Getting Started 
Misconception/Error The student is unable to clearly conjecture a causeeffect relationship between the two correlated variables. 
Examples of Student Work at this Level The student:
 Identifies the two variables (e.g., the number of hours of sleep and reading scores) but does not describe a possible causal relationship.
 Says the negative correlation coefficient means there is no association between hours of sleep and reading ability.
 Describes a causal relationship that is not consistent with the sign of the correlation coefficient (e.g., the student says more sleep leads to better reading scores).
 Interprets the correlation, that is, states that as number of hours of sleep increases, reading scores decrease.

Questions Eliciting Thinking What two variablesÂ are describedÂ in this problem?
What does causal relationship mean?
Does the fact that the variables are correlated mean that there is a causal relationship? 
Instructional Implications Explain what is meant by a causal relationship and provide examples. Explain that when two variables, A and B, are correlated, it might be the case that A causes changes in B, B causes changes in A, or some third variable is influencing both A and B together. Describe two possible causal relationships with regard to the variables, number of hours of sleep and reading scores.
Review the significance of the magnitude of the correlation coefficient (i.e., that it indicates the extent to which there is a linear association between the two variables). Also, review the sign of the correlation coefficient and what it means (i.e., a positive correlation coefficientÂ indicates thatÂ an increase in one variable is associated with an increase in the other while aÂ negative correlation coefficient indicates thatÂ an increase in one variable is associated with a decrease in the other). Remind the student that correlationÂ does not necessarily implyÂ causality. Use the example that a positive correlation between shoe size and scores on an arithmetic test is more likely due to the fact that older children tend to have larger shoes as well as a stronger mathematics background. Ask the student whether the data suggest that more sleep is associated with higher scores or lower scores and if there might beÂ a lurking variable that affects both the duration of sleep and reading scores.Â
Consider implementing other MFAS tasks. 
Making Progress 
Misconception/Error The student does not appear to understand the distinction between correlation and causation. 
Examples of Student Work at this Level The student describes a possible causal relationship between the variables, such as more sleep leads to lower reading scores. However, the student is unable to determine if the correlation coefficient indicates that the relationship between the variables is cause and effect. For example, the student concludes that the relationship must be causal because:
 The correlation coefficient means that amount of sleep must haveÂ an effect on reading scores.
 The data follow a negative slope and is reasonably strong.
 The more sleep you get, the worse you do because sleep takes up study time.
The student concludes that the relationship is not causal because:
 The correlation coefficient is negative.
 The correlation coefficient is not very big.Â

Questions Eliciting Thinking On what basis did you draw your conclusion?
Can one infer a causeeffect relationship on the basis of the correlation coefficient?
Are there other possible explanations for the relationship between these variables? 
Instructional Implications Provide instruction on the distinction between correlation and causation. Emphasize that correlation does not imply causation. Explain that when two variables, A and B, are correlated, it might be the case that A causes changes in B, B causes changes in A, or some third variable is influencing both A and B together. Provide the student with an example of data that include the grade level, number of hours of sleep, and average reading scores for nine children in grades K â€“ 8. Ask the student to represent the hours of sleep and reading scores on a scatter plot. Then ask the student to identify a lurking variable in the hoursofsleep and readingscore data. Prompt the student to explain why the correlation coefficient is negative and not near zero or one.
Provide instruction on how causation can be established (e.g., by conducting a carefully designed experiment in which variables are controlled). Distinguish between an observational study and an experiment, and make clear that observational studies result in correlational data while welldesigned experiments test for a causal link between variables. Ask the student to look for examples in the real world (e.g., in news reports, newspapers, and magazines) of statements of causality and to assess whether a causal conclusion seems warranted.
Consider implementing other MFAS tasks. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student describes a possible causal relationship (e.g., more sleep causes lower reading scores). The student explains that one cannot infer any causation in this example since correlation, in general, does not imply causation. The correlation coefficient of 0.48 means that there is some trend in longer sleeping hours being associated with lower reading scores. The reason might be that younger people need more sleep and tend to have lower reading scores. The age of the students is a lurking variable. 
Questions Eliciting Thinking How could one improve the study in order to better assess the relationship between hours of sleep and reading ability?
Would the correlation coefficient change if the sleep time were measured in minutes rather than hours? 
Instructional Implications Ask the student to formulate another example in which a negative correlation does not guarantee a causal relationship. 