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| Kim | Jane | Jolly | Sid | Tom |Click here to go back to the previous page When carrying, Kim first adds the carried number and then continues multiplying. For example in the first problem, Kim multiplies three by five to give fifteen. She then carries the one and adds this to three to give four. Next she multiplies three by four to give twelve. Kim answered the second question correctly. Kim's method will only get the correct answer for questions where there is no carrying necessary. Kim would complete the problems given in the following way.
Teaching strategies: Kim can efficiently and correctly solve single digit problems. Her errors are due to a mistake in the process she uses. This is often apparent when students are simply rote-taught a set method, but lack the mathematical understanding behind the process. Kim needs to first realise that her method is faulty. This could be done through use of concrete materials or by checking with a calculator. Kim secondly needs to revise multiplying by a single digit with carrying. She needs to write the algorithm in its extended form (as shown below) until the procedure is familiar and the concepts (especially the distributive property) are well understood.
Back to Kim's work
Jane When carrying, Jane retains the carried number from the units column and continues to add it at every stage. If two numbers have been carried into the tens column, she will add both. Jane answered none of the problems correctly. Jane would complete the problems given in the following way.
Teaching strategies: Like Kim, Jane's errors are due to a systematic error in the process rather than in her number facts. Jane needs to first realise that her method is faulty and then revise the correct method. Jane's errors demonstrate that when completing the algorithm, she is simply following a number of recalled steps rather than understanding the mathematical concept. Breaking the algorithm into a series of stages will help Jane master this process. Back to Jane's work
Jolly Jolly's method treats both numbers in the multiplier as units and completes the question in one line. He has used the process for multiplication by single digit numbers to invent a process for multiplying by two or more digits. When multiplying by two digit numbers, Jolly will first multiply the two unit values together and then his second step is to multiply the two tens values together. Jolly answered only the first problem correctly as he can successfully multiply by one digit. Jolly would complete the problems given in the following way.
Teaching strategies: From looking at Jolly's work it seems apparent that he is not familiar with the process of long multiplication by two or more digits. Jolly can multiply by one digit (the first stage of multiplication) and after this, the most important thing is to teach him to multiply by multiples of ten. If Jolly continues to have difficulty with this step, it may be that he would be best to use a calculator for long multiplication. He will still need good strategies for deciding if the calculator answers are reasonable. Back to Jolly's work
Sid When completing the question Sid treats both numbers as units, forgetting to place a zero in the units column before multiplying by the tens. None of the problems completed by Sid were correct. Sid
would complete the problems given in the following way.
Teaching strategies: Sid's error is common with many children when moving from single digit to two digit multiplication. Sid treats both numbers in the multiplier as units. Sid needs to revise the concept behind each step of the multiplication process. He needs to realise that a 3 in the tens column is not a three but rather 30. More work with multiplication by multiples of ten and also writing out the partial products for each step as follows will assist Sid.
Solving problems in small groups orally with the teacher as scribe is also an important step in Sid's development. Being able to orally explain the process will assist students to consolidate knowledge. Back to Sid's work
Tom Tom's Error: a x 0 = a Unlike the other examples where the errors where due to process, Tom's errors are due to a mistake in computation. Tom has a misunderstanding about the properties of zero. Tom will only answer questions correctly if there is no zero in either number. His answer to problem four is correct. Tom
would complete the problems given in the following way.
Teaching strategies: Tom needs to be reminded about the properties of zero. Zero is a difficult concept to demonstrate with concrete materials, so discussing worded examples may help Tom understand his mistake better. An example may include: I have five bags with no lollies in each, how many lollies do I have altogether? Tom may also be typical of students who work too quickly. Although competent with the processes and most of the computation, he makes silly mistakes as he is careless. Students like this can benefit from being made to check their work before submission and rewriting and solving incorrect problems again. Back to Tom's work
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