## What if there was a Google for Math?

What if you could go to a free and readily available website and enter an equation, an expression, a question about math, a request to analyze data, or anything else, and the site would answer your question, elaborate on it, give you all the steps for the mathematical work, etc.?

Did that make you uneasy or excited?

Well, ready or not, it’s going online at 7pm CST today, and I think we ought to pay some attention to this.

http://www.wolframalpha.com/index.html

You can watch a screencast about Wolfram Alpha here.

It does have the potential to seriously wreak havoc on the way we teach math today if students can simply copy all their work from an A.I. website. Whether you think that it’s time that somebody forced a change, or whether you think it’s just hype and not really a threat, I think we should all be aware that after today, it exists.

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I guess we’ll have to start teaching different stuff. If we can easily get a computer to generate human-readable(and followable) solutions to symbolic integrals (for example) then do we really need to drill and kill this kind of problem into our students?

Many people say ‘yes we should’ because it was what they learned at school but some of what you and I learned at school is no longer relevant in today’s society.

I know how to do the kind of symbolic integrals expected of a current undergraduate just fine – I know most of the techniques – BUT I almost always make mistakes. Mainly becuase I am out of practice.

I get a sign wrong, I’m out by a factor of pi or I forget the anti-derivative of some bizarro function. It’ll be MUCH easier to plug it into Wolfram Alpha and check through the results.

If I needed the symbolic result of a complicated integral for a research problem then I would be an idiot to try to do it by hand first when I can try for a computer solution in a second. If the result looks nice THEN maybe I’ll try to figure it out by hand.

The REAL skill we need these days in this area (IMHO) is the ability to check that the computer hasn’t returned a load of rubbish – to not blindy rely on whatever it spits out because even Mathematica gets it wrong sometimes

http://www.walkingrandomly.com/?p=578

I am sure there are math teachers everywhere that disagree with this kind of thinking. Our students simply must be good at the mechanics of symbolic integration because we were expected to be…right?

Wrong! For the same reason that we don’t expect students to be experts at using slide rules and log tables.

I absolutely agree with the previous post. In my research, I’ve always relied on Maple or Mathematica to manipulate complicated algebraic expressions and to evaluate double integrals.However, after having given many talks to college math instructors around the country, I think our voices are in a substantial minority. While I maintain that some facility with doing math by hand is important, I think there is an over emphasis on hand calculations and drill and practice. Just look at the math sequence for dev. math, precalc and calculus. There should be a more balanced emphasis on reading and understanding concepts, and thinking of math as a whole. It is absolutely essential to have enough understanding to know that a computer generated is valid or not.

Instructors should be made aware that sites such as Wolfram’s will pretty much spit out most of the answers in most online quizzes and homework. The amount of grading weight given to these online worksheets should be rethought, at the very least.

It’s not *that* much different than the existence of TI-89 and other CAS systems, which the AP exam has already dealt with nicely for over 10 years now.

Honestly I think we should be moving past graphing calculators now into teaching students Sage et. al. Alpha might make an easy introduction to this.

I’m very excited about the statistical capability. I may be helping develop a non-AP stats course (we now are requiring 4 years of math in HS and need an option for the less mathy types other than Pre-Calc) and Alpha would be a perfect sort of tool for this. We’d likely need the pay version so we can get the actual data. Have any pricing announcements been made as of yet?

Jason,

I have no idea how much stats functionality there is in SAGE but will happily help you work it out if you let me know what you want/need.

Yeah I know we started talking about Wolfram Alpha but SAGE math excites me too

There have always been ways to cheat on assessments. I realize that and hopefully, we always adapt our grading so that “cheatable” assignments are not a major component of the grade.

There is one big difference between Wolfram Alpha and resources like CAS Calculators, Maple, etc. Wolfram Alpha will be freely available to anyone with Internet access. That, in my opinion, makes it a potential “black swan” here (a large-impact, hard-to-predict, and rare event beyond the realm of normal expectations).

What makes me uneasy is the answer to this question, “Why do I need to learn this?” Many of our U.S. math courses focus on symbolic manipulations as

partof the course (for better or for worse). There are skills that students learn by practicing algebraic manipulation and learning when to logically apply old rules to new problems. But I’m willing to guess that what most STUDENTS focus on is memorizing algorithms and spitting them back, and with the introduction of a readily available system that does this, we are going to all have to sit down and think hard about how to get students to see that math is about more than that.In 1994 and 1998, a group of researchers in Australia (Crawford, Gordon, Nicholas, and Prosser) published some excellent research on what they called “Student Conceptions of Mathematics.” Students were found to have either a “fragmented” or “cohesive” view of the discipline of mathematics. The researchers developed an index to measure this (validated and tested for reliability) and were able to show correlation to the way that students study. Students with fragmented conceptions of mathematics tended to approach their learning of mathematics at a surface level. Students with a cohesive conception of mathematics tended to approach their learning of mathematics at a deep level.

We have to find a way to change students’ conceptions of mathematics. It is, from other papers I’ve read from the Trigwell, Prosser, et al. research group, the most important factor in what students walk away with. It turns out to be even more important (in my opinion after a lot of research) than how the instructor teaches the class (research to support this too).

I could go on and on about other interesting things I’ve found from the group of Aussie and U.K. researchers … but this comment field is not really the appropriate place.

So, Wolfram Alpha makes me uneasy – it will deliver another excuse into the waiting arms of students with fragmented conceptions of mathematics … “math is not teaching them anything important if a website on the internet can do it too.” Maybe it doesn’t make you uneasy, in which case, I’m glad someone will be getting some high-quality sleep tonight.

Maybe it doesn’t make you uneasy, in which case, I’m glad someone will be getting some high-quality sleep tonight.I’m in the “excited” camp, but I’m pretty good with dealing with hybrid by-hand/by-calculator class already. My major finals all had both a no-calculator and a calculator-required part. I’ve also gotten good at convincing them they need to understand things by hand in addition to be able to use the electronic tools; I’ve got a ready set of examples designed to make their calculators go haywire.

I’ve refined my ability to ask “insight” questions (why are there an infinite number of graphs that form the same picture as sin(x)?) but I unfortunately still need help refining my students’ ability to answer them. (Suggestions appreciated! I hate having them just sit and stare.)

For what it’s worth, Wolfram has had the Integrator freely available online for a while now:

http://integrals.wolfram.com/index.jsp

Ah but the Integrator doesn’t give you the steps…only the answer.

Wolfram Alpha will give you the steps making most pre university calculus homework obsolete (and many undergraduate ones too).

I’m going to be really interested in seeing how good those steps actually are. Maple has an applet that gives you the steps for many calculus problems but it has a certain style that I think I would recognise in a heartbeat.

Back in World War Two, just before the advent of computers, the term “computer” referred to a human who performed complex mathematical calculations by hand. (The forthcoming documentary “Top Secret Rosies” explores the story of women computers in WW2, http://bit.ly/hS8tn.) Now, a $15 scientific calculator from the drug store can handle a lot of the calculations that once required trained human computers, making the skills once values in human computer obsolete, for the better in general.

What skills will Wolfram|Alpha make obsolete–for the better?

I certainly plan to talk about Wolfram Alpha to my calculus students next term. I already spend part of the first class wowing them with Maple. I would never think of giving them graded homework consisting of indefinite integrals. I do put a few of those in the exam (usually not all by themselves but as part of a more complex problem), and I give students practice lists (as well as separate hints and answers). It’s up to them to do as many or as few as they think they need.

I have colleagues who still think integral calculus should be mostly a course in how to do indefinite integrals by hand. Lots of textbook authors seem to agree.

I agree that integration techniques are worth teaching. Some indefinite integrals can be nice puzzles to develop problem-solving strategies (fuzzy pattern-matching, splitting into subproblems, backtracking), and working integrals by hand can help students strengthen their algebra skills.

Nevertheless, spending too much time on this has two pernicious effects. It means we have less time to spend on the applications of integral calculus, which is really fun (especially the part about modelizing the real world and seeing where the model and reality diverge), and it gives many students the idea that advanced math (which for them is calculus) is mostly concerned about algebraic manipulations.

The math side of Wolfram Alpha reminds me a lot of Mathematica – which it should, as it is driven by the symbolic software developed by Stephen Wolfram. I mention this because I have struggled with, sweated over and occasionally got some stunning stuff out of Mathematica for pushing on 15 years now, and if anything have found that it has challenged and stretched my understanding and use of math. It’s about as far from a crutch as you can get!

Of course, there is a danger that some will use Alpha to mindlessly provide answers that they should be working out for themselves. But my sense is that there is tremendous potential here for the symbolic math base to encourage and inspire people to develop a deeper understanding of the subject.

I think the question this raises, as will all other newer technologies, is not should there be a change in paradigm but how can it be achieved. I would love to teach my students more creative critical thinking but as all the other educators know many students are not jumping on that bandwagon. So even if we want to change, how do we convince the students to change with us? How can we make the by into the idea that they *must* be smarter then the computer/calculator?

I am glad WolframAlpha is here. I hope this will change the math instruction drastically. Here are two similar assignments.

1. Find the derivative of x^2.

Accepted answer: 2x

Students type in WA and they are done.

No learning or understanding needed.

2. Prove that the derivative of x^2 is 2x.

Accepted answer:

y = x^2 , Increase y by a small bit of y and x by a small bit of x, therefore

y + dy = (x + dx)^2, so

y + dy = x^2 + 2xdx + dx^2 … but we can discard dx^2 because it is a negligible quantity in comparison with the other terms ( a small bit of a small bit of x)

so… y + dy = x^2 + 2xdx … but y = x^2 so we can substitute and eliminate…

y + dy = y +2xdx

dy = 2xdx … dy/dx= 2x QED.

It will take some time before WolframAlpha will do exactly this. At that moment we’re in for another paradigm shift. A.I. at it’s best.

Another example:

1.Calculate (-2)*(-3)

Accepted Answer: 6

2. Why is the product of two negatives a positive?

Accepted answer:

The negative sign indicates direction with respect to the origin. The number indicates magnitude, how far from the origin. Par Example (-2)(-3) = 6 … It tells us that we are operating on the left side of the origin (going West) twice 3 units, so the new position is 6 units away West(left) of the Origin. ( Or bellow sea level)

Another Example:

1.Solve:

x^2 + 4x + 4 = 0

Accepted answer: x = -2

(no problem for WolframAlpha here…nor four your students)

2. Given ax^2 + bx + c = 0, derive the Quadratic Formula

and use it to show that the solution of x^2 + 4x + 4 =0

is x =-2 and explain why is x = -2 a double solution.

Accepted Answer:

…well you know what I mean.

Another Example:

Explain why the Area of a Circle is πR^2, or any of the Areas for that matter. These types of problems are not solvable with WolframAlpha.

So… Maria, you should create a page where we can post types of problems that WolframAlpha would not be able to solve it. This is the Revolution in Math we’ve all been waiting for.

@Alison Not only the students. None of us teach in a vacuum. Personally, I would consider making some major changes to my courses to live in the new WA-world, but … those students then go on to some other instructor, or possibly some other school after me. If they have not adjusted too, then have I done my students a service or a disservice by changing?

There’s an interesting study that was done regarding the effect on students when they had to switch from a traditional precalc to a reform calc, or a reform precalc to a traditional calc. I am going to go dig that study up.