## Inferences

An inference is a conjecture based on inductive reasoning. A conclusion is a statement that follows logically from other facts.

We can take a basic statement and assume that the data we see is leading to some sort of conclusion, but we may be misunderstanding the data and are instead presenting a generalization. The use of inductive reasoning, which tends to make conclusions from a set of finite facts or data is not a flawless approach to problem solving because it may allow someone to misunderstand the information. It is only through validation that data can be argued correctly.

Here is a common example: A teacher assigns her students daily homework, and notices that the scores on quizzes and tests improve dramatically. She concludes from the “data” that the homework is making the scores higher. Clearly, this is not an accurate way to compare or analyze any data in order to determine the relationship between the homework and the improved test scores.

What is necessary for this to be done correctly using Algebraic thinking? The teacher should select or create mathematical models for the data sets. These models must accurately describe or represent each of the data sets (the homework and the test results). Only then can the relationship between them be accurately measured.

Until the teacher does this, she is using a conjecture that has not been validated. There are serious limitations for the conclusions based on this approach simply because she may be misrepresenting certain parts of the data through her assumptions and faulty conclusions.

In order to validate the data, the teacher should have done one or more of the following exercises:

• Drafted a “table of values” based on the situation;
• Written a sentence in words along with an equation for the results; or
• Graphed data points.

This would have allowed her to prove that she was not confusing the idea of association of certain coefficients, variables, constants and operators with “causation”.

#### Try these questions

1. Which of the following is a conjecture based on inductive reasoning?
1. Inference
2. Conclusion
3. Causation
4. Association
2. Which of the following tends to make conclusions from a set of finite facts or data is not a flawless approach to problem solving because it may allow someone to misunderstand the information?
1. Iinductive reasoning
2. Validation
3. Causation
4. Association
3. In order to validate the data one should have done which of the following exercises?
1. Drafted a “table of values” based on the situation
2. Written a sentence in words along with an equation for the results
3. Graphed data points.
4. All of the above

1. Answer: A. Likewise, a conclusion is a statement that follows logically from other facts.
2. Answer: A. It is only through validation that data can be argued correctly.
3. Answer: D. An example would be a teacher assigns her students daily homework, and notices that the scores on quizzes and tests improve dramatically. She concludes from the “data” that the homework is making the scores higher. Clearly, this is not an accurate way to compare or analyze any data in order to determine the relationship between the homework and the improved test scores

#### Final Questions

1. Which of the following is a conjecture based on inductive reasoning?
1. Inference
2. Conclusion
3. Causation
4. Association
2. Which of the following is a statement that follows logically from other facts?
1. Inference
2. Conclusion
3. Causation
4. Association
3. Which of the following tends to make conclusions from a set of finite facts or data is not a flawless approach to problem solving because it may allow someone to misunderstand the information?
1. Inference
2. Conclusion
3. Causation
4. Association
4. It is only through which of the following that data can be argued correctly?
1. Inference
2. Conclusion
3. Causation
4. Association
5. In order to validate the data one should have done which of the following exercises?
1. Drafted a “table of values” based on the situation
2. Written a sentence in words along with an equation for the results
3. Graphed data points.
4. All of the above