Is 4 x 6 same or different with 4 x 6 ? This case is popular after a teacher that putting 4 x 6 = 6 + 6 + 6 + 6 = 24 in wrong. In my opinion there is explain that which case is 4 x 6 = 6 x 4 and 4 x 6 not = 6 x 4. This is my explain that I found in kompasiana.com
Krathwoll (2002) said that there are 4 knowledge that must be essential knowledge. There are Factual, Conceptual, Procedural, and Metacognitive [1].
There is 2 kind of knowledge that generally used in Mathematics learning. That is Conceptual and Procedural
Procedural Knowledge of 4 x 6
Ok, if the attainment is what is 4 x 6 = ? This case is include procedural knowledge. Procedural knowledge is procedure that done by process steps to achieve the expected results. Procedural knowledge means mastering process, for example, students can carry out research through a gradual process. We the student done like this
4 x 6 = 4 + 4 + 4 + 4 + 4 + 4 = 24
or
4 x 6 = 6 + 6 + 6 + 6 = 24
Is it right because this question is attain students to what is result from 4 times 6 whatever how students solve it using repeated addition. That must be inline is the question "What is result", right?. But it must given by conceptual knowledge that is why students multiplied 4 as 6 times in addition or multiplied 6 as 4 times
Conceptual Knowledge of 6 x 4
Conceptual knowledge should be explained properly, so when asked matters relating to procedural knowledge, the child is able to answer correctly the questions. Given the importance of these two kinds of knowledge that causes a teacher should instill the basic concepts calculations using correct operation.
Prof. Yohanes gives an explanation by analogy orange and box, "When calculating the 6 x 4, we imagine counting the number of oranges in 6 boxes each containing 4 oranges. So, 6 x 4 = 4 + 4 + 4 + 4 + 4 + 4. When calculating the 4 x 6, we imagine counting the number of oranges in 4 boxes each containing 6 oranges. So, 4 x 6 = 6 + 6 + 6 + 6 ". [2]
Thomas explained the difference of 6 x 4 with 4 x 6 through a story about, "Ahmad and Ali had to move the same amount of brick, 24. Because Ahmad stronger, he took 6 brick 4 times, mathematically written 4 x 6. However, Ali whose body was smaller, only capable of carrying 4 bricks as much as 6 times, mathematical models 6 x 4. So, 4 + 4 + 4 + 4 + 4 + 4 = 6 x 4, a different concept with 6 + 6 + 6 + 6 = 4 x 6, although the result is the same 24 ". [3]
Anecdote:
- Guest 1x24 hours please report means guests have to report as much as once in a 24 hour period, not a must report for 24 hours in a single report.
- Results of a 6x4 size photo print is different from the size of 4x6.
- Buying eggs 5 kg once bought more effective than buying per 1 kg of back and forth as much as 5 times.
Conclusion
If you do not want to receive another answer option, do not give hope that there is another possibility. If the target is they understand the formula and applied, guided in that direction. If the goal for students is easy to understand, then make it easy. Pure math problem (without contextual) does is exercise towards applied mathematics (with contextual narrative), but both are still different and therefore one can not be used as a measure judge others [4].
So, we must give inline when 4 x 6 is same with 4 x 6 or different. In this case, conceptual and procedural knowledge must given to make student comprehend correctly.
Reference
[1] http://gurupembaharu.com/home/15494/
[2] http://sains.kompas.com/read/2014/09/23/10502281/6.x.4.atau.4.x.6.Ini.Penjelasan.Sederhana.dari.Yohanes.Surya?utm_campaign=popread&utm_medium=bp&utm_source=news
[3] http://sains.kompas.com/read/2014/09/22/22195601/Ini.Beda.4.x.6.dengan.6.x.4
[4] http://edukasi.kompasiana.com/2014/09/24/4-x-6-atau-6-x-4-bukan-inti-persoalannya-676114.html
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