Problems with Prerequisites: Part 1

Late last month upon returning from my college 25-year reunion, I decided to reassess my algebra 1 students’ prerequisite knowledge since their performance on earlier assessments repeatedly reflected deficiencies in those skills.  Difficulties ranged from adding and subtracting integers to operations with rational numbers to basic algebraic simplification and solutions.  These misunderstandings of basic numeric and pre-algebra skills limit their success on most algebra assessments.  Until those skills are remediated, my students will continue to suffer in their ability to learn algebra or other higher level mathematics.

My assessment contained fifty questions covering skills needed to succeed in algebra as well as reviewing the most recent topic covered in class.  Each of my three algebra 1 classes took the assessment in the span of a 55-minute period.  For the most part, problems did not require a calculator and consisted of single digit numbers, whether integer or rational.  Example problems from each concept area, the number of questions per area, and summary results of each class on each area follow.

Pre-algebra & Algebra Skills Assessment

As the data in the table show, students did not fare very well on much of the assessment.    Even a class average of 80%, which might be considered impressive by some, signaled challenges in students’ fundamental mathematical proficiency as the complexity of the operations and numbers in each problem merited a higher score.  In contrast to the semi-success with integer operations, student performance with fractions indicated significant struggles. Furthermore, and somewhat surprisingly, the simple presence of the variable “x” with the same integers and mathematical operations led to scores falling considerably, as shown in the following chart.  Performance declined less from rational numbers to rational expressions since those students who struggled with fractions were already removed from consideration, which speaks volumes in itself.

Given these results, it is clear that intervention in these topic areas must occur immediately.  As we wind down this semester’s curriculum, I must devise a plan to give needed support to many students, while moving forward with the district mandated curriculum.  At the same time, student effort in areas requiring prerequisite competency will continue in vain unless they develop needed prerequisite skills.  Intensive intervention in these key areas must precede, or parallel, further algebra instruction, otherwise, students will continue to struggle demonstrating even rudimentary understanding of new algebraic concepts and procedures.  Why this is not obvious to school districts today escapes me.  Hence, the wrongs the current implementation is meant to prevent simply continue to revisit themselves upon students in less obvious forms.  Worse yet, the energy and focus of education reform and policy, associated pedagogy, curricula, and standards today is misdirected at best, and at worst, perpetuates the sins of the past.  Students cannot succeed in learning if they do not have needed prerequisite skills.  Walking is near impossible without having learned to crawl and experiment with balance; running is beyond conception, as much as any mandate may require.

About Dave aka Mr. Math Teacher

Independent consultant and junior college adjunct instructor. Former secondary math teacher who taught math intervention, algebra 1, geometry, accelerated algebra 2, precalculus, honors precalculus, AP Calculus AB, and AP Statistics. Prior to teaching, I spent 25 years in high tech in engineering, marketing, sales and business development roles in the satellite communications, GPS, semiconductor, and wireless industries. I am awed by the potential in our nation's youth and I hope to instill in them the passion to improve our world at local, state, national, and global levels.
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3 Responses to Problems with Prerequisites: Part 1

  1. Eo says:

    I’m wondering how much better their scores are than the beginning of the year.

    How do you know these scores are not just the result of unmotivated it students? Was this graded? If not, how different would the scores be if they were graded?

    It seems to me that almost every problem presents an opportunity to teach pre-algebra skills. In math the work at each new level cements the learning from the last level.


    • Great questions!

      Scores on this would be worse at the beginning of the year primarily due to the last ten questions covering slope and linear equations. At the same time, many of the students that scored high on this assessment took algebra last year as 8th graders so they may have done well all the same.

      In terms of motivation, I believe most students want to learn when in my classroom. I’ve worked very hard to develop a positive, welcoming environment where all students feel supported to learn. I confer status on all students throughout the year and they know it. In return, they generally give me their best effort. I say generally since they are mostly 9th graders after all with developing levels of maturity. Now, their level of motivation declines considerably once they exit my classroom. Few students complete homework assignments, or even study for quizzes or tests. Instilling that level of motivation in students is nigh impossible without the student themselves embracing the necessity to do so.

      Students did know the assignment was to be collected and graded. This is an area where most shine. They give me their best effort on most individual assignments. They falter a little on group work since they have not internalized mechanisms to focus on the task at hand when they have more autonomy. Instead social systems dominate and many wander of task quite rapidly.

      Your last comment really got my attention. While every problem does present an opportunity to re-teach prerequisite skills, it significantly slows down the learning process if students struggle with that re-learning. Worse, if they do not grok the base skill, they never feel mastery over the higher skill. In terms of cement, the newly poured stuff seeps through the cracks in the foundation and never hardens into the desired form. The cracks must be patched thoroughly first before the new cement will take shape properly.

      I invite you to volunteer your time in a local high school math class to expand your awareness of the challenges in our educational systems. You will become a better person for it, as will those students whose lives you touch.

      Thanks for your thoughts, eo.



  2. Eo says:

    When I tried to volunteer with my daughters math teacher in 9th grade she openly discouraged me. She seemed to believe that the barriers to me coming into the class room were too high. But maybe she just didn’t like parents in her class room. This was Los Gatos High. I did come into the class room in elementary school and help with math. In middle school my daughter would have none of it.


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