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Effective Math Intervention

Show me an Effective Math Intervention
Program Looks Like

Author: Apolinario "Sam" Ortega, founder of IN-V-BAT-AI

All Arizona public schools, including district schools and charter schools, are required to properly administer state and federally mandated assessments.

To better understand how an effective intervention math program looks like, a discussion of Null Hypothesis testing is necessary.

In hypothesis testing literature null hypothesis is often designated by variable letter H₀ read as H naught. Null hypothesis means there is zero or null relationship/no difference/no effect that exist between the implemented intervention method and the existing business as usual method. For example, assume your state is Arizona (AZ). You want to run a randomly controlled trial to validate your hypothesis that your proposed intervention program method will improve the mathematics test score of 4th grader in your class when compared to NAEP 2022 published test score which is 232.

Figure - 1

Effectiveness of Reading and Mathematics Software Products: Findings from the First Student Cohort Report to Congress
March 2007


Zero Effect Hypothesis was proven by data as true



Zero Effect Hypothesis was proven by data as true

In order to do that you will need to produce evidence to prove your hypothesis is valid or reasonable. Then you will set up controlled group of 4th grade students that do not use your proposed intervention program method. This is the business as usual or do the existing mathematics teaching program for 4th grader. Then you will set up intervention group of 4th grade students that will use your proposed intervention program method.

The null hypothesis is initially assumed to be true in hypothesis testing unless there is significant evidence to reject the initial hypothesis. In this case the null hypothesis is saying there is no relationship/no difference/no effect/no improvement the implemented intervention program when compared to your published 2022 STATE NAEP 4th grader average score. In order to do the comparison, you will need to use sample previous test questions from NAEP 4th grader mathematics test database. This test questions will be used as a baseline or reference for pre-test to collect the statistical mean value of controlled group μ1 (mean score for controlled group) For example for controlled group of twenty (20) 4th grader students you administered pre-test exam using previous NAEP test questions for 4th grader. Then using NAEP automatic scaling score you will record the corresponding scale score of 20 students. Let's assume your controlled group of twenty (20) students their statistical pre-test mean value is = 232 exactly the same with your state average NAEP scale score. Therefore controlled group pre-test mean value, μ1 = 232. You do the same for your intervention group of students and again let's assume your computed their pre-test mean value, μ2 = 232.

The null hypothesis states that after implementing the proposed intervention program the statistical computed mean score of controlled groups of students (do not received intervention instruction) and the statistical computed mean score of intervention group of students has zero or null difference meaning no effect, no improvement in test score. Shown below is the computation to prove that there is no difference between control group of students and intervention group of students.

The zero is also called null. Hence, it is known as Null Hypothesis. The null hypothesis is the initial assume true statement unless proven by evidence that there is a significant test score improvement after implementing the intervention instruction.

Written mathematically as
H₀ : μ1 (mean of controlled students, not using intervention software) = μ2 (mean of intervention students, user of software)


H₀ : μ2 (MEAN 2, software user) - μ1 (MEAN 1, not user) = 0 The intervention program using software has no difference/effect/relationship to test score improvement. Here the initial hypothesis that there is no evidence or zero improvement in test score, the null hypothesis is retained.

After post - test , you computed again mean score of controlled (not user) and intervention students (software user).

Here mean score μ1 = μ2 are the known population under investigation for example the population of 4th grader in the whole Arizona state. Since it is impractical and expensive to collect all the mean score of known population under investigation. Researcher usually agree to a reasonable sample size mean to approximate the true behavior of the known population.

Let x₁ as our controlled group sample mean score (not user)

Let x₂ as our intervention group sample mean score (software user)


H₀ : μ1 = 232 (μ1 mean score =232) and μ2 =232 (μ2 mean score =232)


H₀ : x₁ = 237 (x₁ mean score =237) and x₂=240 ( x₂ mean score =240)

H₀ : μ1 = μ2 or x₁ = x₂The intervention program has no difference/effect/relationship. Here the null hypothesis is retained.

If the null hypothesis is not true or not valid. It means the intervention group got a higher mean score compared to controlled group of students. Therefore there is an alternative hypothesis, written mathematically as H_a or H₁


H_a or H₁: μ2 ≠ μ1 The intervention program has evidence of difference / effect / relationship. So the alternative hypothesis using limited sample of population appears to be a good alternative.


H_a or H₁: μ2 - μ1 ≠ 0 The intervention program has evidence of difference / effect / relationship. So the alternative hypothesis using limited sample of population appears to be a good alternative to improve students mathematics test score outcome with respect to limited sample from NAEP previous test database.

Disclaimer : I am not trained as a statistician, my explanation below is my best shot explanation on how to interpret p < 0.05 use in null hypothesis testing.
by Apolinario "Sam" Ortega, founder of IN-V-BAT-AI

The randomized post test score of intervention students are shown below for discussion purposes. The researcher usually makes a decision what is the acceptable significant evidence to prove that the intervention program is working. This significant evidence is captured by the alpha value, α . The common alpha value use by industry researcher is α = 0.05. For example, NAEP use p < 0.05 as significant evidence criteria. What does p < 0.05 means? To answer this question, you must understand what is the letter variable "p" means? The letter variable "p" is the placeholder variable for computed probability of student that did not improve even after taking the intervention instruction (null hypothesis). Therefore, the variable letter "p" is associated to the computed probability of null hypothesis.

So, how do you calculate the probability of student that did not improve p < 0.05 ? With probability (p) express in fraction 1/20 = 0.05, it becomes easy to understand and explain the meaning of p < 0.05. It means 1 out of 20 students did not improve their test score even after taking the intervention instruction. But the 19 out of 20 students improve their test score. Wow 19 out of 20 students improve their test score, you will conclude that the intervention instruction is significant enough. Now you understand why NAEP uses p < 0.05 as the significant value criteria which is the industry standard for alpha, α

It means 1 student out of 20 students DID NOT improve his/her test score after taking the intervention instruction. It means 1 out of 20 students has zero effect or no difference to improve his test score or null effect even after using the intervention instruction. But the remaining 19 students out of 20 have clear evidence that they improve their test score. The α = 0.05 significant value means 19 out 20 students improve their average test score compared to the reference average test score.

So every time you see this probability expression ( p < α = 0.05) , it is telling you that only 1 student DID NOT improve his test score but the remaining 19 students out of 20 students did improve their test score.

Now, let's use actual data to see how the p < α =0.05 works. Shown below, only student ID = 4 did not improve his test score even after taking the intervention instruction. We can say that 1 out of 20 students the intervention has null or zero effect to improve his test score. But the remaining 19 students did improve their test score compared to state average score.

Now you understand the meaning of p < 0.05. It means 1 student out of 20 did not improve their test score even after taking intervention instruction that was demonstrated by evidence as significantly effective for improving test score. So what does it mean? It means for AZ 4th graders with total enrollment of 80,780 in FY21, if AZ Department of Education implemented a significantly effective intervention instruction there would be evidence that 80,780 x 0.05 = 4,039 students will not improve their test score, but 76,471 students will improve their test score. That is WOW, a significant improvement.

IN-V-BAT-AI, a personal AI tutor service provider is offering this Math personal AI tutor intervention to AZ schools.

Now Review The Performance of 4 Schools (Hypothetical - not based on actual data)

Here it appears that School A has a significant improvement because only one student did not improve. But statistics has a standard practice of computing probabilities associated with null hypothesis designated by p-value. Upon application of statistical computation, it turns out that School A did not improve significantly. But if you review the detailed test score, you could argue that 19 out of 20 students improve their test score compared to state average score seems significant enough. According to NAEP scaling score AZ students from School A is above basic mathematics and below proficient in their mathematics skills.

Here School B has a significant improvement many students exceeded the state average test score but some has test score equivalent to the state score. But how significant is the improvement? The statistics has a standard practice of computing probabilities associated with null hypothesis designated by p-value. Upon application of statistical computation, it turns out that School B did a p value of 0.0000, an increase of 12 points compared to state average score of 232. According to NAEP scaling score AZ students from School B is very close to proficient in their mathematics skills.

Here School C has a significant improvement many students exceeded the state average test score. But how significant is the improvement? The statistics has a standard practice of computing probabilities associated with null hypothesis designated by p-value. Upon application of statistical computation, it turns out that School C did a p value of 0.0000, an increase of 19 points compared to state average score of 232. According to NAEP scaling score AZ students from School C are proficient in their mathematics skills.

Here School D has a very significant improvement many students exceeded the state average test score. But how significant is the improvement? The statistics has a standard practice of computing probabilities associated with null hypothesis designated by p-value. Upon application of statistical computation, it turns out that School D did a p value of 0.0000, an increase of 62 points compared to state average score of 232. According to NAEP scaling score AZ students from School D are all advanced in their mathematics skills.

Important thing to remember notice school B, school C, and school D all of them have reported p=0.0000. How will you compare which school did perform better using t-test information only? By using t-test score, you can compare which school perform better. School D has the highest t-test score of 9.94 therefore school D is the best in performance. School A t-test = 1.35 , School B t-test = 6.33, School C t-test = 7.81

Understanding the phrase, "None of the effect is significantly different from zero at the 0.05 level", this phrase I believe is related to Null Hypothesis or Zero Effect Hypothesis. Algebra software average license fee per student per year is $15. The 6th grade mathematics software license fee range is $7 to $30. Go to page 25 of 140

Understanding the phrase, "None of the effect is significantly different from zero at the 0.05 level", this phrase I believe is related to Null Hypothesis or Zero Effect Hypothesis. 6th Grade mathematics software average license fee per student per year is $18. The 6th grade mathematics software license fee range is $9 to $30. Go to page 24 of 140

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