Chapter 3 Reflection

What are the key ideas presented in chapter 3? How do these ideas inform your understanding of teaching numbers and operations?

• The two word problems at the beginning of the chapter showed me that addition is not always putting together and subtraction is not always taking away. I realized that addition could be used for a problem in which “gave away” were the key words that I noticed, immediately thinking it would be a subtraction problem when in reality I was wrong. It is important to teach children numerous addition and subtraction structures because when children learn the limited definitions for addition (put together) and subtraction (take away), they will most likely have trouble later when solving problems that do not follow these simple structures.
• The difference between addition and subtraction is that addition is used to name the whole when the parts of the whole are known, and subtraction is used to name a part when the whole and the remaining part are known.
• Using context or story problems is more beneficial for children because they model the real world and help the students develop a more practical sense for math instead of simply solving problems to get the answer.
• It is important to teach students symbolism correctly, and I have seen numerous examples of the symbols taught incorrectly, which can confuse children later on. The minus sign should be read as “minus” or “subtract” but not as “take away,” and the plus sign should be and usually is substituted as “and.” He equal sign is usually thought of as “the answer is coming” and should be taught as “is the same as” because there can be two equations on each side, which doesn’t necessarily represent an answer.
• When using model-based problems for addition, I had never really thought about the importance of leaving the two parts separate, whether by keeping separate piles, using different colors, etc. The students need to be able to identify which ones they started with, what was added, and the whole amount after the action of adding was completed.
• Addition and subtraction are interrelated and should be taught together. When teaching these, teachers should use the same models for both types of problems in order to help the children begin to realize the connection between the two operations.
• When teaching multiplication and division, the big terms for the parts of the problem are not meaningful or beneficial to children. Instead, they should be taught as parts and whole, which is useful in helping the children make the connection to addition.
• Just as with teaching addition and subtraction, it is beneficial to teach multiplication and division at the same time or at least one soon after the other in order for children to learn how they are connected to one another.
• It is important to teach multiplication using contextual problems, just as with addition and subtraction. Students should use whatever techniques they feel comfortable with to solve problems but they need to be able to explain what they did and why it makes sense, preferably in writing which could be in the form of words, pictures, and numbers.
• In multiplication, the distinction between which number represents the sets is not necessarily that important, but it is important that the students understand the concept that 4x8 means there are 4 sets of 8 or 8 sets of 4. Students will use repeated addition equations to show their work for these problems before they learn and understand the symbolism for multiplication, so they will need repeated practice and explanations of it.
• As long as numbers are within the students' counting strategies, they can and should be used in multiplication and division problems. When children are given challenges involving larger numbers and have not learned a computation strategy yet, they will most likely develop one on their own.
• Students should never be taught to look for key words in a problem! Not only can they be misleading as to which computation to use to solve the problem, but some problems don't even have any key words and if a student only relies on looking for the key words, then he or she is left with no strategy to solve the problem. Also, students should be taught to solve contextual problems by analyzing and making sense of the problems rather than looking for an easy and quick way to solve the problem.
• As we have discussed before, asking children to explain their strategies and processes of solving the problems is extremely vital to gaining insight of their understanding and abilities. It is more beneficial when a student is able to describe how they came upon an answer and explain the way that they carried out certain operations, rather than just writing the equation and the answer on their paper.