Ich bin im Rahmen meiner Experimente mit ChatGPT auf ein Repository von Beispiel-Prompts für ChatGPT gestoßen. Das habe ich als Anlass genommen, einen weiteren Versuch zu starten, ChatGPT eine tragfähige Erklärung zu entlocken, warum das Produkt negativer Zahlen positiv ist.

Das ganze fühlte sich nach einem ermüdenden Gespräch mit Hilfe der sokratischen Methode an. Die einzelnen Bausteine einer guten Erklärung sind zwar erkennbar, aber weiterhin fehlt der KI das mathematikbezogene Weltmodell, um eine konsistente und fehlerfreie Erklärung zu generieren.

Erklärungen generieren mit ChatGPT

Auf Basis der Vorschläge des oben verlinkten Repositories habe ich ChatGPT angewiesen, in die Rolle einer Mathematik-Lehrperson zu schlüpfen. Diesmal arbeite ich auf Englisch − in der Hoffnung, dass das Sprachmodell hier stärker ist.

Prompt 01: I want you to act as a math teacher. I will provide some mathematical equations or concepts, and it will be your job to explain them in easy-to-understand terms. This could include providing step-by-step instructions for solving a problem, demonstrating various techniques with visuals or suggesting online resources for further study. My first request is “I need help understanding why the product of two negative numbers is a positive number.”

Certainly! I’d be happy to help you understand why the product of two negative numbers is a positive number.

When you multiply two negative numbers together, the result is positive. This is because the two negative signs cancel each other out. For example, (-3) * (-4) = 12. In this case, the two negative signs are like a pair of opposites that cancel each other out, leaving us with a positive number.

It might help to think about it this way: if you had a balance scale and placed a negative weight on one side and a negative weight on the other side, the scale would balance out and be in a neutral position. This is because the two negative weights are canceling each other out. The same idea applies to negative numbers in multiplication.

I hope this helps! Do you have any other questions about this concept, or is there anything else you’d like to learn about?

Als ersten Ansatz verwendet ChatGPT wieder das Kalkülargument. Diesmal sehen wir aber den Versuch einer verständnisorientierter Erklärung. Leider ist sie vollkommen falsch. Das hier bemühte Waagemodell ist ein Erklärungsmodell für Äquivalenzumformungen und nicht für Multiplikationsoperationen.

Wir haben im letzten Blogeintrag schon sehen können, dass ChatGPT mathematische Kalküle nicht gut auseinanderhalten kann. Neben Addition und Multiplikation scheint das Sprachmodell auch Äquivalenzumformungen hin und wieder in einen Topf zu werfen.

In den nächsten Schritten habe ich versucht, spezielle Erklärungsmodelle einzufordern und die „Lernfähigkeit“ von ChatGPT für den Kontext eines Chatverlaufs zu bemühen.

Prompt 02: Can you explain this again? This time using the permanence principle?

Certainly! The permanence principle is a mathematical concept that states that certain operations preserve certain properties of numbers. For example, the permanence principle tells us that the product of two positive numbers is always positive, and the product of two negative numbers is always positive.

To understand why the product of two negative numbers is positive, let’s consider an example: (-3) * (-4) = 12. In this case, we can use the permanence principle to understand that the product of two negative numbers is always positive.

Here’s how it works: when we multiply two numbers together, we are combining their quantities. For example, when we multiply 3 * 4, we are combining 3 groups of 4 objects, resulting in a total of 12 objects. The same is true when we multiply negative numbers. When we multiply (-3) * (-4), we are combining -3 groups of -4 objects, which results in a total of 12 objects.

So, according to the permanence principle, the product of two negative numbers is always positive because the operation of multiplication preserves the property of quantity, regardless of whether the numbers are positive or negative.

I hope this explanation helps! Do you have any other questions about this concept, or is there anything else you’d like to learn about?

Okay, mein Fehler … „permanence principle“ ist keine adäquate Übersetzung von Permanenzreihen. Aber auch hier ergibt die Erklärung keinen Sinn. Einen schönen Gedanken zum Weiterdenken gibt mir ChatGPT dennoch mit: „the operation of multiplication preserves the property of quantity, regardless of whether the numbers are positive or negative.“

Prompt 03: Can you explain this again? This time using a series of instructive examples?

[…] Example 1: (-3) * (-4) = 12 In this example, we see that the product of two negative numbers is a positive number. When we multiply two negative numbers together, the result is positive because the two negative signs cancel each other out.

Example 2: (-5) * (-2) = 10 […]

Example 3: (-7) * (-9) = 63 […]

Auch hier generiert ChatGPT keine Permanenzreihe bzw. keine adäquate Beispielserie, aus der sich eine Erklärung ableiten lässt.

Wechseln wir zu einem anderen Erklärungsmodell: Operationen am Zahlenstrahl.

Prompt 04: Can you explain this again? This time using operations on the numberline?

Certainly! The number line is a visual representation of the set of real numbers, with negative numbers on the left and positive numbers on the right. When we perform operations on the number line, we can get a better understanding of how negative numbers behave.

To understand why the product of two negative numbers is always positive, let’s consider an example: (-3) * (-4).

If we represent these numbers on the number line, we can see that -3 is located 3 units to the left of 0, and -4 is located 4 units to the left of 0. When we multiply these numbers together, we are combining their quantities. In other words, we are moving -3 units to the left and -4 units to the left. This results in a total of -7 units to the left of 0.

However, when we represent this distance on the number line, we see that it is actually 7 units to the right of 0. This is because the two negative signs cancel each other out, resulting in a positive number.

So, to summarize, the product of two negative numbers is always positive because the two negative signs cancel each other out, resulting in a positive number.

[…]

Wie letztes Mal: ChatGPT verwechselt Addition und Multiplikation. Können wir dem Sprachmodell auf die Sprünge helfen?

Prompt 05: Can you explain multiplication on the numberline representation of numbers?

[…]

If we represent these numbers on the number line, we can see that 3 is located 3 units to the right of 0, and 4 is located 4 units to the right of 0. When we multiply these numbers together, we are combining their quantities. In other words, we are moving 3 units to the right and 4 units to the right. This results in a total of 7 units to the right of 0.

Noch nicht ganz …

Prompt 06: Can you explain the difference of addition and multiplication using the numberline representation of numbers?

Addition and multiplication are two different operations that we can perform on numbers. Here is how these operations differ when we use the number line to represent them:

Addition: When we add two numbers together, we are finding their sum. For example, 3 + 4 = 7. On the number line, we can represent this operation by starting at the position of the first number and then moving a certain distance to the right (or left) to reach the position of the second number. The distance we need to move is equal to the value of the second number. For example, if we start at the position of 3 on the number line and want to find the sum 3 + 4, we would move 4 units to the right to reach the position of 7.

Multiplication: When we multiply two numbers together, we are combining their quantities. For example, 3 * 4 = 12. On the number line, we can represent this operation by starting at the position of 0 and then moving a certain distance to the right (or left) a certain number of times. The distance we need to move is equal to the value of one of the numbers, and the number of times we need to move is equal to the value of the other number. For example, if we want to find the product 3 * 4 on the number line, we would start at 0 and move 3 units to the right 4 times, which would take us to the position of 12.

[…]

Okay! Zumindest für ganze Zahlen kommen wir der Sache näher. Erfragen wir noch ein weiteres Puzzlestück.

Prompt 07: Can you explain why the multiplication with (-1) results in a reflection at the center of the numberline?

[…] When we multiply a number by -1, we are essentially flipping it across the center of the number line. For example, if we multiply 3 by -1, the result is -3. On the number line, 3 is located 3 units to the right of 0, and -3 is located 3 units to the left of 0. We can think of this as reflecting 3 across the center of the number line to get -3.

Similarly, if we multiply -3 by -1, the result is 3. On the number line, -3 is located 3 units to the left of 0, and 3 is located 3 units to the right of 0. We can think of this as reflecting -3 across the center of the number line to get 3.

[…]

Das ist zwar keine Erklärung aber immerhin scheint das „Wissen“ nun in der Chathistorie sichtbar zu sein. Mit dem nächsten und letzten Prompt wurde ich wohl etwas übermütig.

Prompt 08: Now, using our established understanding of the multiplation with (-1): Can you explain again, why the product of two negative numbers is a positive number?

[…]

When we multiply two negative numbers together, we are reflecting each number across the center of the number line (0) and then multiplying their quantities. For example, (-3) * (-4) is the same as reflecting -3 across the center of the number line to get 3, and then reflecting -4 across the center of the number line to get 4. We can then multiply 3 and 4 together to get 12.

On the number line, we can visualize this process as follows:

Start at the position of -3 on the number line. Reflect -3 across the center of the number line (0) to get 3. Start at the position of -4 on the number line. Reflect -4 across the center of the number line (0) to get 4. Multiply 3 and 4 together to get 12.

So, to summarize, the product of two negative numbers is a positive number because when we multiply negative numbers together, we are reflecting each number across the center of the number line and then multiplying their quantities, which results in a positive number.

[…]