The Math I Learned After I Thought Had Already Learned Math

True story: it’s possible to fly through your own secondary math education — honor roll bumper sticker on your mom’s minivan and all — but miss some of the Very Big Ideas of secondary math.

For one example: in our last post on simplifying rational expressions, the process of turning a lengthy rational expression into a simpler one, Bill F writes:

Another benefit of evaluating both expressions for a set of values is to emphasize the equivalence of both expressions. Students lose the thread that simplifying creates equivalent expressions. All too often the process is seen as a bunch-of-math-steps-that-the-teacher-tells-us-to-do. When asked, “what did those steps accomplish?” blank stares are often seen.

Past a certain point, those operations are trivial. But it’s only past a point much farther in the distance that the understanding — these two rational expressions are equivalent — becomes trivial.

For another example: I left high school adept at graphing functions. I could complete the square and change forms easily. I knew how to identify the asymptotes, holes, and limiting behavior of those thorny rational expressions. 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. They look like a line

That’s one reason I’m excited about the Desmos Activity Builder and this activity I made in it last week, Loco for Loci!

It asks students to place a point anywhere on a graph so that it makes a particular relationship true. Then it asks the students to imagine what all of our points would look like if we pictured them on the same graph. Then the teacher can show the results, underscoring this Very Big Idea that I didn’t fully appreciate my first time through high school — what we eventually think of as a continuous line is a picture of lots and lots of points.

Here is what happened when 300 people on Twitter played along:

“Drag the green point so that it’s the same distance from both blue points.”


Trickier: “Drag the green point so that it’s five units from both blue points.”


Whimsical: “Drag the green point so it is the same distance from a) the line of dinosaurs and b) the big dinosaur.” I really couldn’t have hoped for better here.


And then a couple of very interesting misfires.

“Drag the green point so that it’s four units from the blue point.”


“Drag the green point so that a line segment is formed with a slope of .5.”


You could run a semester-long master’s seminar on the misconceptions in that last graph.


Maybe more like ten quick seconds at the start of your Algebra class.

If you’d like to run this activity with your own students, here is the teacher link.

Questions for the Comments

  • Obviously, I didn’t invite hyperbolas and ellipses to the party. Which other loci should have received the same treatment?
  • Which Very Big Ideas did you only fully understand once you started math teaching?

Featured Comment

Jason Dyer:

I find this sort of gap fascinating [my inability to conceive of graphs as a picture of solutions –dm] 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.

Bowen Kerins:

The description you give about graphs is something we have to hit early and often in CME Project, it’s one of the top 3 things to learn in the entire curriculum. It’s amazing how that can get lost in the shuffle, but it does, and where it gets lost is the algorithmic way of graphing a function or equation: the A does this, the B does that, etc – all of this ignores the deeper fact that under the hood, this is all a relationship between two variables x and y.

The other two of Bowen’s top three things to learn in Algebra, according to Bowen on Twitter, are:

  • Variables represent numbers, so test numbers to test ideas and build equations.
  • Rules for new stuff should respect existing rules.

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Amazing, all the people unburdening themselves on Twitter of math they only understood once they began teaching. What does it all mean?

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. As a math enthusiast but not a teacher, I’ll crash the party by talking about matrix multiplication. Here’s what I wish I knew: Matrices describe functions from vectors to vectors. Each number can be understood as how much an input (the column) influences the output (the row). Matrix multiplication is just function composition. And, this one I’ve only partially gotten my head around, matrix multiplication takes what the output of the first matrix would be and then feeds it into the second matrix.

  2. 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.

  3. Brian R Lawler

    August 11, 2015 - 10:55 am -

    My list of learned once teaching is embarrassingly long. What interests me is how as a child I was very certainly mathematically precocious. As schooling went on, the topic dulled me more and more. At the time, I recognized the separation of puzzle and play from what I now experienced in math class.

    I would never any longer be recognized as precocious in maths, but still wonderfully enjoying puzzling over mathematical ideas. Thankfully, my early teaching opportunities provided an opportunity to remarry these divergent experiences of school maths and my mathematical pleasures.

  4. 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?

  5. 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.

  6. And since I stated blogging about school math I have
    a) figured out how to get from a sliced double cone to the actual equation of a conic section (it always seemed like magic)
    b) figured out how to do every arithmetical operation using number lines (they didn’t have number lines when I was at school !)

  7. 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?

  8. Five units from the point does better than four units from the point, because there are other “lattice points” to go to; with four units from the point, people typically pick the four compass directions (as you saw).

    The description you give about graphs is something we have to hit early and often in CME Project, it’s one of the top 3 things to learn in the entire curriculum. It’s amazing how that can get lost in the shuffle, but it does, and where it gets lost is the algorithmic way of graphing a function or equation: the A does this, the B does that, etc — all of this ignores the deeper fact that under the hood, this is all a relationship between two variables x and y.

    Other loci? My favorite is the set of points N times as far from one point as another; or the set of points N times as far from one point as from a line of dinosaurs.

  9. 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:

    (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.)

  10. I can’t even begin to list the things I’ve learned about K-12 math since I graduated from college. I feel like the list gets longer and longer each day. The first one that jumps to mind (probably because of the discussion of functions and graphing above) is just how truly amazingly predictable the graphs of polynomials and rational functions are. Sure we know that (0,0), (1,1), (2,4), (-3,9), even (0.5, .25) are all points on y = x^2. But the fact that we can connect all those points with a smooth curve and hit EVERY OTHER SOLUTION!?!?! Why isn’t it squiggly? Why doesn’t it flatten out? Why should it be so predictable? That is mind blowing to me now, but I took it for granted when I learned it. My 8th grade self thought, how could it be anything else? Extrapolate that to the fact the we can sketch y = (x+3)(x+1)^3(x-2)^4(x-3) with a high degree of accuracy in about 3 seconds. Why should a crazy 9th degree polynomial be so well behaved? Just amazing. (I’m not actually asking that, by the way–I feel like at this point I have a pretty good handle on why, but it is still amazing!)

    With regard to the Loco for Loci activity, I do have one concern. The leap from “find one point that ….” to “consider all points that …” is a huge leap–one that my students have struggled with tremendously in the past. Is is possible to ask students to find 3 or 4 or 5 points? Maybe marking those multiple points on the graph as they go? And then to show them all their classmates points before asking them to consider all points? Desmos activity builder seems very cool, though. I look forward to playing with it.

  11. 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!

  12. The fact that you somehow made it out of high school (taking fairly advanced classes) without understanding that a graph is a way to represent the infinitely many solutions that a two variable equation points to something missing from your mathematical experience at the very beginning of graphing. After all, you’ve been solving one variable equations, theoretically you know that a solution makes an equation true, hopefully you’ve seen some that are a little funky, with no solution or infinitely many solutions or a few solutions. Now you get this new kind of funkiness: two variables. And suddenly instead of looking for all the numbers that make the equation true, you’re looking for all number pairs that makes the equation true. And you find that there are many of them. Like so many that you don’t think you can list them all. Headache. But then there are graphs. Aspirin.

    Yesterday I was tutoring a child who’s going into 7th grade “Pre-Algebra”. (She says it’s the dumb class because some of her classmates are taking Algebra in 7th grade. And she proudly states that they’ve been solving equations since 4th grade. *Lovely*.) And she clearly doesn’t get equations. So we’re going through a few equations, and it becomes clear that she just needs to do some guessing and checking and stumbling on solutions, and “if the sides are 6 apart when I plug in 0 and 10 apart when I plug in 1, and 14 apart when I plug in 2, then I’m going away from the answer….” kind of stuff. And, as the tutor, it is painful to watch her in this inefficient struggle. But sometimes we teachers need to let it happen. (After all, she’s getting some good practice with negative numbers that she needs, she’s reinforcing the idea that the solution to an equation is something that makes it true, I’m hoping that she’s going to catch on to some linear patterns by looking at how far from equal the sides are. . . . that’s all good stuff for her to be doing.) Sometimes we just need to stand with them as they struggle through the hard way and recognise that just because it isn’t the most efficient way doesn’t mean that there isn’t value in doing it that way.

  13. 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.

  14. To Naomi: there’s multiple possibilities but the usual thing that goes wrong is that at the beginning, instead of learning to graph, a lot of students learn to graph *lines*. And the things they learn, usually, are “m is the slope and it does this” and “b is the y-intercept and it does this”.

    The focus is often not on the equation-graph connection or even on x and y. Try asking students to graph x = 2y + 5 or y = 2 + 5x and watch in horror ;)

    But it’s worse: students move on to other graph types (typically quadratics next, sometimes absolute value functions) but without a sense of what rules from graphing lines carry over and what rules don’t: the y-intercept is still sort of there, but the slope is gone completely. This happens every time with every new graph — new parameters, new rules.

    The one big rule underlying it all, graphs as the set of points making an equation true, needs to be pushed to the front of all of this. For CME we decided this meant there should be a chapter on general graphing, with this as the biggest point of emphasis, and that chapter includes examples of lines, quadratics, circles, ellipses, and more — heck, the test has a hyperbola on it. (Which of these points is on the graph of 2x^2 – y^2 = 1?) This helped the fundamental concept push to the forefront, then when each new graph type is explored, it’s explored in the context of that fundamental concept.

    This really requires a high-level oversight of where things are and where they go. If my goal is to teach lines today, it’s hard to justify cramming in a bit about the equation-graph connection. And yet, that is a critical piece to building the larger understanding, it’s a piece that allows students entry into all sorts of new areas, and it’s a piece that can survive to be recalled months or years after its first introduction.

  15. 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.

  16. Belinda Thompson

    August 13, 2015 - 12:40 pm -

    In one of my first “math for teachers” type class, Dr. Ryoti asked us right out of the gate “Why do we invert the divisor and multiply when we divide fractions?” I thought, “Why? IS there a ‘why’ for that procedure? OMG is there a ‘why’ for all the procedures I’m lucky enough to remember? I’m a math impostor!” So, I was lucky in my teacher prep program to realize I needed to learn more math, even the math I thought I already knew!

    Many of the comments seem to be realizations about equivalence and representations. Maybe our teachers thought maintaining equivalence and representations of solutions or relationships were obvious, so we were never asked “What do you think this line represents?” or “Do you think the solution to the original equations is the same or different than all these other equations here in the guts of our equation solving process?”

    Confession time: The first year I was tasked with teaching Algebra 1, I realized why when solving quadratic equations if (ax+b)(cx+d)=0, then one or both of those factors has to be 0. I knew splitting that equation into two equations gave me numbers that made the original equation true, but I thought that was just the next step in the solution process. Obviously.

  17. 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.

  18. Simple linear graphs used to bother me a lot for one specific reason. Take y = 2x as an example and just look at the segment between x=0 and x=1

    For EVERY x in this interval, there is a corresponding y value.
    For EVERY y in this interval there is a corresponding x value.

    But the y interval here is from 0 to 2. Which means that there is a one-to-one correspondence between all the number between 0 to 1 and all the numbers between 0 to 2.

    That used to fry my brain in high school. Wasn’t until university that someone could explain it to me.

    When I was teaching math I used to do exercises similar to these Desmos ones, but by hand. In one example, the students would work in groups and I would ask them to find 4 points that were a distance of 5 from the origin. When they showed me their solution, I would ask them to find 4 more. Then a total of 12 points (which are usually all the easy 3,4,5 points). And then finally if they could find ALL the points. It wasn’t long before they realized that they couldn’t really list all the points, since there were infinite points and that might take a little bit of time to write down. With different amounts of nudging and hints (depending on the group) they could eventually be helped to develop the equation of a circle.

  19. 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 (, 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.

  20. It was only after grad school that I learned (from Lockhart’s book Mathematician’s Lament) to consider natural numbers as stones that can be arranged in various patterns that illustrate the different properties of a number. For example, evens are piles of stones that can be arranged into two equal rows, and square numbers have just the right number of stones to make a square! It’s really fun thinking about various operations in this way, and there are some beautiful proofs based on this technique. For example, why the sum of the odd numbers 1 + 3 + 5… Is always a square.

  21. 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.

  22. Somehow I made it through an entire engineering degree without figuring out how the sine and cosine graphs relate to the unit circle. Wasn’t until I subbed for a high school math teacher and had to teach it myself. Brilliant.

  23. 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?

  24. You might have had better results with the slope question by writing the slope as 0.5 rather than .5.

    Unless maybe you were trying to prove a point about leading zeroes?

  25. 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?

  26. I’ve seen a couple of posts about the .5 versus 0.5 on the line slope exercise. The real question is why present this as a decimal? Ask for all points along a line of slope 1/2 gets the “5” completely out of the picture.

    It’s a minor point compared to some of the other nice replies. I remember being in an honors physics class in high school and being increasingly surprised by trigonometry – and effectively circles – appearing all over the place in the math for periodic motion. The idea that the linear acceleration due to springs was a simple (1 dimensional) case of what we’d see later in circular orbits and so forth becomes sort of an open question in what circles even *are*, which I found fascinating.

  27. 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)

  28. I’m starting a probabilty unit with my 7th graders. After reading this post, particularly Jason Dyer’s comment, I started wondering what the graph of possible dice combinations would look like. I jumped on desmos, and tried it out. Really cool stuff. For two 6-sided dice the number with the highest probability is seven. The cool thing about graphing it, is that number is along the diagonal of the square, always. Blew my mind.