Decide for each statement whether it is true or false.
1. Problems where there is a change over time (that would need to be represented by a movie) are harder for young children to solve than problems where you could just draw one picture to show everything (no change over time).
False: Problems where there is a change over time are easier for young children to solve, because the change prompts the action needed to act out (direct model) the solution.
2. The phrases "in all" and "all together" are ones that some 5-6 year old children don't understand.
True: Children need experiences using these phrases in a context of telling "how many" are in different sets.
3. JRU problems are easier for children to understand and solve than PPW-WU problems.
True: Some children find PPW problems siginificantly more difficult than JRU problems. This is most true of children in early elementary grades (K-1). This is a stage that most children move past fairly quickly with appropriate experiences and instruction.
4. The first step in modeling both SRU and JRU problems is to count out the starting amount of counters.
True: In an SRU problem you count out the starting number of counters, and then take some away; in a JRU problem you count out the starting number of counters and then add some in.
5. Some Join and Separate problems have an amount that changes over time, but some don't.
False. Join and Separate problems have to have an associated action that can change an amount over time, so it is false that some don't have that sort of action.( I guess if the change amount is 0 (so if you take away or add on 0), then the start amount doesn't actually change...)
6. JRU problems are easier for children to direct model than SRU problems.
False. At the direct modeling stage, these problems are equally easy/difficult. It is only when the child starts using more abstract solution strategies that SRU problems become more difficult.
7. The difficulty for children in solving JCU and SCU problems is about equal
False. Unlike JCU problems, SCU problems do not have an easier entry point/way of phrasing to make problem solving easier, and the step where you either repeatedly count the remaining set or count backwards makes SCU direct modeling technically more difficult to carry out.
8. You should avoid teaching compare problems to children until upper elementary because they are so hard to understand.
False. Although compare problems are conceptually more difficult to understand, children in grades 1 and 2 can learn to understand them. Further, the structure of compare problems leads children to important understandings about the nature of subtraction that are not clear from the action (take-away) conception of subtraction.
9. Childrens difficulty with solving SCU problems through direct modeling is mostly because they find the problems hard to understand.
False: While SCU problems are more difficult to understand than SRU problems, the most difficult part of solving SCU problems is keeping track of the diffent parts while repeately counting the set of counters remaining. Modeling SCU problems is almost a guess and check process, which makes it more cumbersome than either the modeling strategy for JCU or CDU problems.
10. Childrens difficulty with solving CDU problems through direct modeling is mostly because they find the problems hard to understand.
True: sentences describing comparisons of amounts (how many more than) have a unique and inherently numerical nature, and children need experiences with using and making comparisons of amounts of items that are already shown and modeled before they are ready to solve problems about comparisons. Some children will enter school with a familiarity with comparison language, and other children will have less experience with it.
11. Part-Part-Whole problems can all be solved using the same direct modeling strategy as Join (result unknown) problems.
False (trick question kinda). Part-Part-Whole-Whole Unknown problems can be modeled using a Join, result unknown strategy, but Part-Part-Whole-Part Unknown problems are modeled using the strategy of either a Separate, Result Unknown, or Join change unknown.
12. Children figure out how to solve Part-Part-Whole problems later than Join and Sparate (result unknown) problems, because there isn't an action taking place in the problem to guide them on how to act out the problem.
True. The easily-modeled actions in the Join and Separate result unknown problems make them the easiest for children to understand and model. After gaining some sophistication with both language and mathematical problem solving, children can apply their knowledge to other problems, including Part-part-whole problems.
13. Some kindergarten aged children don't understand part-part-whole problems, and what the problem is asking for.
True. The notion of combining smaller sets into a larger set is one that takes a little time and practice to get used to for children at this stage.
Answer these short-answer questions
14. What makes CQU problems so much easier than CRU problems?
The referent is known, so the comparison sentence tells what to do to figure out the compared quantity.
Another way to look at this is that the comparison words more/fewer tell what to do to the referent to find the compared quantity: we addi and subtract in the way we indicated by the comparison words (more triggers the idea of getting more--addition; fewer triggers the idea of having less--the result of a subtraction). In a CRU problem these relationships are reversed, so it's harder to keep track of.
15. Is it a good idea to tell children to add when they see the word "more" and subtract when they see the word "fewer" or "less"? Why or why not?
No, it's a bad idea because while that works fine for CQU problems, it's exactly wrong (backwards) for CRU problems. We need to do our best to not create misconceptions along the way if we can reasonably avoid it.