I know the question in the title is very broad so I will try to explain it as succinctly as I can. Half my time is spent on as a researcher on didactics, while the other half is devoted to teaching. The course I discuss below is an introductory course to mathematics for those who only need some mathematics in their further studies, some brands of chemistry students, biology, CS students and so forth (whether they only need a bit is up for discussion). The curriculumn is a litle bit of set theory, algebra, functions, graphs, limits, derivation, integration and the most simple differential equations.

We have done some surveys amongst the students where they have been given a series of simple problems within algebra. Factorize this fraction, simplify this expression, alongside some more involved problems such as evaluating 1 # (2 # 3) when they are given a # b = a/b + b/a. This test is conducted in a closed environment, individually with no aids other than a pen and pencil. According to what we would expect university students to score, the results are abysmal.

However, I also give them big hand ins during the semester, about 3 of them give or take. Each project takes about a month for them to solve these problems in a group, and they are fairly involved compared to the survey we did. These projects or exercises uses "real life" problems to motivate the students.

We also did a post-test (not representative due to covid)


According to what we would expect university students to score, the results are abysmal in the pre and post-test.

On the other hand just looking at the what the students handed in, the results where fantastic.

Possible solutions and explanations

It was clear that several groups did use online CAS solutions to perform the algebra part of the projects. Note that the algebra part in these projects was just a small part of the problem. This was clear by the replies from the teaching assistants and a very small percentage of students even screenshoted WolframAlpha step by step in their hand ins!

The projects were new as of this year, so they could not have obtained solutions from older students. In addition the numbers on every groups project varied, so they could not copy directly another groups solution.

The projects counted indirectly count towards their grade as the plan was to have a individual pass / no pass grading system with a group oral exam at the end of the semester. The oral exam would be based upon the projects, and start by asking how they went about solving some of the problems. With some follow up questions to see if they understood what they did. Again not possible this semester.


I realize that the pre and post-test are conducted in a very unnatural environment to the students. Yet, based on the results from these tests I would not have expected any of the students to be able to solve the complex projects.

If I asked a student at point blank today if he would be able to find the derivative of (1-x)/(1+x) I doubt he or she could do it. However, seemingly in a group, with every resource available and enough time, they are. I think I am just a tad confused what it means to understand algebra.

Do my students know algebra / how do I know whether I have taught them anything?

My gut feeling is that algebra is more than symbolic manipulation, but I do not have sources to back me up on this.

  • 3
    $\begingroup$ Just curious $\ldots$ you wrote pre-algebra but what follows were to my mind clearly algebra topics. To me, prealgebra has always meant numerical fraction and decimal computations, percents, weight measure conversions, numerical ratios and proportions, integer exponents, square roots, and the like, along with evaluating algebraic expressions (plugging numbers into formulas), but NOT algebraic manipulation of any kind. However, upon googling I find the term seems to be more varied now (or maybe I'm just seeing more than just U.S. middle school prealgebra curricula with google searches now). $\endgroup$ Dec 27 '20 at 17:49
  • $\begingroup$ @DaveLRenfro I think It might have to be a language barrier at play here yes, I am based in Europe and here whenever someone mentions algebra, it is always in the context of what you call abstract algebra. But yes, I refer to what you call algebra. Should I change the text, or is the meaning clear from the post as a whole? $\endgroup$ Dec 27 '20 at 17:59
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    $\begingroup$ I think it's probably OK, since you are specific with what the topics are, but maybe change "pre-algebra" to "school algebra". By the way, I completely sympathize with your situation, and despite my great need for a steady job right now (was laid off 3 years ago from my non-teaching job, along with everyone else in my work group, and have been doing part-time contract work since then), I'm really really glad I'm not teaching right now! $\endgroup$ Dec 27 '20 at 18:06
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    $\begingroup$ Something I find confusing about your question is whether you teach algebra in the class. The class seems to be "a mile wide, and an inch deep." It includes a "little bit of set theory, algebra, functions, graphs, limits, derivation, integration and the most simple differential equations." It seems there could be no time for students to learn the pattern matching algebra skills associated to a class that prepares students for success in calculus. If this is a calculus class, then students should have the algebra skills before. If a broad survey course, then why are you focusing on algebra? $\endgroup$
    – user52817
    Dec 28 '20 at 19:23
  • $\begingroup$ During my infrequent Substituting in both Algebra 1 & 2, It was very common to find students doing their substitute worksheets, for lack of a better term, by finding the solved standard worksheet on the web via their laptops and copy what they found. Ironically actually doing it, assuming minimal understanding, would have taken a fraction of the time and effort. $\endgroup$ Jan 2 '21 at 22:32

There are two levels at which someone can understand algebra. (1) They can do stylized tasks using a set algorithm, such as multiplying out $(a+b)(c+d)$. (2) They understand what it means and can apply it to real-life problems.

In principle it is possible for someone to master #2 while still not being competent at #1. The reality is that this never happens.

A CAS will help with #1, but not with #2. It's great that your class is designed around projects, which test level #2. But you seem to be assuming that a CAS is the only method your students have for cheating on these projects. They can instead use online cheating services such as chegg. If you see students who can't do level #1 on the pretest but turn in projects where they have suddenly mastered the more difficult level #2, then this is probably what is happening.

You use an example in which students are asked to evaluate "1 # (2 # 3) when they are given a # b = a/b + b/a." To me this clouds the waters. This is not a test of algorithmic knowledge of algebra, nor is it a test of whether students can apply algebra to real life. This is a test of their ability at abstraction, which is a completely different mental ability.

  • $\begingroup$ Everything in math is abstraction, starting from counters as an abstraction for cows, pigs or people. I think this task is perfectly valid for a high school algebra class. $\endgroup$
    – Rusty Core
    Dec 31 '20 at 3:49
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    $\begingroup$ Why does mastering #2 never happen before #1 ? (This corresponds to my intuition, too.) $\endgroup$
    – Tommi
    Jan 4 '21 at 10:14

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