In this video, a very determined third-grade girl attempts to add large numbers using an Investigations Math strategy. It takes eight minutes to get the wrong answer. In one minute, she finds the right answer by using the traditional “stacking” method she learned at home. (It’s not allowed in school.) Investigations is supposed to teach conceptual understanding. The girl says it’s “confusing.”

Also on Out in Left Field, Barry Garelick wonders how teachers can be evaluated if they’re forced to use ineffective curricula. Check out the lively debate in the comments.

In how many district is non-sensical math used and why in the world would any teacher teach it?

Reason #4,667 why my child does not attend a government school.

@anon: This is hardly a “governmental school” problem. It’s a math curriculum problem. There are many different approaches in use in “government schools.” What we need to do is to show which ones work and which ones don’t without vilifying the teachers who are caught up in curricula they are most often not allowed to change.

Personally, I think that learning how to add big numbers is a bit foolish. Small easy numbers, yes, that should be memorized for convenience and because it is actually faster, but what is the point of learning either method when a calculator is so much more efficient for larger numbers.

I suggest that if we want mathematically literate people, we need to focus on actual mathematics, rather than rote memorization of algorithms to do computations. Clearly the investigation method here is a poor algorithm, and the stack method is a good algorithm. I’d like to see the ideas of Conrad Wolfram and Dan Meyer be implemented on a broader scale and redefine what it means to learn mathematics in schools.

Learning to do what a computer can do faster and cheaper is inefficient. Learning to do what a computer cannot do, which is to set up and initiate problems is much more useful. The problem with the Every Day mathematics curriculum that people vilify so often is that it is measured with by how well students can do calculations, rather than how well they can solve problems.

For those of you who say, well what about when the student doesn’t have access to a calculator, I ask, when was the last time you didn’t have a calculator handy to do a calculation?

DavidWees-

When you can do the calculations without the calculator, you are much more aware of the sorts of opportunities that you seem to be claiming do not exist.

When you have a hammer, they say, everything starts to look like a nail. But there’s a corollary: when you don’t have a hammer, you don’t notice the actual nails as much.

That’s part of why education is so important, after all: it changes the way we

the world. When you’ve got an excellent grasp of poetry and meter, you can start to hear things in people’s speech that you simply didn’t hear before. It’s not that they weren’t there — it’s that you weren’t hearing them. Likewise, there are instances every day of opportunities to do some mental math with a well-mastered algorithm — like watching the first twenty seconds of this video just to name an immediate if unimportant example.perceiveAnd that’s really where the payoff is, by the way — with the mental math that full mastery of the algorithm brings. Being able to do it on paper is an intermediate step.

I might even agree with your point when it comes to the ability to add numbers like: 243,655,009 and 1,768,334. That’s just silly and time consuming and some people don’t have the sort of short-term memory needed to be able to do those sorts of calculations in their head even if they have mastered the algorithms.

But the fact is that learning how to add those numbers together, and learning how to add much more modest sums like 15,467 and 5,899

, and the latter really is both useful and practical.is the exact same skillI was struck by the focus on

understandingat the original argument. I thoink this is the crux of the matter, and not just in mathematics.The proper role of K-12 education is to prepare our students to be successful as adults. The vast majority of students have no need, or desire, to

understandmath, they just want/need to be able todoMath. Those who want tounderstandcan take honors courses, or go to college to do so.“Learning to do what a computer can do faster and cheaper is inefficient. Learning to do what a computer cannot do, which is to set up and initiate problems is much more useful.”

That is a beautiful way to teach dependence on and blind trust in computers. As someone who works in software development, I do *not* recommend either.

I’ll repeat a comment i posted on the other site.

And this new math differs from counting on your fingers, how?

Regards,

JJ

Long time reader, first time poster….

“Learning to do what a computer can do faster and cheaper is inefficient”

Me to student during an Algebra I test: Better check that answer, it’s wrong…

Student: But that’s what the calculator gave me.

Me: 1/3 times 3 is not 0.99 …

Student: But that’s what the calculator has; see?

Me: It’s wrong; THINK

Student: How do you KNOW?

I see some value in learning to do what a calculator can.

I think that a student who understands the process of problem solving should be able to estimate a solution in his or her head so that they can look at the obviously wrong answer on their calculator and go “oops, I made a mistake somewhere.”

In terms of blindly trusting the computer, I have no idea how you take “using a computer efficiently” and turn that into “blindly following a computer”. That’s a complete non-sequitur in my mind. It does not follow that just because we use a computer that must therefore blindly use a computer.

I really don’t see how adding 15,467 and 5899 is useful or practical. If you were dependent on the solution to this problem being correct, and your life depended on it, would you do it by hand? Or would you use a calculator. Be honest.

Michael’s argument is the most compelling simple because he uses some analogies that seem reasonable. However, my response to him is, does he really think that students who are blindly following an algorithm to add two numbers together have a deeper understanding of numbers than they would if they punched the same numbers into a calculator and similarly do not understand the process?

For students who have a natural gift with numbers, they will be able to play with an algorithm and get a deeper understanding of the process, but they could do the same with a calculator. Neither is hampering their ability to play with the computation, they get numbers.

However when a student does not understand the algorithm and struggles with it, I’d argue that messing around with it and experimenting with it doesn’t happen. They gain no deeper mastery of numbers by playing with something they don’t understand. Similarly, those same students will gain no mastery of numbers using a calculator as well, but they will have fewer errors of computation, and be able to spend more time to ensure that they understand the answers the calculator is giving and the process of setting up a problem so that you can enter it into a calculator and get a reasonable response.

Your analogy of being able to hear the poetry in someone’s voice is an interesting one. I think kids should have to listen to lots of poetry, create lots of their own poetry, and explore the idea of words, none of which is accomplished by memorizing stanzas of poetry as an activity. Yes, they are playing with words, and I don’t think you can reasonably ever expect to be able to do that activity effectively with a computer. You can use a word processor to help you organize your thoughts and to make it neater, but you can’t really write with one, it’s just a medium of the same process you would do on paper.

The difference is that when you a calculator, you can focus on the process of problem solving rather than the process of computing, which are very different things. Now, a caveat to this argument is that for teachers who have no understanding of the problem solving process themselves, who are blindly following a curriculum they don’t really understand, you can quite easily end up with students are both poor at solving problems and poor at computation. The only benefit of using the computational approach is that it is more likely that a teacher who is poor at mathematics can handle the same computational approach that they learned by when they grew up.

All too often it seems people do fall into the trap of blindly following a computer. It’s a slippery and easy slope to get on. How many people are using retirement calculators without knowing the assumptions going into the model? I shudder to think how many people are using a 7.5% after inflation rate of return…

Technological aids are great tools, but goals I’ve always maintained for students in my classes; besides mastering the specific content that they should…

1) Develop “spider sense” – that gut feeling that something isn’t right. The example I gave in the prior post illustrates a failure in that. Multiplying two odd numbers together and getting an even might be another.

2) Appreciate the task and knowledge at hand. Becoming facile at routine computation pays big dividends later on when students approach polynomials and factoring. Being good at factoring helps you “get” why roots are important. This is the “poetry” part of math, you see it, you play with it, you may eventually “get” math. Not everybody will be a poet, but the exercises are important and good in themselves. Same with math.

3) Understand physically what the implication of the math is or might be. If you have a picture in your head of this stuff, you can carry it around longer. That might be an artifact of that long ago bachelors in physics; hard to tell.

Sean Mays brings up some very important points.

“Gut” feelings are a person’s subconscious processing available data with their knowledge and experience. These are an absolutely vital tool for telling a person where to look and where to go. Their usefulness, however, is in direct relation to the quantity and quality of knowledge and experience that a person has.

This is why “critical thinking skills” are absolutely meaningless when divorced from content. It’s like teaching a course called “how to use a tool” without ever talking about or using any actual tools.

It is precisely the need to develop skills that makes adding 15,467 and 5,899 important. A good athlete will place greater challenges upon himself or herself in training than they would expect to encounter in competition. The point being to make the competition task relatively easy. Likewise, students need to be practiced with calculations more difficult than they would expect to do in the field, so that: A) they can develop the experience to tell them if they’re on the right or wrong track; and B) they have enough confidence so that they don’t run away crying like a baby every time something just a little bit difficult comes along.

The stacking method works, and lets them get the practice to develop the experience necessary to have a reliable gut feeling. The Investigations method sounds nice in theory, and could even be a valuable component of learning. You cannot, however, teach “understanding” with diagrams too complicated for any student to figure out what in the heck is going on. On that note it fails miserably.

With students going through years of this mess, being told they’re doing fine, and showing up in our classrooms with a sense of entitlement where their knowledge ought to be, is it any wonder that we get nervous about being held accountable while those who force this upon us are shielded?

“@anon: This is hardly a “governmental school” problem. It’s a math curriculum problem. There are many different approaches in use in “government schools.” What we need to do is to show which ones work and which ones don’t without vilifying the teachers who are caught up in curricula they are most often not allowed to change.”

Curmudgeon–no, this IS a government school problem. We already KNOW which approach works for adding numbers. I learned it in first grade in 1956. My parents learned it in the early 1930s. This issue needs NO more investigation. What needs to happen is for government to get out of the business of educating students at all levels of education (from kindergarten through graduate school)and let good teachers who know how to teach math (and everything else) do their jobs. But, this will not happen when government is involved at any level of education.

Automaticity in the use of standard algorithms and understanding of the underlying mathematical relationships are not mutually exclusive. They go together. I imagine that most readers are familiar with the concept of muscle memory, which is what makes athletes able to constantly improve their performance through practice. Well, I think the brain has some of that going on with the smaller tasks we use to build number sense, a sense of ratio, and a sense of place value. Certainly, there needs to be instruction that teases out the “why and how” of the basic operations, but if that’s all a student has, s/he is like a beginner athlete who is expected to improve by watching and analyzing videotapes of accomlished atheletes. Not gonna happen.

“In terms of blindly trusting the computer, I have no idea how you take “using a computer efficiently” and turn that into “blindly following a computer”. That’s a complete non-sequitur in my mind. It does not follow that just because we use a computer that must therefore blindly use a computer.”

Mainly, because you are *not* teaching someone how to use a computer efficiently when you do what you said earlier, you are teaching them to use it blindly. In your model, people would not have any ability to second guess what the computer is telling them because they wouldn’t know any better.

Your model assumes that computers and software are so infallible that users can put stuff in and trust the answer that comes out without any further thought. Sorry, but that’s just not true. Software contains bugs. Software contains bugs in the code that was written to patch over the bugs. Software contains implicit assumptions.

Moreover, when using a program like Excel, the user is actually programming the computer. Once you get Excel workbooks with any sort of complexity, you need to know what answer you expect at each step of a calculation. Why? It’s the only way you can spot errors in the program.

In terms of using a computer, you’re bringing up an overly simplistic use case and wondering where the value is. To use a computer to produce reliable results, you need to know what to expect out of it. Sean Mays and Obi-Wandreas have brought up good points about “gut instinct”, which I would call unconscious mental effort. Skills do not, ever, become available unconsciously until they have been mastered consciously. Personally, I usually see one of three things that makes me question a number:

1. Wrong order of magnitude

2. First couple of digits incorrect

3. Last couple of digits incorrect

The thing is, I just *see* them. By having learned how the computation works to mastery, my subconscious is able to warn me when things aren’t right. In my field of work, this is usually because of a software bug. Doing my job would be much, much more difficult without this ability.

Now, that’s just talking about the addition piece of it. The limitations of digital computers are thrown into full relief when fractions are brought into the equation. We’ve had computers around since the 1950s, yet the simple fraction of 1/3 is still difficult to work with correctly. Without teaching the students what computers are, and aren’t, capable of, they fall into the same trap as Sean’s student above.

A person can only use a computer efficiently when they know what the machine should produce. Saying you’re helping someone use a computer efficiently by denying them knowledge about what it should produce is an oxymoron.

“I really don’t see how adding 15,467 and 5899 is useful or practical. If you were dependent on the solution to this problem being correct, and your life depended on it, would you do it by hand? Or would you use a calculator. Be honest.”

I would do it by hand. It’s easier to check errors on paper. I’d also look really stupid if I couldn’t ballpark 40% of large numbers quickly, without dragging out a calculator.

“For students who have a natural gift with numbers, they will be able to play with an algorithm and get a deeper understanding of the process, but they could do the same with a calculator. Neither is hampering their ability to play with the computation, they get numbers.”

I don’t think there is such a thing as a “natural gift with numbers.” Some children may learn material very quickly, and if you don’t watch them closely, it might seem they have a gift for it. How do you recognize the 3 year old with a natural gift for numbers?

Even if there were such a thing, that would argue for teaching everyone the simplest, quickest algorithms for simple arithmetic.

Every student should learn and become proficient with the basic algorithms that underlie all mathematical operations. Why? Fluency. Just as one only becomes proficient with a language through basic memorization of vocabulary and grammar rules, one can only become proficient with mathematics through the same process.

One of the best lessons learnt from adding/subtracting/multiplying two large numbers is the importance of focus and attention to detail. I have so few students that can correctly follow any instructions that go beyond three steps…they are mentally lazy.

As for the calculator argument, I am rather tech-savvy, and I cannot even begin to list all the situations that I have had to do arithmetic without a calculator in the past month because they are so numerous… and I trust my own calculations as much if not more than a calculator because I am more likely to mistype something than miscalculate something.

Interesting……do we know this child attends a public school? Or, was that an assumption?

Clearly, we (the U.S.) are trying to come up with ways to improve education. I would say that is a good thing. It seems the real problem is that we tend to “try out” different strategies without due diligence. We have been experimenting with the children for a long time..desperate to improve our educational system.

Clearly, the vast majority of people have no need to be able to read or write words with as many as three or four syllables, never mind five. Such complex things can be easily looked up in a dictionary, or Google’d.

There’s a name for people who are not fluent in reading, writing, and arithmetic: Slaves.

This kind of thing has gone on for years. I remember in the late 70s, being kept in at recess so I could “redo” the long-division problems I was assigned in the idiotic “new math” style that was popular at that moment (which took twice as long, as I remember). The previous summer, my mother had taught me the “old fashioned” way of doing long division. I could get the right answer with that method AND I COULD EXPLAIN WHY IT WORKED. But the teacher insisted I followed the “currently approved” method. The old method was apparently forbidden.

Well, until the next year of school, when the math teacher looked at us and said, “I’m too close to retirement for them to fire me. I don’t like New Math and I don’t think it works. So I’m going to teach you the way I’ve been teaching for the past 30 years.”

I’m not sure why throwing out methods that work, and that lots of people understand, is necessarily an advancement.

“I’m not sure why throwing out methods that work, and that lots of people understand, is necessarily an advancement.”

There is money to be made selling new “improved” materials to school districts.

There is money to be made selling new “improved” materials to school districts.Call me cynical, but I suspect that idea is at the heart of many of these “innovative” methods.

I’m not sure why throwing out methods that work, and that lots of people understand, is necessarily an advancement.

There is money to be made selling new “improved” materials to school districts

Money is a likely culprit. A less cynical interpretation is that we’re searching for a method that works for ALL learners, which the classical methods didn’t seem to. In the Harrison Bergeron inspired world of education, we need methods that yield no disparate outcome based on race, ethnicity, socio-economic status or parental involvement and educational attainment.

It could be that the goal of the educational establisment (government schools?) has become the achievement of social justice rather than the inculcation of knowledge and the transmission of culture.

“Interesting……do we know this child attends a public school? Or, was that an assumption?”

I don’t think that I said public school. Instead, I said government school. I teach at a so-called “private” college. My “private” college gets well over 60% of its annual funding from government. It is a government school, not a private school. My college does what the government says it must do, not what the college could do if it were really private.

“Clearly, we (the U.S.) are trying to come up with ways to improve education. I would say that is a good thing. It seems the real problem is that we tend to “try out” different strategies without due diligence. We have been experimenting with the children for a long time..desperate to improve our educational system.”

No, we are not trying to improve education. Education does indeed “try out” different strategies. It introduces these strategies without any evidence of their effectiveness. Years later, education says “oops, doesn’t work. Sorry about that.” See Whole Language. Education also refuses to use strategies that have accumulated evidence of their effectiveness. Imagine if medicine was conducted in this manner.

“Imagine if medicine was conducted in this manner.”

It was

Before the rise of scientifc medicine, medicine was conducted in this manner. Disease was caused by unknown or falsely known causes. Many cures acutally killed. People recovered by false remedies or by accident.

To be fair, I am sure that the girl’s teacher and school both realize that stacking numbers is a much simpler and more efficient method of adding numbers, and ultimately is how the girl will be instructed. Adding by drawing the pictures is intended to improve the girl’s conceptual understanding of place value, regrouping, etc.. Building an understanding of these basic math concepts is critical in elementary school.

http://www.k5learning.com

Calculators are not the answer for simple computations. I’ve had two recent experiences with sales people who couldn’t figure out how to enter the problem into their calculator.

What were the difficult problems they couldn’t do, either without or with a calculator? (1) 5% sales tax on a $10 purchase and (2) change due from a $2.82 purchase, given $3.02. Inexcusable

My older child had the Investigations curriculum for several years. It encourages the most cumbersome ways of doing things. It also has a big emphasis on problem solving methods that may work for some problems, but not for others, for example 99+99 vs. 68+47.

By those of you using the term stacking, I’m assuming you mean traditional algorithms. K5, using traditional algorithms is highly discouraged in Investigations.

From an Investigations point of view they’d rather see a child “invent” a strategy of drawing 68 apples in one group, and 47 apples in another group, and counting them to get the final answer, rather than using a standard algorithm.

@ David Wees

Count me in as another individual who regularly does math in my head and doesn’t want to need to drag out a calculator every time I need to do a calculation. As a matter of fact I often find it quicker to do the calculation in my head than to have to punch it into a calculator.

And then there are times I don’t have a calculator handy like when I’m shopping and want to know what the price of that item that is 30% off. And on Saturday morning when I want to make a batch and a half of pancakes when a batch calls for 3/4 of a cup of water. Kids need to be able to do BOTH — calculate and set up a problem.

And Quincy is right on target about computers — not only is there a problem with software bugs but there is also GIGO — garbage in/garbage out. Data entry errors happen all the time. Knowing that I (or someone else) is likely to have mistyped a number is a good thing.

When my boys were in elementary school they used to love to play the math calculation game in the car — I’d give them a calculation and they’d do it in their heads and give the number back to me. It was a great mental exercise for them — and for me! Luckily they had learned the stacking method!

@K5Learning

To be fair, I am sure that the girl’s teacher and school both realize that stacking numbers is a much simpler and more efficient method of adding numbers, and ultimately is how the girl will be instructed

I’m not so sure. Once they’re done with Investigations, will the next program be Discovering Algebra and Discovering Geometry? Where the emphasis is on paperfolding and origami rather than construction and riqorous proof? I’ve been part of too many conversations where people rave about paperfolding being a perfectly adequate demonstration of bisection and that using a compass is too hard for so many to grasp.

I would suggest having the districts which use Math Investigations charged with child abuse. What else can you call the frustration and mystification these misbegotten ideas cause when inflicted upon innocents? I wouldn’t be surprised if child psychologists have already investigated the effects of such “innovative” schemes, you just need to find the papers for evidence and a prosecutor willing to take it to court.

The idea of going to jail would bring administrators and “curriculum consultants” down to earth with lightning speed.

All this video proves to me is that following ANY curriculum blindly leads to kids who don’t learn. NO curriculum, no matter how fabulous, will be perfect for every kid in the class. As a professional, it is the teacher’s job to figure out how the kids in his/her class learn best and to teach them in a way that will reach them.

With math, if children are taught understanding along with an algorithm, then they can use any algorithm that works for them and that gets the right answer. For a school to say that students aren’t allowed to use a different, though valid, algorithm is pedagogically unsound and somewhat lazy of the teacher or the school.

I had a math methods prof who showed us the algorithm her grandmother taught her for something. It was completely opaque to an outsider but it was quick and elegant and, most importantly, it worked without exception. The schools moved to what we teach today because packaged curricula became the norm and they used a different algorithm. People outside this country do their math very differently and yet they still get the same answer. The trend to forcing students to show their work is supposed to support using whatever works instead of some standard way, but all it seems to do it make teachers more aware of when a student is doing math in a non-approved way. What a shame.

And yes, we do need to teach students how to use technology, because it’s not going away and I’ve got better things to do than add big numbers in my spare time (though I should have a sense if my answer is correct or not). But blind faith in tech doesn’t lead to correct answers any more than forcing a kid to use an algorithm that doesn’t make sense to them when they already know one that works.

How children as individuals learn best is not as important as what is the best method vis-a-vis the subject being taught. “Learning styles” are poorly understood at best and vastly overrated at worst. See http://www.aft.org/newspubs/periodicals/ae/summer2005/willingham.cfm for an article by Dan Willingham regarding the learning modality myth.