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We’d like some clarification about 4.MD.2 , which reads as follows:
4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as the number line diagrams that feature a measurement scale.
Our question relates to the “including problems involving simple decimals” portion of this standard.
The Grade 4 standards in “Number and Operations in Base 10” involve the four operations and multidigit whole numbers. Operations with decimals (to hundredths) do not appear until Grade 5, which has the following standard:
5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Grade 4 does have three other standards involving decimals.
4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. (Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.) For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
N.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.Are students expected to use 4.NF.5 and 4.NF.6 to solve word problems involving the addition of simple decimals? Is the intention something like the following?
Problem: “Four women ran the 4 x 100 meter relay. Their leg times were: 10.6, 11.33, 11.9, and 9.98 seconds. How long did it take the team to finish the race?”
Possible solution strategy:
Rewrite each decimal as a mixed number: 10 6//10, 11 33//100, 11 9/10, 9 98//100.
Express the fraction portion with a denominator of 10 as an equivalent fraction with denominator 100: 10 60//100 and 11 90//100.
Per 4.NF.B.3c, add the mixed numbers with the like denominator 100 by adding the whole numbers together, adding the fractions together, and then combining the whole number and fraction.
10 60//100 + 11 33//100 + 11 90//100 + 9 98//100
10 + 11 + 11 + 9 = 41
60//100 + 33//100 + 90//100 + 98//100 = 281//100 = 2 81//100
41 + 2 81//100 = 43 81//100.
Use decimal notation for the mixed number. 43 81//100 = 43.81 seconds.
It would be helpful if you would confirm whether this is what you intend — because it does appear to us that this is what you want from the way you have written the standards.
Assuming that you do intend something such as the above for addition, how are students supposed to deal with subtraction, multiplication, and division with simple decimals, as called for in 4.MD.2?
Andy
Your method of solving the 4 x 100 meter relay problem certainly fits with the Grade 4 standards, although I also think it also illustrates that the problem is challenging for Grade 4. And it seems to me one could stage things differently; perhaps developing first the understanding that 10.6, 10.60 and 10 6/10 are all different ways of writing the same number, and then adding the numbers in decimal form more directly. Decimals are regarded in the standards as a different way of writing fractions, not as different sorts of numbers from fractions, so the phrase “a fraction with denominator 10” in 4.NF.5 refers equally to 0.6 or 6/10, and an “equivalent fraction with denominator 100” could be written equally as 0.60 or 60/100. Thus, after having become familiar with the meaning of decimal notation one might write the solution to the word problem as 10.6 + 11.33 + 11.9 + 9.98 = 10.60 + 11.33 + 11.90 + 9.98 = 10 + 11 + 11 + 9 + 0.60 + 0.33 + 0.90 + 0.98 and then add the whole numbers and the fractions with denominator 100 as you have indicated. By Grade 5 one might leave out the fraction notation altogether, as indicated by 5.NBT.7.
As to your question about subtraction, multiplication and division, it is a good idea to put the standard in the context of all the other standards at the grade level. Multiplication of fractions is not completed until Grade 5, nor division until Grade 6, so 4.MD.2 should not be construed as introducing additional content. One might make this a general principle in reading the standards; reading standards in isolation can lead to nonsense. It is neither a requirement of English grammar nor in keeping with the weight and focus of the other standards in Grade 4 to produce 48 distinct standards by multiplying the 3 lists
- “four operations”
- “intervals of time, liquid volumes, masses of objects, and money”
- “including simple fractions or decimals” [taking this to mean whole numbers, fractions and decimals].
Thus, word problems involving multiplication and division of decimals are not implied by 4.MD.2.
Subtraction however does not strike me as completely excluded since students since Grade 1 have been encouraged to “understand subtraction as a missing addend problem” (1.OA.4).
Thanks, Bill. I appreciate your response. Here are some follow-up questions. You wrote:
“As to your question about subtraction, multiplication and division, it is a good idea to put the standard in the context of all the other standards at the grade level…One might make this a general principle in reading the standards; reading standards in isolation can lead to nonsense.”
My follow-up questions concern several of the fraction standards at fourth grade. Start with 4.NF.C.5, which reads as follows:
4.NF.C.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
Based on your response to my previous question, we might want to assume that in 4.NF.C.5 the phrase “add two fractions” actually means “add and subtract two fractions,” based on the link between subtraction and missing-addend addition you cite. But if we put this standard in the context of two other Grade 4 fraction standards, 4.NF.B.3c and 4.NF.B.3d, we are puzzled. Given the content of 4.NF.B.3c and 4.NF.B.3d, both of which mention both addition and subtraction explicitly, it is notable that 4.NF.C.5 would specifically state addition while excluding subtraction.
4.NF.B.3c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
4.NF.B.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.Here’s the problem. On the one hand, you say that “reading standards in isolation can lead to nonsense,” which seems perfectly reasonable. On the other hand, we have the Publishers’ Criteria, which requires a quite literal reading of CCSS. See, for example, the following statements from the Publishers’ Criteria:
· “Failing to meet any single focus criterion is enough to show that the materials in question are not aligned to the Standards.” (p. 6)
· “Applications in the material draw only on content knowledge and skills specified in the content standards.” (p. 11)
· “MP.5 does not say, ‘Use tools.’ Or ‘Use appropriate tools.’ It says ‘Use appropriate tools strategically.’” (p. 14)
· “… the Standards were crafted to reward study on multiple levels … Specific phrases in specific standards are worth study.” (p. 20)
These statements certainly seem to require very literal reading of CCSS. So, back to 4.NF.C.5. If we attend to the Publishers’ Criteria’s requirements, particularly the last one above, which recommends attention to “specific phrases in specific standards,” we might now assume that there is a specific reason for not including subtraction in 4.NF.C.5. We might therefore also conclude that any fourth grade instructional materials that included subtraction of tenths and hundredths would not be aligned to the Standards.
So, my questions, one global and one specific, are:
1) How are we to know when a narrow, literal reading of a standard is required and when a more expansive, integrative reading is called for? Or, more to the point, how are practitioners in the schools to know?
2) In regard to 4.NF.C.5, is a narrow, literal reading required (add two fractions with respective denominators 10 and 100) or is a more expansive, integrative reading (add and subtract two fractions with respective denominators 10 and 100) called for?
I look forward to your insights.Andy, I sense a certain frustration on your part here, but thanks anyway for bringing these matters up … that’s what this blog is for.
First, I would say that there’s a difference between the case of 4.MD.2 and the case of 4.NF.5, in that the former was open to more than one interpretation. In my earlier reply I advocated using other standards in Grade 4 to decide the interpretation of 4.MD.2, including the Grade 1 understanding of subtraction. In particular, 4.MD.2 should not be interpreted as creating additional operations not already covered by the other operation standards in Grade 4. And note that I didn’t really need to cite the Grade 1 standard because, as you point out, there is warrant for subtraction of fractions in 4.NF.3. However, none of this interpretation work requires a change in the wording of 4.MD.2; it is a question of whether you interpret a standard made up of lists by multiplying all the lists together or by selecting the combinations that are warranted by the other standards.
The case of 4.NF.5 is different, because it’s pretty clear what the words mean, and it doesn’t say “subtract.” As you point out, other standards do say “add and subtract” and this one doesn’t. I agree with you that it’s unfair to expect curriculum writers to interpolate a word that isn’t there, at the same time as admonishing them to stick to the standards, so I would say you should take this standard as written. Note this does not exclude something like 0.7-0.3, which is already covered by 4.NF.3.
The question remains, why limit to addition in 4.NF.5 and not in 4.NF.3? One possible reason is that 4.NF.5 is the only Grade 4 standard where we are adding fractions with different denominators. Given the extra conceptual demand involved, it is perhaps a good idea to limit to addition.
P.S. I don’t think I am being inconsistent in the level of expansiveness advocated; I am advocating a non-expansive interpretation of both 4.MD.2 and 4.NF.5.
Dear Bill,
We have been working non-stop for almost three years to create faithful enactments of the CCSS — working to translate the standards into a viable classroom reality. Through this work, we have discovered a number of inconsistencies or ambiguities, such as the one we discussed here. I would be happy to share other examples with you if you would like.
Nobody would expect the standards to be perfect documents. However, the bigger issue right now is that there is no formal mechanism to raise these kinds of concerns, short of your blog. This strikes me as not healthy for the standards or for the field. The standards could benefit from a clear understanding of how these kinds of issues can get raised and a process for considering issues raised by the field and for updating (or correcting) the standards as indicated. I personally would like to see the ongoing development of the standards reside with an independent organization, such as NCTM or MAA, or perhaps with a designated coalition of professional organizations. That could open the door to a formal mechanism where practitioners, assessment developers, curriculum developers, and others to raise questions about the standards and get the kinds of clarifications that we have been forwarding to you.
I’d be happy to continue this conversation with you, though I think it preferable to continue the discussion face-to-face, rather than via email or blog. We invite you to visit us at CEMSE, where we can continue the discussion and also illustrate how we have been working on updating EM in response to the CCSS-M. Please let me know if you are interested and available.
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