Insight into the conceptual gap and how to fill it

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I recently started volunteer math tutoring in an after school program in a city near me. This is one of the poorest cities in the state, and the students' math preparation is about as bad as you'd expect. The after school program is not part of the school system, it's run by a non-profit. I work with whoever I'm assigned to on a given day, as the staffing and student attendees fluctuate.

My particular question has to do with one child's homework on the Pythagorean Theorem.

$14^2 + 20^2 = c^2$
$196 + 400 = c^2$

And she seemed to understand that the next step was to add 196 to 400 to get 596, but then she was at a total loss what to do with that number. She hopefully wrote down this:
$596 + b^2$
And then at my explaining why that was wrong, tried this:
$596 + c^2$

I tried explaining that her first equation was saying $c^2$ is the same as something, so her next equation should also say $c^2$ is the same as something. And that $=$ was the sign that meant "is the same as". I could tell her that the next step was $596=c^2$ but I was not getting a feeling that she understood why. And while I was struggling to figure out what she was missing and how to bridge the gap, her ride showed up and she had to leave.

The lady who runs the program tells me that there's nothing like a structured curriculum any more, just a bunch of random topics on any given day, trying to prep them to get good test scores. So all of these students are missing fundamentals of one kind or another. This student might have never been given an equation before.

Anyway, enough of the generalities. About the specifics: Does anyone have any idea what the gap is here and how to teach it?

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kuruman
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You have identified the gap but not necessarily its origin. I think the point here is to bridge the gap. I would have attempted the same approach with the definition of the "two parallel hyphens" meaning "is the same as". Obviously that identification didn't sink in. Failing that, I would investigate how the student has internalized the concept of "is the same as" and try to sharpen his/her understanding with an incremental approach. For example,

You: What is two plus two?
Student: Four.

You: Then two plus two is the same as four. Correct?
Student: Yes.

You: Instead of writing "two plus two is the same as four", we use shorthand notation to make our life simpler. Here "2" stands for "two", "+" stands for "plus", "=" stands for "is the same as" and "4" stands for "four". So we write 2 + 2 = 4 to say the same thing only more compactly. OK?
Student (I hope): Yes.

You: OK, if you understand this, how would you write "Three plus two is the same as five?
Student (I hope): 3 + 2 = 5.

You: OK, now answer me this, what is eleven plus five the same as?
Student: Sixteen.

You: Great! Can you write "eleven plus five is the same as sixteen" in compact form?
Student: 11 + 5 = 16.

You: Excellent. Now if I were to write 11 + 8 = ?, I am saying that "eleven plus eight is the same as some unknown number." Note that I am using the question mark as a placeholder for a specific number that is the same as eleven plus five. What could that number be?
Student: Nineteen.

You: Fantastic! Hold onto that thought. If I were to write 196+ 400 = c2, I am saying that "one hundred ninety six plus four hundred is the same as some unknown number", except that instead of a question mark, I used the symbol c2 as a placeholder. Still, what could that number be?

Just a suggestion.

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gleem
She may not have the same concept of the phrase "is the same as" as you do. We often say something is the same as something else which means one can be replaced by the other without any connotation of quantity, value or size. So you might try saying the equal sign mean that the right side has the came value as the left side providing of course she understands the term value in the context of math.

Another thing that educators have found to be a problem is writing equations with the operations typically on the left followed by the equal sign. A ⊗ B = C with the ⊗ sign referring to any math operations. Students have been found to interpret the equal sign as a command to execute the specified math operation on the left. To avoid this write the indicated operation on the right side of the equation.

Having identified the problem with her equation, sum of squares, you might want to avoid the complication of squares and square roots until you have confidence she understand what an equation is. The exponents may be distracting.

The lady who runs the program tells me that there's nothing like a structured curriculum any more, just a bunch of random topics on any given day, trying to prep them to get good test scores. So all of these students are missing fundamentals of one kind or another
If this is correct then you have a lot of work to do since she is basically working with patterns with little understanding of their meaning.

vela
Staff Emeritus
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When I can't figure out where a student is getting stuck, I often find it helpful to ask them to explain to me what they're thinking is. The thing to avoid is an answer like "I don't know" or "I just guessed" because that leaves you in the same place as before, not knowing what the conceptual roadblock is. There is a reason your student wrote down $596+b^2$ as opposed to anything else she could have written down. For some reason she decided it was more likely to be the right thing to do.

Most students aren't used to articulating their reasoning like this, though, so you might have to help out.

It could be really helpful for you to admit to her that you need her help because you don't know where she's getting stuck. Seek to turn your sessions into more of a conversation where she plays a less passive role in her learning.

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It could be really helpful for you to admit to her that you need her help because you don't know where she's getting stuck. Seek to turn your sessions into more of a conversation where she plays a less passive role in her learning.
Thanks, that's really good advice. I will try to bring this principle to bear next time I'm there.

There are larger issues here of course, with entire populations of students who are being neglected in their early years, creating handicaps which can last for a lifetime. The question is whether you can intervene somewhere in between early childhood and adulthood and make a difference in overcoming those early disadvantages. Obviously it's a huge task. But not, I choose to believe, impossible.

Thanks to all for the valuable input.

Not an educator, but this is really interesting to me.

I have to wonder, have the students been shown what Pythagoras is all about? That is, if you know the length of two sides, you can figure the length of the third? I mean, that's the point, right? Seems like without that, the equation is just hieroglyphs. But if you put it as a puzzle, "I tell you a and b, you tell me c" then the equals sign means "here comes the answer." That kind of goes along with @gleem 's idea to put the c on the left of the equals.

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Not an educator, but this is really interesting to me.

I have to wonder, have the students been shown what Pythagoras is all about? That is, if you know the length of two sides, you can figure the length of the third? I mean, that's the point, right? Seems like without that, the equation is just hieroglyphs. But if you put it as a puzzle, "I tell you a and b, you tell me c" then the equals sign means "here comes the answer." That kind of goes along with @gleem 's idea to put the c on the left of the equals.
I don't even know. I don't see their textbook, I know nothing about the course they're taking, I see only the homework for a given day and their main goal is to get it done. I am assuming that the homework is built on previous things they've learned but I'm being told that I'm being optimistic to think that. So I'm trying to figure out some minimal set of underlying conceptual gaps for the particular assignment based on their approach and fill in those gaps as best as I can.

I'm starting to think of it in terms of carpentry, as if there is a house with only a partial foundation and first floor and some carpenter climbs up the scaffolding and starts building parts of the second floor with nothing underneath. So you have to race to put up just enough foundation and first floor to hold that part up. With luck, over time you'll get a complete foundation and first floor that way.

The thing about the $c$ being on the right as opposed to the left is a really good insight, and something I'll pursue when I get another session with this student.

I would say much the same as has been said before, with one small modification. Rather than say the phrase "is the same as" I was taught and still teach saying simply "is." Thus A+B=C reads as "A plus B is C." To my mind, I think this makes the equivalence and interchangeability more evident.

Many students today have some level of computer knowledge, even before they know much arithmetic. I wonder if the use of the "=" sign as a replacement operation may be part of the confusion? If we think in a strictly mathematical way, the computer statement x = x+1 is nonsense and leads directly to 0 = 1. Could this be a part of the problem?

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I would say much the same as has been said before, with one small modification. Rather than say the phrase "is the same as" I was taught and still teach saying simply "is." Thus A+B=C reads as "A plus B is C." To my mind, I think this makes the equivalence and interchangeability more evident.

Many students today have some level of computer knowledge, even before they know much arithmetic. I wonder if the use of the "=" sign as a replacement operation may be part of the confusion? If we think in a strictly mathematical way, the computer statement x = x+1 is nonsense and leads directly to 0 = 1. Could this be a part of the problem?
Don't know but I remember personally struggling with this, having seen math long before I saw programming (in a school computer club, probably around age 11-12). I thought if you said $y = x^2$, then changing $x$ later should automatically change $y$, and I really struggled to understand why $x = x + 1$ was a valid thing to write.

I really struggled to understand why x=x+1x = x + 1 was a valid thing to write.
So did I!!

kuruman
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I would say much the same as has been said before, with one small modification. Rather than say the phrase "is the same as" I was taught and still teach saying simply "is." Thus A+B=C reads as "A plus B is C." To my mind, I think this makes the equivalence and interchangeability more evident.
I am not so sure, especially with equations that describe physical laws mathematically. For example, $$\epsilon_0 \int_s \vec E \cdot \hat n~dA=q_{enc.}$$ is best interpreted as: If you walk on a closed surface subdivided into elements $dA$, multiply the outward normal component of the local electric field by $\epsilon_0 dA$ and add all such product elements over the entire surface, you will get a number; that number turns out to be the same as as the total charge enclosed by the surface. In my way of thinking, the interpretation of "$=$" as "is" is appropriate for formulas, e.g. $A=\pi r^2$, but not for laws of physics that are an assertion that the expression on the right is the same as the expression on the left.
Many students today have some level of computer knowledge, even before they know much arithmetic. I wonder if the use of the "=" sign as a replacement operation may be part of the confusion? If we think in a strictly mathematical way, the computer statement x = x+1 is nonsense and leads directly to 0 = 1. Could this be a part of the problem?
You raise a good point here and I would not at all be surprised if that were the case. In coding one never sees the sum of two numbers to the left of "$=$" so the appearance of such a thing could very well leave someone not knowing how to handle it. When such difficulties arise, perhaps a diagnostic test would be to ask the student, "Using plain words, describe what $x=x+1$ means to you." The answer should reveal whether the student is algebra savvy, code savvy or both.

I am not so sure, especially with equations that describe physical laws mathematically. For example, $$\epsilon_0 \int_s \vec E \cdot \hat n~dA=q_{enc.}$$ is best interpreted as: If you walk on a closed surface subdivided into elements $dA$, multiply the outward normal component of the local electric field by $\epsilon_0 dA$ and add all such product elements over the entire surface, you will get a number; that number turns out to be the same as as the total charge enclosed by the surface. In my way of thinking, the interpretation of "$=$" as "is" is appropriate for formulas, e.g. $A=\pi r^2$, but not for laws of physics that are an assertion that the expression on the right is the same as the expression on the left.

You raise a good point here and I would not at all be surprised if that were the case. In coding one never sees the sum of two numbers to the left of "$=$" so the appearance of such a thing could very well leave someone not knowing how to handle it. When such difficulties arise, perhaps a diagnostic test would be to ask the student, "Using plain words, describe what $x=x+1$ means to you." The answer should reveal whether the student is algebra savvy, code savvy or both.
That depends on whether the $=$ is used for assignment or for test. In VSBASIC, for example, one could say:
LET $X = 5 LET$Y = 5
and
IF $X +$Y = 10 GOTO 20

In javascript the use of $=$ for assignment and for test is disambiguated by being doubled for test, as in:
a = 10;
if (a == 10) {a++}

I think the child used the plus sign not because she didn't understand the meaning of $=$, but because she intuited that some further operation was in order, but couldn't yet recognize that the 'extract square root' procedure was indicated.

Andy Resnick
Another thing that educators have found to be a problem is writing equations with the operations typically on the left followed by the equal sign. A ⊗ B = C with the ⊗ sign referring to any math operations. Students have been found to interpret the equal sign as a command to execute the specified math operation on the left. To avoid this write the indicated operation on the right side of the equation.
This is potentially very insightful, do you have any references to the literature on this? As a related example, my (intro physics) students can work homework problems and test problems that involve F = ma but are completely confused if I write ma =F.

gleem
This is potentially very insightful, do you have any references to the literature on this?
I ran across this idea quite accidentally some time ago, so I don't remember the exact reference. But I Googled "problems with the equal sign in math" and got these;

https://educationresearchreport.blogspot.com/2010/08/students-understanding-of-equal-sign.html
Summarizing that the problems seems to be an American one.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3374577/#R80discusses the operational interpretation issues of the equal sign.

DrClaude
Mentor
Very interesting. I found the following particularly insightful:
“Students who have learned to memorize symbols and who have a limited understanding of the equal sign will tend to solve problems such as 4+3+2=( )+2 by adding the numbers on the left, and placing it in the parentheses, then add those terms and create another equal sign with the new answer,” he explains. “So the work would look like 4+3+2=(9)+2=11.

“This response has been called a running equal sign—similar to how a calculator might work when the numbers and equal sign are entered as they appear in the sentence,” he explains.
So the problem would not be programming but rather calculators.

kuruman
Homework Helper
Gold Member
So the problem would not be programming but rather calculators.
... excepting the throwbacks that use reverse Polish notation.

Bystander
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Gold Member
So the problem would not be programming but rather calculators.
AKA, "calculator pidgin."

... excepting the throwbacks that use reverse Polish notation.
In the late '70s you could get a $35-50 Casio RPN calculator with the about the same set of functions as a$200+ TI or HP because RPN was easier to implement in terms of the hardware.

Andy Resnick
Very interesting. I found the following particularly insightful:

So the problem would not be programming but rather calculators.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3374577/#R80
The authors present findings like:

"Weaver (1973) also compared how students in first, second, and third grade managed problems where the operator was on the left side (i.e., standard) versus the right side (i.e., nonstandard) of the equal sign. Students experienced a higher success rate for correctly solving standard (i.e., operation-left-side) equations than nonstandard (i.e., operation-right-side) equations. Lindvall and Ibarra (1980) discovered a similar pattern with first- and second-grade students. When the operator was on the right side of an equation, students made many more mistakes than when the operator was on the left. "

and

"Often, students interpret the equal sign as an operational symbol, not a relational symbol (Baroody & Ginsburg, 1983; Kieran, 1981). In terms of viewing the equal sign as an operational symbol, most elementary students believe the equal sign signals them to “do something” or “find the total,” or that the “the answer comes next.” Students viewing the equal sign as a signal to do something look at the left side of an equation and decide the equal sign means to do something to the right side of the equal sign (Cobb, 1987). Other students view the equal sign as a clue to find the total, even when finding the total is inappropriate (McNeil & Alibali, 2005a). "