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We need to do something about these problems, which recur all throughout school Algebra.

The original title of this post called them “horrible,” but they’re truly “tragic” – the math education equal to Julius Caesar, Othello, and Hamlet – full of potential but overwhelmed by their nature.

Here’s the thing about variable expressions: they’re used by programmers and students both, but those two groups hold variables in very different regard.

Ask programmers what their work life would be like without variables and they’ll likely respond that their work life would be impossible. Variables enable every single function of whatever device you’re reading this post on.

But ask students what their school life would be like without variables and they’ll likely respond that their school life would be great.

What can we do?

The moral of this story isn’t “teach Algebra 1 through programming” or “teach computational thinking.” At least I don’t think so. I’ve been down that road and it’s winding.

But in some way, however small, we should draw closer together the wildly diverging opinions students and programmers have about variables. Ideas? I’ll offer one on Monday.

2014 Jul 25. I appreciate how Evan Weinberg has thought through this makeover (now and earlier).

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Dylan Kane restates our task here in a useful way:

In terms of making these problems a little better, students should feel a need for the expression. I think this question stinks in part because the expression it’s asking for is so trivial — it’s extra work, compared to just multiplying by 3/4 or doing some simple proportional thinking.

Jennifer offers an example of that kind of need:

I like to introduce the idea of expressions by having the students playing the game of 31 with a deck of cards!their goal- play until they can predict how they can win every time! This will take less than 15 minutes, and a whole class summary of verbal descriptions on ‘how to win’ are shared. Verbal descriptions become cumbersome to write on the board, so ‘shorthand’ in the form of clearly named and defined symbols are used to make the summarising more efficient. the beauty of this is that the idea of equivalent expressions presents itself.

Kate Nerdypoo:

i think the ability to generalize and write a rule with variables is really important, but you can come to that through lots of nice activities and investigations as well.

for example, i did dan’s “taco cart” with my students with a few notable changes. instead of telling the students how fast dan and ben walked, i had each group decide on the speed of the two men themselves and list that along with other assumptions they made in the problem.

when we did the whole class summary, i told them that i had written down a formula on my paper that would allow me to check if their answers were correct and that i needed that since everyone used their own speed. i should’ve asked them all to take a minute to try to come up with the formula i used, but instead i elicited it at that moment and one student gave me the correct formula. the need for two variables (speed on sand and speed on pavement) was obvious.

in part two of taco cart, when the students were trying to figure out where the taco cart should be so that the two men reached it at the same time, one group did seemingly endless guess and checks. i suggested to them that this wasn’t a good method and asked if maybe it wasn’t better to write an equation with a variable. again, they could see the need. once they started with a variable, the rest of the problem started to fall into place.

here’s the thing: these “write an expression” problem want to train students to learn to generalize and write a rule. they want them to be able to see a situation that would best be tackled with a formula of some sort, write said formula with various variables, and use their formula to solve complex problems. but the issue is that these problems don’t actually train students in that way because they’re so artificial and one-dimensional. what they do is teach students to “translate” from english to math (an important step along the way, i do believe), but not to recognize a situation in which a formula would be helpful or necessary or how powerful it can be.

Dan L:

Two thoughts from a computer science teacher’s point of view:
1) When introducing programming to 15 yrs olds for the first time, we use Python interactive prompt as a calculator first. And the first point is to show the advantages. Variables and funstions (one-liner formulas) simply save work. That is it, that is one of the goals of the whole programming topic anyway. When they do quadratic equations in maths at that time, we are headed that way. And students realize pretty fast: aha, I have to understand how to solve them, but once I do and I describe it properly, I never have to do it by hand anymore. Their understanding of q.e. deepened, they interest in programming increased, and I could naturally introduce a load of important CS concepts on the way. In younger age we do simpler formulas, also like BMI calculation (not only the “area of the circle” kind of stuff, which they, um, do not prefer), but the point is the same: the kids need to see at least some hypothetical benefit for themselves. Having to introduce variables in maths, I would in principle search for a similar approach. (Note: for even younger kids, fun and creativity achieved with Scratch, turtle etc. overweigh the “practical benefits”; but when practicality leads to more fun – win-win!).

2) A good and often forgotten tool between calculation on paper and programming is a spreadsheet. It can store lots of numbers in a structured way and perform basic calculations, what is well understood by kids. And when we want to do anything more complex without getting beards grown, we absolutely need formulas and “variables”. Their advantages are imminent. And the whole time, everything is in plain sight, the level of abstraction is way lower than with programming, making it very accessible for kids. I am of course interested in it “from the other side” – after some decent work in spreadsheets, many more advanced concepts are a step away (for-loop, data type, input-output, function, incremental work on more complicated calculations, debugging etc. etc.). But I believe that thoughtful use of spreadsheets can improve understanding in appropriate topics in maths.

Elizabeth Green compiles the history of math education in the United States from New Math to the Common Core:

Americans might have invented the world’s best methods for teaching math to children, but it was difficult to find anyone actually using them.

She also tours through some of the best ethnographic research you’ll read in math education but doesn’t cite two of them explicitly (that I counted) so I will.

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Daniel Schneider:

It might be worth noting that the paragraph about ‘answer getting’ seems to be referring to Phil Daro and his whole take on answer-getting.

Simon Terrell writes about his trip to Japan with Akihiko Takahashi.

Dan Goldner on his resolutions:

Of all the great things to focus on in this article, this is the one that spoke to me where I am now. Student-initiated in 40%, not 100%. 41% of time practicing, not 5%. Half the time on invent/think, not all the time on invent/think. I’ve been working so hard on making “invent/think” the dominant activity in my room, that practicing, which is also a cognitive requirement for learning, has been de-emphasized. The next paragraph in the article acknowledges that Japan isn’t perfect, either, and these percentages certainly aren’t a perfect recipe. But as my personal pendulum finds its equilibrium it’s great to read this and take from it the encouragement that that all the modes of learning have to have a place during the week.

David Wiley:

For me, personalization comes down to being interesting. You have successfully personalized learning when a learner finds it genuinely interesting. Providing me with an adaptive, customized pathway through educational materials that bore me out of my mind is not personalized learning. It may be better than forcing me through the same pathway that everyone else takes, but I wouldn’t call it personalized.

Held to that standard, most groups that are attempting to personalize learning through software are pretty screwed.

Jai Mehta:

But what I can tell you from visits to blended classrooms and schools, in both traditional public and charter schools, is that students tend to find what exists thus far as fairly dull, lacking both the community and the accountability that comes with good face to face learning. A number of students told us at one highly celebrated blended school that they liked everything about the school except for the online learning!

That last link via Justin Reich, who confidently predicts the results from the 2017 Khan Academy study.

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Jane Taylor:

Another aspect of personalization is the relationship between student and teacher, and I found that as blended learning decreased the amount of face to face whole class instruction in my class last year, I didn’t get to know my students as well and as quickly as I had in the past. When I know my students and find out what “works”, what engages, each particular group of students, as well what works for individual students, then my classroom can better meet individual needs, not just in the way I teach math, but in the way I encourage students to manage their time, to grow in their work ethic and study habits, to overcome math anxiety, and many other things. Whole class interaction is a lot of fun for me and, I believe, for students. Resources, such as videos, are great for motivated students to review or move ahead, and I will continue to provide them, but I am returning to primarily whole class instruction this year.

Two anecdotes about curiosity, followed by a challenge:

1. Nana’s Lemon Water

I facilitated a workshop in Atlanta a few weeks ago and a participant had one of these enormous Thirstbuster mugs. I asked, somewhat nervously, “Whatcha got in there?” She replied “water with lemon.”

I wondered, as I’m sure others might, “Well how much lemon would you need in that enormous thing to even taste it.”

It’s natural for humans to have questions and seek to answer them. Once I heard her answer, though, an unnatural, teacherly act followed. I tried to recapture the question, something like mounting a butterfly in a shadow box or preserving a specimen in a jar, so that a student could experience it also.

That’s this video and the attached lesson.

2. Rotonda West

Another example. It takes very little curiosity to appreciate the gorgeous, curated satellite images from, such as this image of a Florida housing development:


What’s trickier for me is to format that appreciation, that awe, into a question, to capture that question so I can share it with students.


Making that image (and the answer) required a certain technological know-how, sure, but the really challenging part is training myself to probe interesting items for the curious questions they contain. It’s one of teaching’s unnatural acts and it requires practice and feedback.

3. Challenge

Curiosity is cultivated. Curious people grow more curious. These are examples of how I cultivate my own curiosity.

With that said, what curious questions can you find in this interesting story and video about the tallest water slide in the world? How can we capture that curiosity and make it accessible and productive for our students?

Previously: How Do You Turn Something Interesting Into Something Challenging

Bill Gates, via Tom Hoffman:

… the one thing we have a lot of in the United States is unmotivated students.

It’s astonishing to me how many people develop their pet education theories assuming there is little or no interaction between motivation and learning, or that motivation is somehow outside the teacher’s job description. The assumption that motivation is entirely the student’s job leaves us no way to check ourselves for de-motivating pedagogy. If students don’t like sitting in warehouses, watching lecture videos, and clicking away at multiple choice questions, it’s either their own fault, or the fault of Miley Cyrus, social media, or Kids These Days, but not ours. Our theories can’t be impeached. We just need a better class of students.

Related: Rocketship charter schools (which were last seen on this blog here) are abandoning their enormous warehouses where elementary students click away at multiple choice questions:

Teachers — who are at-will employees who can be fired at any time — also criticized Rocketship’s intolerance for dissent, saying it contributed to the disastrous redesign that placed 100 students in a classroom.

“Teachers raised concerns,” said one ex-teacher, “and no discussion was allowed on the subject.”

Those who privately expressed doubt feel vindicated [by the removal of the warehouses] although sad, by the resulting test decline.


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Tom Hoffman:

I was thinking that you can tell a lot about a person’s view of education by exactly when they realize the importance of motivation. From the beginning, in the middle or at the end.

I think one thing that probably strikes teachers about Gates’ quote there is how much it sounds like a cranky old teacher in the break room.

Jay Fogleman:

I find the idea that “today’s youth” are “unmotivated” is bizarre. When teenagers are “hooked” one topic or activity, they are darn near unstoppable.

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