Is the ability to show what all students did available?

Having found matrices arbitrary and confusing in high school, I do wonder what a great presentation of “matrix algebra” would be. The thing is, after high school, in my honors freshman math class in college that included linear algebra, I suddenly saw the light, because this hardcore theoretical class taught linear algebra as being about linear transformations, starting from affine transformations, and we spent weeks on theory (including geometrical intuition) before even mentioning matrices as being a REPRESENTATION of linear transformations (which are functions from one space to another), and beginning to compute with them. I apparently got to college without anyone telling me this.

This story makes me sad. It seemed to me like all the honors courses I took in any subject were the ones that stepped back from incessant calculations and memorizations and talked about concepts and intuitions. My friends who were not on the honors tracks in school in math, science, and writing got material that was so dreary and mechanics-based that if I had been forced to take those classes, I would have failed and/or quit those subjects.

I learned that if multiplication and division are written on the same line, you evaluate from left to right.

I had always used the division bar or parenthesis. Still confuses me to see the divide symbol, why would anyone write down the problem that way?

But it wasn’t until I had graduated university math and was several years into teaching that I really, really understood that the graph is a picture of all the points that make the function true. This was difficult for me because graphs don’t often look like a bunch of points.

I find this sort of gap fascinating, especially because it is likely somewhere along the line you were at least told this fact (you might even be able to track exactly where). But it still didn’t stick! It’s as if being told just isn’t enough.

I’ve done this type of activity before with technology (TI-Navigator) and with physical bodies (going outside and making a human graph) but I always still feel like that particular realization doesn’t sink in even though we both work with it implicitly and I tell them explicitly.

I can think of two gaps I had until well into teaching quite quickly. The first was division. I was only taught/learned that division by three was dividing into three equal groups. This concept is useless for division by fractions. It was several decades into teaching that I understood dividing could be “into groups of size __3__”. It was when we were forced to visualize division of fractions that this lack came to the front. 8÷4/3 is 8 divided into groups of size “4/3” so there would be 6 of these groups.

The second was when I was getting students to see the connections between operations with fractions. The script of the revelation went something like this: “You’ve learned that for multiplying fractions we can do it quickly by multiplying tops and multiplying bottoms. Well powering a fraction, can be turned into multiplying so let’s see what happens if we power the top and power the bottom.” As I’m writing up an example my brain is giving me a self talk, “But dividing and multiplying are related as well, can I divide the tops and divide the bottoms?” Now students were looking bemused as I had stopped writing in mid-example. “Class, it just struck me that we might be able to divide fractions similarly by dividing the tops and dividing the bottoms. Shall we try it out to see if this works?” As examples like
4/15 ÷ 2/3 = (4 ÷ 2) / (15 ÷ 3) =2/5 worked beautifully, and some became complex fractions that were correct with an adjustment, my mind was wondering, “(Dan would insert WTF?) Why did my math teachers not explore this with us? It seems so logical and would lead to other ideas for dividing fractions. It is like there is only one algorithm ever taught. ‘Ours is not to reason why but to invert and multiply’ although my personal favourite was a grade 8 of mine who called it ‘flip’n times!’ (humour only for the British perhaps).”

What else did my teachers, with pure delivery in the acquisition metaphor, not uncover with me that an exploratory, discovery-based approach would? Hmmm?

First: I really like these activities (and agree, RE: “You could run a semester-long master’s seminar on the misconceptions in that last graph” — you have many of what I like to call bergtips, mathematically and pedagogically, here!).

Second: You are dragging green points. I (would) wonder about dragging/rotating green lines (or green line segments). One task could be initially to show a triangle, and have users drag/rotate a green line (or green line segment) to divide the triangle’s area in half. (Maybe the users begin by choosing two points to define the line or line segment; the implementation details I do not specify.)

My own interest is in problem sequences, and so I would be interested in, say, moving from equilateral triangles, to isosceles triangles, to scalene triangles of different sorts. Alternatively, you might move from triangles to quadrilaterals (or within a sequence quadrilaterals).

A specific problem of which I am especially fond:
Depict a shaded rectangle with a smaller rectangle removed from its interior (indicating something like: a bird’s eye view of cake with a single piece removed). How does one divide such a figure into two components of equal area using a single green line (or green line segment)?

The latter problem is written up at:
https://mathwater.wordpress.com/2015/08/03/creative-cake-cutting-divergent-convergent-problem-solving/

(And, in the spirit of generating ideas: Perhaps one could add in a few buttons, for the tasks described above or others; “I think I have found all of the solutions” “I think there are finitely many solutions, but I have missed some of them” “I think there are infinitely many solutions” etc.)

What I learned in grad school — tangent at a point is the slope at that point. (Certainly I could perform the procedure, and this was well before graphing calcs would do it for us!, but never understood that graphical connection until much later).

Second – I think the confusion on the last question about slope of .5 may stem from the decimal is very small and easy to miss. People could have been finding slope of 5. If it were written as 0.5, I think it would be easier to read.

As always, thanks for the posts!

My new math knowledge is quite simple, but something I discovered this summer when doing some real thinking about teaching one and two-step equations.

From as early as I could remember, algebra was EASY. You just do the opposite. For example x-10 = 20. X had 10 added to it, so just “get rid of” the 10 and subtract it from both sides. Voila!

What occurred to me, however, as I studied the properties of addition and multiplication more thoughtfully was that I was not actually subtracting the 10. Rather, I was invoking multiple different mathematical properties. By adding the inverse of 10 (-10) to both sides of the equation (property of equality) I was able to get to the additive identity (0). And now that x was only being added with the additive identity, whatever remained on the right side of the equation (10) was equivalent to x.

I know my explanation is lacking in precision, but what was huge for me was getting to a place where something I once saw as “mathematically” simple was actually the product of numerous mathematical principles. I’ll never look at algebra the same.

I see we all have experienced the lack of depth in our high school learning and it is a good feeling to be in such numerous and qualified company. Yet I wouldn’t say our teachers told it the wrong way. I experienced that the higher the number of different ways to describe a concept I have heard (or tried to express), the better perspective I had of it. So I’d have a faint and flat image of it when I had only a single version at my disposal, while I’d have a solid statue to watch (sometimes from different points of view) when the number available versions was higher. It seems to me that we ought to provide our kids with a good variety of experiences rather than seeking the only good one. By the way: thank you all for sharing your ideas, as they help me see my statues more and more vividly.

My first year of teaching was around the time that computerized grade books were gaining in popularity. Most teachers still had paper grade books and madly averaged away with a calculator each quarter when grades were due. I took the electronic route and thought I understood a concept of weighting grades. It only took me a quarter of a school year to realize that my grades were so horribly weighted towards tests and quizzes that it didn’t matter what my students did on their homework; it wasn’t nudging that grade a bit.

I suspect many non-math teachers still believe that their system admins have set up the weighting scales for them and they don’t have to worry about it – not realizing that if they let one assignment be worth 20 points and another 10 (in the same category like summative or formative) that the first has double the weight of the second in that category.

Not an algebraic concept, but certainly an aha moment for me.

I didn’t study math in college–barely studied it in high school, although I faked my way through AP Calc. So I don’t have those kind of stories. Quite the opposite, I’m often the teacher whose kids go crap, I never knew why that was true, as I explain it.

I’m not congratulating myself, because it’s not enough to *tell* kids, or even *show* kids.

For example, Naomi’s first paragraph is, verbatim, what I show and tell kids throughout my algebra 2 classes. I mean, literally after I do an exercise something like this (https://educationrealist.wordpress.com/2013/02/16/modeling-linear-equations-part-3/), I go through the idea, thoroughly, that the line is all number pairs that make the equation true. I reinforce this with the function activities. I do it again when I go through parabolas. Time and again, I show–through activities, through talking, through conversations–that all graphs are the representation of all points that make the equation true.

And yet it wouldn’t surprise me at all if some top kid, years later, just like Dan, said “wow. That’s what a graph is? Holy crap.”

Which is why it’s probably not a good idea to blame teachers. (I call this the myth of they’ve never been taught…)

They have to remember, not just for a test, but for the next math concept. Transfer of knowledge. Which is a bitch. After six years, I realize that maybe transfer never happens for a lot of kids. In which case, creating engagement and understanding in that moment might be enough, even as I break various parts of my body reaching for more.

All that said, I remember the shock I felt when I really grokked the remainder theorem. That’s friggin cosmic.

Thank you Dan! I was that kid that flew through math in high school, giving me this false sense that I was good at math, when in fact, all I was really good at was following what the teacher did. Nothing came together for, as I saw all math as disjointed. I still remember the day graphs started making more sense…freshman year take 2 (after an 11 year break) in my Calculus class and my prof says, “a graph is simply a set of inputs generating a set of outputs.” What??? My whole world was shattered, I finally started to see the fluidity between everything in math. I am still having “aha” moments after 5 years of teaching and I am not afraid to tell my students when they occur. I think it is helpful because I explain what I thought I knew and tell them what I know now…it’s almost a shared “aha” moment.

Some of the problems are easier than others. One leads to a parabola, but that’s kind of hard to see when just dragging and placing a single point. What if the students could place multiple points?

That last one looks less like misconceptions and more like a lot of students missed the decimal point and described a slope of 5 instead of .5. It’s an easy mistake to make, and not reflective of actual understanding; you should generally use a leading 0 (“0.5”) to avoid that.

Speaking of confusing punctuation, the checkboxes under the Submit Comment button include both of these:

Notify me of followup comments via e-mail
Notify me of follow-up comments by email.

Are they different in some way?

Area of a parallelogram. I had always taught the formula – ie average length x perp distance or TWO triangles. About 6 yrs ago after having taught for near to 20yrs. I was shown that if you draw a rotated SECOND trapezium and put it with the first, you will get a parallelogram ie (a+b)xh then half it! This was a humbling moment for someone who think they are quite good at maths.
Also if you teach parallelogram straight after rectangle, trianlges become easier. Any triangle is half a parallelogram. I think the usual order for teaching is rectangle, triangle, then papallelogram.
Non right angled triangles are then more difficult to understand using only the rectangle idea.
In my first yr of teaching, I was shown by HOD the way of generating pythagorean triples, something I had not been taught. Rule is easier to express as two rules depending on a being odd or even
eg 13, go for two numbers which are one apart but sum to 13squared….13,84,85
eg 14 go for two numbers which are two apart but sum to HALF of 14squared….14,48,50 (which happens to be a repeat of 7,24,25 I know)