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72

A surprisingly large number of students don't know what the equals sign means. Their understanding of the symbol "=" is essentially operational, not relational — they think it means "the next step" or "the answer" or is an instruction to perform some operation. Knuth et al. ("The importance of equals sign understanding in the middle grades", Mathematics ...


65

This is a really interesting question, because similar issues---the question of how demanding to be about formatting of answers---come up a lot, at all levels, and the answers aren't always straightforward. In this case I think your colleagues are straightforwardly wrong, for the following reasons: They're wrong about what students who write "x=1" think. ...


55

I suspect that there are several different and interacting things going on here. It's likely that these students don't understand the "equals" sign in the same way that you do. An extensive body of research over several decades has documented that for many students the symbol $=$ does not indicate a relationship (that the thing on the left is identical in ...


54

Some of your students will become engineers, and engineers use complex numbers all the time, e.g., to represent impedance. This kind of thing is by far the most common application. Complex numbers are also used in quantum mechanics. after algebra II, they never use complex numbers until pretty much complex analysis. I assume you mean "they never use ...


42

The problem you describe is well-known in mathematics education research. I cite the paper of De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50(3), 311–334. and give some ...


41

As a personal tutor, I’ve been teaching algebra to kids from ages 8 to 16 for many years. Mostly I find myself in the position of picking up the pieces when the kids are failing and fearing more failure. The root of the problem, in my experience, is the way algebra is taught as something alien, and in particular, different from arithmetic, which it really ...


38

I don't view these common mistakes as 'universal linearity' assumptions. The mistake that $(a+b)^2=a^2+b^2$ is just a visually appealing statement. It is mistaken to be correct because it looks nice. Our brains tend to like things that look nice. Similarly, $\sqrt{a+b}=\sqrt a+\sqrt b$ is visually appealing and it resembles the correct formula $\sqrt {ab}=\...


32

In a comment, the OP has suggested that he actually wants a practical example convincing students that the product of two negative numbers is positive. This is related to, but psychologically distinct from, the question asked. In particular, students often have different mental models for multiplication (a binary operator) and negation (an unary operator). ...


32

As someone who teaches calculus to college students, I expect my students to have seen point-slope form. We just start using it (because it's the right way to talk about tangent lines and linearization) without teaching it, because we consider it part of the standard algebra curriculum, so students who haven't seen it are at a disadvantage. Further, ...


30

But, when teaching adults, I've found that I can't just tell them "this is the way it's done, get used to it." Good! Students (at any age) should never be satisfied with "This is the way it's done, get used to it", and teachers should never give that as an explanation. More than any other academic subject, everything in math should make sense. ...


29

This is what really happens during education (not in the mind of a mathematician): The teacher introduces $4^9$ as just a way to abbreviate $4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\times 4$. So, the $a^n \times a^m=a^{m+n}$ and othere rules can be shown to be true, simply via thinking to the meaning of this abbreviation. Then, someday, $a^...


28

One of the reasons your students are putting the $\pm$ on the 10 is probably because someone told them that when you remove a pair of absolute value signs that the $\pm$ goes on the number after the $=$. They are simply doing what they thought they were told to do. So I try to avoid telling them this sort of thing. I like to tell them is that $|x|$ is equal ...


27

Foremost: It depends on what the lead-in lesson/topic/direction was. If this was the essential point being exercised, then I would interrupt ASAP and refocus them on the lesson/direction that just occurred. Otherwise I would let them complete their process (unless obviously totally infeasible) and then compare to a faster way afterward.


26

We owe students a presentation of the Fundamental Theorem of Algebra -- that every nonconstant polynomial has a root; or, equivalently, the marvelous fact that every polynomial of nth degree has precisely n roots (including multiplicities). When made, it serves as a capstone and culmination of all the work that the student has done in elementary algebra. Of ...


25

Just before dividing, you can reason "Either $x=0$ or I can divide by $x$." This creates two separate cases to be analyzed. This works for dividing by anything. You want to divide by $\sin(x)$? You need to make two cases: $\sin(x) \neq 0$ and $\sin(x)=0$. And then analyze each independently.


25

Yes, of course it is. You've identified many of the important reasons in your question. In practice in quantitative fields, we are all estimating as a first-pass on whether problems are soluble all the time. Estimation is incorporated throughout Common Core standards (e.g., Grade 3, Grade 4, Grade 7; there are more). And it was part of official state ...


24

Ask your pedantic colleagues to reconcile their formatting expectations with the context "solve $F=m\cdot a$ for $a$". Would they prefer to see just $\frac{F}{m}$, or maybe this expression jazzed up with set brackets somehow? This formatting of the answer looks jarring to me. I think most educators and scientists expect to see the answer presented as $a=\...


23

Personally, I'd say "no." The students who don't write out the steps in their algebra classes appear in calculus thinking that they should be able to write down all the answers without any intermediate steps. I even have some students in Calculus 2 who think that there is some kind of value in not writing down the steps. None of these students can complete ...


23

Absolutely! In fact, in my opinion, the most important "math skill" that should be taught in conjunction with, and using, word problems is checking whether the answers make sense. This is an absolutely invaluable part of making any practical use of mathematics, as opposed to just blindly applying formulas for the sake of passing an exam. There are several ...


23

$$3x^2 -14x - 5 = 0$$ Multiply through by A or here, 3 $$9x^2 -42x - 15 = 0$$ Now, use substitution, u=$3x$ (3X is the square root of this first term, and by using the u substitution, we now have an 'a' of 1. ) $$u^2 -14u - 15 = 0$$ factor to $$(u-15)(u+1)$$ Substitute back u=3x $$(3x-15)(3x+1) $$ last, divide out that 3 we multiplied by - $$(x-5)(...


23

Point slope form emphasizes the actual meaning of slope. Literally, $$ y - b = m(x -a) $$ Says "The change in the outputs ($y-b$) is equal to the slope ($m$) times the change in the inputs ($x-a$)". Translating between a verbal statement like this and an equation is essential. Understanding slope is essential. Point slope form of a line is essential. ...


22

For some students, the difficulty with solving $3x+5=14$ is even more basic than figuring out what operations to do in what order in order to reach the goal. Before getting to that, they need to know what the goal is. "Everybody knows" that, when solving an equation with one variable $x$, the goal is to end up with a statement of the form $x=$ some ...


21

The usage you object to is, in fact, the original meaning of "linear". "Linear" means "having to do with lines". The notion of "linear" in the sense of "linear transformation" is a more modern, restricted notion.


21

I spent some time looking for information that might provide some kind of reliable, evidence-based answer to my own question. I came up with the following, which is not perfect, but I thought it would be worth posting as an answer. I tried two methods of probing this: standardized test scores of students with high socioeconomic status (SES), and heritability ...


21

I like your second option the best: ...wait for them to finish the calculation, or even finish the entire exercise, before I casually tell them there was a more natural way to work out that part? Instead of just mentioning the easier way, however, you could first applaud them for using good algebra to find the right answers, then remind them of the new ...


20

This is also borderline not-an-answer, but it might be a nice broadening of your students' worldview to know that the "$m$" and the "$b$" are not universally accepted. Showing them this map (even though I do not know its original source, so it may not be accurate in its details) could help their thinking out a bit: Substantial edit: I now no longer believe ...


20

The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous. If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2𝑥). If implicit multiplication has a higher priority than ...


19

First: I do not think this is really an issue with a lack of understanding of the square root function. When someone writes $\sqrt9 = \sqrt3$ it means they are not thinking about what the equals sign even means. I have had a small amount of success with the following method, which relies on the students "believing in their calculator" as a source of truth. ...


19

This became to big to be a comment. Layman's opinion. Where does it come from? It comes from the fact that universal linearity is useful to move forward in calculations even if it's wrong. Psychologically this is very attractive. The other option is being stuck. Moving forward has the added incentive that it can be right, that maybe the student can get ...


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