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/
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