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(tl;dr: skip to the question)

Context

For the last few years I have been pondering about the use and disuse of calculators in secondary school level mathematics. Now, after coming across a 'hot network question' regarding a student enquiring why calculator shouldn't be used and the extend to which Math Educators seemed to feel obligated to defend it. On the other hand, for most (not all students) I am inclined to argue that it's more valuable to teach them 6 years of using calculators and things like WolframAlpha, r/statistics and programming than it is to teach so much paperwork (thus allowing also harder questions to be solved). Now, that's just my opinion and one I believe I can defend, however not the crux of this question.

Question

What I am wondering about is whether there are any curricula that do allow calculator use (including better calculators such as de TI98) on all papers/examinations and are build around extensive technology use (doesn't matter if it it's from some far away random country).


What I do know

The two I know best: the IB program and the VWO program both allow the use of calculators on part of the exam, however contain questions explicitly designed to be unusable with a calculator or have entire sections where a calculator may not be used. Additionally both limit the 'power' of the calculator significantly (simply put, nothing above the TI94).

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    $\begingroup$ You might be interested in computerbasedmath.org $\endgroup$ – Steven Gubkin Jun 6 '14 at 14:00
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    $\begingroup$ @StevenGubkin: Love you! That's amazing from the looks of it! And I do believe that would be a valid answer to the question too :) . Nobody might be using it it, but it is a calculator centered curriculum (or as close as we might get today). Why didn't I know about them... :O $\endgroup$ – David Mulder Jun 6 '14 at 14:05
  • $\begingroup$ Sorry to hijack your question, but I want to know what concrete experiences there are with such an approach? Any level, be it primary school to undergraduate studies. $\endgroup$ – vonbrand Jun 6 '14 at 20:47
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    $\begingroup$ From the question, it isn't clear to me what it is that you want. Do you have some intellectual content in mind, with software as a way of avoiding tedium while the students learn that intellectual content? If so, then what intellectual content is that? Or do you see learning software as and end in and of itself? You use Wolfram Alpha as an example. WA is a big, proprietary system that includes natural language processing. Do you have in mind teaching students how to phrase queries for WA successfully, e.g., "what is the population of Nigera?" Or are you thinking of it just as a CAS? $\endgroup$ – Ben Crowell Jun 6 '14 at 22:17
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    $\begingroup$ @BenCrowell: I have a lot of things in mind myself, but what I was looking for was others who had taken the time to investigate and develop curricula around it. Personally I don't care for the specific ways WA takes queries, but indeed, I am thinking of it as a more advanced CAS capable calculator, not a data analysis tool (for that R and/or pure programming and/or spreadsheets should still be taught). $\endgroup$ – David Mulder Jun 7 '14 at 0:12
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This doesn't answer the part of your question about calculator use, but having worked for Jim Kaput I feel the need to say that yes, there have been curricula designed for extensive technology use.

One such combination of technology and curriculum is the SimCalc research, which produced SimCalc MathWorlds and the curriculum that goes with it.

One version of SimCalc made use of calculators (but as programmable handheld graphing and motion simulation devices rather than as calculating devices. That software can be found here.

The Algebra 1 and Algebra 2 curriculum, which can also be used with the Windows/Mac version of the software, can be found here.

What this combination of curriculum and technology were used for is summarized here:

The mathematics content of SimCalc is the mathematics of change and variation (Mathematical Sciences Education Board and National Research Council 1990), which is a core strand within US school mathematics (National Council of Teachers of Mathematics 2000) that is representationally demanding and studied at many levels (Kaput 1994). We have systematically tested SimCalc across a series of mathematical topics and have begun to produce a set of curriculum materials that build on research and experimentation. Such materials exploit the power of both parallel software and the new, classroom-based communication infrastructure, to facilitate rich inquiry-based classrooms, fueled by group based or whole-class collaborative efforts. (Hegedus & Penuel, 2008)

A recent book also summarizes the research contributions of nearly two decades of SimCalc project work.

Bonus:

It was one of Kaput's views that developing educational software was pointless without developing curriculum along side of it. And that it was important to the future of education to think critically about the inclusion of technology. There is a difference between a technology that someone decides to use for math education and a technology that is designed for mathematics education. We shouldn't be surprised when a technology is plopped in the classroom and fails to make a difference.

Tying it back to calculators, I say "plopped" intentionally, because some references to calculators in classrooms do little to talk about what students might do with them that they couldn't do before. There are clearly two important aspects of technology use in mathematics classrooms that are missing when the conversation is simply about whether or not we use the technology. 1) How is that technology designed to transform a student's capabilities, opportunities, thinking, and participation in the mathematics classroom? 2) How is that technology used to transform the mathematics classroom (capabilities, opportunities, thinking, and participation), whether or not we are using it for its intended design? And when I refer to transformation, we must consider what the use of technology is replacing (what activity of the students, or other aspect of the classroom, has changed).

We should probably ask at least these questions of any technology used in math classrooms, including calculators. And if I'm going to call for that critical eye, I feel the need to point out that only part of the many possible answers to the "problem of calculator use in math classrooms" lies within the technology itself. It may be sensible to reject calculators if mathematics instruction equates to giving students practice doing arithmetic. To use a crude metaphor, if the point of a particular driving lesson is to teach a student shifting gears, it doesn't make sense to do that in a car with automatic transmission. However, if the point of the driving lesson is to teach a student to parallel park, or to drive to a friend's house, you may take whichever car you like.

Something to think about, anyhow.

EDIT: Adding a description of SimCalc Activities to clarify the use of this technology. I realize that has made this answer long. The answer proper is above this point. Below here is only for interest in SimCalc, to clarify what this sort of technology means for the mathematics classroom. Works cited are at the very bottom.

As I described in the comments, SimCalc MathWorlds is an environment for mathematics education, meaning that it contains tools used to manipulate linked mathematical representations. Students use these tools for various purposes in the context of the curriculum activities. The educational aspect of the environment comes from the students use of tools acting on mathematical representations, interpreting mathematical representations, having discussions about their own contributions and the contributions of other students, etc.

Early research on SimCalc (late 1990's) focused on what could be done in classrooms with the linked mathematical representations. Later, classroom networks were created using TI-Navigator, and ultimately school wifi between laptops (running on the mobile laptop labs some schools were buying). The connected aspect of the software now allowed students to work briefly in private before submitting a mathematical contribution to the group discussion -- all student work would be aggregated by the teacher and projected in a public display.

Often, students were given parameters that were to be used in their contributions which caused "families" of related functions to appear on the public display when all the student work was seen simultaneously. Students and the teacher could then discuss the patterns and what mathematical meaning could be made from them. "Why are these lines all parallel?" (to use a relatively simple example)

Some specific activities using SimCalc MathWolrds are described on this page.

My description of two of these activities:

The background story of these activities is that students are involved in an exciting race (a potato sack race is sometimes used). The technology is going to allow them to edit functions that represent their own performance in the race. The results are displayed on their handheld device (or laptop, depending on which software they're using) as a plot on a position vs. time graph. They also have an "executable representation" which is essentially a simple animated environment where a character will act out the race, as defined by the function.

[The software itself has a number of ways to edit and represent functions and information related to functions (graphs, tables, "y=" symbolic representation, "number sentence", animation world, and something called "marks".) Many activities depend on creative use of restricted access to these representations or how they are edited. For example, you might be able to see a character moving in the animation but you are not given his position graph. This makes sense if you are trying to determine for yourself what the function that defines his motion would look like on a graph.]

In the first "race," students are shown the teacher's function (Sack A), which starts at 0 and goes 3 ft/sec for 5 seconds. Each student will be in Sack B on their own devices. Each student has been assigned some number and they are told:

Your job is to write a linear function y = mX + b, where b = "your NUMBER" so that you will go at the same speed for the same amount of Time, but starting at your NUMBER.

They have an input mode that allows them to edit the y = mX + b function representation. They're given "b" explicitly, but have to determine start time, end time, and "m" from the description (or their observations) of Sack A.

Once they have created their functions (position as a function of time) and seen their graphs, the functions are collected by the teacher and displayed together. Since they had a bunch of different "b" values, you can see why this activity is called "staggered start, staggered finish." At this point, the teacher leads a whole class discussion which should include the meaning of the variation students see in the functions.

Next, students are asked to go back and create a new race. This time, they'll edit the function graphs directly, adding "pieces" to create a piecewise-defined position function. Their objective is to create an exciting race that begins at position=0, time=0. It must also end in a tie with a constant velocity racer who finds himself at position=12, time=10 seconds. However, along with their own version of the race they're asked to write a narrative about what happens during the race. Of course, their own races will have stops and starts, backwards motion, extreme speed, extreme slowness, etc. They're essentially given free rein.

Typically, students come up with an outlandish and personally interesting narrative. After the different funcitons are collected by the teacher, they are displayed and invariably are a sort of mess of non-systematic variation. Except that they all begin and end at the same point. Now students are asked to give their narrative of how their own race went. The discussion can focus on a number of things: "How much mathematical language does a student use in his or her narrative?" "Is the narrative descriptive enough that we can actually pick out the student from among all the class' contributions?" "What does the shape of the graph mean at various points?" "Are student stories consistent with the graph shape?" "How does using more mathematical language contribute to other students being able to find your motion from among all other students' motions?" There are many other interesting questions that arise, but these are some of them.

We've found, also, that students can get into mathematical arguments about the meaning of the graphs, how to modify them, the best way to describe them, etc. Rather than the teacher mediating such disagreements, there can be productive use of the representations, with students trying to convince their peers. The personal contribution aspect, I believe, is part of what draws students in to identifying with the mathematical representation they have had placed in the public space. And having a stake contributes to the motivation to argue for how that representation should be interpreted, including mathematically.

Students sometimes violate the rules of the activity, creating outliers. These are generally not disruptive, as they can form the basis for a discussion about the parameters of the activity and why it is impossible to hide the outlier among the rest of the class' contributions. Representations can also be edited or hidden by the teacher, if necessary. Students can even be allowed to update their contribution on the fly, from their seat, while the class looks on. All changes in the public display (or access to the ability to change) are controlled by the teacher. The teacher can even pause the activity, which locks out all screens immediately with a message directing attention back to the teacher.


Works Cited:

Hegedus, S. J., & Penuel, W. R. (2008). Studying new forms of participation and identity in mathematics classrooms with integrated communication and representational infrastructures. Educational Studies in Mathematics, 68(2), 171–183. doi:10.1007/s10649-008-9120-x

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  • $\begingroup$ From cursory glancing over it, do I understand correctly that technology here was used as an educational tool or was it meant as a conclusive part of the mathematical exercise? $\endgroup$ – David Mulder Jun 9 '14 at 3:09
  • $\begingroup$ It is an environment for mathematics education, meaning that it contains tools used to manipulate linked mathematical representations. Students use these tools for various purposes in the context of the curriculum activities. The educational aspect of the environment comes from the students use of tools acting on mathematical representations, interpreting mathematical representations, having discussions about their own contributions and the contributions of other students, etc. $\endgroup$ – JPBurke Jun 9 '14 at 8:34
  • $\begingroup$ @DavidMulder I added a long edit that gives a taste of what a SimCalc MathWorlds activity is like. I have conducted SimCalc activities, often for workshop groups of middle or high school students. Some of the activities are easy enough to explain that they can be accomplished in much less than a class period. Last month I had middle school students using motion detectors and SimCalc in order to challenge other teams of students to interpret position graphs and then physically act out motions. $\endgroup$ – JPBurke Jun 9 '14 at 9:37
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As Steven Gubkin didn't come back to post his answer from the comment at least I will write it out into a short answer:

ComputerBasedMath.org

The importance of math to jobs, society, and thinking has exploded over the last few decades. Meanwhile, math education is in worldwide crisis—diverging more and more from what's required by countries, industry, further education... and students.

math ≠ calculating: Computers are key to bridging this chasm: only when they do the calculating is math applicable to hard questions across many contexts. Real-life math has been transformed by computer-based calculation; now mainstream math education needs this fundamental change too.

They work with this very nice concept where they split up the default math process into four steps:

  1. Define Questions
  2. Translate to Math
  3. Looooots of paperwork or Compute Answers
  4. Interprent

And their argument is that step 3 is taking extremely much time in the current system, whilst in reality all steps deserve a lot of attention.

Their progress

Right now they don't have an entire curriculum developed, but they have developed a framework and sample materials and are campaigning for policy makers to build upon this framework and collaborate on creating a fully functional curriculum.

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