It has recently come to my attention that many the proponents of the multiple strategies approach to mathematics education are completely hypocritical. Allow me to explain.
Let's take an example to discuss exactly what is meant by the multiple strategies approach. Consider the question 24 x 3. A student approaching this question might use base-ten blocks to model the question as an area, might try to use a 'doubling-halving' approach to alter the question to a basic fact that they already know 24 x 3 = 12 x 6 = 72, or might try finding three groups of 20 and three groups of 4 using the distributive property 24 x 3 = 20 x 3 + 4 x 3 = 60 + 12 = 72. All of these strategies to tackle this problem are taken as equally valid solutions to the posed question.
The strategies above seem fine; however, there is one glaring problem. I haven't included the use of the standard algorithm as an acceptable way of solving the posed question above! What I see in practice is that proponents of multiple strategies dismiss the standard algorithms under the guise that students do not "understand" the mathematics if the algorithm is used. They instead favour visual and tactile models, and mental calculations - all of which are cognitively heavy on our students, and only work in contrived scenarios. Try to use the doubling-halving method on the multiplication 15 x 15. It won't work because one of the numbers must be even to use this "trick"! I have written about this particular "trick" before, stating that it would be more beneficial for students to learn prime factorization (at least prime factorization comes up often in practice and works in more situations than doubling-halving does). Even better than prime factorization is to teach the standard algorithm! When performed correctly, it is much more reliable than mental calculations, and much more efficient than visual or tactile models. The amount of time spent on the inefficient multiple strategies listed above could be more efficiently used to teach the understanding of the mathematics behind the algorithm. Problem solved!
Despite what I have mentioned above, proponents of the multiple strategies approach continue to word assessment questions in such a way so that students are forced not to utilize the standard algorithms. Or if the student does utilize the standard algorithm where appropriate, they are severely punished. So much for being able to use "multiple" strategies to solve a question!
To give some illustrative examples, I have taken some screen shots of an assessment for Grade 7 teachers.
The first figures show the probing question "Write everything you know about fractions." (Aside: What an AWFUL question. How can you determine what is a good answer and what is not a good answer? There are no guidelines for the student to follow!)
The first student shows that he/she knows some basic fractions and that he/she knows what these fractions look like as a part-whole diagram. Yet this student is "not meeting" the intended grade level objective. How can the student know that by looking at the question? The student answered the questions just perfectly! He/she put down knowledge of fractions.
The second student, who can model fractions using diagrams and algebra, and who even knows the ideas of mixed/improper fractions doesn't even meet the standard! The student must connect fractions to decimals to get a "meeting" grade level objective. But was that clear in the posed question?
The third student here demonstrates that he/she can clearly represent fractions in about 2000 ways, so he/she must be meeting the standard. This is exactly what the proponents of multiple strategies want: a student doesn't "understand" fractions unless he/she can model them in many different ways. However, how can our students be expected to recall ALL of this information with such a poorly designed open question? Would it not be better to have them recall specific ideas, rather than overload them cognitively with such a daunting probing question?
Continuing on, let's see another goodie. Here students are asked to order the following fractions, decimals and percents:
But students might use the standard algorithm to convert fractions to decimals. So, to make sure students don't use it, we will insist that students have to use multiple strategies to get to the answer! Again, a multiple strategies approach was not asked for in the question, so now they are simply punishing students for utilizing a perfectly good strategy.
And, finally, here is my favourite clip from that document. In this question, students are asked to calculate 36% of 25.
This student calculated the correct answer using the standard algorithm for multiplication. Very nicely done! But I am sorry to say that the algorithm is not allowed, so you are "not meeting" expectations. What we actually wanted from you is a contrived mental math strategy given below:
So, I only hope that this gives you some insight into why I am very wary of proponents of multiple strategies. These individuals are secretly removing the standard algorithms from our curricula under the guise that their "methods" (pictures, mental strategies, etc.) promote "understanding." When, in actuality, their methods are archaic, inefficient and non-transferable to many situations.
I leave you with two links. One is to the Heinemann blog on the Standards for Mathematical Practice (SMP) for the US Common Core. In particular you should read standard #5, in which John SanGiovanni clearly states that "In many situations, pencil and paper are inefficient and using them is not strategic." He is promoting use of multiple tools to solve problems, yet you can infer that he would not be happy about students using pencil and paper to solve the question using the algorithm for subtraction (he doesn't even mention the standard algorithm as a viable strategy).
The other link I wish to share is a piece written by Michael Zwaagstra inThe Chronicle Herald, a Nova Scotia newspaper. In this piece, Zwaagstra comments on how Marian Small, a prominent figure for the multiple strategies approach, hypocritically states that different strategies to solve questions are good, but to leave out the standard algorithms from the curriculum.
When working with students, I want them to be flexible in their thinking. Sometimes the standard algorithm, isn't necessary, where at times it could be. Kids should be able to look first and then decide what makes sense to use. Some kids might be able to multiply 15x15 mentally, while others may need to use the standard algorithm. I find students who only use the standard algorithm have a limited understanding of the beauty of number and being able to compose and decompose number.
ReplyDeleteHey Daniel,
Deletewhat I am finding in the schools in Manitoba is that teachers are not devoting enough time scaffolding towards the standard algorithm (multiplication for example). If teachers were allowed enough time to properly scaffold towards the algorithm, all of the "mental strategies" would be hit along the way: distributive property, partial products, even perhaps an area model. Students would see along the way in which situations we can simply utilize distribution to solve multiplications mentally.
In our schools too much time is being spent on methods like "halving-doubling" (this is a major strategy in Gr 4 and Gr 5 here) when the situations are contrived, and the time would be better spent learning prime factorization, or scaffolding towards the algorithm. If taught properly, students will naturally be shown how to decompose/recompose numbers as they make their way toward the standard algorithm.
15*15 is 225...I did 10 15s (150) and know 5 of those is half of that (75). Put it together... 225. Didn't need to use a standard algorithm for that. I can look at the numbers and realise that I can use my number sense (ability to decompose and recompose numbers, understanding of operations) to get the answer.
ReplyDeleteJust out of interest, can you do 200-198 for me using the standard algorithm. Now explain to me how you would really do 200-198. Now explain which of these is 'cognitively heavy'.
What about if you go into Tim Hortons and your bill is $7.85 and you pay with a $20 bill and the till crashes and the cashier needs to figure out your change (it happens, believe me). Would you use the standard algorithm to work this out? Or would you use the alternate strategy of 'adding up' i.e. add on 15 cents from the $7.85 to make $8 and from there add another $12 to make $20?
Or suppose you are in a shop and buy something for $59.99. You know HST is 13% so on the way to the cashier you are trying to figure out the total. Would you get out pencil and paper and use an algorithm to do 59.99*1.13 (or variation thereof)? Or would you use your number sense to use an alternate strategy? I would call it $60, recognise that 13% means an extra $1.30 for every $10 so $7.80 for $60 giving a total of $67.80. I'm not fussed about the 1 cent that I rounded up. This is not a contrived situation.
Teaching alternate methods is not the same thing as not teaching the standard algorithm. It is getting students to realise that good mathematicians look at the numbers first before they decide which is the most efficient way to get a solution. I hope that you would agree that the standard algorithm is not always the most efficient way of getting the answer to a computation. Teaching only the standard algorithm will not guarantee that students develop good number sense. I have had too many students who can do long division but cannot look at a basic word problem and know that it is a division problem. Knowing how to do long division does not mean to say that a student knows what division means; about how it is connected to multiplication (or subtraction even). Unfortunately, too many students are taught standard algorithms too soon. Have a look at what they do in Singapore. Here they understand the importance of the Concrete-Pictorial-Abstract approach to math (based on US Educational Psychologist Jerome Bruner's research from 1960!) They know that jumping to the abstract (standard algorithms) is problematic which is why they spend a lot of time in earlier years getting to know numbers concretely (through the use of ten frames, rekenreks, Cuisenaire rods etc.) before learning such techniques as bar models and arrays to represent operations diagramatically before they are finally ready to effectively use the abstract algorithms. They spend a lot of time getting to understand fractions (and decimals) concretely before representing them pictorially (bar models again) before they are ready to learn and effectively use strategies such as simplifying fractions or finding common denominators. Singapore, as you will be aware, does not too shabby on the PISA and TIMSS tests.
So I hope you see the point: good mathematicians do indeed use multiple strategies, some of which are standard algorithms but they know which strategies to use in certain situations. A good, balanced math program shouldn't leave out standard algorithms but should recognise that developing students' number sense is a lot more complex than simply getting them to memorise sequences of rules.
Hi Mike. You're explaining so-called "strategies" for money handling that your grandparents used continually and effortlessly. And they are better than any of the kids today at it.
ReplyDeleteAnd guess what your grandparents learned in school?
The standard algorithm.
So who taught them these "strategies"?
Nobody. They may have been discussed ... on the side after regular work was done. And picked up casually in the course of working in a store, etc. But they are essentially fluff skills, thinks of little educational value.
You see, a student who learns conventionally in a bottom-up fashion with properly scaffolded will grasp the standard algorithms and also pick up valuable short-cut strategies through multiple and varied experiences in the use of arithmetic (soccer coaches often talk about the importance of ball-touches. Practice tends to be ineffective unless the coach works to ensure students make many, and varied, touches of the ball each practice. It enables automaticity so that over time students stop needing to think about what their foot is doing with the ball. Same in math education. This business of doing one arithmetic calculation slowly, labouriously, in multiple ways, and then writing essays about them does exactly the opposite. It is not consolidation that leads to mastery to allow focus on the big picture -- it is unnecessary obsession with detail. I see the texts, and the assessments that not only encourage students to obsess over details -- they force students to regurgitate a record of that obsession, and mark them down if they fail to do so. That is anti-mastery, and it does not serve students well when they move into higher subject matter!)
Students who learn conventional math ... pick up a variety of useful "strategies" for free, as a bonus.
Students who learn only strategies ... never gain a unified procedure for solving arithmetical problems, and certainly don't simply "pick up" the standard algorithms. They become slaves to special cases and their pocket calculators.
There are some things that are learned as a consequence of having a strong foundation. Shortcuts for mental calculation are among these things. A solid foundation consists not only of familiarity with the rules and skills of arithmetic but with sufficient practice in them that they become automatized. With automatization the working memory is freed up for the essential work of understanding -- instead of obsessing over mechanical skills one can see the big picture.
You describe kids who can do long division but cannot do read word problems and understand that they involve division. I'm sure there were many in the past. There are also many today who can recite, and even use (in cases where they apply) multiple "strategies" but have the same weakness with word problems. This is nothing new, and multiple strategies has done nothing to fix it. There has also always been poor teaching, and there still is today. Well-scaffolded texts 40, 60, even 100 years ago scaffolded learning well, through a spectrum of mechanical skills, through interpretive and problem-solving skills, providing students a range of exposure and helps them make precisely those connections needed to pick out things like that a word problem can be solved by division. A good place to explore some of the great aspects of classical texts is Barry Garelick's series on teaching Common Core math using conventional methods, over at the Heartland institute.
http://news.heartland.org/newspaper-article/2014/11/12/whos-say-teachers-cant-modify-common-core-no-one
Here's the whole series
http://news.heartland.org/barry-garelick
There is this foolish caricature of conventional and classical instruction as having been "pure" rote, meaning "without understanding" (a misrendering of the word "rote", but it is common for it to be used this way nowadays). It's simply false, a historical prejudice. I have a text that was in use in Manitoba in 1948. In the preface the author declares that in the past math was always taught by rote, but now in "today's world (i.e., 1948)" students must be taught with understanding, and this is what his text does.
ReplyDeleteThen I found another book written in the 60s, New Math era .. and guess what? The preface said *exactly the same thing!*. I showed this to a friend, and he showed me commonly used textbooks from the 1920s that said exactly this. Then I learned that there were texts used in the 1800s that purported to teach arithmetic for the first time with ... you guessed it, "understanding".
My point? Every generation thinks it has invented understanding of arithmetic in classroom learning. And every generation is wrong about this. There has always been excellent, well-structured education that taught understanding and skill together (and on the flip side, there was always bad instruction to hold up as a negative example. So what?).
The difference today? Larger numbers than in recent memory are coming through their elementary school education with no skill. Zero. One (university) student whose work I saw recently, asked to perform a two-digit subtraction -- I forget the exact numbers but lets say 87 - 56 -- in a test of proficiency ... marked 87 tick marks in the margin and crossed 56 of them out, counting the remainder. And, incidentally, got the answer wrong through a minor error. But the point is ... basically no skill. And clearly inadequate mastery for the demands of post-secondary math, science or engineering. That is not an isolated example, but it is a good illustration of what goes wrong.
You're right that teaching strategies does not imply *not* teaching standard algorithms. That is why I do not oppose the teaching of strategies. However, it is essential that all in the educational enterprise understand that standard algorithms are the *main course* of arithmetic. The strategies are an optional albeit tasty *side dish*.
ReplyDeleteWe (at WISE Math) made this point with the ministry back in 2012, and they returned mention of the standard algorithms back into the Manitoba curriculum -- the first province in the WNCP agreement to do so. But the algorithms still are not given their proper place. At least now when teachers tell us that their district consultant has told them they are not to be taught, or some workshop leader told them "research shows" that teaching the algorithms is harmful, we can point them to the curriculum and remind them that they are curricular content, so it is perfectly okay to teach them. That is a small victory, but we are thankful. It is something.
But I have not yet even touched on the primary reason that standard algorithms must be taught. It is simply this: Let's take addition of numbers. Addition is not three things. It is not 10 things. It is not a grab bag of random things.
Addition of numbers is ONE thing. It is one large over-arching concept. A big idea. A single, albeit multifaceted structure that is systematically arranged so that addition of any two numbers fits into that structure, whether it is 2+3 or 71.359 + 211.0032. It is all one thing, and students should look at addition in this way.
But students who are taught that addition is a grab bag ... in this case method A applies. Then in that case method B applies, and in this other case methods C, Q and W must be combined. Then in this other ... hold on ... you'd better pull out your pocket black-box and just get it to tell you the answer ... What are they learning about addition?
They are not learning that addition is a single thing. They are not learning to conceive of it through what psychologists called "blocking" -- in which you look at something and you grasp it at once, with coherence and unity. They are learning that it is ad hoc. They are learning that the pathway from question to answer is full of clutter and unpredictable twists and turns. It is a jungle, not a well-layed out streetwork, with known main avenues, and a few neato side streets that you can use as shortcuts ... it is a tangled mass of side streets, that you know how to use to get to some destinations, but others ... well, that's just too complicated.
There is, by the way, nothing "21st Century" about cluttered thinking. That is just poor learning. We have always had students who lacked a coherent conceptual framework for arithmetic. but the unique aspect of making "strategies" the principal focus of arithmetic skills development: it appears to have been designed precisely to optimize that mental clutter.
Singapore, which you mention, does quite well on PISA and TIMSS. They have a great way of scaffolding and many innovative features in their curriculum. They do not teach cluttered thinking, but large, coherent ideas and procedures. But one thing they do exceedingly well ... and share with all high-performing jurisdictions on these tests: they support structured learning of standard curricular outcomes such as memorization of math facts and the learning of standard algorithms.
Presumably you support these things. You appear to think highly of Singapore's math education. Very well then, I have a proposition for you. Would you like to join me in writing a petition to have the province of Manitoba adopt the Singapore curriculum as its framework of outcomes, and to make Singapore Math (which is available in a North American addition, and used in many U.S. schools) a Manitoba Recommended Resource? I would be happy to do so with you. How about it?
Thanks for sharing your thoughts on this. I agree with much of what you say and disagree on other bits.
ReplyDeleteFirstly, it's not just my grandparents (and their generation) who use the adding up method for change (or whatever). I am a Math Consultant (former High School math teacher) and I work with teachers in a variety of schools in our Board and I regularly see students use this method when appropriate, and not just in money situations (I've even seen it used in elapsed time situations). It is not because they have somehow transferred their skill of the standard algorithm: it is because they use their Number Sense to see a better way of figuring out the difference. I'd be very hesitant to refer to this as fluff.
I'm interested in your claim that "Larger numbers than in recent memory are coming through their elementary school education with no skill. Zero." Is this your observation or is there any solid data to back this up? I mention this because at a recent conference, some research was shared from PISA that suggest the opposite: kids today are better at Math than their parents. I'll be honest, I am skeptical about claims either way: I would love to see a longitudinal study that makes accurate comparisons possible. I wouldn't be surprised if such a study shows something that you alluded to: in the past (as in the present) there has been some exemplary teaching and some not-so-exemplary teaching. Wrt the university student who couldn't do 87-56, yes that is troubling. But do you know anything about this student? Maybe she had a Math LD such as dyscalculia which makes it incredibly difficult for her to do computations (either mentally or algorithmically). Anyway, as a counter to that one example, may I offer the example of the many Grade 3 students (even Grade 2) who I have worked with who could do this in their heads? Here in sunny Ontario, it is actually an expectation that they do addition and subtraction of 2-digit numbers mentally.
I do agree with you that just knowing some random strategies is not the same as knowing the big, multi-faceted idea of addition. However, I am not convinced that on its own, the standard algorithms help develop that big idea. As I stated earlier, how does knowing how to do long division help you understand when to do division?
I also agree with your point on the importance of 'blocking' or 'chunking' and how this can lead to mastery. But for me, part of blocking involves looking at 2001-1999 and seeing the 'chunk' as "Oh, those numbers are just 2 apart, so the answer is 2." To me, using the standard algorithm in this case is the antithesis of 'chunking' and is inefficient.
Which leads me to my final point (I hope I've not rambled on too long ;-)) How do you define arithmetic? I only ask as different people say different things. Do you understand arithmetic to be 'Computations'? Or bigger than that? I think of Number Sense as a Big, Big Idea that includes (amongst other things) Counting, Quantity, Operations, and Computations. For example, Operational understanding isn't just knowing your facts (which is incredibly important) but knowing when to use them; knowing how division relates to multiplication. Computations includes standard algorithms (but not exclusively) so as such I don't agree that they are the pinnacle of arithmetic: this would imply that there is nothing beyond!
Wrt SIngapore, I do think they have an impressive system. It strikes me that they have worked hard to get everyone to understand the importance of balance in the curriculum between problem solving and practice. However, as I don't live in Friendly Manitoba, I feel it would be wrong for me to sign your petition.
atb
Mike