Traditional vs. Friendly Number Algorithm-A False Dichotomy

Let me start off this post by saying that I do not think that this is necessary for all students but it may be necessary for some. After seeing a couple of students who still cannot not carry out the traditional division algorithm (including writing down the acronym D.M.S.B.-divide, multiply, subtract, bring down), I feel it is necessary to present to these students a different way of looking at it. Now I believe that these students are beyond the repeated subtraction phase of division but have yet to grasp the underpinnings of the traditional algorithm (which is suppose to be faster).

So let us take a look at the traditional algorithm, using the problem 3672 divided by 8.

As you can see the first thing we would do is to see how many times 8 goes into 3 (which is really 3,000) and it doesn’t so you move to the next digit and figure out how many time 8 goes into 36. We determine 4 times, which is 32. You then subtract 32 from 36 and are left with 4. We are then taught to bring down the next digit (the B in “D. M. S. B.”) which is a 7. We then begin the process again-determine how many times 8 goes into 47, which is 5, 8 times 5 is 40, and 47-40 is 7. We bring down the last digit, 2, and repeat the process one more time, which leaves us with no remainder. This is fine as long as the student understands what is going on.

For example, I would want them to realize that when we are talking about 36 divided by 8 we are really talking about 3,600 and the choice of 4 is 400. When we bring down the next digit, that is what it is not a number. According to Merriam-Webster OnLine a digit is; “a: any of the Arabic numerals 1 to 9 and usually the symbol 0 b: one of the elements that combine to form numbers in a system other than the decimal system” and a number is; “ a sum of units : total c (1): a unit belonging to an abstract mathematical system and subject to specified laws of succession, addition, and multiplication.” For more information on digits and numerals you can go to Wikipedia.

Let’s take a look at the friendly number method. As you can see this method helps build on the strengths students already posses (rounding, multiplying by powers of 10, basic facts). It also looks similar to the traditional algorithm which would make the transition from the friendly number method to the traditional algorithm easier.

As point of emphasis, when working with struggling students, or using this for the first time, I would begin with 10 instead of 100. Students are usually strong with the 10-facts and they could then see how cumbersome and time-consuming it would be to use 10 over and over again. You could move them from 10 to 100, using their knowledge of powers of 10, then to any basic fact like in the second step (300 x 8 because they know 300 is 3 x 100 and therefore they can multiply 3 x 8 and 100 x 8).

As they become more comfortable and use this method they will begin to see the connection between the rounding and the numbers they choose to solve. At first they will use the rounding to see if their answer is “in the ballpark.” You can get them to see that they can use the estimate to help them right away. By that I mean that instead of starting with 100, they could choose 400 because they already know that 500 x 8 is 4,000 (inverse relationship between multiplication and division). They would therefore be as efficient as they can be and only need three steps to solve the problem as opposed to the 4 (above) or more.

Once they have became efficient and effective with this method you could transition them the friendly number to the traditional, if needed. The students can also compare and contrast the two methods to discuss the similarities and differences.

As you can see when dividing 36 by 8 you are really dividing 8 into 3,600. The same applies to the other two steps. The students can see how the 0’s have been eliminated in the traditional algorithm to further the comparison.

I think that for students who do not understand the traditional algorithm, this strategy can be beneficial.

* Division pictures written on ArtRage 2.5 Starter Edition

Number Strings

One technique or tool that I have been using this year with the students I educate, is the mini-lesson. Usually 10 minutes in length, they are designed to develop efficient mental math strategies. They are also designed to highlight certain strategies such as the distributive property.

Mini-lessons are generally done with the entire class. There are usually 5-8 problems and they are displayed one at a time. As each problem is revealed, students are asked to solve them with their hands (they can use pencil and paper to keep track or use a model). We want the students to examine the numbers and use them to efficiently solve problems. Each problem is related in that they are helping to promote an understanding of a mathematical idea (i.e.-commutative property, associative property, etc.).

I would like to take a look at an example of a number string that we used this year with a fifth grade class. We had some students who still were not fluent with the basic multiplication facts. Facts such as 6 x 8 were not be fluent and were not automatic. As I began to use the number strings, I noticed students were able to use what they knew and apply it to what they didn’t and many have a better recall of the basic facts.

One string we issued this year involved the following number facts (remember I would only put one problem up at a time, then students would solve and discuss):

3 x 4

3 x 8

6 x 8

12 x 4

24 x 2

48 x 1

3 x 16

As you can see, the first three problems are doubles of the previous problem. Students knew that 3 x 4 was 12 and that 3 x 8 was 24. Some did not recall 6 x 8 and had to resort to a counting-on strategy. There was not much discussion here as most had recall of these facts, so I would model next to the problem an array that depicted the problem.

The next three problems help promote the doubling/halving strategy. They all have the same answer, which helped begin a discussion about equivalence. The students I was working with did not see the doubling/halving relationship between the 6 x 8 and 12 x 4. They either knew automatically or used another strategy to solve the problems. When I put the 24 x 2 on the board to solve this is when the students began to make the connection. They could see that 24 was double 12 and 2 was half of 4 and each problem had the same product. A discussion then ensued about why 6 x 8 also had the same product. Students could see either through the numbers or the arrays that 6 was half of 12 and 8 was double 4 and that they had the same products.

I had facilitated the discussion about the arrays to help them see what was happening visually to what they were explaining to the class. As you can see, in the 12 x4 and 24 x 2 problems the 12 was doubled to 24 doubling the width of the array and the 4 was halved cutting the length of the array in half as well.

The last problem was no longer following the doubling/halving strategy but tripling and thirds which led to some good discussions.

Using these types of number strings has really helped students develop fluency and efficiency when computing.

Proportions

According to A Maths Dictionary for Kids 2008 a proportion can be defined as:

-a part to whole comparison

-the equality of two ratios, written as an equation

How do we know when to use a proportion and when not to. Consider these two problems:

Memphis and Texas were shooting an equal number of free throws. Memphis shot first. When they had shot 9 free throws, Texas had shot 3 free throws. When Texas shot 15 free throws, how many had Memphis shot?

My in-laws just returned from Spain. The exchange rate for 3 U.S. dollars with the euro in Spain is 3 dollars for 2 euros. How many euros for 21 U.S. dollars?

Both would seem to be proportion-type problems. They each have 3 numbers and one piece of information that is missing. So you could set up a proportion in the a/b = c/d format for each problem. There is a difference though. In the first problem there is no multiplicative relationship between the numbers whereas in problem two there is a relationship. For every 3 dollars you can receive 2 euros (3:2, 3 to 2, or 3/2). If you are exchanging 21 U.S. dollars you would get 14 euros. You could set up a proportion such as:

In order to solve this problem you can treat it as an equivalent fraction/ratio type problem and see that 21 is 7 times bigger than 3, so x has to be 7 times bigger than 2 (which is 14). As well, you could cross multiply

and end up with 3x = 21 x 2.

3x = 42
3 3
x = 42/3
x = 14

So why wouldn’t the Memphis (who at half did actual shoot 9 free throws!) /Texas problem be solved proportionally? The number of free throws that Memphis shot compared to Texas can be explained with addition or subtraction. Memphis free throws equal Texas free throws + 6 or Texas free throws equal Memphis free throws -6. If Texas shot 15 free throws, then Memphis must have shot 21. The difference between the Memphis and Texas free throws will be + or – 6 (depending on which way you look at it).

In the in-law trip problem, they were exchanging 3 dollars for 2 euros. Therefore we could find any number of dollars for euros by multiplying the dollars by 2/3 (21 x 2/3= 42/3 which equals 14). And the opposite can be calculated as well, we can multiply the euros by 3/2 (3/2 x 14 = 42/2 which equals 21).

In order to understand proportions, one needs to understand ratios (comparing two things) and rates (a ratio that compares quantities of different units). Ratios can be look at as part-to-part or even part-to-whole. For example to make a banana smoothie you might use 1 cup of milk and 2 bananas (1:2). What if I wanted to make smoothies for myself and three friends? Since that is 4 people, keeping the rate the same I would need 4 cups of milk and 8 bananas.

The Distributive Property

Students should have a deep understanding of the distributive property. By understanding the distributive property, students will be able to manipulate the traditional algorithm and understand what is happening when using it.

The distributive property states:
When a number is multiplied by the sum of two other numbers, the first number can be handed out or distributed to both of those two numbers and multiplied by each of them separately. Here’s the distributive property in symbols:
a * (b + c) = a * b + a * c.

By approaching multi-digit multiplication in this way, one can assume that you would help to eliminate the confusion regarding where to line the partial products up in the traditional algorithm. Let’s walk through a problem together.

As you can see, students should be able to see the connection between the distributive property and the traditional algorithm. When using the distributive property students are also working on place value, multiplication by 10, as well as strengthening their knowledge of basic facts. Once students have become comfortable with the distributive property and move into the traditional algorithm, educators can show them why the “zeros” can be eliminated.

But where do we begin? Third grade is where we formally begin multiplication. Students are developing the concept of multiplication and mastering their “facts.” Fourth grade is where students will move into 1-digit by 2-digit multiplication. This is a great time to begin to use the distributive property. Beginning the year with problems such as 14 x 6, students can be working on the structure of the distributive property (14 x 6 = (10 x 6) + (4 x 6)) as well as their mental math. As the year progresses, and students become more facile with the ideas being developed, they can work with more difficult numbers.

Whether or not it is fourth or fifth grade, students should then move to 2-digit by 2-digit multiplication. This is likely to take more time to develop then one-digit by two-digit multiplication. I also thinks that once students comprehend the 2 by 2 digit multiplication, they will easily be able to apply their knowledge to 3 by 3-digit multiplication. Like the above illustrations show, work with the distributive property first, then relating it to the traditional algorithm demonstrates what actual happens in the traditional algorithm. These types of experiences help to bridge the distributive property and the traditional algorithm.

I am working on a presentation for next week on a mathematics intervention program called Knowing Mathematics and I was thinking about the “Across-10 facts.” If students leave first grade not knowing these facts because they have not internalized them at that time do we just repeat the same procedure next year and the year after that and the year after that (more of the same facts on a worksheet, over and over again until they do internalize)?

One would hope not and this is where the differences in the approach to the teaching of mathematics will arise. As the motto of the blog, Educational Insanity states: “Insanity: doing the same thing over and over again and expecting different results.” (quote by Albert Einstein) having students do the same thing over and over again whether it is from a conceptual or procedural standpoint makes no sense. How do we provide students with the tools necessary to master what they don’t know?

So let’s assume we have a student in second grade or up who has not yet mastered the single-digit combinations of nine. Let us also assume that they have had their share or worksheets and activities that support his development. A different approach we could is to have the students look for a pattern within the numbers.

What is it that we would want the student to notice in column 1? The digit in the ones place of the sum is always one less than the second addend. Columns 1 & 2 are related facts (a.k.a-Commutative Property of Addition. So if we know one “fact” we really know two).

How is the third column is related to the first column? We would want to bring out in the students that subtraction is the inverse of addition (important to understand, not just know as students move into algebra and solve equations) and that each equation in each row is related to each other.

As you can see what goes on in the early grades supports what is happening in the later grades. Therefore we cannot rely on any one approach to teaching, but we need to use what is going to help that particular student.

By working through these patterns and applying them in different situations (i.e.-practice, problem solving, at the store. etc.) students very well may have a better understanding and mastery of these particular facts. In the end we want students to not only internalize these “across-10 facts” but to know how to apply their understandings to new situations. If we are computing 39+ 27, we know it is the same as 40+26 because we can compose 39 + 1 more from the 27 to make it 40 and 26 (all should be done mentally) which is 66. If we have 37-29, we know that we could subtract 30, which leaves us with 7 and add 1 for a difference of 8 (again done mentally).

Solving 2-Step Equations

(Clip art licensed from the Clip Art Gallery on DiscoverySchool.com and free-clipart.net.)

It has got to be about the order of operations. Otherwise it will not work. Does that apply to all situations?

You Need Concept and Skill

This is part of an actual conversation during an “AIS” math class. If you are unfamiliar with “AIS,” it stands for Academic Intervention Services. The AIS math class is designed to help students who are struggling with various math concepts (we have ELA AIS as well). Most students end up in AIS because of a below proficient score on the previous years state exam. I am not necessarily opposed to testing per se, as long as it is used appropriately. If we are teaching using best practice and developing students at their level, the tests are something we will just deal with as the time comes. One would think that just by listening to the conversation, that these particular students do not need remediation for this particular topic.

I believe that the teaching of concept and skill are linked and cannot stand alone in isolation. One without the other would be like peanut-butter without the jelly. Each decent on their own, but together a classic. As we pilot, we must look at what will be the best fit for our students. We need to take what we know is best practice and incorporate the new program with that.

For example, when developing the idea of division of fractions. Some teachers will just teach “invert and multiply.” Others will just teach it through a picture representation. Well what happens when students move on to more difficult fractions and algebra? Why aren’t we discussing the generalized formula and the inverse operation of division, multiplication. What do I mean?

The focus needs to be on the students and teaching them what they need, not a test, a program, or a workbook.

About one week ago, and due out this week, the National Mathematics Advisory Panel has created a report that claims to have found the way to fix the math problems in the U.S.

The answer: “A laser-like focus on the essentials.”

It appears as if the recommendations include:

• mastery of arithmetic in the early grades and fractions in middle school
• teachers focusing on a smaller number of topics
• textbooks should be shortened to cover fewer math topics.

Mastery of arithmetic, according to the panel, means that students should be proficient with the addition and subtraction of whole numbers by the end of the third grade and with multiplication and division by the end of the fifth grade.

My questions include:

What does the panel mean by mastery? by proficient?

What do they mean by multiplication and division of whole numbers? Are they talking about dividing a three-digit divisor by a six-digit dividend?

With regards to fractions, I am assuming mixed numbers as well, but are they referring to benchmark fractions? fractions such as 17/33 and 62/87?

What about decimals and percents? And the relationships between the two and fractions?

I will be interested to hear more.

The second and third bullets are ideas that have been thrown around for some time now. If you are from New York and have some relationship to the teaching of mathematics, that sounds just like the Curriculum Focal Points. The Curriculum Focal Points were designed for pre-K through 8th grades. The C.F.P. are the most important topics at each grade level. They include the major concepts, skills, and procedures that will produce mathematically literate students.

Here is one of the Curriculum Focal Points from Grade 5:

Number and Operations and Algebra: Developing an understanding of and fluency with division of whole numbers.
Students apply their understanding of models for division, place value, properties, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multidigit dividends. They select appropriate methods and apply them accurately to estimate quotients or calculate them mentally, depending on the context and numbers involved. They develop fluency with efficient procedures, including the standard algorithm, for dividing whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems. They consider the context in which a problem is situated to select the most useful form of the quotient for the solution, and they interpret it appropriately.”

As you can see, this is asking students to go well beyond being able to recite 12 divided by 4 equals 3 (which they must know anyway). Students must be able to apply the basics to different contexts and be able to reason why their answer is reasonable.

The third bullet is directly related to the second bullet, which for me means we need to take a long, hard look at the Curriculum Focal Points and how we implement whatever pilot program we are testing now. With testing just about over here in POB, we turn our attention to the Post-March topics and the pilot programs. As we move forward and get closer to choosing a program, we need to spend some time considering and discussing the points that have been suggested in the draft of the National Mathematics Advisory Panel.
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