The latest informational video for parents posted to the SCASD K-6 Elementary Math web page walks the viewer through 4 possible strategies for solving 2-digit addition problems typically done in Grade 3. It’s a nicely done video that clearly lays out some of the potential strategies that third graders might use to solve math problems.
Referenced in the video is the Investigations’ Student Handbook that parents can refer to (pp. 20-24). Perhaps now each student/family now gets this handbook for reference? (This was not something I was even aware of when my son was in third grade, and each child definitely did not receive one). Or perhaps this is something that is available in the classroom for reference only?
Different approaches and strategies can certainly be appropriate for solving various math problems. The video highlights that students need to select the most efficient approach for solving the problem. Just for fun, I categorized the number of steps required and time to completion for the simple word problem presented in the video, “Bridget went to Sticker Station. She bought 47 horse stickers and 74 space stickers. How many stickers did she buy altogether?”
Strategy A – Adding by Place: 4 separate steps (write the problem horizontally, write the tens addition 40+70=110, write the ones addition 7 + 4=11, write the combined addition 110+11=121), 26″ seconds to complete.
Strategy B – Changing the Number: 6 separate steps (write the problem horizontally, change 47 to 50 by adding 3, change 74 to 75 by adding 1, add 50 and 75 to get 125, calculate how much you added [4], and subtract it from 125 to get 121), 51″ seconds to complete.
Strategy C – Making an Equivalent Problem: 4 steps (write the problem horizontally, subtract 6 from 47 to get 41, and add that 6 to 74 to make 80, then add 41 and 80 to get 121), 31″ seconds to complete.
Strategy D – U.S. Standard Algorithm: 4 steps (write the problem vertically, add the ones, carry a ten, add the tens), 25″ seconds.
This is actually the first time I’ve seen the standard algorithm truly be embraced and I’m glad to finally see it. As for the other approaches, yes, it’s kind of neat to be able to tinker with numbers and shift them around. Fun stuff. Should kids have to solve problems three different ways, though? Repeatedly? Some Investigations homework sometimes seems to require it.
I hope that kids are no longer being penalized for using the standard approach as occurred for my third grader. If students can calculate effectively and get the right answer, I don’t care how they accomplish it as long as the strategy the use works for all problems, and works efficiently. That begs the question – how much more complicated would some of these approaches become if the problem was 89 plus 96, or 426 plus 789, or 7397 plus 4887? At some point, it would seem that ‘adding by place’, ‘changing the number’, and ‘making an equivalent’ might become a bit unwieldy and impractical.
Mastery and accurate recall of basic math facts is an essential component of mathematical problem solving. If we take that mastery as a given, what’s the most efficient method as far as fewest steps and quickest calculation? It may depend on the problem. In the example above, adding by place and the standard algorithm seem equivocal.
But this makes me wonder — which of those two approaches would be most efficient for a 4-digit plus 4-digit problem? I pitted Strategies A v. D for that 4-digit problem above: 7397 + 4887 =
Strategy A – Adding by Place (all done horizontally as per the example: 7+7=14; 90+80=170; 300+800=1100; 7000+4000=11000) left me with a final calculation of:
11000+1100+170+14 = 12284 (definitely not so easy to do written like this. And yes, I DO know my math facts 
). It took me 1:14″
Strategy D – US Standard Algorithm (done vertically, adding from right, carrying, etc). It took me 18″.
It seems pretty clear which is more efficient, doesn’t it?
I agree that this is the kind of video that we should be producing for the parents. I love the Kahn Academy stuff and we should include some of those at least until we can make our own on those topics.
With regard to the four approaches, I’m not sure strategy A, B, or C are designed for efficiency or speed. Rather they are designed to introduce concepts which will be useful to students at other levels of math.
I suspect method A emphasizes the concept of “place value,” i.e. the value of a digit changes depending on its place in a number. Both method B and C introduce students to the concept of balancing the equation which they will encounter in elementary algebra:
y = x+4
y – 4 = x +4 – 4
y – 4 = x
When going to the “benchmark” number by addition, the students have to remember to subtract what they added to balance the equation.
Clearly as the digits increase from 2 to N, the U.S. standard algorithm is the winner for speed, but I don’t think that’s what the purpose of multiple strategies is.
With regard to the Students’ Handbook, this came with my son’s Investigations 2 curriculum. My daughter doesn’t have one, so I suspect this is a I2 feature. It is very helpful.
All good points. A direct link to the Kahn Academy videos is a great idea! (Why reinvent the wheel?) I’m glad to hear that the Handbook is available to parents now, as it would appear to be beneficial. Given what I know about this curriculum, however, such a resource fails to solve the underlying problems with this curriculum and approach.
I absolutely agree that being able to flexibly manipulate numbers, understand how they relate to one another, and how various approaches can be useful in solving equations is important. Having third graders spending time spinning their wheels with unwieldy approaches may not be as helpful though.
The fact that Investigations and the homework it includes actively discouraged my son from using the US Standard Algorithm (and NEVER, heaven forbid, TAUGHT this approach) is a major and serious failure that will hinder many kids’ progress.
I have to agree with both previous comments. It is great that kids learning to manipulate with numbers, digits, placement of digits and so on.
One thing I cannot agree with here. Kids in elementary school suppose to learn how to manipulate with real numbers. Without basic skills it would be irrelevant if they know how to manipulate digits and “remember to subtract what they added to balance the equation” they simply would get in trouble when, let say need to do next:
1/17 x + 3/21 x = 126 59/125
Maybe all this “manipulating” is helpful for elementary algebra, but what they are going to do, when strong basic skills would be necessary, like in complex algebra and trigonometry and later in Calculus. Math is not over with elementary algebra. In addition, I think if “introduce students to the concept of balancing the equation” mindlessly, without explanation of real use of it in future, it is not going to be linked later. That I know from own experience.
I had conversation with math support teachers in SCASD about this method. I was really surprised and confused. Mrs. Taptich and M.J. Kitt told me that they discovered this method of manipulating numbers later in life. They both amassed how interesting it was. They had basic skills established by the time of their “discovery” and could “connect” it to these new ways to add. Should not our kids have the same opportunity, to have skills first, and then build different approaches to numbers on it?
I think it should be next step. Child cannot “chew” on such a mess of approaches right away. We cannot expect child to be ready for Algebra at the same time as for arithmetic.
Nevertheless, regular algorithms can emphasize all this adding by digits also. Teacher just has to explain to the child clearly, what s/he has to do. That is all. I do not think that “secretly” teaching child elementary Algebra in place of strong basic skills is the way to go.
@Anastasia:
You say:
Maybe I’m taking your notion of real numbers too literally, but with the younger kids, I think we start with whole numbers and then work our way to real numbers. For your expression, it is necessary to keep track of products and quotients, but the idea behind solving it is still balancing the equation.
I don’t think anyone is “secretly” teaching algebra, just introducing concepts that can later be demonstrated to be part of elementary algebra.
With regard to my use of the term “elementary algebra” perhaps I am using the term incorrectly. I use “elementary” to distinguish it from things such as “linear algebra,” or “group theory (abstract algebra).”
@Barb
You say:
I have two children in elementary math in the school district and I have not found that to be the case. For homework where a technique is not being taught, the children are encouraged to use whatever they are comfortable with. As they say, “Your mileage may vary.” It appears that yours has, but I have not had that experience with my kids.
Teaching these concepts is not unlike when I learned logarithms. For the first week or two we used log tables. Sure the more efficient way was to use my TI-30 calculator, but the point of doing it via tables and adding, subtracting, multiplying and dividing logarithms was to understand how to manipulate them. I never use anything other than a calculator now, but I learned by doing it the “long way.” That led to a more fundamental understanding of logs, but certainly not one I’d use in daily practice.
No, I wouldn’t trade my SAS or SPSS statistics software for the hand-calculated ANOVAs that I did in graduate school either, and yes, I certainly understand analyses better as a result of doing them (painfully) by hand a few times. But I had to master those skills and calculations, and get the right answer on my quizzes and exams.
My son was instructed on more than one occasion to redo his homework when he used the most efficient mechanism for him (the US standard algorithm) and came up with the correct answer. In essence, it was tedious busy work and he learned nothing from it. What is the point of that? On other occasions, he was told directly “not to do it that way”, and it confused the heck out of him (and us) since we had taught him that approach the year before (while he was still drawing bazillions of circles to meet the “show you work in numbers, pictures and words” problem requirements.). Sigh.
Dear Jim Leous,
Good to see you are posting here! I am impressed.
Your kids are really lucky. Through my two kids, I had plenty of experience with math homework that require answers with “show you work in numbers, pictures and words”. Both my husband and I (both math professors) scratched our head in embarrassment, knowing for sure we would both fail these homework. One dinner table discussion last week tops all. Our younger daughter contradicted us when we said a line is one dimensional. She said it is two dimensional. She said she already learned it in 1st grade.