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Background

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)

Results

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.

Question

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.

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    $\begingroup$ 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). $\endgroup$
    – Dave L Renfro
    Dec 27, 2020 at 17:49
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    $\begingroup$ @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? $\endgroup$
    – N3buchadnezzar
    Dec 27, 2020 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! $\endgroup$
    – Dave L Renfro
    Dec 27, 2020 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, 2020 at 19:23
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    $\begingroup$ 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. $\endgroup$
    – user10216038
    Jan 2, 2021 at 22:32

2 Answers 2

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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.

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    $\begingroup$ 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, 2020 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, 2021 at 10:14
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    $\begingroup$ @RustyCore Yes, but some abstractions are primary topics at school, and some aren't. The example here requires the notion of abstract binary operations, or in other words, it overloads the symbol #. Or alternatively, a student can parse the problem as computing $f(x,f(y,z))$ with $f(a,b) = a/b + b/a$, and then the problem lies in composing two functions properly. These are more abstract than the usual school problems. $\endgroup$
    – Michał Miśkiewicz
    Mar 8, 2022 at 10:33
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Nebuchadnezzar:

Your post contains two related questions. (1) Do they know basic algebra? (2) Does it matter. They should be considered separately.

Topic 1: Do they know basic algebra?

You know the answer to this in your heart. You tested them before and after the course and found them weak.

Really you shouldn't even be surprised at even the entrance score performance, just based on that they are weaker STEMs and are taking a course that sounds like college algebra and "business calculus". Not the standard Calc 101, but something a little remedial. Remedial students will not just be "behind", they will be weaker at the basics, have lower SATs, etc. ON AVERAGE, of course, class to class...there are always outliers...but we are discussing group differences.

And then pedagogy (fancy, projects, group, offline, etc.) was not really oriented to helping them get better at the basics either. You fell into the "project trap". Look at what we are doing? How big, how fancy! But then the basic learning did not happen.

Btw, there is some discussion (as there usually is) about the dichotomy of "understanding" versus "drill". I personally think this dichotomy is overemphasized. But in this case, it's tangential. Your kids can't perform the skills regardless of if theory or drill would enable it. And both tricky question understanding and demonstrate drill questions would be tough for them (with in class, unaided performance).

Now, you do need to be reasonable. You're not going to simultaneously teach them biz calc while giving them the manipulation skills of a 750M SAT, AP BC 5, IQ 140+ kid. You just aren't. All that said, you can still cover the ground. And do a lot of GOOD! And do some en passent strengthening of basic algebra. And God knows freshman chem IS an applied algebra class! But you need to follow a more standard, less ambitious plan. Chop the stuff into simple, week-long segments. Give drill homework (sure, explain it also, but give drill HW). And then do an in-class test every end of week, with problems selected at random from the drill work. sound of hands wiping past each other in that "done" meme

Topic 2: Does it matter?

Does it matter? I think so, but this is much more debatable.

If you had to go to a dictionary every time you needed to know a definition, how fast could you read? These kids are going to see math in derivations and in homework problems in their chem/CS/etc. classes. Freshman chem ESPECIALLY is almost just one giant algebra word problem (so is a lot of business consulting, accounting, etc.) And guest troll knows these two subjects.

I think even your CSes will be helped from having some ability to do simple trouble shooting calcs for algorithms, loops, etc. And nurses and doctors and pharms need to know how to dilute meds.

It's not that there's no value to a "term paper", but it's not the right priority. Especially with weak students in an entrance course.

P.s. I probably mixed some 1 and 2 in my responses, not perfect disaggregation. But hope the attempted issue analysis helps.

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