The Support of Big Ideas toward Mathematical Strategies
Addition and subtraction problems potentially create many strategies for children to solve. The strategies which can be used are, for example, adding on or counting up, counting back, splitting, removing, taking away, etc. The use of strategy by students depends on the numbers which are available in contexts or questions. Moreover, it is mainly influenced by the understanding of students toward the mathematical idea behind the contexts. When talking about contexts of subtraction, there are mostly three kinds of them which often appear in classroom: subtraction in the form of distance or difference, removal, and comparison. The three kinds of them are categorized as mathematical ideas that students are expected to grasp. In this report I will make report and use the examples in the video Addition and Subtraction Minilessons, Grade pre-3. In the video, Michael Galland, as a teacher and his students are involved in discussion of addition and subtraction problems. The students use many different strategies of some problems which are given by Michael.
B. Observational Question
On this occasion I elicit a question to answer next: How the understanding of mathematical ideas and the numbers available in questions develop mathematical strategy of children to solve problems?
There are three problems given by Michael shown in the video: 272 – 14, 146 – 12, and 283 – 275. In the first problem, Michael asks Samantha and Louis to answer it. Samantha uses splitting strategy in which she splits 12 becomes 10 and 4, then takes away 10 from 272 resulting 262 and then 4 which finally results 258. Louis also uses similar way as Samantha does; however, he has different answer that is 256. The answer is derived from taking away 2 first, then 10, and 4. However, many other students disagree since the total of number Louis has split is 16. Immediately, Louis changes his answer which is the same as that of Samantha. In the second problem, 146 – 12, there is a student answering the result is 134. Then Maria gives explanation by giving strategy splitting up: removing 10 first and then 2. In the third problem, there are three children participating to solve it, Ian, Daniel, and Colleen. Ian uses strategy counting up in which he jumps from 275 to 280 and uses jump of three to get 283. He comes up with answer 8. Similarly, Daniel also uses Ian’s strategy. On the other hand, Colleen uses jumping back strategy by also splitting up the number. She comes up with answer 8 after removing 275.
In the first problem, Samantha and Louis seem to know that the idea of subtraction is removal. That is why they take away 14 from 272 and get result 258. The understanding of Maria also in the second problem is the same as that of Samantha and Louis in the previous problem. However, I personally am not quite brave to say that they only know the subtraction as a removal. They choose taking away since 14 and 12 are small numbers which are easy to remove. The most interesting thing is the discussion of the third problem. Ian and Daniel seem to know that mathematical idea behind subtraction can be categorized as distance or difference. Unlike many children, they use adding up from 275 to 283 instead of removing 273 from 283 as Colleen does. The understanding that subtraction also means difference helps Ian and Daniel quickly solve the problem more than Colleen does since 275 is so close to 283. Another mathematical idea which probably exists in Ian and Daniel is subtraction as comparison in which they think to use the same strategy.
The understanding of students of big ideas of subtraction really helps them in using effective strategy to solve problems. When students only know that the meaning of subtraction is taking away or removing, they will find easy to count easy question, for example 172 – 14. Because in the example, 14 is such a small number that students will is likely losing their tracks. However, when finding a question, for example 283 – 275, in which they want to remove 275, students are more likely losing their track because 275 is high number. In contrast, when students know the meaning of subtraction can be also distance or difference and comparison, they will use adding on strategy in which it simplifies them to count. For instance, 283 – 275 can be solved by adding on 275 to 8 which results 283. The answer is absolutely the same as the answer when students remove 275 from 283.
Antonio Chameron et all. Video of Young Mathematicians at Work. Constructing Number Sense, Addition and Subtraction. Heinemann