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My students (grade 7, age 12/13) have learned how to solve equations, I introduced them to the scale model and the learned equivalent transformations in order to solve equations of the type $$ ax+b=cx+d $$ for reals $a,b,c,d$. What they have not learned yet is the case of no solutions and infinite solutions. In one math class I wanted them to understand these cases visually so I said that we work as scientists or veterinarians analyzing the weight of ice bear babys. They filled in some tables with numbers and then they had to put two points in a coordinate system and make a line. They were able to interpret the intersection point of the lines and to answer the questions when the ice bears were equally heavy. Then they got what is called a cognitive dissonance, they thought that their algebraic manipulations made no sense (as there are no or infinite solutions) and we talked about it, it went alright. But they somehow did not grasp the idea that the function I plotted with GeoGebra and the term $6x+7$ have something in common, they resemble the same mathematical object, one is symbolic, the other is iconic. I did let them draw the lines on their own in the coordinate system and afterwards I used GeoGebra to illustrate and secure their new insights.

Now of course we need a function for this, not a term, but they did not have the concept of a function and I still wanted to teach them the different solvability cases visually (its also recommended in their book, but not how to teach it).

My questions are:

  • What went wrong with my approach?
  • What could be an alternative
  • Why did they not get the concept, and are these two representations of the mathematical object identical in a mathematical and maybe even philosophical way?

I thought that my main problem was that I did all that too fast, I assumed they would get that the line is the same as $x\mapsto 6x+7$ but maybe they dont even know that mathematical objects have more than one representation.

My idea: Show them a few points in Geogebra from the table, then showing them twice as much points and so on until there is a line. Maybe they understand that a line is a set of points and a weight rate as well, thus the same.

Thanks for any hints to this matter!

Post edit: Thanks, the reason I like to include GeoGebra is that it is a semi-enactive way to teach, which I usually like as the enactive part is really hard to accomplish in maths class compared to symbolic and iconic.

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  • $\begingroup$ How old are the pupils? $\endgroup$ Commented Oct 7, 2024 at 11:15
  • $\begingroup$ @Tommi Between 12 and 13 $\endgroup$ Commented Oct 7, 2024 at 12:39
  • $\begingroup$ How can you Ask about all that detail, while stating you're not aware even whether they know that mathematical objects have more than one representation? When you assumed they would get that the line is the same as 𝑥↦6𝑥+7, how many of them did you ask? In my limited experience a statement like that would be beyond the understanding of most 15-16-year olds, let alone 12-13s. $\endgroup$ Commented Oct 18, 2024 at 21:03

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First, I would make absolutely sure they have solidified their understanding of how to handle the non pathological cases. Not sure if this is an issue or not. Just I see people sometimes go too fast with neophytes. Make sure they have learned the "basic" (non pathological) case, before confounding them with that it doesn't work everywhere (figuratively pulling the rug out, from what you just taught). [Humans have limited processing power, so when learning new things if you throw in too much complexity at the beginning it is hard from them. When they have solidified the primary concern, then you can move to new content, the secondary concerns.]

Second, my advice would not be to start with the Geogebra, but with the attempted solutions, algebraically. Kids have a method that they have learned to use. And now they see how it doesn't work, as they manipulate the sides of the equation. So, you are using something they know, used recently, vice an analytic geometry (Cartesian coordinates) view. Kids are developing both symbol manipulation and comfort with graphing at this stage...realize they are not at the same comfort level with either as 10th graders or the like. But in this case, the most recent and applicable learning was from symbol manipulation side of the house. Also, you are kind of jumping into algebra 1 (vice pre-algebra) with the implicit y=mx+b graphing as opposed to just linear one equation, one unknown solving.

I would also demonstrate this on the board, manually, rather than with a computer. And have your kids do manually on graph paper. Not with the black box of the Geogebra. (However, I am still a little concerned about the implicit jump you are making from pre-algebra to algebra, given the y=lineqn you are doing.)

Maybe show the graphs later, after just doing several examples of the pathological cases, as revealed from attempted algebraic solution. After, not before.

This is minor, but perhaps helpful. I actually didn't understand it at first, from reading you, what you wanted. Thought you were talking about a, b, c (or maybe x) being infinite or zero. But I get now you are talking about linear intercept peculiarities (same line or parallel lines, versus a crossing). Personally I would say "no" rather than zero and "single" versus "1", for the number of solutions. Again, this is minor, but maybe it helps. And yes, they mean the same thing...just maybe slight clarity.

[Edit, added content] I'm also not crazy about the ice bear babies thing, when showing new content. It's a "word problem". And word problems are hard. Says "guest Barbie". ;-) I would stick to algebraic familiarization and as the demonstration of new content. Word problems come in later. It is just an added strain to move from word world, translate to equations, manipulate equations, then translate result back to world. The same thing applies when teaching second order linear diffy Qs. Do it with symbols first. Then the word problems are added, slightly harder drill, after the new content is known. [And this is a major criticism I have of almost every partial diffy Q book, using physical examples to introduce new methods...and I say this as a very applied guy.]

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  • $\begingroup$ Thanks, I orientated myself on their math book. We had at least a few weeks only discussing equations and how to solve them. I think they had plenty of time to process this concept. I also edited the question and added details. $\endgroup$ Commented Oct 7, 2024 at 14:11
  • $\begingroup$ Sounds like you are being thoughtful. Does the book also use a graphing approach, example? $\endgroup$ Commented Oct 7, 2024 at 14:13
  • $\begingroup$ And I would still avoid Geogebra for neophytes and linear equations. They need to learn comfort with Cartesian coordinates, from now and over the next few years. And the pathway to that is by hand ("grinding the pigments"). Once it is old hat, an automated tool is OK. But before, not only are you losing the opportunity for "by hand" learning, you're also bringing in something that is mysterious and a black box. (Heck I would check it myself manually, if it were something that bothered me.) And I'm not anti tech later. Just anti tech EARLY. $\endgroup$ Commented Oct 7, 2024 at 14:14
  • $\begingroup$ Yes the book shows a table with the three cases and the visual explanation with parallel, identical and intersecting. But it does not a lot on the didactics. $\endgroup$ Commented Oct 7, 2024 at 14:25
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    $\begingroup$ Your professor has expert bias and you are trying to look good for an expert, versus be most pedagogically effective. It's a common issue and even many textbooks have this issue (since their audience is not the kids, but the selection committees). It takes real confidence to "let it swing in the wind" and just be simple. Maybe you can sort of thread the needle, by doing word problem drill, after, not before. I.e. now is the time for ice baby (and money) drills for non pathological cases. And don't tell him what you think of him. ;) $\endgroup$ Commented Oct 7, 2024 at 14:46

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