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p.s. Sorry about the format. All the pieces that are followed by page numbers should be shown as block quotes but there’s some problem with my browser and the editing interface.
Here are comments on two issues: Item format and assessment guidelines.
Item format. I notice that the format used for test items is vertical (e.g., one addend written above another) rather than horizontal (e. g., expressions or equations). The CCSS do not mention this format for K. Vertical format is not used in many countries until multi-digit addition. It appears in the NBT progression at grade 1.
More explicitly, K.OA.1 says:
Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
Assessment guidelines. There are guidelines for assessment in the National Research Council report Early Childhood Assessment: Why, What, and How which includes ages 0 to 5 years old. It can be read online here: http://www.nap.edu/catalog.php?record_id=12446. These say:
Those selecting assessments should be familiar with the assessment standards contained in the standards document produced by the American Educational Research Association, American Psychological Association, and National Council on Measurement in Education (1999). Important questions to ask are: Has this assessment been developed and validated for the purpose for which it is being considered? If a norm-referenced measure is being considered, has the assessment been normed with children like those with whom it will be used? (p. 350)
Implementation:
problems with implementation can pose a challenge to the validity of the data obtained. A poorly trained assessor or a child so distracted that she does not engage with the assessment fully, for example, can lead to questionable data. (p. 281)The following criteria, developed by the committee, operationalize . . . principles in important characteristics that child outcome measures should have if they are to provide useful evidence for the improvement of early care and education systems.
- A clearly articulated purpose for the testing.
- Identification of why particular assessments were selected in relation to the purpose.
- A clear theory connecting the assessment results and quality of care.
- Observation of quality of instruction and specification of what would need to be focused on for improvement.
- A clear plan for following up to improve program quality.
- Strategizing to collect the required information with a minimum of testing.
- Appropriate preparation of testers to minimize disruptive effects on child responses.
(p. 338)
I think that part of the issue is that students need to build their understanding of number line before they get to fractions in later grades. They may be able to function without it earlier if they don’t count 0 as discussed below.
Here are some relevant quotes about from the National Research Council report Mathematics Learning in Early Childhood. This has some discussion of the difference between a “number path” and a “number line” (which CCSS calls a “number line diagram”). This can be read online here: http://www.nap.edu/openbook.php?record_id=12519&page=167
From page 167:
children in kindergarten and Grade 1 are using the number word list (sequence) as a count model: Each number word is taken as a unit to be counted, matched, added, or subtracted. In contrast, a number line is a length model, like a ruler or a bar graph, in which numbers are represented by the length from zero along a line segmented into equal lengths. Young children have difficulties with the number line representation because they have difficulty seeing the units—they need to see things, so they focus on the numbers instead of on the lengths. So they may count the starting point 0 and then be off by one, or they focus on the spaces and are confused by the location of the numbers at the end of the spaces.
From page 168:
The number line is particularly important when one wants to show parts of one whole, such as one-half. In early childhood materials, the term number line or mental number line often really means a number path, such as in the common early childhood games in which numbers are put on squares and children move along a numbered path. Such number paths are count models—each square is an object that can be counted—so these are appropriate for children from age 2 through Grade 1.
Duane, I think that your earlier comments are partly about the meaning of “the” in 4.MD.3:
Apply THE area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
It’s fairly easy to see l x w and w x l as being two variants of the area formula, if you’re aware of the commutative property.It may be less easy to see the various forms of the perimeter formula as the same because they involve the distributive property. That’s also complicated by the way in which the word “formula” is used in the text and discussion of “various formulas” on p. 21. (Sorry about this! My job is to try to catch these things, but remember the drafts of the progressions are called drafts for a reason.)
Note that the text on p. 21 of Geometric Measurement progression draft talks about “specific numerical instances” of the various different forms of the perimeter formula. That’s one reason to write the equation with the different forms as an adult would, without the multiplication symbol. Anyway, your comment suggests that the progression needs to give more detail, perhaps with an example in the sidenotes.
Here are suggestions from an earlier post of Bill’s (http://commoncoretools.me/2011/08/12/drafty-draft-of-fractions-progression/):
“If you are reading it with Adobe Acrobat, you will need the latest version (10.1). If you are using a Mac, you can also use the native Mac pdf reader Preview, or the open source pdf reader Skim.”
