Friday, December 2, 2011

Measurement in K-2

K.MD Describe and compare measurable attributes
1.MD Measure lengths indirectly and by iterating length units
2.MD Measure and estimate lengths in standard units

I wrote about teaching of measurement in primary grades almost 3 years ago (Dec. 2008), In the post, I stated that there are 3 related yet distinct goals while teaching measurement:
* understanding the attribute being measured
* process of measurement
* how to use measuring instruments.

Also, there is a general consensus that teaching of measurement should proceed along the following instructional sequence:
1. Direct comparison
2. Indirect comparison
3. Measuring with non-standard units
4. Measuring with standard units

From this perspective, I wrote that the GPS was unclear about steps 2 and 3. In contrast, the Common Core follows the suggested sequence explicitly, at least with the attribute of length. For other attributes like (liquid) volume, mass, area, angle, etc., the CCSS appears to jump right in with step 4, measuring with standard units. For example, 3.MD.2 states, "Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l)." Although I certainly do not advocate going through the 4 steps of measurement instruction with every attribute, I am not sure if experiences with one attribute (length) is enough for children to generalize the process of measurement. In the typical Japanese curriculum, children study length and capacity (liquid volume) in Grades 1 and 2 (the first 2 years of elementary schools in Japan), and they go through these 4 steps with each attribute. They will also include some direct comparison activities with comparison of areas before they study how to calculate area. As we move ahead with the implementation of the CCSS, we may want to include some comparison activities as well as measuring with non-standard units for some of the attributes. Furthermore, explicit discussions on the process of measurement (selecting a unit, using a unit to iterate/cover the object, count the number of units) so that with the later attributes, we can start with the question of what we should use as a (standard) unit.

Sunday, October 9, 2011

Number and Operations in Base Ten (1.NBT.4~6)

In the Grade 1 Number and Operations in Base Ten domain, you will see the following cluster.

Use place value understanding and properties of operations to add and subtract.
4. Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, 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. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
5. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
6. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), 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.

As I discussed briefly in the previous post, these standards are based on students' understanding of ten as a unit, which are the major focus of 1.NBT.2. While in Kindergarten, students considered 10 as a "benchmark" or a "mile marker" to understand numbers 11 through 19, in Grade 1, students learn to count ten, that is, ten as a unit. Thus, extending the idea of composing and decomposing number further, instead of thinking 64 as just 60+4, we want students to think of 60 as 10+10+10+10+10+10, or six 10's. Being able to coordinate two units, i.e., ones and tens, is not a trivial task for young children. Many children at this age can recite the sequence of decade number words correctly, "ten, twenty, thirty, forty, ..." However, some children will have to count by ones to answer, "what is 10 more than 36?" For many young children, the decade number word sequence is just a memorized set of words with no numerical significance. They do not yet understand when they go from "thirty" to "forty," the number increased by 10. Thus, these standards, although they are based on 1.NBT.2, are not necessarily something that comes after students master 1.NBT.2. As students think about addition of 2 multiples of 10 or subtract a multiple of 10 from another multiple of 10, we want to encourage students to think in terms of 10's. So, questions like, "how many 10's are in __?" and "how many 10's are we adding (or taking away)?" must be an important part of teachers' questioning repertoire.

Monday, September 5, 2011

Number and Operations in Base Ten (K.NBT.1 & 1.NBT.2)

Number and Operations in Base Ten (K.NBT.1 & 1.NBT.2)
At the end of the last post, I briefly touched upon the idea of composing and decomposing numbers 11 through 19. This idea is discussed in both Kindergarten and Grade 1.

K.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones—called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

What is important to note here is the slight difference in these two standards. In Kindergarten, students are thinking of numbers 11 through 19 as "ten ones and some further ones" while in Grade 1, students need to develop an understanding of 10 as "a 'ten.'" In other words, in Grade 1, students need to develop 10 as a unit - at the same time it is a collection of ten ones. Research has shown that this understanding is a major shift, and some might argue that this expectation is not developmentally appropriate for most first graders. Children can easily learn to recite the number word sequence, "ten, twenty, thirty, ... ninety," but just as simply reciting "one, two, three, four, ..." does not necessarily indicate an understanding of numbers, the ability to recite the decade number words in order does not indicate the understanding of ten as a unit (1.NBT.2.c).

In historical numeration systems, the idea of grouping by 10's, 100's, etc. appears fairly early. In those systems, 20, 30, 40, ... were recorded with multiple symbols of 10's instead of saying how many 10's. Even in the systems that utilized place values like the Babylonian System, 20, 30 , 40 were recorded with multiple symbols of 10's just as simpler additive systems did. Thus, even in those systems, 30, for example, meant 10+10+10, not three 10's (or 3x10). This shift, although it might look rather simple for those of us who already understand the base-10 numeration system, is not that obvious for children. For them, "10" doesn't naturally mean 1 tens and 0 ones. Rather it is just like a word "cat" spelled with multiple letters. "10" is just "ten" spelled with 2 numerals "1" and "0." Thus, it is not logical that twenty should be spelled as "20" - even if they understand twenty is made up of 2 tens. After all, there is no logical connection (in how they are written) going from 1 to 2 ones. Although this standard puts this understanding of ten as a unit in focus, we should keep in mind that students will not develop this understanding in one single lesson. In fact, this understanding will probably take months to develop - perhaps stretching into Grades 2 and 3. We should keep this in mind as we look at other NBT standards in Grade 1 - they are, in part, serving to achieve this standard even though their focus may be elsewhere.

Monday, July 25, 2011

Kindergarten: Operations and Algebraic Thinking (2)

Kindergarten: Operations and Algebraic Thinking (2)
In the previous post, I discussed the difference in the meaning of subtraction between the CCSS and the current GPS and its potential implications. In this post, I would like to begin the discussion of the five specific standards in the cluster. Those standards are as follows:

1. Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
4. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
5. Fluently add and subtract within 5.

There is a footnote for the term "drawings" in Standards 1. The footnote states, "Drawings need not show details, but should show the mathematics in the problem." It is easy to read this statement and think that we are making things simpler for children since we are asking them to do less (not showing details). However, for those of us who work with primary students know that this is not quite as simple as it may sound. In fact, many of us have experiences of watching children draw very detailed drawings when they are asked to "draw pictures" to help them solve word problems. In order for children to draw pictures that "show the mathematics in the problem," children must understand first what features of problems are and are not relevant to the mathematics in the problem. If children are drawing pictures to help them solve word problems, they may not understand what the mathematics in the problem is. If so, how can they know what features are or are not relevant to the mathematics? Thus, helping children become able to draw pictures that "show the mathematics in the problem" is itself a major teaching goal in Kindergarten. At the same time, we also want to help students develop an understanding/disposition that drawings are useful thinking tools. So, how might we achieve this goal? One potentially useful strategy used in many Japanese elementary school mathematics textbooks is to use problem contexts in which objects in the problems are fairly simple objects. Thus, when children draw their pictures, drawings will not be overly complicated. Moreover, it will be useful for children to share their drawings. By examining and reflecting on different drawings their friends made, children can begin to think what features of their drawings are essential for doing mathematics.

Another major change from the GPS to the CCSS is the idea of representing with equations. The GPS does not emphasize the formal representations with numerals and mathematical symbols in Kindergarten. However, the CCSS begins the use of the formal/symbolic representations in Kindergarten. Some people may disagree that such an expectation is developmentally appropriate. However, the expectation is there, and we must teach Kindergarteners about the formal representations. As we do, I hope we will emphasize both representing and interpreting. Thus, not simply asking children to represent addition or subtraction situations using equations, we should ask them to come up with different situations for a given equation. Moreover, even from the beginning, we should remember that the "=" sign indicate that the two quantities on both sides are equal, not "calculate." Thus, from time to time, we should write "5 = 3 + 2," not just "3 + 2 = 5."

Wednesday, February 23, 2011

Kindergarten: Operations and Algebraic Thinking (1)

In the domain of Operations and Algebraic Thinking in Kindergarten, there is only one cluster - "Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from." This cluster statement makes it quite clear what meaning Kindergarteners are to give to the arithmetic operations of addition and subtraction. The current GPS (MKN2a) states, "Use counting strategies to find out how many items are in two sets when they are combined, separated, or compared." Table 1 of the CCSS explain what is meant by "putting together," "adding to," "taking apart," and "taking from." In my previous post on MKN2a (link), I discussed how the GPS's classification was based on the framework developed by the Cognitively Guided Instruction (CGI). The categories used by the CCSS are comparable to the CGI categories as well, but labeled differently. Thus, "adding to" is equivalent to "combine," "taking from" is equivalent to "separate." "Putting together" and "taken apart" are the "part-part-whole" category of the CGI - with the "putting together," the whole is unknown while in "taken apart," a part is unknown.

At this point, one major difference between the CCSS and the current GPS should be obvious. In the CCSS there is no comparison meaning of subtraction is addressed in Kindergarten. Instead, the CCSS includes the part-unknown case of the part-part-whole structure for subtraction. How significant is this difference? This might turn out to be a pretty significant difference. One of the findings from the CGI research is that children approach these word problems using different strategies - usually counting and/or direct modeling of the problem situations as the first step. Gradually, children will move toward the strategies that involve more advanced counting or the use of previously learned facts.

An example of "taken apart" problem included in the Appendix is this:
Five apples are on the table. Three are red and the rest are green. How many apples are green?From an adult's perspective, we might think it is simple to use counting or direct modeling for this problem. You just need to count on from 3 till you reach 5, or start counting from 5 down to 3. However, I discussed in the previous post that counting-on requires a major cognitive development. Moreover, the CGI research seems to show that "counting down to" is a more advanced counting strategy (than simply counting back 3 times). In order to model the situation, children must be able to anticipate the result - you can't start with 5 because it is made up of the known quantity and the unknown quantity. Thus, that is a more advanced thinking as well.

Comparison problems, on the other hand, are easier to model. Children can model both quantities, and they can make one-to-one correspondence between the two groups. The ones without matches are the difference. So, from a developmental perspective, comparison situations seem to be more "primitive" type. When mathematics educators discuss subtraction, we often talk about three different ways we can think of subtraction: subtraction as a take away, subtraction as comparison, and subtraction as missing addend. The "taken apart" (or part-part-whole with part unknown) seems to relate more to the last type, and we can see, from other standards, the CCSS emphasizes that way of thinking subtraction. Perhaps a careful and thoughtful teaching with that focus might help students make the necessary cognitive advances. But, I think it is critical teachers are aware of the non-trivial challenges students are expected to overcome.

Friday, February 4, 2011

Kindergarten: Counting and Cardinality (K.CC)

Kindergarten: Counting and Cardinality (K.CC)

In the Counting and Cardinality domain in Kindergarten, there are 7 standards in 3 clusters (Know number names and count sequence; Count to tell the number of objects; and Compare numbers). Those standards are as follows:1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).
4. Understand the relationship between numbers and quantities; connect counting to cardinality.a. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
c. Understand that each successive number name refers to a quantity that is one larger.
5. Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.
6. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.
7. Compare two numbers between 1 and 10 presented as written numerals.

Compared to the current GPS, there are some similarities, but there are some differences, too. Like the current GPS, the CCSS expects students to write numbers up to 20 and be able to compare two (or more) sets - the CCSS has an additional expectation that students be able to compare two written numbers (between 1 and 10) without actual objects. One difference that might stand out is that the CCSS expects students to be able to count up to 100 by ones and tens while the current GPS expects students to be able to count up to 30 objects in Kindergarten. In the current GPS, the range of numbers is expanded to 100 in Grade 1, as well as counting by ones and tens. In contrast, in the CCSS the range of numbers are expanded to 120 in Grade 1. On the surface, this difference (up to 30 or up to 100) appears rather significant. On the other hand, there is an obvious number word patterns in counting from 20 through 99. So, from a language perspective, this difference might not be too significant - other than learning additional number words for 40 through 90 and 100.

Perhaps a bigger question is what is meant by the phrase, "by ones and tens." The CCSS does not provide any elaboration, but if this is limited to simply knowing the decade number words (ten, twenty, thirty, ... ninety) in sequence, it is probably not a major concern. However, the CCSS expects students to be able to count beginning with numbers other than 1. If this expectation also applies to counting "by tens," then that may not be developmentally appropriate. This idea (start counting from number other than one, or counting on) involves a major cognitive development. For many young children, numbers exist only as a result of counting. Thus, numbers do not exist without counting from 1. In order to start counting from numbers other than 1 meaningfully, or to count on from a given number, require a different way of understanding of numbers. Moreover, research seems to be clear that understanding of ten as an iterable unit is a major step that even some 2nd graders are not ready to make. I hope that there will be further elaboration and articulation of what these standards are expecting in terms of children's understanding of ten.

The CCSS seems to articulate various aspects of counting much more explicitly and in details (Standard 3). These ideas are implicit in the GPS as I discussed this matter previously (here). However, the CCSS does not appear to place much emphasis on counting (other than expanding the range of numbers to 120) in Grade 1. However, I believe counting is not something children just "master" in one grade level. Rather, it should be an important activity in primary grades for children to build number concepts. Although we do not want children to become dependent on counting to complete simple arithmetic, counting is nevertheless an important foundational activity for children to construct their number concepts. So, I hope primary grade teachers will continue to engage their students in appropriate counting activities.

Saturday, January 8, 2011

Mathematical Practice

Mathematical Practice

As I mentioned previously, the State of Georgia has adopted the mathematics standards developed by the Common Core State Standards Initiative. These Standards will become the new state standards starting in the school year 2012-13. So, in this blog, I will try to discuss the specific CCSS standards and compare/contrast with the current GPS.

In the GPS, there are two sets of standards: content standards and process standards. The process standards are the five standards that are discussed at the end of each grade and relate directly to the process standards discussed in the NCTM Standards - Problem Solving, Reasoning, Connection, Communication, and Representation. The CCSS mathematics standards, in contrast, include a set of standards on mathematical practice. According to the CCSS, mathematical practice is a variety of "expertise that mathematics educators at all levels should seek to develop in their students," and the eight expertise are:
1. make sense of problems and persevere in solving them
2. reason abstractly and quantitatively
3. construct viable arguments and critique reasoning of others
4. model with mathematics
5. use appropriate tools strategically
6. attend to precision
7. look for and make use of structures
8. look for and express regularity in repeated reasoning

Some of the items in this list sound very similar to the current GPS process standards while others appear to be new and different. For example, the idea of persevering to solve problems is not explicitly stated in the current GPS, but if students were to learn from problem solving, it is essential that students persevere. On the other hands, some of the current GPS process standards are much more obviously related to the eight expertise while others may appear to be forgotten. However, a more detailed look at the mathematical practice does suggest that even those standards are still important. For example, the connection standards seem to be absent from the list of mathematical practice. However, the description of "modeling with mathematics" include the following:
Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
These descriptions of "mathematically proficient" students clearly suggest students must be able to connect their understanding of mathematics to things both within and outside of mathematics, and both within and outside of classrooms.

One of the main concern as we move forward with the CCSS is that these standards on mathematical proficiency will receive less attention just as the process standards of the current GPS do. In some ways, it is understandable as it is rather difficult to imagine these mathematical practice standards in action. Moreover, it is not quite clear how these standards will be assessed. Thus, it is natural for some teachers to focus on things that will be assessed. The authors of the CCSS, however, offers a suggestion that can guide us as we grapple with the content standards:
Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. ... In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.
In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.

As I continue discussing the specific standards, I will try to keep this suggestion in mind. I would also like to encourage you to keep thinking about the mathematical practice standards as we go through this time of transition.