Computer Based Maths ( https://www.computerbasedmath.org/about ) wish to consult you on our list of educational outcomes. These are the long-term goals that we want students learning mathematics to achieve through their schooling.
As valuable members of our community, we would like your feedback, to critique, compliment or suggest improvements upon the fundamentals that drive the initiative.
So please take the time to step through the list of outcomes, including the details and provide some feedback on what you think (comment / post below).
This link is for the online list of the outcomes, found on that same page is a link to download the outcomes in PDF format should that prove useful!
Whether computer-based or not, surely one of the goals should be something like: "Be able to construct a logically correct argument to establish a mathematical fact." (i.e.: constructing a proof).
Your statement packs together a lot of ideas that we have tried to break down into sub-concepts.
All of AM (abstraction) outcomes are a first step, IN5, much of the CV (critiquing and verifying) and the CCV, CCD, (communicating and collaborating). A logical argument in this breakdown is, creat an abstraction infer something beyond the immediate observation, make sure that you understand assumptions that you have made and their impact, and then explain all of that to someone else.
I wonder if those outcomes miss a vital component of being able to construct a proof?
"I wonder if those outcomes miss a vital component of being able to construct a proof?" --
Exactly the crux of my question!
Let me re-word that last line to "I beleive that the outcomes that I listed cover the key elements of proof construction do you disagree?"
One of the things that we debated at length as we worked on these was the role of proof in maths. If you are, or are headed towards, being an academic mathematician, then being able to construct proofs is the central goal of the educational path. However, I beleived, that if you look at the wider application of maths in other research, and practical problem solving, there are different levels of "making a case" that are still informed by mathematical thinking. These range from 'showing plausability', through proviiding a strong argument' with 'proof at the top. Our outcomes were trying to cover the range, rather than focus only on proof.
One need not be headed towards being an academic mathematician in order to benefit from constructing proofs. Constructing a proof is one way, of often a very good way, to be sure you really understand the precise meanings of the mathematical concepts involved. For example, for the meaning of "eigenvalue", to prove that if λ is an eigenvalue of a square matrix A, then λ2 is an eigenvalue of A2, and to give a proof without using determinants. Or to use a little epsilontics to prove that a convergent sequence is bounded. These are at the college 1st year to 2nd year level. And to prepare the way for that, at the school level, to deduce simple properties of integers or rational numbers. And to do such things carefully and precisely, and not just by computational examples or making analogies or drawing diagrams, however "convincing" such means may seem.
I'm having trouble figuring out whether "Mainstream Maths" is meant to be the same as mainstream mathematics, OR whether it is meant to replace mainstream mathematics, OR whether it is proposed as something valuable to learn along side mainstream mathematics, while being distinct from but related to it. The lead sentence of the linked page seems to claim they are the same thing. Thus the claim seems like the first alternative, but to quote IF, "something just 'smells' wrong." Basically, I don't think the outcomes, taken as "axioms" defining an undefined object, "Mainstream Maths," to cast my point in a mathematical framework -- the outcomes do not define mainstream mathematics as I understand it/them. So maybe it's the second alternative (a replacement)?
When I clicked through the linked, I was under the assumption it would list the outcomes of CBM that are beneficial over traditional math programs. Indeed, this is the sort of information I'm constantly on the look out for. What I found instead was quite a lot of information (well laid out mind you) that was more focused what a general math curriculum would be if CBM was implemented? Is my understanding of your intentions correct? I have bookmarked previously from that site a description of CBM's explicate benefits over traditional math but a running total of all the research to date might be more helpful to people like myself interested in influencing policy. People seem to resist the idea letting the computer or calculator do the computation lets students focus to a much greater extent on the math concepts. One can't even get them into the computer if they don't understand the math.
I've watched Conrad Wolfram's TED talk and also watched "Making the Case for Computer-Based Math". There is much valuable here. I think the principal message is: Shift the "plug and chug grunt work" to the computer and spend more time thinking about the subject or problem and methods.
Observing some of the examples in the videos, which flashed by pretty fast and gave no chance for any kind of analysis, gave me the impression that the material mainly consists of well designed apps and questions that are appropriate to each particular app. This is a valuable skill in its own right and that's what most people who are using math will actually be doing. The world is being filled with apps - good and bad. Designing apps, as opposed to just using them, is another skill in itself, not so easy and with many ways to go wrong.
They advocate a four step process for solving problems with mathematics. 1) Define questions, 2) Translate to maths, 3) Compute answers, 4) Interpret results. It sounds almost like a military battle plan and we know how those often work out. Then there are the "Required Outcomes", which I interpret as skills to be learned. There are 11 dimensions, each with a number of particular skills (shown in parentheses): CT(6), IF(5), DG(4), AM(5), CM(4), TM(3), MC(10), IN(5), CV(8), GM(4), CC(10). Sixty-four skills in total. Each of the skills is a worthy skill. For example CCD "Distilling or explaining ideas through written description" is in the "Communicating and Collaborating" dimension.
All to the good. But is this actually the best approach for students who want to work in the hard-core fields requiring advance application of mathematics? Is this the way real mathematicians and innovators work? Faraday and Darwin were quite poor at mathematics and would have miserably failed most of the "outcomes" but Faraday invented and visualized spatial fields that dominate modern mathematical physics and Darwin laid the foundation of all of modern biology, which now has a high mathematical content. Different people bring different skills.
How do mathematicians and other innovators operate? I would argue they do a lot of "mucking around". Most things fail. Euler shows some of his failed attempts. Gauss, along with most mathematicians, hides them. Maybe a question can't be stated precisely at the starting point. Maybe new axioms or structures are needed. Often a solution comes from bringing in something from the outside that is not initially visible at all. Sometimes a problem that seems like it should have a solution has no solution with present ideas, or maybe none ever.
In other words the world is messy and so are STEM disciplines. That makes Mathematica all the more useful! How about this approach: just give an intelligent student Mathematica and a blank notebook and let him muck around. Change the 64 outcomes in 11 dimensions into tips or suggestions. Maybe have a set of short, simple and elegant notebooks that might serve as examples. Mentor the student (using humans!)
I agree with Murray Eisenberg's comment: "Whether computer-based or not, surely one of the goals should be something like: "Be able to construct a logically correct argument to establish a mathematical fact." (i.e.: constructing a proof)." Mathematica offers a lot in constructing or understanding proofs. Even if a student is only trying to understand an established proof there is much that can be done in the way of clarification and presentation of a proof. Book proofs are often a mess and discouraging to students. How many times have you finished reading a proof with the thought: "Well, I suppose so." The various axioms and theorems used in the proof can be brought to the point where they are used. Structures might be defined for the objects in the proof and the axioms turned into rules that operate on the objects. Often the entire proof can be calculated actively. That adds a lot of credibility. Get rid of missing steps that leave the student at a loss. Diagrams can use Checkboxes to turn on or off various parts of the diagram used in various steps. There are all kinds of things that can be done with Mathematica to make proofs clearer. It is value added and the student is actively engaging with the proof.
One thing that seems to be missing in the Outcomes list is a skill at writing Mathematica routines. The outcomes all seem to be centered on understanding and using existing tools the students are handed. That's what makes me think that they are primarily handed apps and questions that funnel into the app. If that is so I think it's a serious deficiency.