Comments

This link points to an article addressing how Everyday Math teaches multiplication. This procedure is called the lattice method.

http://www.lewrockwell.com/taylor/taylor78.html

-- LoneRanger - 12 Sep 2005

The problem with the lattice method is that it's difficult to discern how the algorithm is based on the distributive property.

Ironically, it sheds no light on the deeper meaning of the multiplication algorithm.

-- CarolynJohnston - 13 Sep 2005

Hi Lone Ranger!

How are your daughters progressing??

-- CatherineJohnson - 13 Sep 2005

As to the Everyday math lattice algorithim: Notice it is always a problem with two digits. I have never seen anyone demonstrate the lattice method with a three digit times a three digit problem.

Erin's teacher this year is a big fan of Everyday math. Maybe I'll ask her to demonstrate this for me!!!

I must admit that she does a good job of supplementing with basic skills. She has already started the timed drills for addition and subtraction facts.

-- AnneDwyer - 13 Sep 2005

Actually you can do any number of digits with the lattice algorithm. You can even do decimal multiplication with the lattice algorithm!

There are only two things wrong with it. The first is what I said above; it's harder to see how it proceeds from the distributive property (which you need to learn for algebra). The second is that it takes a longer time to do it that way than the standard way, because you have to draw the grid and the diagonals.

-- CarolynJohnston - 13 Sep 2005

Just the other day, in the comments section for KTM's MiddleSchoolPart3 item, Charles H included a link to Illinois Loop. Upon following that link, I found this link to a specific school's online demonstration of various Everyday Math algorithms. Anne, you will find the product of two three-digit numbers there. Whew! You can also see an example of division by partial products. I think the worst part of this algorithm is that the partial products aren't even found by real multiplication. They are found by addition, except for multiplying by 10. The first partial product is one times the divisor. Add that to itself to get the second partial product (2 x divisor); add that number to itself to get another partial (4 x divisor); get 3 x divisor by adding 1 x divisor + 2 x divisor. Next jump to 10 x divisor by appending a zero to the divisor; then compute 20 x divisor by adding 10 x divisor to itself; etc.

Perhaps this deepens one's understanding of multiplication as repeated addition, but I think it is more likely just avoidance of multiplication.

The opinion I formed from the link above is that the students are learning more about authoring web pages than they are about performing math operations.

I hope I entered those hyperlinks correctly.

-- DanK - 13 Sep 2005

Nuts! I need to learn about authoring on this web page!

-- DanK - 13 Sep 2005

What you did was almost perfect; you just had to reverse the order of the items inside a link. The address comes first, and the text you're linking comes second.

-- CarolynJohnston - 13 Sep 2005

I'm getting up to speed on some of my ability to grasp & teach basic concepts of elementary math, and I think it's nuts to use partial products division to demonstrate repeated addition.

I don't know if I can express this well, but, as a lot of you pointed out in the thread on 'teaching the same procedure more than one way' you have to be VERY careful, when teaching concepts, not to confuse the issue further.

I would bet real money that the time & place to demonstrate multiplication as repeated addition is in the teaching of multiplication only.

To bring multiplication-as-repeated-addition into the teaching of the long division algorithm is NUTS.

-- CatherineJohnson - 13 Sep 2005

It looks to me like "forgiving division" is an interesting alternative, but if a kid just solves division only using the multiplication facts he can remember, what exactly becomes of the ones he can't?

Am I getting that right? It's "forgiving" because if the kid only remembers certain mulitiplication facts he can use those instead, adding up the partial quotients for the answer?

Doesn't this encourage him/her to not bother learning all of his math facts? Well, addition and subraction are still pretty critical here.

This doesn't seem terribly wise to me.

-- SusanS - 13 Sep 2005

btw--it's so cool to be building a body (however tiny) of actual teaching experience--it takes a LONG time for the distributive property to sink in.

Saxon 6/5, which is the 5th grade book, teaches the distributive property explictly, using numbers not letters:

25 x (10 + 2) = (25 x 10) + (25 x 2)

The book has numerous problems requiring the student to write out a multiplication problem in this way; he revisits the topic over and over again.

Christopher still didn't have it by the end of the book.

Now, I should add that we did 6/5 in maybe 5 months, not the normal 9 months.

Stil....I was surprised at how hard it was for Christopher to get. I told him I was going to paint the distributive property on the dining room wall, a la CHEAPER BY THE DOZEN, and he got really mad.

-- CatherineJohnson - 13 Sep 2005

I think the best way to teach the distributive property is mental multiplication.

-- CatherineJohnson - 13 Sep 2005

When I saw your graphic on EM division it caught my eye. I just happened to see that same graphic in my son's EM Reference Guide - not the workbook that they mostly keep at school. He is starting 4th grade. Up until now everything has been reasonably fine because the school supplemented EM. With their funny multiplication, division and unknown coverage of fractions, I think trouble is looming. In this same reference book, I found that the section on how to use the calculator was longer than the section on fractions. It looks like the book is a reference for multiple grades, but I need to verify that. The basics of some math topics were described well, but stopped way short of a full explanation. The fractions section, for example, didn't do a full job of explaining how to add, subtract, multiply, and divide fractions. I haven't seen the workbook yet.

"Isaacs and others defend the alternative algorithms by explaining that they teach students how math works. The partial product method of division, for example, is a lot more transparent to students than the long division method."

Of course they are wrong. It's all about doing something different. Not better, Just different. They just can't seem to do anything the traditional way. They find all sorts of excuses to do things differently. There are lots of good ways to show how multiplication works mathematically and the lattice method isn't one of them. However,how many kids really have conceptual problems with multiplication?

Speaking of Isaacs, you need to look at the following PDF file I found long ago. It gives their contorted justification or rationalization of EM. (We want to do something different, so let's come up will all sorts of reasons to really trash the traditional approach.)

Algorithms and Everyday Mathematics by Andrew C. Isaacs

http://everydaymath.uchicago.edu/educators/references.shtml

Besides, I thought that constructivist math really liked estimation. The traditional approach to division requires a good ability to estimate. The EM technique does not. I find it more ROTE and requiring less understanding of math. Isaacs once said on a math forum that EM was not for "elite" students.

"I haven't tried it with any children just learning long division, but if I ever get a chance to, I'll take notes."

I will let you know how it goes with my son. For division, using the example you show, there is not much difference, except that the traditional approach requires more ("elite"?) skill and understanding than the EM approach. I would just require my son to be able to guess the correct numbers, viz. 40 and 8, rather than 20, 20, 5, and 3. In that sense, their technique is fine because it lets the student know that they don't have to be perfect the first time. This seems to me to be a focus on making math simpler, rather than adding more understanding. One can always substitute a slower pace and simpler algorithms for a faster pace and greater coverage of material. A slower pace should allow for better understanding, but that is no guarantee. However, a slower pace will drop the student off in nowhere-ville in high school.

The lattice method, however, is just weird and ROTE. You can't tell me that it adds anything more to understanding than the traditional approach. It's different just to be different.

"The problem with the lattice method is that it's difficult to discern how the algorithm is based on the distributive property."

Exactly, but isn't EM supposed to focus on understanding? (ha ha) I find the method to be very ROTE! With the traditional method, I can break it down with the distributive property and show where each number comes from. Perhaps you can do this with the lattice method, but why would you do that? Different just to be different.

"Perhaps this deepens one's understanding of multiplication as repeated addition, but I think it is more likely just avoidance of multiplication."

Exactly. It's about simpler, rather than more understanding. The overriding attributes of all modern, fuzzy, constructivist math curricula are the simpler algorithms and slower coverage of material. Not for the "elite". How about not for what can be expected from typical children around the world.

-- SteveH - 13 Sep 2005