D is a bit noisy. She sometimes made a story and talked out laud in the class. However, I respect a person who can make a new story. So, I don't want to discourage her to do that. On the other hand, she disturbs other children. So, I try to ask her to concentrate the problem.
Last time, D practiced plus calculation with Zahlenhaus. In the beginning, Zahlenhaus has plus and minus examples. Figure 1 shows a Zahlenhaus example.
A practice example is the following:
\begin{eqnarray*}
4 + 3 &=& \\
3 + 4 &=& \\
7 - 3 &=& \\
7 - 4 &=& \\
\end{eqnarray*}
Another typical practice example is the following:
\begin{eqnarray*}
1 + 6 &=& \\
6 + 1 &=& \\
7 - 1 &=& \\
7 - 6 &=& \\
\end{eqnarray*}
This practice book tries to explain that the plus calculation is commutative, i.e., we can add numbers in different order, i.e., \(1 + 6 = 6 + 1\). For the minus calculation, we can remove one of the room from the Zahlenhaus, then the other room remains. This book develops the question as the following:
\begin{eqnarray*}
\begin{array}[t]{cccc}
1 & + & 6 & = \\
6 & + & 1 & = \\
& - & & = \\
& - & & = \\
\end{array}
\end{eqnarray*}
D never made a mistake.
However, I am interested in how she understand this. I less care the answer is correct or not. Therefore, I asked her, ``Wie hast du dies gerechnet? (How did you calculated this?)'' This is the most interesting time for me. Most of the children can not explain well. But she explained it me very clear.
The question was:
\begin{eqnarray*}
\begin{array}[t]{cccc}
1 & + & 6 & = \\
6 & + & 1 & = \\
7 & - & & = \\
7 & - & & = \\
\end{array}
\end{eqnarray*}
She told me the first answer is in the third row as shown in the red.
Therefore, she just copied it.
The next answer is at the 4th row (blue). Let's copy it again.
Others are the same. The same color position has the same value.
I said, ``Ausgezeichnet! (Excellent!)''
One of the most important thing in mathematics is finding a pattern. When the human being found a patten about things, ``We can count things.'' Then mathematics was born. Finding the patterns in the world, and formulate them, that is mathematics. But, we need to check the patterns that when the pattern hold and when it does not. Long time, I didn't understand the mathematics proof checked such a pathetic cases. Because we want to know when we can use it. If we know the limit, we also know when we can use it.
This practice book has the same pattern for next pages. And the book continues Zahlenhaus 8, 9, 10. She will never make a mistake without understandings. (I sometimes see such kind of research. Nobody knows how it works, but, it seems working. I don't want to call it a research because no understandings. But, maybe they are still fine since they add some experimental knowledge to the society.)
I told her, ``Everything is correct. It is good to find a pattern.'' But I thought a while. Her method is great, but it will fail at some point. How can I persuade her there is a case it fails?
``Mathematics is a kind of language. So there are meanings all of these lines. You can express something with mathematics.... Here, there are plus and minus. What is the `plus' meaning?''
``Plus means more.''
``Yes, ... but, why 1 + 6 is equal to 7?''
``???''
``1 + 6 is not equal to 10. But, 10 is more than 1 and more than 6. Why 1 + 6 is not equal to 10? If plus means just more, is it OK that 1 + 6 is equal to 8, 9, or 10?''
I usually need to care not overwhelming children. Too much explanation is not good. That I learn from other teachers here. So I slow down.
I came back to the Zahlenhaus and asked her to count the 1 and 6 points in the Zahlenhaus. She can count. So, she realized the total number of the points is 7. I explained `plus' means `together'. If something and something together, then they are more. Therefore, `more' is correct, but, it is better to say `together' for plus's meaning.
Next my task is to explain when her method fails. Now she understand 1 + 6 has a meaning. Therefore, this line itself has an answer. Therefore, this 4 line set of Zahlenhaus practice is not always presented. She can not always have four questions. In that case, she can not copy the answers.
This actually takes around 20 minutes. But, I think she realized this.
This time, I was surprised that 8 year old child found, even partially, the following pattern.
\begin{eqnarray*}
\begin{array}[t]{cccc}
x & + & y & = z \\
y & + & x & = z \\
z & - & x & = y \\
z & - & y & = x \\
\end{array}
\end{eqnarray*}
Computer stores variables in a memory. A memory cell has an address. If a computer language didn't hide the address, like C or C++, the variable has an address. An address is a name of location. Therefore, we, programmers, need to develop the idea, a value is associated with a location as she explained me. I see she has such ability: associating a location and a value. I wonder if I taught a computer language to her. Well, first she should master what is plus, minus, and more calculation.
Last time, D practiced plus calculation with Zahlenhaus. In the beginning, Zahlenhaus has plus and minus examples. Figure 1 shows a Zahlenhaus example.
 |
| Zahlenhaus: 7 = 4 + 3 |
\begin{eqnarray*}
4 + 3 &=& \\
3 + 4 &=& \\
7 - 3 &=& \\
7 - 4 &=& \\
\end{eqnarray*}
Another typical practice example is the following:
\begin{eqnarray*}
1 + 6 &=& \\
6 + 1 &=& \\
7 - 1 &=& \\
7 - 6 &=& \\
\end{eqnarray*}
This practice book tries to explain that the plus calculation is commutative, i.e., we can add numbers in different order, i.e., \(1 + 6 = 6 + 1\). For the minus calculation, we can remove one of the room from the Zahlenhaus, then the other room remains. This book develops the question as the following:
\begin{eqnarray*}
\begin{array}[t]{cccc}
1 & + & 6 & = \\
6 & + & 1 & = \\
& - & & = \\
& - & & = \\
\end{array}
\end{eqnarray*}
D never made a mistake.
However, I am interested in how she understand this. I less care the answer is correct or not. Therefore, I asked her, ``Wie hast du dies gerechnet? (How did you calculated this?)'' This is the most interesting time for me. Most of the children can not explain well. But she explained it me very clear.
The question was:
\begin{eqnarray*}
\begin{array}[t]{cccc}
1 & + & 6 & = \\
6 & + & 1 & = \\
7 & - & & = \\
7 & - & & = \\
\end{array}
\end{eqnarray*}
She told me the first answer is in the third row as shown in the red.
Others are the same. The same color position has the same value.
One of the most important thing in mathematics is finding a pattern. When the human being found a patten about things, ``We can count things.'' Then mathematics was born. Finding the patterns in the world, and formulate them, that is mathematics. But, we need to check the patterns that when the pattern hold and when it does not. Long time, I didn't understand the mathematics proof checked such a pathetic cases. Because we want to know when we can use it. If we know the limit, we also know when we can use it.
This practice book has the same pattern for next pages. And the book continues Zahlenhaus 8, 9, 10. She will never make a mistake without understandings. (I sometimes see such kind of research. Nobody knows how it works, but, it seems working. I don't want to call it a research because no understandings. But, maybe they are still fine since they add some experimental knowledge to the society.)
I told her, ``Everything is correct. It is good to find a pattern.'' But I thought a while. Her method is great, but it will fail at some point. How can I persuade her there is a case it fails?
``Mathematics is a kind of language. So there are meanings all of these lines. You can express something with mathematics.... Here, there are plus and minus. What is the `plus' meaning?''
``Plus means more.''
``Yes, ... but, why 1 + 6 is equal to 7?''
``???''
``1 + 6 is not equal to 10. But, 10 is more than 1 and more than 6. Why 1 + 6 is not equal to 10? If plus means just more, is it OK that 1 + 6 is equal to 8, 9, or 10?''
I usually need to care not overwhelming children. Too much explanation is not good. That I learn from other teachers here. So I slow down.
I came back to the Zahlenhaus and asked her to count the 1 and 6 points in the Zahlenhaus. She can count. So, she realized the total number of the points is 7. I explained `plus' means `together'. If something and something together, then they are more. Therefore, `more' is correct, but, it is better to say `together' for plus's meaning.
Next my task is to explain when her method fails. Now she understand 1 + 6 has a meaning. Therefore, this line itself has an answer. Therefore, this 4 line set of Zahlenhaus practice is not always presented. She can not always have four questions. In that case, she can not copy the answers.
This actually takes around 20 minutes. But, I think she realized this.
This time, I was surprised that 8 year old child found, even partially, the following pattern.
\begin{eqnarray*}
\begin{array}[t]{cccc}
x & + & y & = z \\
y & + & x & = z \\
z & - & x & = y \\
z & - & y & = x \\
\end{array}
\end{eqnarray*}
Computer stores variables in a memory. A memory cell has an address. If a computer language didn't hide the address, like C or C++, the variable has an address. An address is a name of location. Therefore, we, programmers, need to develop the idea, a value is associated with a location as she explained me. I see she has such ability: associating a location and a value. I wonder if I taught a computer language to her. Well, first she should master what is plus, minus, and more calculation.
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