common core and positional notation.

[mathwonk]

A mother told my wife, who is a math tutor, that her 2nd grade daughter is struggling to add 2 and 3 digit numbers like 23+48 = ?  without arranging them in columns over and under each other in the usual way, but that she is expected to do so at school because "it's common core".  I never heard of this and I taught under the supervision of a well known expert in elementary math ed who wrote some of the common core.  Has anyone else encountered this from schools?

[forty-two]

No idea about common core, but Singapore Math does all of grade one that way (add/sub up to 100), using mental math techniques, and even after teaching the standard algorithm has occasional chapters using mental math techniques.

[SparklyUnicorn]

..

Edited January 8, 2016 by SparklyUnicorn

[athomeontheprairie]

We use rightstart, and I'll agree with the above. I would even say there is a deliberate focus in the book to write the problems this way to help "force" the mental math.

 

I do not think this is a cc issue

[eternallytired]

Yep, in RightStart the kids would be taught to add up the tens, then add up the ones and find the total.  20+40=60, 8+3=11, and 60+11=71...so 28+43=71.  It's taught that way because you're more likely to need to add numbers mentally like that in real life, when you can't sit down and write up a nice column.

 

On the other hand, I've noticed my sister's girls coming home from school without having things adequately explained--particularly in math, since curricula often mandate that the child be taught several ways of solving each problem--or simply with misunderstandings of the explanation.  If no one has explained it in a way the child understood (and given a good reason for it, for pity's sake--since there IS a good reason!), no wonder the poor kid is confused.

[mathwonk]

Gosh, I cannot think of any reason to prevent kids from accessing the benefits of traditional positional notation, developed over centuries.  I mean in second grade, you learn it the easy way, then later maybe you branch out.  But what do i know?  I just live here.

[PinkyandtheBrains.]

 

 

Mental math techniques are not new, and they are helpful as one progresses to more challenging math.

[Laura Corin]

I didn't learn (or perhaps didn't absorb) many mental maths techniques from school.  I learned them through Singapore Maths when teaching Calvin, and I use them very frequently in real life.

[SparklyUnicorn]

I didn't learn (or perhaps didn't absorb) many mental maths techniques from school.  I learned them through Singapore Maths when teaching Calvin, and I use them very frequently in real life.

 

I didn't learn them as a student, but did with my kids and I like them.

[Sherry in OH]

Yep, in RightStart the kids would be taught to add up the tens, then add up the ones and find the total.  20+40=60, 8+3=11, and 60+11=71...so 28+43=71.  It's taught that way because you're more likely to need to add numbers mentally like that in real life, when you can't sit down and write up a nice column.

 

 

 

MEP teaches this way first also.  Next the child learns to use a place value table, then progresses to column addition.    That way by the time the child does column addition, he knows what those columns represent.

[Storygirl]

The school my children attend uses Everyday Math. Although the problems are presented horizontally on the homework pages, our teachers allow the students to rewrite the problems vertically before solving, if they desire. This is our first year at the school, so Common Core math is new to us, but our teachers' philosophy seems to be "teach multiple methods, then let the student choose what works for them."

 

OP, I would say that the math tutor should find out what this particular teacher's requirements are, so that she can practice those particular techniques with the child.

 

 

[FO4UR]

I've not encountered that in schools as we've always homeschooled, but I agree that adding numbers under 100 mentally is a skill that a 2nd grader should be growing.

 

The problem is probably that K was not spent mastering number bonds and finding tens, and 1st was not spent mentally adding/subt. numbers through 20.  Part of the reason that CC math is not working in the USA is b/c it's in an atmosphere of teaching to the test and passing the buck (the students!!!) to the next teacher (the next year).  Kids can be passed on up to the next grade without real understanding as long as they perform party tricks on the test.  At home, we homeschoolers make darn-tootin' sure our students understand one thing before the next comes b/c we don't want to be teaching 2nd grade math in the 7th grade.  kwim.

 

 

For the 2nd grader in question, play a lot of Go to the Dump.  (It's Go Fish with numbers that make tens.) Build up the problems like  7+15=   17+15=   27+5=  47+25=....and so on...that has helped mine click with the big numbers. When she brings home math homework, do this with those problems. 

 

And, until my kids were confident whipping out the answers mentally I built the problem in cuisenaire rods in front of them.  We've never struggled with learning the algorithms. All 3 of my school-age kids have either learned them quickly (less than a week), or intuited the algorithm before I had a good chance to teach it.  Exception: Long Division. 

 

 

 

 

[SevenDaisies]

Gosh, I cannot think of any reason to prevent kids from accessing the benefits of traditional positional notation, developed over centuries.  I mean in second grade, you learn it the easy way, then later maybe you branch out.  But what do i know?  I just live here.

 

In Singapore Primary Math kids are not prevented from using positional notation, but are encouraged to also learn mental math techniques. What I like is that they teach several different strategies and leave it to the kid to choose which one works for them. One that is foreign to my daughter works for my son and vice versa. If a method doesn't make sense to the child and the child has a working method, we don't dwell on it.

 

I have no experience dealing with this in public school, but when I hear public school parents complain about methods I recognize from Singapore Math, it is usually because students are being forced to use a particular method. In my (non-expert) opinion common core math is a poorly executed attempt at Asian math.

[MinivanMom]

In Singapore Primary Math kids are not prevented from using positional notation, but are encouraged to also learn mental math techniques. What I like is that they teach several different strategies and leave it to the kid to choose which one works for them. One that is foreign to my daughter works for my son and vice versa. If a method doesn't make sense to the child and the child has a working method, we don't dwell on it.

 

I have no experience dealing with this in public school, but when I hear public school parents complain about methods I recognize from Singapore Math, it is usually because students are being forced to use a particular method. In my (non-expert) opinion common core math is a poorly executed attempt at Asian math.

 

Yes, these are the complaints I hear. At our local elementary, kids will get correct answers marked wrong if they use the "wrong" method. Yet they get partial credit for incorrect answers if they attempted to solve it with the "right" method. It's confusing for children and frustrating for parents.

 

I agree that much of what I'm seeing with the actually teaching of common core math is a (somewhat garbled) attempt at Asian math. But our teachers don't appear to understand the approach themselves and are teaching it very poorly. We continue to have lots of problems with implementation here.

[mathwonk]

Thanks for all the feedback.  It sounds as if most of your kids have had lots more preparation for this mental math than the child in question, who is still counting on her fingers.  I guess I myself, even after a roughly 50 year career of research and teaching pure math at all levels from second grade through graduate school, and having learned the traditional way myself, tend to go with the "whatever works for the particular child" approach.  I.e. no matter what theory I brought to the classroom in all those years, there was always some child for whom it did not work.  I guess the only general principle I have is "start where they are".  I just assumed that "common core" referred to the body of material that should be known, not the method of teaching it.  Thanks again!

Edited January 8, 2016 by mathwonk

[purpleowl]

 I just assumed that "common core" referred to the body of material that should be known, not the method of teaching it. 

 

It does, but people (including many teachers) regularly misuse the term.

 

Math Mammoth teaches both methods (adding horizontally with mental math techniques and adding vertically in columns), and my DD actually really prefers the horizontal method.  She can do either way, but she finds it takes longer and feels more tedious to use columns.  She does find columns useful for adding large numbers or many numbers together.

[mathwonk]

This example will perhaps seem goofy, but it is drawn from an actual experience in my life of training in math.  My university teachers insisted on the conceptual approach to every topic, and at one point I just lost contact with reality.  Here is the example, still burned into my brain after more than 50 years, of a subject I could not deal with until I finally saw the computational approach years later.  It is called "exterior algebra".  The first approach is the one I was taught.  Afterwards I could not do the exercise.  In the second approach I easily did it correctly.  See which one you prefer.  i encourage tackling approach II first.  i believe everyone here can do it with a little effort.  approach I , i doubt any of us can do.  if you can, contact me for referral to a grad math program with free ride support.

 

I.  Abstract approach:
Let V be a finite dimensional vector space spanned by the linear forms x1,...,xn, and consider the symmetric algebra S(V) they generate.  In that algebra consider the ideal J of finite sums of all products involving a repeated factor.  Define an equivalence relation on the algebra such that two elements are equivalent if and only if their difference belongs to that ideal and denote the set of equivalence classes by S/J = E(V), called the “exterior algebra†on the space V.  Then this algebra is naturally graded from degree 0 to n, according to how many linear factors occur in a representative for a given equivalence class of products.  

Ex:  compute the dimension of the graded piece of degree n.

II. Concrete approach ("wedge" multiplication):
Or, just consider all sums of form adx + bdy + cdz, for all numbers a,b,c, and multiply them with the usual rules except require that dx^dy = - dy^dx, dx^dz = -dz^dx, dy^dz = -dz^dy,  and dx^dx = dy^dy = dz^dz = 0.  Thus also dx^dx^dz  = 0, etc... 

E.g.  (2dx + 5dz)^(4dy - 6dz) = 8dx^dy -12dx^dz +20dz^dy -30dz^dz = 8dx^dy -12dx^dz -20dy^dz.

Ex:  show that any triple product, e.g. (3dx-2dy+4dz)^(dz-2dy+7dx)^(3dy+6dx-2dz), can be rewritten as a numerical multiple of dx^dy^dz.

Edited January 9, 2016 by mathwonk

[debi21]

There is a connection to the Common Core which I haven't seen referenced, so I'll mention it. I can't remember if this was in another thread or an article I read. There is a progression in the wording of the addition math standards as you go from grade to grade (either grades 1, 2, 3 or 2, 3, 4) and roughly paraphrased in the first year it says add using strategies, in the second it says add using strategies and algorithms, and it isn't until the the third year that it says, add using strategies and algorithms, including the traditional algorithm (i.e. positional notation). So the Common Core is implicitly suggesting that students be taught these mental math and other strategies earlier and positional notation later.

 

ETA: from http://www.corestandards.org/Math/

 

Grade 1:

CCSS.Math.Content.1.NBT.C.4
Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

 

Grade 2:

CCSS.Math.Content.2.NBT.B.5
Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

CCSS.Math.Content.2.NBT.B.6
Add up to four two-digit numbers using strategies based on place value and properties of operations.

CCSS.Math.Content.2.NBT.B.7
Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method.

 

Grade 3:

CCSS.Math.Content.3.NBT.A.2
Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

 

Grade 4:

CCSS.Math.Content.4.NBT.B.4
Fluently add and subtract multi-digit whole numbers using the standard algorithm.

 

 

 

Edited January 8, 2016 by debi21

[mathwonk]

Thank you!  I went to that link and found explicit descriptions of methods to be used to teach addition and subtraction at each grade level.  No wonder there is controversy over this stuff.  I had no idea.  It is very hard for adults who learned one way, and who have used that knowledge profitably for decades, in my case even at the international mathematics research level, to accept someone's theory that we should teach it another way.  Of course I could always be quite wrong, but I do have a lifetime of experience that to me recommends allowing a good deal more flexibility in approaches.

 

I did find there the statement that conceptual understanding and computational facility are both important, which I agree with.  I would suggest however after a very long career of practice, with thousands of students of almost every level of ability, that for many students, indeed most, computational facility should usually come first.  I would go out on a limb and suggest that in real life, even research level math, being able to compute accurately, even without complete understanding, is more useful than understanding a principle that one cannot carry out computationally.  I wonder how many practicing engineers, who use calculus and differential equations every day, remember the theory of differentiation and existence and uniqueness of solutions of d.e's?  I guess I could ask my brother the EE, but I already know he has always been very dismissive of overly theoretical math.

 

 

I confess too that having been a lazy student, who contented myself with abstract formulations of many mathematical ideas, without taking the time and effort to compute examples, I was always hampered in my research and highly dependent on other stronger mathematicians for help.

 

Anyway I think I see the difficulty, namely telling people not only what result is desired but exactly how to go about it, without flexible accomodation of different learning styles, is asking for problems.  Another challenge is that it takes more teacher training to produce teachers who can teach in a variety of ways with different children, and who are not dependent on the specific method they are given.  Good luck to us all!

Edited January 8, 2016 by mathwonk

[mathwonk]

I won't go on forever about this, but I want to give an example where computation is universally agreed to be more important than understanding why.  In my field of algebraic geometry the most famous and important theorem is the "Riemann-Roch" theorem.  Every algebraic geometry student and researcher knows the statement of this theorem, but few know the proof.  Here is a quote from the standard graduate text on the subject by Robin Hartshorne, then a Professor at Berkeley:

 

"If a reader is willing to accept the statement of the Riemann-Roch theorem, he can read this chapter at a much earlier stage of his study of algebraic geometry.  That may not be a bad idea, pedagogically, because in that way he will see some applications of the general theory, and in particular will gain some respect for the significance of the ...theorem,  In contrast, the proof... is not very enlightening."

 

As a matter of fact, I audited the course from this professor at Harvard in the 1960's that preceded his writing the book, and he taught it in exactly that way, stating and using the theorem, but never proving it at all.  As another famous example, Archimedes used ideas of integral calculus to make correct and useful calculations that were not theoretically justifed clearly for centuries, as did Newton.

 

Forgive me, the trouble with philosophy of teaching is it is undecidable, and the discussions can become lengthy and sometimes argumentative.  I certainly do not insist on any of my own opinions, and I greatly appreciate all the help I have received here,  thank you!

Edited January 8, 2016 by mathwonk

[mathwonk]

Uh oh.  I downloaded the common core standards and read them.  What a depressing read.  It seems like a lot of blah blah and it is not really clear to me what it exactly means.  I would have preferred a nice clearly written book actually explaining the math in plain language, over a tract talking abstractly about desirable math skills.

 

[edited] When I did find an explicit discussion of a topic, like the quadratic formula, it said to be able to derive it in the standard way from completing the square.  This is one of my pet peeves since when I was a high school student that particular derivation was hard for me to follow and completely unmotivated.  Decades later as a teacher I read algebra books written by some of the greats, like Diophantus, and Lagrange, they don't do it that way at all, but make the idea much clearer as I hope to explain next:  the basic idea is you are given the sum and product of two numbers and you want the numbers.  From the identity:

(a+b)^2 - 4ab = (a-b)^2, you see that the sum and product of a and b, determines also the difference a-b.  Now if you know a+b and also a-b, you can find both a and b.  That's all there is to it.  Try and dig that insight out of the common core, or any modern algebra book.  In more detail:

 

QUADRATIC FORMULA:
To solve a quadratic equation X^2 -bX + c = 0, first assume the solutions are X=r and X=s, and note that then (by the root-factor theorem), the equation factors as X^2 -bX + c = (X-r)(X-s) = X^2 -(r+s)X + rs, so that we must have b = r+s and c = rs.  Thus we know in particular the sum b of the roots.  Hence if we only knew the difference, say d = r-s, of the roots, we would be done, since then we would have 2r = b+d and 2s = b-d, so we could get r and s by dividing by 2.

So we assume the roots are expressed as a sum, namely r = b/2 + d/2, and s = b/2 - d/2.  Plugging in X = (b+d)/2 and eliminating denominators we are trying to solve (b+d)^2 - 2b(b+d) + 4c, for d, since we already know b and c.

Expanding gives b^2 + 2bd + d^2 -2b^2 -2bd + 4c = d^2 - b^2 + 4c = 0, or
d^2 = b^2 - 4c.  Thus 2r = b + sqrt(b^2-4c) and 2s = b - sqrt(b^2-4c), the usual formula, if you notice we had a minus sign in the linear term of our original equation.
 

 

Notice how the reason for the goofy square root expression (b^2-4c), becomes completely clear here as opposed to the completing the square method.  Note also that the basic idea here is already given in the common core standards, in a more complicated form, as a stand alone trick for producing Pythagorean triples, i.e. it is based in the identity  (a+b)^2 - 4ab = (a-b)^2, (which they give with a = x^2 and b = y^2).  So the common core contains the germ of this method but does not make the connection between it and the quadratic formula, as the great ancient mathematicians did.

 

[edit:  I just looked back in the book by Lagrange on Elementary Mathematics, and found his account of the derivation by Diophantus, much simpler than mine above.  Namely he says indeed that in the equation X^2 - bX + c = 0, that b is the sum of the two roots, so we only need to find their difference d, since then the roots themselves would be (b+d)/2 and (b-d)/2.  However then he observes that since their product is c, we have [(b+d)/2][(b-d)/2] = c, i.e. [b^2-d^2]/4 = c, or d^2 = b^2-4c.  hence d = sqrt(b^2-4c).  Rather shorter and simpler then my version, and makes more clear the use of both coefficients b and c, and their meanings, as the sum and product of the roots sought.  It always helps to consult the great authors.]

 

[edit: Let me admit however that it is a useful skill to know how to derive the formula by completing the square, as in the common core.  Indeed, after checking I found this method in Euler, contrary to what I said yesterday.  I just advocate also giving the more transparent method here for the benefit of those to whom it may be found preferable.  The point is not just to make one certain approved presentation and quit, but to try different strategies until the child "gets" it.]

 

So someone who knows more and tries to make these connections runs the risk of being out of compliance with the core.  This is why those of us who think we understand something about the subject, or at least think of it as an open topic for experimentation, have trouble when we are told it must be done a certain way.  Now I know some of the authors of those standards personally and they are well trained research mathematicians, so I don't know why the core standards read as they do, without any visible benefit of some of these historical insights.  Sometimes there is a power struggle among the various authors as to what should be there.  It is also possoble I would agree completely with how one of them actually implements the standards if I saw them at work.  I have noticed people argue about teaching in the abstract, mostly out of poor ability to communicatre what they mean.  In practice they often actually do the same things in class.  But if I were a teacher and were handed this document, I would not be a happy camper.  Apologies to my friends who worked so hard to help produce it.

 

Actually an example of this puzzling dichotomy, is that my friend who participated in writing the common core standards which I find disheartening, wrote her own book on teaching math which I recommend to everyone as absolutely wonderful:

 

http://www.amazon.com/Mathematics-Elementary-Teachers-Activities-4th/dp/0321825721/ref=sr_1_1?s=books&ie=UTF8&qid=1452295687&sr=1-1&keywords=sybilla+beckmann

Edited January 9, 2016 by mathwonk

[mathwonk]

Going out on a limb here, I want to try to give Euler's explanation of the cubic formula, since again the idea is that once you realize you should write the solution as a sum of two other numbers, the formula plops right out.  I think you will not find this explanation in any high school math book today anywhere, unless someone somewhere uses Euler's great opus, Elements of Algebra, written for his butler about 250 years ago.  (I have taught this method to bright 10 year olds.)

 

CUBIC FORMULA:
To solve a cubic equation, we start with a simplified one of form X^3 -3bX - c = 0, and again assume we want to find X as a sum X = (p+q).  Plugging in gives (p+q)^3 = 3b(p+q) + c, and expanding gives p^3 + 3p^2q + 3pq^2 + q^3 = p^3 + q^3 + 3pq(p+q) =
3b(p+q) + c, and for this to hold means that pq = b, and c = p^3+q^3.  Cubing the first of these gives p^3q^3 = b^3, and p^3+q^3 = c.  Since we know b and c, we know both the sum and the product of the cubes p^3 and q^3.  Can we find p^3 and q^3 from this?  If so, then we could take cube roots and find p and q, and finally add them and get our root X = p+q.

Just recall in a quadratic equation of form X^2 - BX + C, that B and C are precisely the sum and product of the desired roots, and we can find those roots from B and C.  I.e. we can find any two numbers when we know their sum and product, by solving a quadratic.

Since p^3+q^3 = c and p^3q^3 = b^3, the numbers p^3 and q^3, which can be used to give a solution X = p+q of our cubic, are solutions of the quadratic equation
t^2 -ct + b^3 = 0


e.g.  to solve  X^3 = 9X + 28, we have b = 3, c = 28, and so we solve t^2 -28t + 27 = 0.  Here B^2-4C = 676, whose square root is 26, so we get t = (1/2)( 28 ± 26) = {27, 1}, for p^3 and q^3, so p,q are 1 and 3, and hence X=1+3 = 4 solves the cubic.  Of course if we know about complex numbers, there are two more cube roots of 1 and 27, and we get two more complex roots.  (Only two more because b = pq, so we must always have q = b/p, i.e. the choice of the cube root q is determined by the choice of p.)

Finally, one can translate the variable in any cubic equation to change it into one with zero quadratic term, so this process works in general.

Edited January 9, 2016 by mathwonk

[mathwonk]

By the way if you are teaching positional notation and grouping, I encourage you to make up your own real life concrete examples.  One I came up with, while teaching Professor Beckmann's class, involves coca cola bottles.  I.e. they are organized into cartons (sixpacks), then cases of 4 sixpacks, then maybe flats holding 4 cases each, then maybe trucks holding say 48 flats.  Ask a kid to compute how many trucks, flats, cases, and cartons would be used to store say 10,000 bottles. 

 

Or if you know about British money, it gives groupings into pennies, or I guess pence, shillings, pounds, and I don't know what else.  Ask a kid to say how to rearrange say 1,000 pence into shillings and pounds.  Be original! 

 

After they get the idea, then the abstract grouping of 1's. 10's. 100's, .... may make more sense.  I recommend in teaching these ideas to just use your own good sense.  To teach from, and for, real creative understanding, not according to any strict set of marching orders from on high.

Edited January 9, 2016 by mathwonk

[Targhee]

Gosh, I cannot think of any reason to prevent kids from accessing the benefits of traditional positional notation, developed over centuries. I mean in second grade, you learn it the easy way, then later maybe you branch out. But what do i know? I just live here.

This isn't a pro common core post - but I definitely AM pro multiple strategies. I am happy that Singapore, RightStart, BeastAcademy, and others teach standard algorithms after they have taught conceptual building blocks and alternative strategies, and I am so greatful to have discovered these strategies myself.

 

I do think it a disservice to all when teachers lack conceptual understanding of the strategies or lack time to thoroughly teach them - and when they lack support from parents because the parents don't understand it/it is foreign to them.

Edited January 9, 2016 by Targhee

[Tanaqui]

At our local elementary, kids will get correct answers marked wrong if they use the "wrong" method.

 

This is nothing new, though. I had it happen to me as a kid. My mother had it happen to her as a kid. In the US, there have always been math teachers (especially in the younger grades) with weak math skills who didn't understand the methods they used and were very confused by novel methods. And those teachers tend to react badly to kids doing things they don't understand.

 

[hepatica]

I didn't learn (or perhaps didn't absorb) many mental maths techniques from school.  I learned them through Singapore Maths when teaching Calvin, and I use them very frequently in real life.

 

Yes! Yes! to this. Don't fight it, embrace it. I am on my third child working through Singapore math and I, myself, am so much better at mental math now than I was as a student. If you accept that it is good to teach both the traditional algorithm and mental math techniques, then this child just needs more practice with mental math. But, she probably needs to be required to do it in order to get that practice (I have the opposite problem with my 7 yo DS - he wants to solve all problems mentally and I allow him to do it, and then have him check his answers by also solving them using a positional algorithm. He doesn't always want to do this, but I require it because I think it is important to be proficient in both). It might have been ideal if she had been working on this for a couple years, but it's definitely not too late to start now. She's in elementary school for goodness sake. And, it's ok if a kid struggles a bit while learning this new skill. That's probably why teachers are giving partial credit for attempts even when the child does not get the correct final answer. They have to change the motivation. When a teacher is teaching a new way of thinking, you want the kids to try that new way of thinking without being so caught up in the idea that the be all and end all is one single right answer. You have to make room for mistakes, so that they can practice. In the end, the better conceptual understanding will lead to fewer mistakes. 

 

It's hard to make big changes to curriculum, so anything like common core was always going to be a rough ride to implement. Teachers, too, have to gain proficiency in new methods and that will take time. And, the testing regimen complicates the implementation. But, overall, the move to more conceptual understanding is a good thing.

[mathwonk]

After consulting the book by Lagrange where I learned Diophantus' solution of the quadratic equation, and the book of Euler, I have edited/corrected the discussion in post #22 above.  It turns out Euler does complete the square, but the original solution by Diophantus, in the 3rd century AD, seems to me still the simplest one.  Indeed I am reminded that the great modern algebraist Fermat [thanks purpleowl!] used the book of Diophantus, and wrote the famous marginalia "I have found a truly marvellous proof.... but the margin is too small to hold it" in his copy.  (Referring to the assertion that X^n + Y^n = Z^n has no positive integer solutions X,Y,Z for any n > 2.)

 

A quick search reveals that the method of completing the square occurs first(?) in the work of the famous Arab mathematician from Baghdad, Al - Khwarizmi, in the 9th century. 

 

I just want to remind that in the 18th century elementary math books were written by the greatest mathematicians, i.e. Euler and Lagrange, and for some purposes those books are still the best.

Edited January 9, 2016 by mathwonk

[purpleowl]

After consulting the book by Lagrange where I learned Diophantus' solution of the quadratic equation, and the book of Euler, I have edited/corrected the discussion in post #22 above.  It turns out Euler does complete the square, but the original solution by Diophantus, in the 3rd century AD, seems to me still the simplest one.  Indeed I am reminded that the great modern algebraist Galois used the book of Diophantus, and wrote the famous marginalia "I have found a truly marvellous proof.... but the margin is too small to hold it" in his copy.  (Referring to the assertion that X^n + Y^n = Z^n has no positive integer solutions X,Y,Z for any n > 2.)

 

A quick search reveals that the method of completing the square occurs first(?) in the work of the famous Arab mathematician from Baghdad, Al - Khwarizmi, in the 9th century. 

 

I just want to remind that in the 18th century elementary math books were written by the greatest mathematicians, i.e. Euler and Lagrange, and for some purposes those books are still the best.

 

Pretty sure that was Fermat, not Galois.

[mathwonk]

"Pretty sure that was Fermat, not Galois."

thank you!  corrected.  I was off also in time by some 200 years! Galois lived in the 1800's, and the first new case of Fermat's theorem (after Fermat's n=4?), i.e. for n=3, was due I believe to Euler in the 1700's.

 

This also reminds me of how old the math is we teach high schoolers and younger.  The quadratic equation seems to go back at least 1700 years, probably more.  Euclidean geometry in a form more sophisticated than we teach goes back over 2000 years.

 

The common core does suggest using the basic identity above to produce Pythagorean triples, but does not classify all of them, nor use it to deduce the case n=4 of Fermat's last theorem which Fermat himself knew.  I haven't checked to see what they say about irrational numbers and their approximation by rationals, again treated geometrically in Euclid.  But basically I have thought of nothing yet that we teach that was even discovered in the last thousand years.  Maybe complex numbers, are they covered?

 

Ok, I see the core mentions complex numbers and also vectors and matrices it seems, but I cannot find how complex numbers are defined.  It just says students should know they exist, but what it means for them to "exist" is not explained.  (They should probably be defined as ordered pairs of real numbers.)   It also says they should know a few basic facts about irrational numbers, but I cannot find the idea that even the existence of irrational numbers requires proof, already given by Euclid in the case of sqrt(2).  In particular I cannot find any discussion of the meaning of "real" numbers.  I also did not find any mention of the key "rational roots" theorem in algebra, an essential tool for finding rational solutions of higher degree p[olynomial equations.  In geometry it says students should be able to prove various theorems about triangles and circles, but it does not say what axioms are to be given, without which "proof" is meaningless.  It does say one should use geometric transformations such as rigid motions and similarities, but it is not clear to me here either just what precise definitions of these are assumed.  In particular a notion of length seems to be assumed, which requires a precise definition of real numbers, that I did not find.  The principle of similarity for triangles is to be proved, but it is not stated how this is to be done.  An arithmetic proof requires precise understanding of approximation of real ratios by rational ones, and the geometric proof in Euclid uses the theory of area.  Perhaps they intend a transformation approach, assuming some properties of similarity transformations.  There are some nice topics on volume included like deriving the volume formula for a sphere from Cavalieri's principle, but no mention that this is due not to Cavalieri but much earlier to Archimedes.  For some reason the requirement is only to give an "informal argument", whereas the assumption of Cavalieri's principle allows a rigorous proof using Pythagoras.  This is very clear in Harold Jacobs' Geometry, where I first learned it.  The laws of sines and cosines are mentioned but not that the latter occurs in Euclid in a simple geometric form, as a natural generalization of Pythagoras.  So to me there are a lot of good topics in the core standards, but many good ones omitted, and no completely precise description of how the topics are to be covered.

Edited January 9, 2016 by mathwonk

[Kiara.I]

I think there's an appreciable difference between conceptual understanding and completing a proof.  There's also an appreciable difference between a conceptual understanding of the meaning of whole numbers, and of basic arithmetic functions, and the conceptual understanding of integral calculus and Riemann-Roch theorem!

 

I would question whether coping with Riemann-Roch is possible for a child who does not develop number sense or a real understanding of what arithmetic functions *mean*, rather than how to compute them.

 

Common core is (at the early elementary level) not discussing 4th year math, and I think addressing early elementary comprehension of concepts by comparing it to the proofs require of upperclassmen is not a useful comparison.  In early elementary, concrete thinking is important and conceptual understanding of the math is pretty concrete.  In university, students are developmentally well into the abstract stage, and the conceptual understanding required is abstract.

[mathwonk]

This discussion of the usefulness of mental math reminds me of an alzheimers test I heard about recently, where they ask you to start at 100 and subtract down by 7's successively.  It dawned on me to thwart it by repeatedly adding 3 and subtracting 10.

 

I agree conceptual understanding of numbers and grouping is useful, I am just trying to come up with strategies to help a kid who is struggling with it, and all I have thought of is either to preface it by computational practice, or to teach grouping in more concrete examples.  For instance, do you think my coca bottle grouping or British money examples could help anyone learn the idea of "grouping" numbers?  Can you suggest some more such examples of real life grouping?  I guess in the military we have squads and platoons and companies and regiments and armies...

 

My usual approach when someone does not get something is to back up, make things easier and then gradually come back at the original topic more gently.  And I have also changed my viewpoint a lot as time has gone by.  I used to always teach everything in full detail, with all theorems completely proven, partly because I myself enjoyed that more and learned from it.  But sometimes my students complained they came out a little weak on using the material in practice.  My idea now of conceptual understanding is less rigorous proof and more of an informal explanation that still reveals the reasons behind a phenomenon, as I think you referred to in your first sentence.  Like that quadratic formula derivation by Diophantus that I like so much.  (I also read Riemann's own proof of the Riemann-Roch theorem in his [and his student Roch's] works, because I wanted to know why it was true, and I found he makes it very clear.)

Edited January 9, 2016 by mathwonk

[wapiti]

I agree conceptual understanding of numbers and grouping is useful, I am just trying to come up with strategies to help a kid who is struggling with it, and all I have thought of is either to preface it by computational practice, or to teach grouping in more concrete examples.  

 

FWIW, this topic comes up often on the Learning Challenges board.  Ideas to try include cuisenaire rods and also a program (involving dots, I think) called Ronit Bird.  I find the following post about developing number sense to be fascinating:

http://forums.welltrainedmind.com/topic/533786-can-we-talk-mathdyscalculia/?p=6028445

[mathwonk]

thank you, i'll check that out when i have a minute.

[Janeway]

Yep, in RightStart the kids would be taught to add up the tens, then add up the ones and find the total.  20+40=60, 8+3=11, and 60+11=71...so 28+43=71.  It's taught that way because you're more likely to need to add numbers mentally like that in real life, when you can't sit down and write up a nice column.

 

On the other hand, I've noticed my sister's girls coming home from school without having things adequately explained--particularly in math, since curricula often mandate that the child be taught several ways of solving each problem--or simply with misunderstandings of the explanation.  If no one has explained it in a way the child understood (and given a good reason for it, for pity's sake--since there IS a good reason!), no wonder the poor kid is confused.

The teachers are generally not trained it in. I think they need to go to dedicated math teachers in the schools even at the elementary level.

[hepatica]

 Can you suggest some more such examples of real life grouping?  I guess in the military we have squads and platoons and companies and regiments and armies...

 

 

 

No need to re-invent the wheel here. There are a number of longstanding math programs that teach this brilliantly - Singapore, Right Start, Math Mammoth etc... If you really want to understand how to teach this way, then spend some time looking at the first and second grade levels of these programs.