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In the recent past, I've come across a pedagogical strategy for teaching/learning algebra that is sometimes called "Indicated Arithmetic" or "Delayed Evaluation". However, I've been unable to find any literature on it, and only know it by name from a former colleague. I've searched resources and have come up empty. Is this a term anyone is familiar with, and if so, do you have any source materials?


In the interest of clarity, here is an example where Indicated Arithmetic (or Delayed Evaluation) is used.

  1. Rommel brought $15 to a school fundraiser and spent it all on hot dogs and game tickets. Game tickets were 25 cents each, and hot dogs were 3 dollars each.

A number of standard algebraic questions can be asked following this set up, but they all generally get to the desire for an algebraic equation which describes the situation. A valuable tool for students who struggle with finding this equation is Indicated Arithmetic. The key attribute in IA is writing out all operations to be performed, but not performing any binary operations. Combined with the ideas of trying some explicit values, keeping track of your work with a table of values, and organizing your work, you might be able to create a visual like this.

enter image description here

The right column of the table builds towards the algebraic expression, and it is easy to see in this organized way, what different roles the different numbers play in the scenario and the expression. It is much easier for struggling students to see pattern that the $h$ variable follows, rather than trying to find a pattern in the sequence of values 0, 3, 6, 9, 12, 15.


Edit: 1/29/19

In my searching I found a footnote:

The term “indicated arithmetic” was shared by Nicholas Branca in 2004. It is used to be explicit about making used work visual so that emerging patterns can be captured.

Branca was teaching at San Diego State University until his passing in 2008. Still, this doesn't quite satisfy my needs. If anyone has more information, I'd be forever indebted.

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    $\begingroup$ What is the goal of such an exercise? Where # of tickets in the second column comes from? Considering that this is a two-variable equation, or alternatively, a function, where number of tickets depends on the number of hotdogs and vice versa, you cannot pull # of tickets from a hat for a specific # of hotdogs. So you need to calculate it from the function. This defies the point of the exercise. $\endgroup$ – Rusty Core Dec 18 '18 at 18:07
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    $\begingroup$ I'll note that this kind of thing is useful for students in late algebra or early calculus courses who need to compute $f(x+h)$ on their way to a difference quotient. Asking them to calculate $f(7)$, $f(13)$, $f(☺)$, and $f(TOMATO)$ before asking them to calculate $f(x + h)$ is very effective. $\endgroup$ – Chris Cunningham Jan 23 '19 at 16:48
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    $\begingroup$ @ChrisCunningham I am sorry, still not getting it. Is this formula supposed to be an intro to derivatives? "h stuff"? Could you explain the usefulness and validity of the table shown in the original question. I suppose the idea is that students can play with numbers and get various results. What I am saying is that they cannot get the resulting amount without having the formula: tickets times ticket price plus hotdog times hotdog price is the total. When they have it, the exercise is pointless unless they calculate hotdogs from tickets or vice versa, which is not clear from the table. $\endgroup$ – Rusty Core Jan 25 '19 at 19:48
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    $\begingroup$ @RustyCore For beginning students, the formula "tickets times ticket price" is easy if "tickets" and "ticket price" are both numbers, and there is a significant difficulty spike when one or both of the two is represented by a variable, which is one level of abstraction higher. To you, the formula "the total is tickets times ticket price" is just as good as "the total = TP where T = tickets and P = ticket price" but that is simply not the case for students with weak algebra skills. $\endgroup$ – Chris Cunningham Jan 26 '19 at 15:16
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    $\begingroup$ @ChrisCunningham I am not questioning extra complexity caused by letters instead of numbers. My question was about the specific table listed above. If I am to read it top to bottom left to right, then I can see very specific combinations of # of hotdogs and # of tickets, and my original question was: where these combinations come from? Why for 2 hotdogs 36 tickets is chosen? This cannot be a random value, it clearly comes from the formula, hence whoever obtains these pairs already knows the formula, hence the table is utterly useless to teach algebra skills, and is just a boring time sink. $\endgroup$ – Rusty Core Jan 28 '19 at 17:48
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To provide a source... I brought this term to the LINCT professional development team (Andrew Sanfratello and others) and the New York State Transition curriculum in 2014. I learned it from the late Nicholas Branca who had led professional developments in New York City back in 2003 and 2004. It can be a powerful tool in a wide range of problems. Al Cuoco and others will label it as "delayed evaluation."

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