you are within a percent or two. Add in 13 and you have the exact result. This extrapolates to any two numbers.

I would love to see schools do a lot of this kind of practice, but that’s not what they are doing, even with the odd technique they show here. Their goal is understanding (undefined) and not mastery of a skill. They continue to unlink mastery with understanding. Understanding is only anecdotal (as shown above), not formal.

• Educationally Incorrect says:

One thing that I think people are missing is that “number sense” comes as a CONSEQUENCE of adequate practice and not as a result of the lack some explicit strategy.

In addition, ed people seem to think that the only things people ever need to add are numbers. Small numbers at that. As students advance through math levels they increasingly need to able to add various non-number expressions together, or extremely large numbers together, and some “number sense” — especially one coming from some ed school professional development session — isn’t going to cut it anyway.

• Michael E. Lopez says:

Amen to that, E.I.!!!!!!!

It’s a common example in Philosophy that while you can “know” what 54,576,125,009,128 + 852,776,391,505,006 is, in the sense that you can generate a mathematically correct answer, you don’t REALLY know what that means in any practical sense of the word “know”, except in the vague way you know that 54 quadrillion is like 1/16 of 852 quadrillion because you know that 54 is around 1/16 of 852.

But if you’re good with a cheap algorithm, you can get the right answer when it counts regardless of whether you have a “sense” of the particular number or not.

I might even go further, and say that “number sense” emerges when you’ve fully internalized a set of memorized facts to the point that you no longer are *consciously* aware that you’re using them.

If someone asks me what 13+14 is, I don’t have to perform any additional operations. I just know it’s 27, because I’ve added those two numbers together so many g*d**** times it’s like asking me how to spell my name. Ditto with really basic geometric principles.

It’s not mysterious. It’s practice. “Drill, baby, drill” was one of only two unqualifiedly true statements (to the extent it is taken as an actual statement and not the mere recitation of a political slogan) from the “Tough Teachers” post.

• Educationally Incorrect says:

I’ve recently bought a used digital piano and I’m trying to revive what I learned during piano lessons decades ago. I managed to purchase a copy of Bruce Hornsby’s “Every Little Kiss” and I’m trying learn the opening piano intro. So how do I know that I’ve hit a wrong note? I’ve listened to the tune hundreds of times so I know what it’s supposed to sound like. Yesteryear’s practice sessions (such as they were) didn’t hurt either.

I’m waiting for the ed people to come up with a “sour note strategy” for music students who don’t bother practicing.

(Let’s hope they’re not reading this and getting any ideas)

• Quincy says:

As a musician, I hope the ed people don’t start getting ideas about music education at all. There’s too much at stake.

Along with Phys. Ed. and the Industrial Arts, Music is one of the few subjects where irresponsible teaching can physically hurt students. Push a macho 16-year-old trumpet player beyond his physical ability with screaming jazz charts? A distinct possibly that damage to the embouchure results. Miss the signs of vocal overuse in your star singer? Possible damage to the vocal cords.

Fortunately, two other factors weigh against edu-fads in music:

1. Nearly all music teachers are themselves musicians.
2. It is immediately obvious when the instructor isn’t getting a result. Take, for example, this YouTube video of a high school band deliberately playing only 90% of the right notes in a piece: http://www.youtube.com/watch?v=yd-mkMq2aM4

That said, as a conductor myself, I do have and use something that could be termed a “sour note strategy”, though it’s not for musicians who don’t practice. In areas where the ensemble is getting too caught up in the notes, I’ll tell them to not worry about right notes and focus on rhythm (or dynamics or articulations) instead. That changes the center of attention, allowing the musicians’ unconscious mind to handle the notes. Those who practice produce very few wrong notes in this situation, while those who don’t produce more.

• SteveH says:

I’ve seen this over and over. Educators hide behind trivial ideas of understanding. They don’t see the need for practice. All of the students I have taught or tutored understand concepts. The problem comes when they have to apply them to even the slightest variation. Their “understanding” falls apart. I’ve harped about this for years. I will sit with a student and help them with their problem sets. They “understand”. However, I tell them that they really don’t understand until they can do all of the problems by themselves, not just in a group.

Perhaps it’s easier to claim drill and kill if you talk about practicing long division by hand (when was that last done?), but as soon as you hit decimals, percentages, and fractions, homework is no longer speed drills, but development of flexible understanding. This can’t be reliably achieved with group work in class. This wastes too much time and allows many students to slip through the cracks. Proper results require individual practice. Group work could be effective, but only if all students are fully engaged and at the same level. Educators don’t seem to be too concerned about that. If they see a few students engaged in “active learning”, they are happy as clams. They hope to define a process that works by definition so that they can blame all poor results on the students, parents, and society. “Trust the spiral” and blame everyone else. Trust the ideas of understanding but place the onus on the students.

• Educationally Incorrect says:

The nuns used long division as punishment As far as I was concerned it was preferable to praying on your knees.

• Kiana says:

Steve: I’m not familiar with the text itself, but given the two numbers you gave, my response would be:

327 + 476 = 325 + 2 + 475 + 1 = 325 + 475 + 3 = 803.

• Ann in L.A. says:

Our kid loved doing long division. We always brought paper and pencils to a restaurant so the kids could entertain themselves with drawing. He would write out an impossible long-division problem that crossed the length of a page and then proceed to work on it until our dinner came–sometimes he’d bring it home and finish it there. He liked being able to do something hard.

18. cranberry says:

Given the original problem, I would opt for: 6+7 = 13, 20 + 10 = 30, 13 + 30 = 43. Fewer steps, and fewer items to keep in memory.

I doubt the “number sense” approach teaches number sense; I think it washes out the child whose short-term memory can’t hold nine items (Those items being 26, 17, 20, 6, 10, 7, 13, 30, 43).

Average short-term memory has been shown to be 7 items, plus or minus 2. That means the MAJORITY of children won’t be able to do this exercise “in their head.” Extra practice won’t improve their performance–short-term memory seems to be fixed, i.e., it’s not possible to increase how many items you can remember. The children whose short-term memory is 6 or 7 might be perfectly fine if they could write the numbers down.

To me, this sort of thing falls under the heading of “parlor trick.” The people who are adept at this (and I am one of them) are usually adept at manipulating numbers, because having a large short-term memory makes _everything_ easier.

• SteveH says:

My point is really about what number sense is and how one formally goes about developing it. As Cranberry said, I doubt their technique develops number sense. They unlink practice and understanding. They see practice as only adding speed. They are wrong. This is especially wrong when you get to anything above basic arithmetic.

My left-to-right approach may not develop number sense, but it works for any set of numbers, it provides at least an estimate, and some can go further towards an exact number than others. One only has to remember one number at a time, since you sum as you go.

300 + 400 = 700 – remember 700
20 + 70 = 90; add to 700 to get 790 – remember 790
7 + 6 = 13; add to 790 to get 803

This is harder to do when the numbers get longer because even though you only have to remember one number, there are more digits.

I agree that the example in the post is a “parlor trick” and not a formal way to develop number sense. Perhaps they expect kids to be able to figure out some variation of this for any set of numbers, but is that a proper or direct approach to number sense? These examples just hide the fact that expectations are lower.

Their assumption is that these examples develop number sense better than the standard algorithms. There is no proof of that. What they are missing are formality and mastery.

• cranberry says:

SteveH, I think it makes a difference whether the student’s allowed to refer to the problem in writing while figuring it out, or if he has to remember the starting state as well.

The entire exercise reminds me of Everyday Math’s approach to division, the emphasis on looking for “friendly numbers.” As my kid had mastered the multiplication tables, there was nothing more “friendly” in insisting on creating multiples of 10 or 100. It did make the division process longer, which added opportunity for silly mistakes.

19. EB says:

It’s possible to explain and demonstrate (to kids who are interested and moderately able) how and why the long division algorithm works. Whether they want to remember the explanation is another issue! But that algorithm is certainly the most efficient way to perform long division.