Recently, after carefully reviewing the K-12 curriculum documents that are now being used in my province (as early as 2006), I wrote to some of the curriculum developers and asked whether it was indeed the case that the standard algorithms are no longer taught in our schools. The answer I received was that teachers do not have to teach standard algorithms (they are not mentioned anywhere in the curriculum). When I pressed a little more, I was given the impression that students in the advanced streams, headed for university, would be taught them. My understanding is that this applies to a select group of students in junior high or high school (again, there is no mention of any of the standard algorithms in the curriculum documents). I am extremely concerned about this. The new algorithms that kids are learning only work well with small numbers (2-digit multipliers for multiplication, 1-digit divisors for division, 3 digits for addition & subtraction, for example). Some will tell you that it is only necessary to have kids work with numbers they will encounter in everyday life. What about your salary, your house price, your car price? Should students not be provided with a method that works for all numbers? The answer is simple: When numbers get too big, reach for your calculator. Are we OK with this?
There is a huge focus on mental math. While this is a good thing and certainly develops understanding of the concepts, it cannot be the entire focus! This is nothing short of preposterous! Concepts should be developed and practiced with smaller numbers – students can still learn to do mental math with these types of numbers. However, educators need to go beyond that and teach kids the most efficient standard algorithms, that have been used for centuries, so that they have efficient methods for working with larger numbers.
Someone sent this YouTube video to me about a year ago. It was posted 4 years ago in response to the use of certain reform math curricula in the US. She displays some of the different approaches to multiplication and division that are now being adopted in our schools. When you watch the video, note that the “cluster problem” and “partial products method” for multiplication are two approaches our kids are now learning in place of the standard algorithm. The partial products method really just uses what we call the distributive law – you may see it in many forms. At a recent math meeting for parents that I attended at my daughters’ school, a child was shown using this method but was putting the numbers into a tableau and then adding them up. There is nothing wrong with this method and it is a very nice application of the distributive law (so is the standard algorithm) but for numbers that are longer than two digits, it is extremely cumbersome to use (even for a 3-digit x 2-digit) and no one in their right mind would use it in such a situation. For the life of me, I cannot understand why some educators think they should stop here. Why not move forward and teach them the standard algorithm for multiplication so that they have a method for working with larger numbers? Why not teach both methods?
My greatest concern is the approach used for division. It is confusing, time-consuming, and again only works for small numbers. In the curriculum documents, you will see phrases like “for numbers that are not 1-digit divisors, the use of technology is expected”. The repeated subtraction method that is shown in the video is a method that is suggested in our curriculum documents (and, again, was demonstrated at the meeting I attended). There are a number of reasons that kids should be learning the long division algorithm in addition to these methods. Again, the methods being taught become unwieldy (even impossible) once the numbers get larger. In addition, later in their education, students need to learn how to perform polynomial division. If they do not know how to do long division with numbers, this is going to be a very difficult concept for them. Ironically, curriculum developers tell me that these new methods are better for leading kids into algebra. I disagree and, believe me, I know a lot about algebra. I will make sure that my daughters know how to perform long division, but what about all the other kids out there?
The video is about 15 minutes but, if you have time, I hope you will watch it. She does a very good job of explaining the different algorithms and one can see just how cumbersome some of them are. Some of these non-traditional algorithms are precisely those that are being taught, while standard algorithms are excluded, in my daughters’ school and in schools across Western Canada. I would really like to hear comments and opinions about the video.
At the end of the video, she mentions the Singapore Math series. I also highly recommend JUMP Math. Either of these series would be of great benefit to children whose parents are seeking math materials to supplement their education.
I am astounded that it is suggested in a provincial math curriculum that “for numbers that are not 1-digit divisors, the use of technology is expected”. Who will create the technology? Who will write the software the technology requires and who will manufacture the devices that require precision instrumentation? Both require significant mathematical skill and understanding, beyond the use of a calculator.
Imagine not being able to add, subtract, multiply or divide numbers larger than 2 digits without a calculator. Imagine not knowing how? This is the equivalent of being illiterate.
I have no issues with using methods that show the inner workings of the algorithm to increase students understanding, but for the love of math, don’t stop there. Teach the algorithms. They work, and they’re efficient. I can answer a question on paper in less time than it takes someone to find a calculator and do it. And let’s not forget the “use it or lose it” principal. If we don’t use our brains, well…
I for one find this very sad and do not understand the province’s justification for changing the math. It seems they are treating ALL students as if they are only capable of operating a calculator and not as smart as they are. The minds of children should be trained for the very best from the beginning as they are all capable of greatness.
I’ve seen this video before and it troubles me. Any algorithm taught in the absence of conceptual and contextual understanding is an exercise in futility. I teach many of the methods this woman (not a teacher, by the way) scoffs at. Teaching the expanded form of multiplication is simply a scaffold for understanding what is happening in a multi-digit multiplication problem. The standard multiplication algorithm is simply a shortcut. Don’t teach the shortcut until students understand what steps that short cut is eliminating, as well as the big mathematical ideas (such as the distributive property of multiplication) that support those steps. Teaching the lattice method without attending to the underlying big ideas that make it work is ridiculous and no more meaningful to students that the traditional algorithm taught as a meaningless series of steps, disconnected from the beautiful mathematical ideas that support them. Beware people who try to make you believe that these strategies are the end. In many cases, they are the means to an end.
An algorithm taught without “conceptual/contextual understanding”. An unpopular idea among most educators, yes. Futile, no. I was taught algorithms without either of these two things as a child and went on to get a PhD in math so I would not say this was futile. Most adults I know were taught in this fashion and do not have problems with arithmetic.
This doesn’t mean that we need to continue teaching children algorithms without explaining the meaning behind them (and you have probably gathered that I am a strong advocate for teaching with understanding and lots of practice, among other things). However, educators all too often try to push the idea that teaching algorithms at all must be at the sacrifice of understanding. This is absolutely false.
I didn’t take the video to mean that algorithms should be taught without understanding. I understood her point to be that many of the other algorithms shown in the video do not work as efficiently as the standard ones (which is true, especially for larger numbers). In fact, I thought she said at one point “why not go on, and teach the traditional algorithms too”. I know of many other algorithms besides the ones shown in the video that accomplish the same goals and I wouldn’t have a problem with any of these, or the ones in the video, being taught in schools. However, I do have a problem with standard algorithms not being taught at all.
You noted that she is not a teacher. That is OK – parents, and other members of society can all have meaningful contributions to the discussion. In fact, there are many members of society who have much stronger mathematical backgrounds than some teachers and can easily see the flaws in some of the new teaching methods. People in many different professions (eg. engineering, manufacturing) also see the fall-out from unsatisfactory math education.
The video does a good job of pointing out that many children are no longer developing competency with arithmetic. The video is not by any means suggesting that concepts behind mathematics should not be adequately explained. Unfortunately, a common argumentative technique that I have seen used again and again is: Someone points out that the current educational practices are cheating children of their number skills by not providing them with adequate opportunity to practice these skills by using – for example – the standard algorithms for arithmetic. The math consultant then responds with the accusation that the first individual does not support “conceptual understanding” and “exploration”. This is not an either / or situation. If I say that I enjoy reading the newspaper, it is erroneous to conclude that I don’t also enjoy reading novels. If I say I like chocolate cake, it is erroneous to conclude that I don’t believe in the importance of eating vegetables. It is surprising to encounter such illogical conclusions from people who claim to have a deep understanding of mathematics. There is no reason why a child cannot – through practice – attain competency and speed with arithmetic calculations and, at the same time, develop an understanding of the properties of numbers on which their techniques of calculation are based. This was, I believe, the intended point of the video.
The multiplication algorithm certainly is a nice application of the distributive law. But is the distributive law that complicated and difficult to teach? Absolutely not: it simply says, for example, that 3 x (4 + 6) = 3×4 + 3 x6. This could be taught, using several examples, and exercises in 45 minutes. One can then move on and develop the multiplication algorithm, in similar 45 minute sessions, within the period of one week. After which, the students can spend a week or two actually working with the algorithm so that they develop speed and accuracy with multiplication. Alternatively, the kids can be taught to work quickly with the various algorithms in the lower grades (K to 4) with less emphasis on understanding, after which the properties of the real numbers system could be taught in the higher grades (I would suggest the fifth grade before they learn about fractions). This could easily be done very thoroughly in one month – there aren’t that many properties of the real numbers that they need to know! – in a manner that prepares the children for algebra in a much better way than the current system. My point is that it is completely unnecessary to beat the kids to death with different illustrations of the properties of the real numbers. It is boring and the properties themselves are really not very difficult for children to grasp. Furthermore, although understanding is important, it should never be stressed at the expense of competency (such as the ability to actually add, subtract, multiply and divide large numbers with speed and accuracy).
This mother has every right to question educational practices even though she is not a teacher. (I am not a Member of Parliament, but is it not my right to question the policies of my government?) As someone who has taken a calculus course at the University of Washington, she almost certainly knows more math than the majority of those teaching the subject in elementary schools: the mathematics prerequisite for entering into an early/middle years education degree program is – shockingly! – only high school Consumer Mathematics (what would have been called MATH 104 in my day).
Should we teach the standard algorithms for arithmetic? Absolutely, but they shouldn’t be the only algorithms kids learn.
Why exactly is the ability to add, subtract, divide and multiply large numbers so critical? It seems clear to me that these are useful skills for numbers we will encounter in our day to day lives, and that it is useful to know that algorithms exist to work with larger numbers, but your other connections seem tenuous to me at best.
You’ve argued that without practice using algorithms, students will not be able to remember them to use them later, and this I agree with. It is a basic tenet of education that spaced repetition helps students remember how to use knowledge.
The question is, what type of knowledge is critical for students to remember? Does knowing how to multiple 39835 by 2338383 or any other arbitrarily large number assist the typical person in their life? Does it even contribute to a greater understanding of advanced mathematics? Has the number of people completing advanced mathematics degrees dropped? Statistics Canada data from 2007 suggests that it has dropped very slightly (see http://www.statcan.gc.ca/pub/81-004-x/2009005/article/11050-eng.htm) but not by an alarming amount.
Regarding your achievements as a PHD in mathematics, don’t forget, the plural of anecdote is not data. You can’t generalize from your one experience to what is useful for all of society.
Understanding how to use the algorithm seems sensible to me, but I think it is even more important that people understand algorithms (emphasis on the plural) which is probably lacking in the current curriculum as it is constructed.
One problem is that all across our society, at many different age groups, we have a lack of people using any advanced mathematical thinking to solve problems. If you look at how people solve problems similar to what they learned in school, but in a different context (see Jean Lave’s work), you find that it is rare for people to use the standard algorithms they learned in life, despite the fact that the standard algorithms are much more efficient than the various algorithms people construct for themselves. This suggests that even though the standard algorithms are more efficient, they may still not be the best algorithms to teach.
It seems to me that if over the course of a lifetime, some knowledge is going to be forgotten, the skill of learning is more important than what specific knowledge is learned.
Hi David,
Thank you for your comment. I am happy that you have decided to weigh in on this discussion.
At no point did I say that children shouldn’t learn other algorithms besides the standard algorithms. Nor did I say that I wanted kids to be multiplying numbers like 39835 by 2338383 for practice (this is quite an exaggeration). I simply want kids to have efficient methods in their back pockets for computation. This is in addition to understanding – you won’t find a mathematician who doesn’t want kids to UNDERSTAND math concepts or to be good problem solvers. And, yes, the standard algorithms do have theoretical significance that lead into more advanced mathematics. Integer long division generalizes to polynomial division, distributive properties for multiplication which show up in the multiplication algorithm are useful in algebra (in fact, the standard algorithm for multiplication is a beautiful and powerful application of the distributive property – kids should be given the opportunity to see this!) Mathematicians have written several articles on this. Here is one: http://www.nychold.com/ocken-aaa01.pdf.
The entire point here is that I want standard algorithms in the curriculum. (Again, that doesn’t mean that I don’t want other mental math techniques or mini-algorithms in the curriculum.) In fact, I haven’t talked to a mathematician who isn’t in favour of kids being taught standard algorithms (some of us even had a meeting about exactly this and other curriculum issues last week!) and I think we should have a strong say in what goes into math curricula since we see the end result of all this. I do have a PhD in math but I am also a Grade 13, 14, 15, 16 math teacher.
You seem to agree that kids should be taught standard algorithms so I’m not entirely clear on where our opinions on this matter differ.
I guess I saw an over-emphasis in your article on the move away from the standard algorithms. On a reread, it does seem that I over-reacted a little.
I have a sort of question; what do you think of this movement? http://computerbasedmath.org
Hi David,
Thank you.
I am in the middle of making up an exam for my students and have not read through the link you’ve given above carefully. However, I did watch Conrad Wolfram’s video awhile back and you can probably guess what I think.
As I recall, the argument is for removing computation (leave that to the computers) in favour of developing problem-solving skills. After all, computers can take care of all those tedious calculations for us, can’t they?
Computers can also read for us – they can also make music.
I am very much in favour of placing more emphasis on critical thinking and problem-solving skills. I am not in favour of removing algorithms and human computations altogether. My colleague, Rob Craigen, has made a solid case for this and I won’t repeat what he has said.
(Please also click the link, in Rob’s comment, that links to a letter to the editor written by a math teacher in Winnipeg.)
Of course, I use computers to do a lot of laborious calculations when I need to. In fact, I often spend large portions of my time writing computer programs to work out complicated examples for me in my research. (I wouldn’t be able to write those programs, by the way, if I didn’t have a good understanding of how algorithms work.) I don’t use computers to do minor calculations, though, and I surely don’t want to see kids reaching for the calculator every time they have to multiply 9×7. And, I don’t want to see kids using calculators to do simple arithmetic problems. I think that kids shouldn’t be using calculators and computers to do arithmetic problems until they know how to do those arithmetic problems themselves. They’re crippled otherwise.
Why does this have to be an either or argument? Why can’t we a) teach kids arithmetic AND b) develop problem-solving and critical thinking skills?
As for this business about math not equaling calculations…Who ever said that math=calculations? By calculations, I presume we mean arithmetic and I want kids to be able to do arithmetic. Arithmetic is a branch of mathematics, so is problem-solving, so is logical reasoning. I want kids to have all of these things and understanding to boot.
So I wonder then, is it worth learning how to use your computer to do calculations as part of the process of learning math from k to 12? There are parts of the existing k-12 math curriculum which are surely critical to understanding in future math, but are there parts which are less critical to learn, for which we could substitute some of the computational approaches you use on a regular basis?
I think that one of the goals of the computer based math approach is to find the calculations which themselves lead to understanding (which we agree includes arithmetic), and an ability to problem solve, and ensure that these are kept in the curriculum, and find areas in which we can ensure that kids learn how to use these powerful tools which are all over the place in our society. If that isn’t the intention, I will surely push for it when I attend the summit in London.
It is frustrating to me that I have students in 11th grade who have never used a spreadsheet effectively. Think about how much effort and time is saved, and how much more complicated the problems can be that one can solve, if one just knows how a spreadsheet works.
I think part of the issue here as well, is that our mathematics education system has two goals, which in some sense compete with each other.
We have the goal to produce people who want to enter mathematical fields, like engineering, science, etc… We also have the goal to produce people who use mathematics as a thinking tool in their day to day lives.
Is it possible to achieve both of these goals successfully? Or is there some polarization between the goals. When I speak to adults, and let them know I am a mathematics teacher, I get one of two reactions typically: “you must be so smart” and “I was never very good at math, and I hated it in school.” We must recognize that if someone finishes school and says “I hated math” that it is less likely that they will use it in meaningful contexts later in life. See this comment for an example of someone who is highly frustrated with the way she is learning math: http://davidwees.com/content/fundamental-flaw-math-education#comment-6964
Hello David. From your reply and your earlier response to Mike Zwaagstra’s report a couple of weeks ago, I infer that you are simply unaware of the changes happening in Math content in many jurisdictions across North America.
Or, you ARE aware of these things but being disingenuous. I’ll assume the former.
If you are ignorant of what’s happening you must become aware. You are, after all, the moderator for the Assessment group at Edutopia, and often blog yourself about math instruction. You should not continue to speak from a position of ignorance.
You say the standard algorithms “absolutely” ought to be taught.
Assuming you’re serious about this, what would you think about a Math curriculum that not only DOESN’T teach any of the four most important standard algorithms of elementary arithmetic — but strictly militates AGAINST the teaching of them?
Misguided, you’ll say? I hope so.
Now what if I told you that not only is there such a curriculum, but that the proponents of this curriculum have managed to get it implemented across a region encompassing millions of children, before the general public was aware of what was happening?
Welcome to the WNCP curriculum in Mathematics.
The acronym stands for Western and Northern Canadian Protocol. It is a concerted political device for creating a common curriculum to be used in EVERY public school across every province of Canada to the west of Ontario.
I know of what I speak — I sat for several years on the committee charged with implementing this curriculum in the Province of Manitoba, under the auspices of the Ministry of Education.
One of the trainers who went out to instruct teachers in the use of this curriculum was reporting to us and told of a grade 5 teacher in a rural school who was confused: “I don’t get it — where do I introduce long division?” The answer: “You don’t!” “Oh, is it now moved to grade 6?” “No, it is not taught”. “Not taught until Junior High School?” “No, not taught at all.”
I rudely interrupted because I wanted clarification. “You mean we no longer teach long division — AT ALL?” The rest of the committee looked at me with bewilderment, like I — the only actual Mathematician in this large group — was desperately ignorant. “Don’t you get it, Rob? How long have you been on this committee? You should know by now — NO STANDARD ALGORITHMS!”
Then I seriously dove into the framework documents, the hundreds of pages of learning outcomes, looking for the four basic algorithms.
None. They’re all gone. Hypatia’s anecdote here is not hyperbole, not exaggeration, or misinformation, and her experience is not an anomaly — it is now the norm over a region of our country 2000 miles wide.
This year the first students having been taught their entire lives in this system have entered university (only a few schools are early adopters). Full implementation in all grades begins in the 2012-2013 school year.
Your examples of when students might “need to use” long division belies yet another ignorance: apparently you don’t know the integral nature of mathematical and scientific knowledge: These algorithms are pieces of a very large and complex puzzle.
And they are not minor pieces — they are critical for understanding the whole. When the precursor algorithms came to the West from the East via some great Persian mathematicians 1000 years ago, it changed our world and helped bring us out of the dark ages into the enlightenment and ultimately the great western civilization we have built. Without these pieces, there is a big hole in the entire picture.
These are not pieces of a puzzle only for math and science geeks, but for anyone who needs to function in a world of highly technical knowledge. Without great swaths of educated people who grasp concepts built upon these algorithms at a very deep level — a level only attainable by intimate familiarity — our society is seriously harmed in comparison to others which do not hamstring the educational process in this way.
I polled my honours calculus class this term, and learned that two of the students are from early-adopting schools. Neither one of them has EVER seen long division before my class.
One of the first things we did was examine geometric series. We saw did the following example: the real number x = 0.123123123… can be thought of as a geometric series (I won’t go into the details — I assume you’re familiar with them). If we apply that fact, a simple derivation gives us that x = 123/999, and it is evident from the example that the same technique shows that every repeating decimal is equal to a rational number.
An important fact.
But more important, as I pointed out to the class, is the converse fact: that every rational number has a repeating decimal expansion. They agreed that this is common knowledge.
So, I asked them — HOW DO YOU KNOW this fact? After a moment of silence, one student raised his hand. The answer was easy — he could prove it in two words: “long division”.
Every head in the class nodded. Except two. Two students were blinded to this elementary argument, which is often taught to students in elementary school, because … it’s elementary! But here were two honours calculus students struggling to see the “obvious” because they lacked the basic tool for understanding. Try “seeing” this without invoking something equivalent to the standard algorithm.
So…you may think… Students may have trouble with ONE question in a calculus class some time in their future. Big deal.
But if this is what you think, you miss the point. This is not about calculus class, or “missing a question”. It is about having a giant blind spot at a critical point in the development of the concept of a real number line. When you conceive of the number line, what does it represent? What concept do we have of this line? Where do the rational numbers fit? Where are all the other numbers? Without long division some of the ingredients are reduced to hocus pocus or pure dogma.
Remember what a mystery it was to the Pythagoreans that some numbers were not rational. These great ancient minds were hampered by not having anything equivalent to the modern decimal representation for numbers. The gaps between the rationals are understood because we understand the decimal representations of the rationals themselves…because of long division. Pythagoras, Euclid, perhaps even Archimedes would be awestruck to learn that 10-year-old children one day could grasp the basic key to understanding what to them was a deep mystery.
Later in that class we were doing (infinite) power series, and the importance of quotients of series. But how do we see these exist, in general? Long division. We can even do the first several terms, which gives us useful approximations for physics and engineering, with this technique. But its key value here is to provide a critical step in the development of this important tool in engineering an science. It’s role is to provide UNDERSTANDING, not (merely) a number at the end.
And it doesn’t stop there. When integrating rational functions — a very important type of problem in economics (for example) — the basic technique of partial fractions is indispensable. But the very FIRST step of partial fractions involves … LONG DIVISION of polynomials.
How can this be done by a person who doesn’t know long division of ordinary numbers?
The remainder theorem in elementary algebra, again critical for developing the Fundamental Theorem of Algebra. Needs long division. I could go on.
I’ve been counting, and in one of my advanced courses here we have invoked knowledge of long division explicitly as a critical step in understanding something six times since the term began. And in every course I have taught we have required long division (the knowledge of how it works — not necessarily ever actually dividing numbers like 3576 and 47).
How are these kids going to function in any technical subject beyond high school? Why are they missing one of the centerpiece developments of Western Civilization? They are illiterate on a point that has hundreds of unanticipated, but very real, consequences.
One anti-algorithm advocate told me “So, if your students need long division — just teach it to them. What’s the big deal?”
The big deal is that students need more than knowing how to STRUGGLE through these algorithms and blindly get an answer. They must be so familiar with the algorithms that they are like breathing; so that when I say such-and-such is a consequence of doing something “with this method” they see it right away because of long familiarity. Seven, ten or more years of familiarity.
You don’t tell someone the brake is on the left, the accelerator is on the right, and here’s how you pull on the steering wheel, then put them into busy traffic at highway speed. Think of your first experience driving. Similarly you cannot teach long division today and expect students to breeze through a problem that invokes the basic technique, but with infinite series.
Now I am quite capable of teaching grade 5 arithmetic. In fact, I just did — I gave a private tutorial because those calculus students would likely flounder in our program without long division. I can impart the knowledge. But I can’t give them the understanding that only comes with long familiarity. And it begs the question: why should a university professor have to remediate holes in a student’s basic knowledge that should have been filled 10 years ago in an elementary school?
Now, I HOPE You’re thanking your stars, by this point, that your children are growing up in the USA and won’t be taught under this crippled curriculum. I hate to be the one to break it to you, but many large jurisdictions in use in the U.S. use curricula with exactly the same flaw. To shorten this comment I’ll just point you to the two videos prominently featured on this educational site:
http://nychold.com/
Invest half an hour of your time in this, and get informed!
Now, as to your patronizing tones about how yes, yes, the “standard algorithms are fine and dandy but they shouldn’t be taught in isolation” — you are evidently not familiar with either Hypatia’s or Mr. Zwaagstra’s views on breadth of content. I happen to know them both personally and neither one of them advocates the teaching of standard algorithms in isolation and to the exclusion of all else. You cannot derive such a position from anything they have written, and speaking as if you think they do just makes you sound like another anti-algorithm zealot with a knee-jerk reaction to any mention of the pillars of arithmetic. But I sincerely hope that this impression I’ve gotten from you is wrong.
Additional note: I must have penned my long rant while you and Hypatia had the above exchange, Dave. I’m glad to see you recognize that she holds a quite reasonable position on the matter.
Here is a letter to the editor by a friend of Hypatia’s and mine (a highly respected senior math teacher here known both locally and nationally for his work):
http://www.winnipegfreepress.com/opinion/letters_to_the_editor/greed-is-good-in-math-130616248.html
that articulates a both/and approach.
This is the same approach advocated by Mike Z, Hypatia and me. At first it shocked me that Mr. B seemed to believe in the existence of these mythological teach-math-by-rote-with-no-understanding advocates. Then I remembered that, despite his great knowledge, he is steeped in the world of education, and this is one of the paradigms used to sell the no-algorithms approach: “THEY are against teaching understanding but WE are are all about understanding”
But the purist teach-math-skill-without-understanding beast is a relative of Bigfoot and Nessie: I have never met one. It’s only a bogeyman, a straw-man to convince you to crowd over to the other side of the balance. However, the purist teach-math-understanding-with-no-actual-algorithmic-skill element, unbalanced in the extreme on the other side, is very real. It took a whole army of them to engineer the adoption of the WNCP math curriculum across four provinces and three Canadian territories.
So, if you’re going to spend your life prowling the internet jumping on educational extremists who advocate harmful approaches, I’ll advise you to learn to identify them a little more reliably.
Laura writes:
“Teaching the expanded form of multiplication is simply a scaffold for understanding what is happening in a multi-digit multiplication problem. The standard multiplication algorithm is simply a shortcut. Don’t teach the shortcut until students understand what steps that short cut is eliminating, as well as the big mathematical ideas (such as the distributive property of multiplication) that support those steps”
While I agree with most of this statement it rankles to see these beautiful algorithms described as (mere) “shortcuts”. That is not a good way to put it. These algorithms are optimize efficiency, which is key to understanding why they are so important. The ancients multiplied too. Ever try multiplying with Roman Numerals? Then you understand what a revolution it was when Al Khwarizmi (from whose name we get the word “algorithm”) introduced the Hindu numbering system to the west through his book on arithmetic. The methods being taught in the “no-algorithms” approach as if they were the bread and butter of the curriculum are merely reversions to the 2000-year-old less efficient methods.
I agree that it is pedagogically useful to teach multiplication by laborious application of the distributive law. One of the principal reasons I think that should be taught is that anyone who’s tried it on a reasonably realistic problem immediately understands the value of the more efficient standard algorithm. Remember when you first learned calculus, Laura, doing derivatives from first principles? Then you were exposed to the rules of derivatives and suddenly it became easy, mechanical, and about 100 times more efficient. There’s a point to “forcing” students to use first principles — it is a great motivator to move on to the better, more efficient approaches.
Or when you learn Gaussian elimination — first by heaving around equations bulked up with so many extra symbols. Then you are taught to do it in matrix form and there’s a quantum leap in efficiency, which frees you to concentrate on the essential aspects of the task instead of all the extraneous structural details.
One junior high school teacher wrote me recently complaining about the sudden decrease in certain skills as soon as the new curriculum (sans algorithms) was implemented, and he had a good way of putting this: these students lack “economy of thought”. And that’s the point. These “shortcuts” (as you put it) are not mere tricks — they are essential tools for sharpening your critical thinking skills in preparation for more advanced conceptual development.
It is more efficient to use the standard algorithm when you want an exact answer, and you have paper and a writing tool on hand.
It is faster to use the distributive method when you want an approximation. While both are useful, I’d like to see more emphasis on the ability to approximate answers, since this is often more useful in a day to day context.
I agree about approximation being a useful skill, David. But not as a replacement for proper use of algorithms. Remember that is what we’re talking about here. What alarms me is the wholesale replacement of standard algorithms with (frankly) poorly-designed fluff of this sort written by people who appear to have less sense than the general public.
My son’s school in California in his grade 5 year was using a curriculum of this type. Now, for the previous 4 years they had been learning scattered estimation techniques but someone in their wisdom decided that HALF the grade 5 curriculum would be “how to estimate the results” of addition, subtraction, multiplication and division.
Talk about rote learning! They were given hard and fast rules to MEMORIZE with NO justification for every calculation under the sun. Then they were given word problems and sample toy calculations to learn the techniques. Many, many of the problems themselves were extremely poorly formed, but often it was the inadequacy of the recommended solutions that revealed the real problem — the methods provided, as instructed in the teachers’ manuals, were simply WRONG in the general case.
My favourite problem was on a take-home worksheet halfway through the year. It came home with a note from the teacher (who knew I was a math professor) saying “HELP!”
The problem was to estimate 15 x 15 using one-digit arithmetic. So, obviously, one should round one number up, and the other one down, to compensate… this is understood by 95% of the population, though very few have ever been taught such a thing in school. You just KNOW because it makes good common sense.
Except that this would violate the one hard-and-fast rule throughout the course: ALWAYS round numbers ending in 6,7,8 or 9 UP. With this rule, of course, you get 400 as your estimate of the actual answer, 225. You don’t need to be a rocket scientist to know that this kind of inaccuracy is what leads to space shuttles blowing up and lives being lost.
Using the common sense rule most people have without having wasted half a year of their elementary school math education on this nonsense, you get 200, which I would say is a reasonable approximation, and clearly the BEST one-digit estimate.
The teacher had doubts whether “the rule” had to be strictly adhered to, but found no support for any other approach, so she looked at the answer in the teacher’s manual: 400, with a fully worked solution showing the above misguided method.
Note even the approach I suggest is not a hard and fast rule: in other contexts one requires even more common sense. For example, rounding up/down for 5×5 gives an answer of 0 (which is clearly bad, but actually CLOSER than the text’s answer, which would be 100). But it is a “strategy” that reasonable people familiar enough with arithmetic can modify as needed.
I pick on this one example but there were dozens of clear violations of common sense throughout that text, which was in wide use in California. The poor teacher was in tears. She had little math background and thought she was stupid because half the time she felt she would have approached the problem differently than recommended by the text. We had many talks. Usually she was right, and the text was wrong.
I’m all for teaching (in some tiny corner of one’s education) strategies for estimation. Largely this will come as a byproduct of students getting very familiar with numbers and how operations work. Ask anyone who’s accustomed to estimating the cost of the groceries in their cart and they’ll tell you they violate the hard and fast rules all the time. Add the dollars, forget the cents, round up and down about equally often, unless you see a lot of “.97″‘s and “.98″s, in which case, round up almost all the time. The strategies adjust according to common sense, and there are no hard and fast rules. It’s not “mathematics” as much as intelligent decision making, informed by experience (through “drilling” and other strategies for familiarization) and insight (obtained through systematic development, grounded in standard algorithms).
The “new” curriculums in which estimation figures prominently is harmful because it subordinates concise math to fuzzy math instead of the other way around. You can’t build a solid house on a squishy foundation, but you can put up a squishy tent on solid ground.
And if they must feature estimation in textbooks, let it be written by someone with half a clue as to how real estimation works. Like a PhD in mathematics. Or perhaps a rancher, carpenter or grocer. But, I’m sorry to say, the “math education consultants” who get these contracts seem to have little grounding in reality.
Yeah, I can see how if you teach estimation without knowing how to find the exact values, it could lead to students memorizing a different set of algorithms without understanding how the algorithm works. I suspect that people who are competent in math are able to address this issue within the curriculum, and that people who are poor at math are just going to blindly follow a curriculum (and end up with students who learn math poorly as a result).
I think what you are speaking about (re)implementing is basically how I learned math when I went through school. I learned the standard algorithms, we spent some time estimating, and we spent (a very small amount) of time learning some of the non-standard algorithms. We spent basically no time learning how to communicate our results, and we spent a lot (a LOT) of time doing practice exercises. I did fine with this curriculum, but the vast majority of people who learned in this way I think were done a disservice because almost all of my friends tell me how much they hated learning math. That’s not to say that the current curriculum doesn’t do a disservice as well, I think we both agree that change needs to happen, but to what, is where we disagree a bit I think.
I wonder about your comment about how people will naturally learn estimation via practical experience. Are you familiar with Jean Lave’s anthropological work from the 1970s where she looked at adults and their use of mathematics? Some of her findings are absolutely unintuitive. For example, she found that people could solve one set of math problems in a supermarket with a very high level of accuracy (93%) but when faced with the exact same problems in a “school math” context, they scored much poorer (only 57%). She further found that the types of successful strategies people employed to do supermarket math were essentially unrelated to the highly efficient procedures they had learned in schools. I’m getting this information second-hand unfortunately, via Keith Devlin’s book “The Math Instinct.”
I wonder if part of the problem is that we are attempting to use a single system to address all of the needs of everyone in it? It’s clear that students come to school with very different needs in terms of literacy development, for example, and schools respond to this need using levelled readers, additional support for the learners, and the early literacy programs. What do we do to support these issues in mathematics? I like Maria Droujkova’s work with Natural Math (see http://www.naturalmath.com/blog/functions-3-4/) for example. I also think that the work that Mind Research has done on “math without words” where they attempt to address the issues that people who struggle with text have with learning math (see http://www.youtube.com/watch?v=7odhYT8yzUM).
Another issue is that I’m not convinced we have enough people who understand mathematics teaching it. I read a suggestion on Reddit that we should use computer based assessment for some undergraduate courses, and then use the freed up graduate students in mathematics to fill some of the need in schools where they could act as math coaches and tutors. I do think that more training of elementary school math teachers would be a good idea (one course – ugh!). I also find that often grade 8, 9, and 10 mathematics is taught by someone with a background in science instead of math (or no background at all) and that this too does a disservice to students.
I agree completely with your commenter “Jamie”, who had this to say:
“I have taught everything from high school math down to grade 3. I think that it is very important to teach both sides of this mathematical coin. The most frustrating times as a math teacher was when I was teaching 6th and 7th grade and could not teach the bigger concepts because the kids were stuck on the basic facts. They didn’t know them, became frustrated, and gave up before they got to the problem solving thinking. Now I teach grade 3 and feel frustrated by my math consultant and my new math program that both seem to disregard basic math facts and algorythms as something kids don’t need to master because they can use a calculator. If we were to focus more on these in the primary end, I think it would open the door for middle years and high school math teachers to delve deeper into the problem solving and bigger math concepts.”
http://davidwees.com/content/should-we-teach-standard-algorithms-arithmetic#comments
Give kids a solid foundation so that they don’t get hung up on simple number facts when they’re trying to solve more complicated problems.
The problem I see that if we just focus on arithmetic skills in the early grades in an effort to improve the skills for later, we run the risk of turning kids off of math completely early on, and there is a strong relationship between how much you like doing something, and how well you learn it.
So while we all agree here that a strong foundation is important for students, similarly, we should make sure that students do a lot of mathematics which is interesting and compelling for them in the younger grades. Grade 3 and 4 are the grades when many, many students start to loathe mathematics, and we almost never get those students to enjoy it again, even though many of them are perfectly capable of doing well in advanced mathematics courses later in life.
In my experience, students get turned off math when they have trouble with it. (When they get lost, are behind, have no idea where to start with a problem and don’t have the tools to get them through a problem.) That’s why foundation is important. Kids like things that they are good at. Give them lots of praise (and practice!) and help them to be good at math. Math is fun if it’s communicated in the right way. There are many fun things we can do, just with numbers, that build basic skills.
Again, you can provide children with a solid foundation and still make math exciting and interesting. Give kids passionate teachers who love the subject and communicate it well – this would go a long way to eliminating the dislike of math that sometimes (unfortunately) develops.
I work with 12 Grade 4 kids and they LOVE math so I’m surprised to hear you say that students start to loathe math in Grades 3 and 4.
I think that the experience that YOUR 4th grade students get in math is not necessarily the experience that most 4th grade students get. How many 4th grade students have a mathematician teaching them math? Not many. Your passion for the subject, and your ability to find connections is pretty key in how well your students will understand math. I suspect that a knowledgeable and passionate person can likely teach in almost any style somewhat successfully.
Like you said “give kids passionate teachers who love the subject and communicate it well” which is unfortunately not often the case. What can we do for the students whose teachers loathed math? How should we build our systems so that the typical elementary school teacher is able to inspire interest and passion about mathematics?
I forget where I read that most math loathing begins in 3rd and 4th grade unfortunately. I’ll have to see if I can dig up that research again.
“What can we do for the students whose teachers loathed math? How should we build our systems so that the typical elementary school teacher is able to inspire interest and passion about mathematics?”
I am quite aware that math teacher training in Canada (at least in MB, where I work) is not ideal. I have seen this firsthand and I think that a lot of things need to be changed in this regard.
What can we do for teachers who may be struggling with the math curriculum or may not feel the passion for math that we do?
I think the key would be to provide them with excellent instructional materials and teacher training that helps them to teach the material that they need to teach in the classroom. I would recommend JUMP Math, for instance, to any K-8 teacher. I love this program and this is what we use, for the most part – not always – with our Grade 4 kids (whom we see once every two weeks) and they really love the stuff in JUMP Math. What impresses me most about this program is the success stories for teachers and kids. There are many stories coming out of this program about teachers who didn’t like math or had math phobia before using the program and now find that this is their favourite subject to teach! The teachers’ guides are thorough and do a good job of explaining the why (not just the how). I think it would really help the kids in K-8 if teachers started using a program like this.
I read John Mighton’s book, and wasn’t very impressed with it (the book that is) but I found the chapter on JUMP math illuminating. Not so much because it changed my own practice, but because it occurred to me that the practices that he describes seem so obvious, that I realized that if he is sharing them in this way, they must NOT be obvious to many people. I’ll have to get a copy of the teacher resources, and share them with my colleagues here.
Which book did you read? The Myth of Ability or The End of Ignorance, or both?
The Myth of Ability. I was frustrated that I had to wait until the end of the book to find out how JUMP Math worked, and that the first four chapters seemed so repetitive. I’ve just signed up for the Jump Math website itself, and have started to review their materials.
I read them both. I did not like The Myth of Ability that much either (I can’t remember why now – I read both books over 2 years ago). I loved The End of Ignorance and regularly go back to it. You should read that one if you have time. It is excellent – this book is, in fact, what got me interested in math education. (Of course, I love teaching, and have always been interested in education but I hadn’t thought much about K-12 education before I read that book.)
Hypatia, you are absolutely correct: “I think that kids shouldn’t be using calculators and computers to do arithmetic problems until they know how to do those arithmetic problems themselves. They’re crippled otherwise.”
I see this all the time in my tutoring business.
Students who do not understand how the calculations are computed do not know what answers to expect. They are given calculators at such an early age, (“Just give me the decimal answer”, says the elementary teacher) that they think the fractions are something to fear. They can’t even figure out how to use fraction functions if they pick up a different brand of calculator. They seldom learn to reason on the correctness of their answers, to verify their answers, or to find their mistakes because they don’t even know when their answers make sense. If they are cashiering, don’t let the electricity go out or the register fail because most kids today can’t count change if you give them cash! If the numbers don’t make sense, then the variables seldom will either.
Standard algorithms absolutely should be taught and along with that, how the Babylonians, Egyptians, Greeks, Pythagoras, Euler and others did mathematics.