Common Core Mathematics Methods

[newjerseyrunner]

I've seen a lot of complaining on the internet by parents about the new way addition and multiplication is taught in American schools. I looked at it, and have to admit, at first, I thought it was insanely overcomplicated and worse than the way I was taught where you did one digit at a time and worked right to left.

Then I started to realize that the way they were doing it was actually what I do in my head all day every day and I think anyone in a mathematical job would agree.

If you haven't heard of it yet, addition is done like this: 37542 + 34726 used to be done by adding the 6 and the 2, then moving to the left and repeating with the carry if there is one. In algorithm terms: recursion. The new way is to do it via recursion too, but starting on the other side and using rounded values instead. 37K + 34K, then 500+700, then 42+26. (In school they'd actually teach 30K+30K then 7K + 4K, but we're not children so we can handle two digits at once.) The obvious advantage is that you have a reasonably precise estimate after only the first step.

Multiplication looks really weird. 432 * 286. They're taught to built a table with the modulus' of the placeholder's position in the table across one axis, then the same across the other.

____|_400_|_30_|_2_
200_|_____|____|___
_80_|_____|____|___
__6_|_____|____|___

Then they fill in all of the boxes and add them up. I have to admit, it looked really weird to me, but then I just decided to do the big calculation in my head and see how I got there. I realized that I do the same splitting up into boxes in my head, but I do it to scale, where I envision my 400 much larger than the 30. So I quickly understood that this is using geometry to solve the equation by breaking the multiplication of arbitrary numbers into adding areas of nice round numbers.Do you think there is a specific advantage to teaching children to think this way? I developed my way of doing arithmetic this way over years of working with hexidecimal numbers. Do you think this will be a passing fad or a paradigm shift for how children are taught math.

 

[Bystander]

https://en.wikipedia.org/wiki/Lattice_multiplication

 

[Andy Resnick]

I<snip>

Do you think there is a specific advantage to teaching children to think this way? I developed my way of doing arithmetic this way over years of working with hexidecimal numbers. Do you think this will be a passing fad or a paradigm shift for how children are taught math.

The math-education colleagues I know think lattice multiplication is far superior because it more naturally leads to algebra. However, thas little to nothing to do with the question of it being a passing fad or a long-term shift.

 

[gleem]

The math-education colleagues I know think lattice multiplication is far superior because it more naturally leads to algebra

Is there any reference that shows or discusses how this multiplication algorithm leads to algebra or a better understanding thereof?

 

[Andy Resnick]

Is there any reference that shows or discusses how this multiplication algorithm leads to algebra or a better understanding thereof?

That's a fair question- I'll ask around.

 

[Andy Resnick]

Is there any reference that shows or discusses how this multiplication algorithm leads to algebra or a better understanding thereof?

I'm still waiting to hear back, but I was surprised at how many studies are already out there. For reference, lattice multiplication is part of the NSF-funded "Everyday Mathematics" K-6 curriculum:

http://everydaymath.uchicago.edu/

A survey of the relevant studies can be found on pages 46-50 here:

http://www.google.com/url?sa=t&rct=...LdX5HB4wPpUh2YxK-2cBTg&bvm=bv.103388427,d.dmo

The conclusions of this survey are clear. However, I don't have the time or inclination to closely read all the original studies; hopefully asking my colleagues is a more efficient method.

 

[gleem]

Thank You for your effort, I to do not wish to plow through all sorts of studies. Hopefully someone can give a relative straight forward explanation as to why this new approach works.

 

[jack476]

Multiplication looks really weird. 432 * 286. They're taught to built a table with the modulus' of the placeholder's position in the table across one axis, then the same across the other.

____|_400_|_30_|_2_
200_|_____|____|___
_80_|_____|____|___
__6_|_____|____|___

Then they fill in all of the boxes and add them up. I have to admit, it looked really weird to me, but then I just decided to do the big calculation in my head and see how I got there. I realized that I do the same splitting up into boxes in my head, but I do it to scale, where I envision my 400 much larger than the 30. So I quickly understood that this is using geometry to solve the equation by breaking the multiplication of arbitrary numbers into adding areas of nice round numbers.Do you think there is a specific advantage to teaching children to think this way? I developed my way of doing arithmetic this way over years of working with hexidecimal numbers. Do you think this will be a passing fad or a paradigm shift for how children are taught math.

That's how I learned to multiply out polynomials in "(x+a)(x+b)(x+c)..." kind of form like 10 years ago, I didn't know what "F.O.I.L." was until very recently. I especially like that method because you can do it in reverse to very quickly factor second-order polynomials, and some 3rd-order with a little creative tweaking. Like if you have x² +4x +3 = 0, you can do

|_x²_|___|
|____|_3_|

Then all you need to do is fill in the last two sections by knowing that 3 and 1 are the numbers that multiply to 3 and sum to 4, so therefore you have that x² + 4x+ 3 = (x+3)(x+1). Some algebra or creative reasoning can become necessary if the roots turn out to be complex or not integers, but overall I find it to be a more intuitive and "agile" method (lacking a better word) than just plugging away into the quadratic formula. When I show this method to my classmates it kind of disappoints me that it isn't more well-known, because it's a much easier way of working out factoring and multiplying computations than whatever that F.O.I.L BS is. It also makes a good starting point for developing mental arithmetic, like being able to just glance at a table like that and know which numbers both sum and multiply to a given number, and in general it's an easier method to use to multiply numbers in your head than the brute force multiply and carry method.

You can also use a similar process to find prime factors. My understanding is that Common Core includes some new emphasis on pure math, so that might be why we're seeing lattice multiplication come back.

 

[symbolipoint]

About the post #1,
If that is how most of elementary-school Mathematics is being taught, then most of it is good. The long-multiplication method shown there is a form of lattice method and this can be shown and coordinated graphically, and it should make good sense for young students. Note that often enough, a lattice-type method was shown to students a few decades ago, also.

 

[symbolipoint]

Again, about the discussion in post #1, years and years ago, students in first grade were - yes, WERE - taught about place-value, and this was used for learning about "carrying" when doing multi-digit addition of numbers. That knowledge and the included explanations were very helpful; and if still used in instruction today, it is still good.

 

[gleem]

The math-education colleagues I know think lattice multiplication is far superior because it more naturally leads to algebra.

I have been inclined to dig just a little bit deeper into the Common Core curriculum because It presumably promises to improve math maturity. I think everybody should including the philosophy of this approach, how it came about, its implementation, ongoing controversy, and results.

Dr. James Milgram, Professor of Mathematics at Stanford University in an opinion article for FOX News, provides an authoritative critique of the Math standards of Common Core..

http://www.foxnews.com/opinion/2013...re-massive-risky-experiment-on-your-kids.html

The US educational system Dark Age (1970 - present?) has been characterized a a dumbing down of the curriculum is that still a problem?

In the new report on economic competitiveness from the 2015 World Economic Forum the US is ranked for the quality of its education system (18th),and for math and science education (44th) p 361

http://www3.weforum.org/docs/gcr/2015-2016/Global_Competitiveness_Report_2015-2016.pdf

 

[Bystander]

Dark Age (1970 - present?)

More Post WW II - present.

a dumbing down of the curriculum

For the next steps to follow "Common Core Curricula," Google "Smarter Balanced" and "PARCC."

 

[gleem]

I probably should have said the "current" Dark Age of Education" not fully knowing all the educational ups and down through our history. However prior to 1970 educators where noting a constant and significant improvement in freshmen preparedness for college.

During the 60's while I was in grad school there was a call for relevancy in college curricula. Student lobbied actively for courses that could be use to solve world problems. Environmental concern was a big emphasis at that time. An the colleges obliged with all sorts of interdisciplinary programs and new courses not necessarily bad but it did dilute ones concentration on the fundamentals I was somewhat disturbed at this as a student as I believe in the wisdom of the established educational system. The teaching system could see beyond my ability to see my future and was better able to provide proper preparation. As a student what did I know. Was I right in assuming they knew better what I needed? .I cannot say I was wrong though. But that was before the " feel good " curricula and the emphasis on self actualization

One more point, I attended private schools through my entire education. As today, private schools, were (are) more likely to provide a better education unhindered by state supported educational pundits who continually tinker with the public educational system ( as of yet) to no avail. Even home schooling is found on average to be better.

Is this new mathematics paradigm just another wasted effort to cure a problem that never existed?