Looking back on the 2013-2014 school year for my three algebra 1 periods, my third year as a full-time teacher, I would like to think that the hard work, crazy hours, and stress filled moments were worth it. If I am truly honest with myself, they were not, especially for my family. I sacrificed too much for too little. Yes, students improved in their mathematical proficiency, of that I am sure. Some students made huge gains, many slight gains, and a few appeared to recede due to unfortunate personal situations. Yet, few humans could endure, year after year, heck – maybe not even for a week, the level of effort required of teachers to move the needle in similar circumstances; specifically, if moving the needle means that most, or all, students pass the course with scaffolding or other interventions.

I believe these circumstances, better known as initial conditions in the world of mathematics and science, are central to perpetuating what many call the achievement gap, or the “fall” in our national standing worldwide in academic achievement; whether either, the achievement gap or the fall in our nation’s worldwide ranking, are truly manifestations of what they purportedly represent is debatable. Nonetheless, rearranging the deck chairs, such as the shuffling and tweaking of content standards à la the Common Core, will not alter the fate of our faulty course in any meaningful manner as there may not be a solution given the constraints and initial conditions no matter what methods are applied.

In my view, revamping our “one size fits all,” “damn the torpedoes” approach to public education to give students, and their families, more influence over their path forward offers our greatest hope. But that is for a future post. For now, let us look in the mirror a moment to see the situation facing typical Title I high school algebra 1 classes.

## Years Behind in Prerequisite Understanding

The immense range of student prerequisite understanding, mostly years below grade level, made it nearly impossible to teach students the district specified Common Core curriculum for algebra 1; while earlier cohorts had similar levels of understanding, I had much more influence with them over the curriculum: content, pacing, assessments, and etcetera. [*See the figure titled “Final Exam versus Readiness Test Scatterplots” via the earlier link for a clear comparison between cohorts spanning three academic years.*]

The following figure serves as a proxy for student readiness with each student’s scores on two different diagnostic tests represented as a diamond. The horizontal axis represents their score on a thirty question assessment with mostly fifth and sixth grade level topics. The vertical axis represents their score on a fifty question assessment with some overlapping topics along with straightforward algebra 1 topics, as most students to date took algebra 1 at least once beforehand; this is changing with the onset of Common Core.

Red diamonds represent students who transferred out of my class, for a variety of reasons to include being moved up to geometry given their scores to leaving the school or district. [1] Green diamonds represent students who transferred into my class at some point in the school year whether from another algebra class in our school or another school or district. Blue diamonds represent students who started and finished in my class. Notice how scores vary over a range of seventy to eighty percentage points on each test. This variance, along with the below passing means on each, underscores the disparity in student readiness for algebra 1.

Furthermore, while the new Common Core based curriculum included the traditional linear- and quadratic relationships and functions covered with earlier cohorts, it expanded to include exponential functions, complex numbers, and recursive notation, among others. All of these topics needed to be presented against a richer backdrop of mathematical practices, which compounded student difficulty as they suffered more than prior cohorts’ who were mostly limited to procedural misunderstandings.

Hence, at the end of the first semester, realizing the folly of continuing to teach all students the same district-specified curriculum, I requested that my three periods of algebra 1 be split into two concurrently run sections per period: one algebra 1 and the other algebra intervention. While this measurably helped students, as the data below show, it came at too great a cost for me, so much so that I requested not to teach algebra this coming fall. My workaholic tendencies brought me the debilitating effect of burnout. But enough of that for now. Let’s look at some further data.

## Improvement

The following data show improvements of 14% and 20%, on average, of students’ scores on two diagnostic exams given at the outset of the school year as a pretest and at the end of the school as a post-test after an intensive second semester intervention program.

Using a one-sided, paired t-test, student gains (n=56) after the intervention are considered statistically significant at a p-level far less than 0.01 with associated effect sizes of just over 0.5, computed as Cohen’s d, suggesting a moderate to high level of practical significance. Students who were not present for both the pretest and post-test were not included in this analysis.

The following figure illustrates gains using a percentage point basis.

Note that the preponderance of scores in both figures show positive improvements on both assessments. The two prominent negative outliers in the data represent students who struggled with significant personal issues throughout the school year; I believe the scores belie their true ability, or understanding, as they each scored highly on both pretests. Removing their contribution from the data shifts the average improvement upwards to ten percentage points for each assessment.

## Comparing Pre-test to Post-test Score Distributions

Comparing pre-test and post-test distributions of student scores on the math lab diagnostic as box and whisker plots below shows a clear upward shift in scores. At the same time, a significant percentage of students still scored below basic in their understanding of these foundational mathematics topics.

Likewise, comparing pre-test and post-test distributions of student scores on the algebra 1 diagnostic as box and whisker plots below shows a clear upward shift in scores. With this assessment, the higher initial median score coupled with the intervention gain indicates that student readiness improved significantly, hopefully leading to success in a later algebra 1 course.

## Conclusion

Clearly students improved in their scores on each of these diagnostic assessments. While many students still failed the course second semester, whether algebra 1 or algebra intervention, most improved in their mathematical proficiency nonetheless. Whether the improvement is enough for those students to pass algebra 1 next school year, or as credit recovery during the summer, only time will tell. I believe most are now more able to pass, and likely will if they apply themselves. Whether they do or not is one of the mysteries of student behavior we have yet to unravel, however.

[1] One student was a misplaced senior who failed second semester algebra 2 the prior year: he scored near 100% on both tests – I was sad to see him go as he, nearly singularly, possessed the necessary skill to do well in the course.

I hope this does not come across as spam*, but I am honestly curious if you think it would help the readiness problem if students did all their work in an interactive environment that checked their work as they went? This way, even though their fundamentals are deficient they will find out immediately that they made a mistake, fix it, and keep going. More slowly, of course, until (being optimistic) their fundamentals improve thru practice.

* I am developing a tool that does this.

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Hi Kenneth.

I really like your site (tiltonsalgebra.com) and resonate with your pedagogical philosophies. As a quasi-techie myself, in my heart, and my gut, I know there is a much better way to help students overcome misunderstandings, misconceptions, and mistakes learning arithmetic and algebra and your approach aligns with how I would consider doing so. As with anything in education, the student must desire the knowledge before he/she will develop mastery. If the student is ready, your site appears to offer him/her a great opportunity to succeed. Way to go! If you are ever in Silicon Valley, let me know so we can meet to talk more.

Dave

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Thanks! Probably my biggest mistake was not moving to Silicon Valley thirty years ago! But there is talk of an ed-tech show-and-tell in the valley in November, if they invite me I’ll give you a shout. Cheers, Ken

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Great. Looking forward to meeting you and talking shop.

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