This entry will be a bit different from others. Instead of providing my commentary on specific GPS statements, I want to talk about the challenge of creating a focused and rigorous school mathematics curriculum. According to the Executive Summary of the GPS online, the state Board of Education asked the Department of Education (DoE) to write a new standards (GPS) with four specific charges. Those charges were,

(1) The curriculum needs to be rigorous…

(2) The curriculum needs to be focused…

(3) The curriculum needs to be clearly understandable by teachers…

(4) Instruction needs to be student-centered…

So, the GPS is supposed to be standards for a focused and rigorous curriculum that is easily understood by teachers and implementable (is this a word?) through student-centered instruction. There are many questions we can ask about the GPS based on these charges, but today, I want to focus on a “focused and rigorous curriculum.” In fact, much of my attention will be on a “rigorous” curriculum and how difficult it is to actually create one.

What does a rigorous curriculum look like? In our everyday language, the word “rigorous” is often used interchangeably with the words like “difficult” or “challenging.” Clearly, we want a curriculum that is (appropriately) challenging to our students. However, speaking of mathematics and mathematics education, the word “rigor” or “rigorous” should also mean something else. When mathematicians use the word “rigor,” what they typically refer to is logical cohesiveness of their arguments. Thus, a rigorous mathematics curriculum must be logically cohesive and coherent. There must be logical sequencing of topics.

This idea seems to be so obvious and easy to accomplish, but it is not that simple because it requires us to know the standards very well. It is very easy to overlook something while creating a curriculum. A good example can be found in the Mathematics Framework, a document that was created by the DoE “to be models for articulating desired results, assessment processes, and teaching-learning activities that can maximize student achievement relative to the Georgia Performance Standards.” So, these documents are supposed to be illustrative of a rigorous curriculum. Unfortunately, there are some oversights.

For example, look at the “Quotient if Greater Than One” task found in the Grade 3 multiplication and division unit. The task gives the following expression: []/^ > 1 (it's supposed to be a box over a triangle), and ask students questions such as “If the dividend is 10, what is the largest number the divisor can be? Why is that true?” and “If the divisor is 6, what is the smallest number the dividend can be? Tell why.” In the discussion section, the Framework states the following:

“This activity requires that students prove their answers to be true. Just a simple numerical answer is not enough. Help students articulate their thinking with correct math vocabulary and make sure they understand the connection between division and fractions. Students should be able to write any division problem as a fraction and for any division problem that has a remainder they should be able to write the remainder as a fraction also.”

Although these goals are very important goals for our students, looking at these goals in Grade 3 is totally inappropriate within the GPS. In the GPS, understanding that a fraction indicates the division of the numerator by the denominator is a Grade 5 topic (M5N4a). Although it is not quite clear in the GPS when students should learn to express remainders as fractions, certainly we can’t expect them to understand this idea until they have made this connection between division and fraction. Thus, this task, though it may be perfectly appropriate and useful task in Grade 5, is inappropriate for Grade 3 because it requires students to use something they have yet to study. It is a good example how difficult it is to create a rigorous curriculum, that is not only challenging but also logically cohesive.

## Monday, September 17, 2007

## Friday, September 7, 2007

### M5N4 (b) - Equivalent Fractions (2)

M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them.

a. Understand division of whole numbers can be represented as a fraction (a/b= a ÷ b).

b. Understand the value of a fraction is not changed when both its numerator and denominator are multiplied or divided by the same number because it is the same as multiplying or dividing by one.

In the previous post, we looked at the idea of equivalent fractions. According to the GPS, students are expected to become aware that two fractions that look different may represent the same number, i.e., the concept of equivalent fractions, in Grade 4, while they are expected to understand how to create equivalent fractions by multiplying or dividing both the numerator and the denominator by the same number in Grade 5 (M5N4b). In this post, I would like to look at the reason why this procedure works.

In M5N4b, it is stated that this procedure works “because it is the same as multiplying or dividing by one.” This explanation makes perfect sense for us since we already know how to multiply or divide by fractions. For example,

However, this explanation is inappropriate for Grade 5 students because they have not studied how to multiply fractions by fractions. In Grade 5, students are expected to “model the multiplication and division of common fractions” (M5N4d). However, learning the procedures/algorithms for multiplying and dividing fractions is a Grade 6 expectation. Furthermore, even if students were to study the algorithms for multiplication and division of fractions in Grade 5, most likely they will be studying the procedure of creating equivalent fractions before they study multiplying and dividing fractions because they need to have this procedure to add and subtract fractions with unlike denominators. So, how can students understand that why this procedure for creating equivalent fractions work using only what they have already studied? Or, do they simply have to accept this procedure for now and justify it later?

Although there may be some topics in school mathematics where students may have to accept a formula or an algorithm just because teachers tell them it works, this is NOT such an occasion. Students do have something they have studied to justify this procedure. They key for understanding this procedure is M5N4a, which expands the meaning of fractions from simply a part of a whole or a collection of unit fractions (see my earlier post on the meaning of fractions). According to M5N4a, 5th grade students are expected to understand a fraction represents the answer for division of a whole number by another whole number. For example,

Moreover, in Grade 4, students are expected to “understand and explain the effect on the quotient of multiplying or dividing both the divisor and dividend by the same number” (M4N4d). Therefore, if you multiply or divide both 3 and 5 in the above example by the same number, the quotient does not change. For example,

But, this last expression, according to M5N4a, is the same as because a fraction represents the quotient of a whole number divided by another whole number. Or, in general,

, for any whole number .

The Georgia Performance Standards emphasizes that students should construct new mathematics understanding based on what they have previously studied. However, this is much easier said than done. Because we are already very familiar with mathematics students are still learning, we can easily overlook the fact that we slipped in something students have not studied yet. M5N4b is a very good illustration how difficult it is to create a rigorous, i.e., logically cohesive, curriculum.

a. Understand division of whole numbers can be represented as a fraction (a/b= a ÷ b).

b. Understand the value of a fraction is not changed when both its numerator and denominator are multiplied or divided by the same number because it is the same as multiplying or dividing by one.

In the previous post, we looked at the idea of equivalent fractions. According to the GPS, students are expected to become aware that two fractions that look different may represent the same number, i.e., the concept of equivalent fractions, in Grade 4, while they are expected to understand how to create equivalent fractions by multiplying or dividing both the numerator and the denominator by the same number in Grade 5 (M5N4b). In this post, I would like to look at the reason why this procedure works.

In M5N4b, it is stated that this procedure works “because it is the same as multiplying or dividing by one.” This explanation makes perfect sense for us since we already know how to multiply or divide by fractions. For example,

However, this explanation is inappropriate for Grade 5 students because they have not studied how to multiply fractions by fractions. In Grade 5, students are expected to “model the multiplication and division of common fractions” (M5N4d). However, learning the procedures/algorithms for multiplying and dividing fractions is a Grade 6 expectation. Furthermore, even if students were to study the algorithms for multiplication and division of fractions in Grade 5, most likely they will be studying the procedure of creating equivalent fractions before they study multiplying and dividing fractions because they need to have this procedure to add and subtract fractions with unlike denominators. So, how can students understand that why this procedure for creating equivalent fractions work using only what they have already studied? Or, do they simply have to accept this procedure for now and justify it later?

Although there may be some topics in school mathematics where students may have to accept a formula or an algorithm just because teachers tell them it works, this is NOT such an occasion. Students do have something they have studied to justify this procedure. They key for understanding this procedure is M5N4a, which expands the meaning of fractions from simply a part of a whole or a collection of unit fractions (see my earlier post on the meaning of fractions). According to M5N4a, 5th grade students are expected to understand a fraction represents the answer for division of a whole number by another whole number. For example,

Moreover, in Grade 4, students are expected to “understand and explain the effect on the quotient of multiplying or dividing both the divisor and dividend by the same number” (M4N4d). Therefore, if you multiply or divide both 3 and 5 in the above example by the same number, the quotient does not change. For example,

But, this last expression, according to M5N4a, is the same as because a fraction represents the quotient of a whole number divided by another whole number. Or, in general,

, for any whole number .

The Georgia Performance Standards emphasizes that students should construct new mathematics understanding based on what they have previously studied. However, this is much easier said than done. Because we are already very familiar with mathematics students are still learning, we can easily overlook the fact that we slipped in something students have not studied yet. M5N4b is a very good illustration how difficult it is to create a rigorous, i.e., logically cohesive, curriculum.

## Sunday, September 2, 2007

### M4N6 (a); M5N4 (b) & (c) - Equivalent Fractions (1)

M4N6. Students will further develop their understanding of the meaning of common fractions and use them in computations.

a. Understand representations of simple equivalent fractions.

M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them.

b. Understand the value of a fraction is not changed when both its numerator and denominator are multiplied or divided by the same number because it is the same as multiplying or dividing by one.

c. Find equivalent fractions and simplify fractions.

In an earlier post, I discussed different meaning of fractions. The fact that fractions may be interpreted in many different ways is a major reason why fraction is so difficult to teach and learn. Another major difference between fractions and whole numbers is the fact that fractions that look different can represent the same number – we call those fractions “equivalent fractions.” In the Georgia Performance Standards for school mathematics, the idea of equivalent fractions first appear in Grade 4, M4N6 (a). However, it should be noted that the goal in Grade 4 is that students become aware of the fact that two fractions that look different may represent the same number. Students are to understand this idea and be able to demonstrate their understanding by representing those fractions to show their equivalence. Understanding how to create equivalent fractions by multiplying (or dividing) both the numerator and the denominator by the same number is a Grade 5 standard, M5N4 (b) and (c). Consequently, answering problems involving fractions with the simplest form should NOT be a focus in Grade 4. Thus, for example, if students calculate 3/4 – 1/4 in Grade 4, the answer should be written as 2/4, not 1/2. Clearly, some students will understand that 2/4 and 1/2 are the same based on their study of equivalent fractions. Therefore, if they do present their answers in the simplest form, that is ok. However, we should not penalize students who do not present their solutions in the simplest form. That emphasis should begin only after students understand the procedure for creating equivalent fractions.

a. Understand representations of simple equivalent fractions.

M5N4. Students will continue to develop their understanding of the meaning of common fractions and compute with them.

b. Understand the value of a fraction is not changed when both its numerator and denominator are multiplied or divided by the same number because it is the same as multiplying or dividing by one.

c. Find equivalent fractions and simplify fractions.

In an earlier post, I discussed different meaning of fractions. The fact that fractions may be interpreted in many different ways is a major reason why fraction is so difficult to teach and learn. Another major difference between fractions and whole numbers is the fact that fractions that look different can represent the same number – we call those fractions “equivalent fractions.” In the Georgia Performance Standards for school mathematics, the idea of equivalent fractions first appear in Grade 4, M4N6 (a). However, it should be noted that the goal in Grade 4 is that students become aware of the fact that two fractions that look different may represent the same number. Students are to understand this idea and be able to demonstrate their understanding by representing those fractions to show their equivalence. Understanding how to create equivalent fractions by multiplying (or dividing) both the numerator and the denominator by the same number is a Grade 5 standard, M5N4 (b) and (c). Consequently, answering problems involving fractions with the simplest form should NOT be a focus in Grade 4. Thus, for example, if students calculate 3/4 – 1/4 in Grade 4, the answer should be written as 2/4, not 1/2. Clearly, some students will understand that 2/4 and 1/2 are the same based on their study of equivalent fractions. Therefore, if they do present their answers in the simplest form, that is ok. However, we should not penalize students who do not present their solutions in the simplest form. That emphasis should begin only after students understand the procedure for creating equivalent fractions.

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## Creative Commons

Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.