On Oct. 7, I wrote about the difference between division done via a traditional method, and division done via a reform method. The reform method that I illustrated doesn’t work in all situations. It doesn’t handle decimals well, or efficiently manage larger numbers. A pro-reform-math professor who used the reform method to divide 396.3 by 16 was never able to get the correct answer despite several attempts and a white board filled with numbers.
This week, I’m explaining the difference between multiplication done via a traditional, vertical method, and multiplication done via two, different reform methods. The traditional method is clear and efficient; the reform methods are neither. First, we’ll review the traditional method.
Vertical Multiplication (traditional math): 1,642 x 849 = ?
The traditional model quickly and efficiently provides a complete and correct answer: 1,394,058. This traditional model works well in all situations, including problems containing decimals or large numbers. With this method, it’s easy to keep track of one’s work and to check for errors.
Now, let’s look at the reform approaches.
1. Cluster Method (reform math): 1,642 x 849 = ?
If I were to use the Cluster Method, I might begin by saying this:
“I know that 1,000 + 600 + 40 + 2 = 1,642. And, I know that 800 + 40 + 9 = 849.
So, I could say that (1,000 x 800) + (1,000 x 40) + (1000 x 9) + (600 x 800) + (600 x 40) + (600 x 9) + (40 x 800) + (40 x 40) + (40 x 9) + (2 x 800) + (2 x 40) + (2 x 9) = ?
Using ‘mental math’ to simplify, I can say that 800,000 + 40,000 + 9,000 + 480,000 + 24,000 + 5,400 + 32,000 + 1,600 + 360 + 1,600 + 80 + 18 = ?”
In most reform classrooms, I would be given a calculator. I would plug in the numbers, and I might get a result of: 1,365,258. If I were a student, I would turn in this answer, and it would come back marked incorrect. I would have no idea of why it’s incorrect, and neither would my teacher. I might never find out that I plugged in 3,200 instead of 32,000.
Clusters also can be done after factoring the multipliers, but I doubt reformers would use the Cluster Method to solve problems containing this many digits. They would recommend using a calculator. The entity doing the calculating, in that case, would be the calculator and not the student. Little actual learning or practicing would take place.
A serious limitation of the Cluster Method is in the decimal. If either multiplier contains a decimal, how does this method handle it? The children would be stumped, and out would come the calculator.
2. Lattice Method (reform math): 1,642 x 849 = ?
With the Lattice Method, children are asked to draw a grid, with diagonal lines intersecting each square in the grid. Students write one of the multipliers along the top, and the other vertically down the right side. Students then multiply each integer of the multiplier on the top with each integer of the multiplier down the side – placing answers in the intersecting squares. (The tens of each answer are written in the upper part of each square, and the ones of each answer are written in the bottom part of each square.) The students then add diagonal columns, beginning on the lower right of the grid and moving left, writing the ones along the bottom and carrying the tens always to the left. Like this:
You can see that 1 x 8 = 8, 6 x 8 = 48, 4 x 8 = 32, and 2 x 8 = 16, and so on. The answer is alongside the left and bottom of the grid: 1,394,058. However, it’s difficult and time-consuming for children to accurately draw these grids and diagonals. Any mistakes made in drawing the form, filling in the squares, or adding the diagonals will muddy their understanding. Checking one’s work can be done with this method, but with so much going on in the grid, it’s more difficult to do.
And again, what if the problem contains a decimal? Where on the grid does a decimal go? If a problem contains larger numbers, the grid also must be larger, resulting in more squares and diagonals, and more possibility of error. What if the multiplication problem resides within an equation, or within a division problem?
Let’s take a moment to examine that last one. Using traditional math methods of division and multiplication to divide 6836 by 98, it would look something like this:
The traditional methods of division and multiplication efficiently provide us with a correct answer. These methods also are flexible. If a remainder is required, the remainder is there after the second step. If a fraction is required, the fraction is easily gleaned from the remainder. And if a decimal is required, the work continues through the desired number of decimal places.
Now, let’s use a reform method of division and the Lattice Method of multiplication. (If the student can do multiples of 10, it might look more like “a.” If the student can’t do multiples of 10, it might look more like “b.”)
a) b)
Remainders are handy when students begin learning about division. As they progress, however, the format of the answer also should progress. Division problems that don’t divide evenly should be completed with a mixed fraction (reduced to its simplest form), or with a decimal (rounded to two or more places). The mixed fraction can be gleaned through this reform method. 69 R 74 can be written as
which is reduced to
What if there is a decimal in the dividend (the number being divided) or in the divisor (the number doing the dividing)? (It's true that students can temporarily move or remove decimals, but it’s easy to forget to put them back, as the pro-reform-math professor unwittingly demonstrated.)
When I mention these limitations to reformers, I’m told that students should use a calculator. “Everybody’s going to use an electronic device anyway,” I’ve been told. Who do those reformers see as the builders of the electronic devices? (Obviously not anyone who went through an exclusively reform math program.) This flippant referral to calculators is ironic, since reformers are always claiming that traditional instruction turns children into “little computers.”
If only using calculators for everything at least produced math proficiency. Calculators are everywhere in reform programs. They permeate public education, all the way to kindergarten in some districts. If calculators were sufficient for producing good results, we would have good results, but we don’t.
Forget the numbers we get from the education establishment, and look at the sinking abilities of students and graduates. Most of these students frequently or always use calculators in place of paper and a pencil. Look at the weak pass rates on college entrance exams for which calculator use is allowed. Calculators are handy tools once skills have been learned, but an over-reliance on calculators during the learning process inhibits learning. Over-reliance turns into dependence; dependence prevents students from developing skills and necessary number sense. At that point, the much-vaunted Holy Grail of reformers – i.e. “deeper conceptual understanding” – is out of reach.
You can see why so many Americans struggle now with division and multiplication. The current incarnation of reform math has been around for about 30 years. Many students and graduates are now math-illiterate and math-phobic – panicked at the thought of doing simple calculations. The failure of reform is obvious to all except the reformers and the unaware.
If any other institution operated this way – pushing failed products and ideology around, long after their failure was proved 700 ways from Sunday – the people would rise up, consumer groups would be up in arms, there would be inquiries and a class-action lawsuit, and the media would slice and dice those responsible for the mess. Unfortunately, most of the media remain stubbornly ignorant, their eyes closed to the children’s misery.
Sadly, the Common Core initiatives are bringing back reform math to many districts that had managed to kick it out. The media are again bleating – as they did in the 1980s, 1990s and 2000s – “Oh, look! A new way to teach math! It doesn’t look like the math you had as a child, but it will improve conceptual understanding and be more fun!” Blah, blah, blah. It seems that every time math advocates manage to get somewhere in a district, some idiot with a BA in English and a doctorate in education brings bad process right back in – often assisted by the local newspaper.
It’s shocking that reformers continue to get away with damaging the children like this. Reformers still love reform – after 30 years of failure. They refuse to see, appearing to care more about their pet theories and their ego than about the children.
Until the de facto federal takeover of public education manages to block all escape, parents can still walk away from reform math by finding different schools, by hiring tutors, or by homeschooling. The children get one shot at a good K-12 education. At some point, the rubber must meet the road. At some point, the students need that math. Parents must make sure their children have it.
Please note: The information in this post is copyrighted. The proper citation is:
Rogers, L. (October 2012). “In defense of vertical multiplication: Reform methods stumble over decimals." Retrieved (date) from the Betrayed Web site: http://betrayed-whyeducationisfailing.blogspot.com /.
This article also was published on the Education News Web site at: http://www.educationnews.org/k-12-schools/in-defense-of-vertical-multiplication-reform-math-stumbles-on-decimals/
15 comments:
You might be interested, Laurie, that Leonardo de Pisa (Fibonacci) used the lattice method. He and others were also using the vertical algorithm. After some time he abandoned the lattice method because it was less efficient and did not support understanding very well. Fuzzies love to show this as "a different algorithm" even though you can easily map each of its steps to that of the standard algorithm. It is a different bookkeeping method, that's all. Next time a fuzzy shows this to you as if it were the most modern innovation ever, remind them of this history and then challenge them to explain why it works and what THIS number (in the middle of the lattice represents (ie show understanding) and what role it is playing in the algorithm.
Yes, good points, Robert.
It's silly for people to point to reform methods as being valuable because they were done a long time ago. Many things were done a long time ago -- horses and carriages, wringer washers, surgery without anesthesia, 8-track tapes ... We don't do things that way now because we have something that's more efficient and more effective.
It's also true that place value is not as obvious in the Lattice Method as in the vertical method.
Along similar lines I like to suggest that because it is so tiresome for students to draw all those rectangles, squares and cubes for doing "Pictorial" arithmetic, why don't we improve it by making simpler symbols for 1, 10, 100 and 1000 that don't tax the artistic skill or take up so much space on the page.
My recommendation:
I = 1
X = 10
C = 100
M = 1000
Then we can perform these same manipulations with accuracy and much more efficiently than with the pictures -- a great improvement.
Wait a minute! I hear some "reformers" saying. This is just doing arithmetic with Roman Numerals. I agree, except that if you're paying attention it is a bit more crude than Roman Numerals. Sort of "Roman Lite".
Then I congratulate them for setting back the art of arithmetic by over 2000 years.
John Bolton was being interviewed on TV this a.m. about the political mishandling of the Libyan tragedy. He said that ideology blinds individuals to reality and that it is more dangerous than any actual coverup. Ideologists simply will not accept that their beliefs are wrong and often disastrous. It made me think of our education system and how dangerous the leaders are, not only for our children but for our country. I find it all very scary.
Dabbling in "constructed reality" IS dangerous whether you are in Diplomatic Corps and at the mercy of an ideologist President. Or, if you happen to be passenger on a new airplane designed, built and flown by past students from "constructivist math" era classrooms. Sorry combinations, both examples.
In case someone hasn't seen it: M. J. McDermott's video "An Inconvenient Truth" remains the best expose of this fraud. On YouTube.
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Now, I dispute this: "The failure of reform is obvious to all except the reformers..."
Precisely because the failure is so obvious, I've concluded that the reformers must see it, and therefore must want it. They designed Reform Math so kids would not learn math.
It's an exact parallel with Whole Word, which guarantees that children will not learn to read.
Out here in Greenwich CT, we have been fighting to get rid of Everyday Math for the last year and one-half. As part of this effort, I have become familiar with the Teacher's Manual. One of EDM's favorite tools is the Lattice Method. I wrote a blog post disputing EDM's contention that the Lattice Method is efficient. Enjoy.
http://www.blogger.com/blogger.g?blogID=2284208209386081584#editor/target=post;postID=2037965003296928268
Brian, your link might be in editing mode. I don't have access to see it.
Would you mind sending a different link for the article?
Thanks.
Just as a note, the Common Core does not mandate the adoption of student centered instruction. In fact, the last version I read of the the Common Core actually pointed out that it was not to suggest one method of instruction over another but to give guidelines on what material should be covered. It would work to teach the material directly as well. So blaming it for the methods used is probably a stretch logically.
In some areas, the Common Core is stronger than what was in place. In subjects such as trigonometry, Washington had no real guidelines for educators which meant departments teaching various topics for reasons as frustrating as "Our teachers know this topic better than that one so we include it over the other". In that sense, it could be useful to, in theory, make schools cover important concepts.
In some sense, the ideas of student centered education could be useful as well. For example, students working together on a long, difficult project can be incorporated successfully into direct instruction curricula. However, I do agree that it has been taken to an unhealthy extreme.
One idea about the reason for that which I want to toss out there is that it requires less knowledge to teach advanced math in this manner. In many districts, including SPS, teachers without much experience in math are being forced to teach it. Saying "Here is a worksheet, work together on it to derive the Pythagorean Formula" is easier than acquiring the knowledge to effectively teach it and be able to answer any number of student questions about it as well.
Re: "Precisely because the failure is so obvious, I've concluded that the reformers must see it, and therefore must want it. They designed Reform Math so kids would not learn math."
Exactly. Reformers send their own children to private schools that teach the traditional algorithms.
Interesting article about reform
http://www.csun.edu/~vcmth00m/nsf.html
I have a PhD in an STEM field from a top-10 university and have also taught high school, though admittedly I have not taught elementary school, nor do I have children. I stumbled upon this post after learning about the lattice method the other day.
There are many things wrong with how math is taught in school these days, foremost the permissive use of calculators. But these two multiplication methods are not among the wrong things. Nor are they better than the vertical method. It depends on the student and the situation and, most of all, how well the methods are taught and explained.
Though this may not be your intention, you come across as opposing the methods because they are "reform" methods without actually understanding them or thinking about how they might have some advantages over the vertical method.
You also seem to confound your opposition to calculators, which I share, with your opposition to unfamiliar methods. You write, "Clusters also can be done after factoring the multipliers, but I doubt reformers would use the Cluster Method to solve problems containing this many digits. They would recommend using a calculator. The entity doing the calculating, in that case, would be the calculator and not the student. Little actual learning or practicing would take place."
If calculator use is so freely permitted, why would it matter whether they were doing it the vertical way or the cluster way?
The cluster method is the best way to do mental math. It's cumbersome when you have two huge numbers like 1642x849, but it works better than the vertical method if you want to mentally multiply two manageable numbers like, say, 122x33.
The cluster way is disadvantageous in that it's cumbersome to write out. It's advantageous in that it teaches the distributive property of multiplication better than the vertical method.
As for the lattice method, whether drawing the grids causes more or fewer mistakes is debatable. Using the vertical method, I've certainly made my share of mistakes by putting something in the wrong column. I'm still prone to doing that, so I've decided to adopt the lattice method going forth if more than three digits are involved.
Though I haven't seen it presented in this context, I worked out some quadratic problems with the lattice method and find that it's easier than the method I was taught (and probably you were taught too). Expanding something like (a+b)^2 is easier with the lattice method.
As for decimal places, it's simple. In the cluster method, it's a non-issue because you've already broken it down into tenths, hundredths, etc.
In the lattice method, just count how many digits there are to the right of the decimal place, and adjust accordingly.
Your example shows that 1642 x 849=1394058. Suppose instead that it were 1.642 x 8.49. There are a total of five digits to the right of the decimal place, so make your answer have five digits to the right of the decimal point (13.94058). Visually, in the lattice, this would be the intersection of the two lines. It's hard for me to describe that in words, sorry.
Here's another method for multiplying two multi-digit numbers. I present it not because I think it should be taught as the primary way to solve multiplication, but rather to demonstrate that there are many ways to multiply, but they are all based on partial products and the distributive property.
www.youtube.com/watch?v=85Vd0NpL32k
By the way, what would you think about teaching the soroban method? Japanese kids learn it in elementary schools.
On the one hand, I oppose calculators, and the soroban is kind of like a calculator. It also takes considerable time to learn and should not be taught as the only multiplication method. On the other hand, it's a great skill you can use anytime because it can be modified to the fingers or to mental imagery, and it allows you to do complicated calculations quickly.
I suspect that "anti-reform" people (of which I am neither a member nor a non-member) would mostly have a kneejerk opposition to teaching this method because it looks weird.
Thank you for your comments, MM.
I'm not opposed to certain methods out of habit or because they're unfamiliar.
I'm opposed to certain methods of calculation if they:
a) are inconsistent
b) are inefficient
c) don't work effectively for the long-term (such as division methods that must be abandoned when doing more complex problems).
d) are not the best, most accurate, and most efficient way to get the job done
e) tend to produce weak habits and weak procedures in the children
f) tend to produce incorrect answers
g) confuse and/or frustrate the children
The lattice method is complex for children to draw, and it isn't sustainable for the long term. It is not the best way to do things.
As a tutor, I accept that every child is different and requires different instruction. My hat goes off to teachers who can manage classes of 25-30 students. What a challenge. I admire those who do it well, in spite of all that they face.
However, the goal should always be to get children doing the math in the most efficient, most accurate, and most long-term-useful manner possible.
I'm not opposed to calculators, but they should be used judiciously, and almost never in any math level below Algebra.
I like the abacus. I own one that has been in my family since I was a child. It can be used to teach certain concepts behind the math, but that can also be done in other ways that are more useful.
Teaching math isn't about the teacher. It isn't about you or me or the superintendent, the board, Bill Gates or Arne Duncan. It's about the student, and what the student will be able to use later in life, after he or she leaves the classroom.
The tools provided to the student should always be the best tools available.
Math doesn't have to be constantly reinvented. It just has to be taught, learned, practiced, and then used. That's all there is to it.
Many public schools, colleges of education and administrators don't see it that way. That's why we have the national math problem that we do.
As a 3rd grade teacher, I'd like to explain why I teach grid method of multiplication and lattice method of multiplication prior to teaching the standard algorithm. I start with the grid method because I want my students to fully understand that when they are multiplying 346 x 78 they are in fact multiplying [(300 x 70) + (300x8)+(40x70)+(40x8)+(6x70)+(6x8)= 26,988] and I agree it is a long and arduous method. I don't spend a long time with this method, just long enough to ensure the students fully comprehend what numbers are being multiplied. Also when I teach this method we round both factors and multiply. This ensures when we come up with a result, the students will quickly see if the answer is reasonable. So prior to starting the above problem we would have multiplied 300x80=24,000. When we get the solution of 26,988 we see that our answer is reasonable. After grid multiplication I teach the lattice method, and after doing the grid method the student absolutely LOVE lattice. After all it is much less time consuming and requires less writing. Again when doing the lattice method, the students round their factors and have an idea of what a reasonable answer should be. When I teach lattice method I also stress place value of the columns being added and factors being multiplied. When we finish the lattice method the students can and will tell you the value of each column. Finally after the lattice method I teach the traditional algorithm. If I were comfortable with other methods and I thought they'd be of some value, I'd also teach them. My goal when I teach my students is to ensure they fully comprehend the math procedure they are performing. Once I've taught the various methods I really don't care which method they use as long as the student can consistently produce accurate results. In the 6 years that I've been teaching all three methods, I've found my students settle about 50/50 between the traditional algorithm and the lattice method. I've only had one student who preferred the grid method. This student was one of my weaker math students and she stated that she felt the most confident with that method. The purpose of the teaching should be to teach the students and provide them with as many ways as possible to arrive at a correct solution. (YES they must come up with the correct solution and NO they cannot use calculators!)
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