Question

Can moving their hands help children learn math? This question was investigated in the paper "Gesturing Gives Children New Ideas About Math" (Psychological Science [2009]: \(267-272\) ). Eighty-five children in the third and fourth grades who did not answer any questions correctly on a test with six problems of the form \(3+2+8=+8\) were participants in an experiment. The children were randomly assigned to either a no-gesture group or a gesture group. All the children were given a lesson on how to solve problems of this form using the strategy of trying to make both sides of the equation equal. Children in the gesture group were also taught to point to the first two numbers on the left side of the equation with the index and middle finger of one hand and then to point at the blank on the right side of the equation. This gesture was supposed to emphasize that grouping is involved in solving the problem. The children then practiced additional problems of this type. All children were then given a test with six problems to solve, and the number of correct answers was recorded for each child. Summary statistics are given below. \begin{tabular}{lccc} & \(n\) & \(\bar{x}\) & \(s\) \\ No Gesture & 42 & 1.3 & 0.3 \\ Gesture & 43 & 2.2 & 0.4 \\ \hline \end{tabular} Is there evidence that learning the gesturing approach to solving problems of this type results in a significantly higher mean number of correct responses? Test the relevant hypotheses using \(\alpha=0.05\)

Short answer

Based on the steps above, the result will depend on the calculated P-value. If this is less than the significance level (0.05), then there is evidence to suggest that gesturing leads to a higher average number of correct responses. Otherwise, there is not enough statistical evidence to support this claim.

Step-by-step solution

Formulate Hypotheses

Our null hypothesis \(H_0\) is that the average number of correct responses for both groups is equal (i.e. the gesturing doesn't affect the results), which is mathematically \(\mu_{gesture} = \mu_{no-gesture}\). The alternative hypothesis \(H_a\) is that the children in the gesturing group have a higher average number of correct responses, mathematically represented as \(\mu_{gesture} > \mu_{no-gesture}\).

Conduct the Test

We will use a two-sample t-test for the means of two independent populations at a significance level of \(\alpha=0.05\). The test statistic is given by the formula: \(t = (\bar{x}_1 - \bar{x}_2) / \sqrt{(s^2_1/n_1) + (s^2_2/n_2)} \) where \(\bar{x}_1, \bar{x}_2\) are sample means, \(s^2_1,s^2_2\) are sample variances and \(n_1,n_2\) the sizes of the two groups. Substituting given values: \(t = (2.2 - 1.3) / \sqrt{(0.4^2/43) + (0.3^2/42)}\).

Determine the P-value

For our calculated test statistic, we now need to determine the corresponding right-tail P-value using the t-distribution table with degrees of freedom \(df = n_1 + n_2 - 2\). The P-value gives us the smallest significance level at which we would reject the null hypothesis. If the P-value is less than our selected significance level, \(\alpha=0.05\), we reject the null hypothesis.

Conclusion

If the P-value is less than \(\alpha=0.05\), we may conclude at a 95% confidence level that gesturing does lead to a higher average number of correct responses. If not, we do not have enough statistical evidence to support this claim. We would simply interpret the result in the light of the data and our statistical analysis.