Does Textbook School Mathematics really exists? Absolutely! Can you spot why the equations in the “Follow the path” question in the picture lead students to conclude that the equal sign means “compute” rather than its true meaning? Why is this wrong? I look forward to your answers below!

Hint: What does the equal symbol mean?

CHANNEL: *Engineering School Mathematics*

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## About Scott Baldridge

Distinguished Professor of Mathematics, LSU. Geometric topologist: gauge theory, exotic 4-manifolds, knot theory. Author: Elementary Mathematics for Teachers.

It is easy to abuse something one doesn’t understand … and good intentions are no protection. Something inside me dies a little bit when I hear my high school students say “I want to teach elementary school so I don’t have to deal with hard math.” Even a good teacher can’t teach what she doesn’t understand herself.

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I taught a content course for El Ed majors in grad school.

Never had a class completely revolt against the idea of learning

anythingnew before. Or since.LikeLiked by 1 person

How were your evaluations?

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I get the same response from my El Ed majors. Good grief!

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When I saw this (through my mathematician’s eyes), I thought, “Wow. These ‘Follow the Path’ questions are at a much higher level (i.e., algebra) than the computational problems above them.” Why? Because I naturally read the equal signs as actual equal signs and therefore interpreted the empty boxes (as variables) to be filled in so that the equalities become valid. That is, so that the first completed “Path” would be: 4+6=12-2=6+4=10. BUT, based on how people have responded, I am presuming this is not what was intended(?). Was the intention of the question really to have students work “stream of conscious” computations? If so, … oy!

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I love your reply! If this thinking does not show up in any of my student’s work, it will definitely be “Part 2” of my lesson on this set of problems! 🙂

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Sadly (and dangerously), the answer is “oy!”

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This helps explain why students write their work this way (and are sometimes surprised when they lose points) in my university math classes. They were (implicitly) taught that this was OK.

I gently explain to them that they just told me that 10 = 12 (for example) and they seem to understand (but only if they lose points).

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My sentiments exactly!

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This can be done as linkage or correctly without violating the sanctity of the equal sign. However, I would be surprised if the student doing it correctly would get credit/recognition for their work! I see the example I would have provided has been posted above since I first read this.

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This is something that I blame calculators for. On most calculators used in Elementary and Middle school the button to perform the computation is labeled =!

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My solution is to have the copy the thing to be reused onto a new line

4+6=10

10-2=8

Etc.

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But that would mean using more than one page! 😡

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I don’t think it’s the electronic calculator’s fault . . . I prefer to call it an “operator problem” – for not understanding the input and output relationships involved in using calculators. This would be a great example to show students the proper way of recording calculator work.

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I wanted to edit my previous response, but cannot . . . so I will just add to it here:

In addition to showing the “proper way of recording calculator work” accurately, this would be a great opportunity for helping older / higher level students understand the occasional dual role of input / output values (related to the powerful iterative functionality of the calculator), as well as the important role of = in their work! 🙂

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Did you try following my blog? You might be able to after you sign up. (Although I must admit, I’m not sure.)

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I enjoyed reading all of the replies so far. Here is what I wrote in reply to Scott’s post on fb:

No wonder students come to my class (Alg 2 or higher) writing that garbage! I mark every = that does not represent an equality wrong and take off a point for each one! After they write corrections for the first couple of quizzes, most of my students stop doing that! I make them write out corrections, after first explaining why it was wrong! I call them “run-on false equations”!

[My kids get a kick out of my relating it to what they hear in their English classes!]

Thanks for the post! I plan to have my students “solve” and then analyze those problems next week! Maybe this will be the first year I won’t see that mistake showing up on their quizzes and tests!

[I will definitely make sure that Paul Friedman’s logic is incorporated into this lesson!]

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The introduction of the equal sign was one of the most important steps in the advancement of mathematics. So it is distressing that so many of our freshman mathematics students do not understand it. They frequently place the sign between things that are not equal and fail to place it between things that are. I have no data, but I suspect things don’t get much better in college since so much of post-secondary mathematics is still very procedural. I’ve even had graduate students misuse this symbol and protest when marked down.

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The biggest problem with letting equal sign mean compute is later when students start solving equations. In algebra students will need to understand equal to as well as inequalities. Without the true understanding they will become confused. Graphing equal to and inequalities will be next to impossible to understand. The equal sign is in essence a foundation of math.

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I do like the “identities”. That does have some redeeming qualities.

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