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Hello. Thank you so much for providing this forum and being so accessible.
I’m curious to trace the development of the table of addition/subtraction situations on p. 88 (that also make up much of the K-2 OA standards). The table is listed as being adapted from the NRC’s 2009 Mathematics Learning in Early Childhood publication. That document (which does not contain a table itself) cites Adding It Up as a reference. Similarly, the table on p. 88 of the CCSSM appears to resemble Table 6-1 on p. 185 of Adding It Up, which itself is based on Children’s Mathematics. I do recognize that both Adding It Up and Children’s Mathematics are listed among the works cited in the CCSSM.
There are some subtle but mathematically significant distinctions between the classifications on p. 88 of the standards and those in the CGI research. Obviously, the names of the problem types are different and the “both addends unknown” problem has been added, but also the scheme for classifying the comparison problems is different. Carpenter’s research classifies the sets in comparison problems according to the linguistic structure of the problem, not the relative size of the unknown quantity. The statement of difference assigns one set (its subject) as the compared set, and the other as the referent set. Problems where both pieces of information (the known quantity and the difference) are assigned to the compared set (leaving the referent unknown) are typically more difficult for children than those where the compared set is unknown.
I’m interested to better understand some of the thinking around these adaptations, as it will enable me to better answer the questions of my teachers and colleagues in providing professional development to support the implementation of the CCSSM. I’ve come across related (but still different) problem type tables in Fuson’s chapter in the 2003 Research Companion to the NCTM Standards and Clements & Sarama’s 2009 Learning Trajectories publications, but neither of these provide explanations for how the terminologies/classifications differ from Carpenter’s original research. Apologies if this has been clarified elsewhere and I have missed it, or if I am totally off the mark in my understanding.Thank you for your consideration,
NickNick, in the meantime, you might want to check out the elaboration of these tables in the Operations and Algebraic Thinking Progression at http://ime.math.arizona.edu/progressions/
This progression mentions pp. 32-33 of Mathematics Learning in Early Childhood, which does not give a table, but does give the problem types and some of the example problems.Thanks for your productive question and for seeking more information to help your teachers and colleagues. That is how we all learn. I learned more in writing this response. Thanks.
The history of research on word problem solving using categories for addition/subtraction roughly relatable to those in Table 1 of the CCSS-M on page 88 is long and complex. This is a huge area of research done internationally over decades. In this country such research was done by researchers in mathematics education, cognitive development, and cognitive psychology. People used different terminology and made different subcategories. The distinctions primarily focused on the mathematical and action structure of the situations, linguistic variations, and specific contexts or order of the sentences that might affect performance. A group of such researchers met informally in St. Louis many years ago to focus terminology and distinctions. But minor variations have continued to occur since then. The NRC’s 2009 Mathematics Learning in Early Childhood did have a simple table as referenced on page 88 of the CCSS (6Adapted from Box 2-4 of National Research Council (2009, op. cit., pp. 32, 33). The NRC report did not give the same detail as did the CCSS Table 1 because the NRC report was focusing on young children, who solve only the simplest types in Box 2-4.
Most category systems do use the three major categories reflected in Table 1 of the CCSS-M on page 88: Add To/Take From (called Change Plus and Change Minus in the NRC report), Put Together/Take Apart (sometimes called Collection or Combine), and Compare. All problems in these three major categories involve three quantities. Each of these quantities can be the unknown quantity. This is the most fundamental distinction in the research literature.
The second most important distinction is between addition and subtraction situations. The three major categories differ considerably in how this distinction is made and how fundamental it is. Addition and subtraction actions in the Add To/Take From (Change) situation are quite different. These are the earliest meanings of addition and subtractions for children. For this reason many categories including Table 1 of the CCSS-M on page 88 make two subtypes and give them names: Add To and Take From. These two subtypes have also been called Change Plus and Change Minus or Join and Separate. The equal sign in the equations for these types have the action meaning “become” as an arrow. The action for Put Together/Take Apart is more subtle and may be only conceptual (in the apple problem in Table 1 of the CCSS-M on page 88: considering the apples by color and then disregarding color to make the total). Also, for this major type there is not a fundamental distinction between the situational role of the addends, although one addend must occur first in the word problem. In contrast, in Add To/Take From problems, one addend is first in the situation and the other addend is added to or taken from that first addend. For this reason some researchers including CGI at some times only distinguished two cases for this second major type—unknown total and unknown addend—but this can be confusing in understanding that all major types involve three unknown quantities and either addend can be unknown. The special case in which both addends are unknown was used in Table 1 of the CCSS-M on page 88 because this is one of the prerequisites for the important make-a-ten strategy; these prerequisites are K.OA.4, K.OA.3, and K.NBT.1.
Compare situations have no situational addition or subtraction action. In fact, such situations only have two quantities in the situation: a bigger quantity and a smaller quantity that are compared to find how much bigger or smaller one quantity is than the other. This quantity must be conceptually constructed by comparing the two given quantities; this quantity is the difference. There are always two opposite but equivalent ways to state the comparison: Using “more” or “less/fewer” when the difference is unknown (and other linguistic variations of this distinction). Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? OR How many fewer apples does Lucy have than Julie? When the difference is known, the language used to state the comparison can suggest the solution operation: saying “more” for a bigger unknown situation, where you need to add the difference to the smaller quantity (Lucy has two apples. Julie has three more apples than Lucy.), or saying “less” (or fewer) for a smaller unknown situation, where you need to subtract the difference from the known bigger quantity to find the smaller unknown quantity (Julie has five apples. Lucy has three fewer apples than Julie). Such problems are easier for students than problems in which the comparing sentence suggests the wrong operation, for example, Lucy has two apples. Lucy has three fewer apples than Julie. Table 1 p. 88 in the CCSS distinguishes this language variation as subtypes within compare. The version of this table in the OA Learning Progression on page 9 is more specific about how this language variations affects problem solving, and it indicates that this is the reason that the two subtypes suggesting the wrong operation are not for mastery in Grade 1.
The CGI problem types used a category system that focuses on whether the quantity in the comparing sentence is unknown (Compare Quantity Unknown vs. Referent Quantity Unknown). Compare Quantity Unknown is easier than Referent Quantity Unknown because using the unknown quantity as the subject of the comparing sentence means that the comparing action can be done by starting with the known quantity. But this linguistic analysis does not reveal the essential underlying mathematical situation: two quantities are being compared, and one is smaller and the other one is bigger. The linguistic analysis also is not the fundamental way in which children and adults (for 2-step problems) solve comparing situations. The key to success is deciding which quantity is the bigger and which is the smaller; making a drawing to show this can be very helpful even for adults. It is important for classroom discourse to focus on the most important issues and to decide which quantity is the bigger and which is the smaller. This is the basis for equations that children write to show comparing situations. Children write many different kinds of equations for compare situations, and teachers and programs should not focus on a subtraction equation as the most important or only equation, as some textbooks have done in the past (children are more likely to write an unknown addend equation). Also, the CGI linguistic categories mean that a problem switches types when the equivalent comparing sentence is used. So the same situation in the world changes types when the comparing sentence changes its subject. This is also problematic because one major problem solving strategy for the more difficult comparing language is to say the opposite comparing sentence to avoid the misdirecting language. This strategy seems more straightforward to teachers if using an equivalent comparing sentence does not change the problem type.
A decision that had to be made for the CCSS problem type terms was whether to choose names for the types that were easier for children or for adults (teachers and researchers). The choice was to choose terms that were easier for children (with the thought that they might also be easier for teachers). Add To/Take From and Put Together/Take Apart are action words that are easier for children than the older category names Join/Separate (or Change Plus/Minus) and Part-Part-Whole (or Combine). Terms for the main problem types do not have to be used in the classroom, but such use can facilitate discussion about the types. But as this summary has indicated, these terms are not part of the official mathematical vocabulary. Rather, they are descriptions of the situations whose structure children need to understand. Any useful names can be discussed including those elicited from children. Choosing terms that were easier for children also meant that the CGI terms Compare Quantity Unknown and Referent Quantity Unknown were not the best. These terms focused on problem difficulty (a major focus of the research being done at the time) rather than on the representation and solving of the problem. We wanted to help teachers understand children’s representation and solving of problems, so the focus on the Bigger and the Smaller was more appropriate.
The distinctions between the types are often easier for children than for teachers to make, especially between the Add To/Take From and Put Together/Take Apart types because children in the younger grades tend to pay more attention to the situation while teachers tend to be more abstract and just think add or subtract. For example, children understand commutativity earlier for Put Together/Take Apart problems than for Add To/Take From problems because the roles of the addends are not so different.
I do want to be sure that readers understand the world-wide emphasis on word problem solving and the huge research literature on this topic. GCI has helped many people in this country understand the different problem types and the learning paths in methods children use to solve problems. But sometimes people do not realize that this research extends far beyond CGI. This research is summarized in articles in the First and the Second Handbook of Research on Mathematics Teaching and Learning. I’ll just mention two examples of the international nature of this research. Jim Stigler and I analyzed the problem types in the Russian Grade 1 and 2 textbooks translated by the University of Chicago project and found that the types were equally distributed across the 12 subcategories in the CCSS Table 1 on page 88 and that 60% of Grade 2 problems were two-step problems [Stigler, J., Fuson, K. C., Ham, M., & Kim, M. S. (1986). An analysis of addition and subtraction word problems in Soviet and American elementary textbooks. Cognition and Instruction, 3, 153-171.] And a writer of Japanese textbooks once told me that they used a binder of information about word problems that was 4 cm thick in deciding which kinds of word problems to use when, how to phrase them, etc.
I do have two questions about what you wrote. You say “Similarly, the table on p. 88 of the CCSSM appears to resemble Table 6-1 on p. 185 of Adding It Up, which itself is based on Children’s Mathematics.” I do not know what you mean by Children’s Mathematics.
You also stated that “Problems where both pieces of information (the known quantity and the difference) are assigned to the compared set (leaving the referent unknown) are typically more difficult for children than those where the compared set is unknown.” I do not see that the difference information is ever assigned to the compared set. The difference is the third quantity in the situation. It is just an artifact of English that the comparing sentence is so complex and includes the size of the difference and the direction of the comparison: “Julie has three more apples than Lucy.” is easier for children to understand if the information is separated into two sentences: “Julie has more apples than Lucy. and The amount more is three.” Initially many children do not even hear the three in the comparing sentence. Some other languages do separate this information, making comparison problems easier to comprehend and solve.
Tom Carpenter is visiting me on Wednesday. I’ll see if he has anything to add to this. Thanks again for such good questions.
Thank you so much for taking the time and effort to craft such a thoughtful, detailed response. I appreciate the insights into some of the decision-making in writing the standards, and I am grateful for the opportunity to deepen my understanding of the comparison situations. This is such a rich topic for learning!
To address your questions:
1) Apologies, I should have been clearer. Children’s Mathematics refers to the first CGI book.
2) My thinking around the difference being “assigned” (maybe not the best word here) to a set is connected to how I expect children would directly model the situation, and also what is linguistically necessary to fully describe the situation. Let’s consider a comparison where Bill has 5 pizzas and Ellen has 2, so the difference is 3. To write a clear problem situation, the statement of difference must be made in relation to a particular set. If Bill has 5 pizzas, (and Ellen’s set is unknown), stating the difference as simply “3 more” is ambiguous. I have to name a set as the subject of the difference, “Bill has 3 more,” before I have enough information to solve the problem.
I see this – the “perspective of the difference” if you will – as influencing how children are likely to try and model the situation. In a comparison problem where one of the sets is unknown, a child might construct the known set and the difference quantity. But what to do with the difference? In “Bill has 5 pizzas, and Ellen has 3 more than Bill,” the difference information is in relation to the unknown set (CQU). In these situations, once the known set and the difference have been constructed, it’s a little easier to see what do to solve the problem – place the “3 more” with the unknown set (as the language in the problem suggests), and then match Bill’s set.
Bill’s set Ellen’s set 0 0 (difference: Ellen has 3 more) 0 0 0 0 } match this 0 0 However, when the difference information is in relation to the known set (CRU), “Bill has 5 pizzas, he has 3 more than Ellen,” it’s not as clear what to do with the difference to solve the problem. A child might try to put together the “3 more” with Bill’s set, rather than seeing that the “3 more” is, in one way of thinking, already contained within Bill’s set, and either putting the 3 more on top of Bill’s set and matching the remaining part of Bill’s, or just setting aside the “3 more,” seeing it within Bill’s set, and matching only the remaining two.
Bill’s set Ellen’s set
0
0
0
0 } match only this part of Bill’s set
0 }Thus, (in my thinking) it is the subject of the difference information in the problem that makes the two comparison situations feel different to a child (though a child might do so, I don’t see that s/he will necessarily need to think about which of the two sets is larger to correctly solve the problem). It is interesting to note, however, that both the CCSS and the CGI classifications would be the same for the two problems described above (if for different reasons). It is only when switching to a “fewer” situation that the results of the two classifications would be different. I need to think more about why this is so and what significance it might hold…
Of course, there are other factors that are going to influence how a child might solve the above problems. These are particularly difficult problems to direct model; I would guess that by the time most children can solve these problems through modeling they are likely to have also developed some number facts and part/whole thinking that they might draw upon in making sense of these situations.
Again, thank you so much for the opportunity to broaden my understanding of the CCSSM problem situations and to share my thinking. This will be of great value to me in my work with teachers.
Drat! Messed up the formatting on the first diagram. The first three circles should be moved over to the right to be underneath Ellen’s set.
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