SL2RL-Math247

22 February 2009

Goodie of the Month – Fun and Learning with Quadratics

Filed under: applets,education,ICT,math — Tags: , , , , , — admin @ 3:23 am

Goodie: “A technique/question that can be applied in many places and teaches thinking.”

Baddie: “A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

February 2009 Goodie of the Month – Real Fun and Learning with Quadratics

In the second half of Algebra 1:

A typical standard is: Apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

A typical question for this is: A ball is thrown straight down with a speed of 20 [ft/s] from a height of 80 [ft]. When will it hit the ground?

A typical application of technology is:

Tell the student that the function is y(t)=-16t2-20t+80. They know they can’t graph with t so they switch to x, which they graph on their graphing calculator.

They see that parabola crosses the x-axis. They find the intersection and write x=1.7 and get their points.

Now ask them "Where does the ball hit the ground?". They will point to the intersection point – totally forgetting that this is vertical motion and that the ball hits the ground at (0,0)!

Ask them "What is the units on your answer?". You will be lucky if they give you [seconds] and not [feet]!

So why is this a “Goodie of the Month”?

The problem isn’t the standard. Nor is it the question. Both are excellent. The problem is the technology – it is undoing the learning.

Let’s change the technology!  If the animation below doesn’t work – open this link: Vertical Motion

 

 

 

 

 

 

 

 

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Here are some “good problems”.

  • Set ho = 80[ft] and vo = -20[ft/s]. Run the animation. Point with your finger to the place where the ball hit the ground. Now find the place on the graph where it says when it hit the ground.
  • Set ho = 80[ft] and vo= 20[ft/s]. Run the animation. Notice that the ball goes up before it goes down. Why is this? Reset the animation and using the step forward + and step backward – buttons, stop the animation when the ball is at its highest point. Point with your finger to the place where the ball is at its highest point. Now find the place on the graph where it says when it is at its peak. What time is this?
  • Set ho = 0[ft] and vo= 100[ft/s]. Find when the ball hits the ground. Do this via the animation and algebraically using the function. When is the ball at its highest point (remember – parabolas are symmetric!)? What is this highest point? Do not forget units!

Here are some “good questions” for the function: h(t)=ho+vot-16t2

  • The function h(t) gives height in [ft]. So each member of this function must give [ft].

    • ho is (initial) height. So its unit is [ft]. It is all by itself so this member is in [ft].
    • vo is (initial) velocity. So its unit is [ft/s]. How does this member give [ft]?
    • What do you think the unit of “16” is so that this last member gives [ft]?
  • In what part of the function is gravity playing a part? In which of the above problems is the only force gravity?
  • Why do you think there is a plus sign in front of vo and a minus sign in front of 16? That is, what does it mean in mathematics/physics for an object to have a positive velocity? Does gravity increase this velocity?
  • Make up a problem that describes this situation: ho = 0[ft] and vo= 100[ft/s].

Links for this interActivity (worksheets, downloads, etc.): Open Metadata

25 January 2009

Goodie of the Month – A Good Question for Algebra 1

Filed under: algebra,applets,education,ICT,math — Tags: , , , , — admin @ 2:01 am

Goodie: “A technique/question that can be applied in many places and teaches thinking.”

Baddie: “A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

January 2009 Goodie of the Month – A Good Question for Algebra 1

Two ships are sailing in the fog and are being monitored by tracing equipment. As they come into the observer’s rectangular radar screen, one ship, the Rusty Tube, is at a point 900 mm to the right of the bottom left corner of the radar screen along the lower edge. The other ship, the Bucket of Bolts, is located at a point 100 mm above the lower left corner of that screen. One minute later, both ships’ positions have changed. The Rusty Tube has moved to a position on the screen 3 mm left and 2 mm above its previous position on the radar screen. Meanwhile, the Bucket of Bolts has moved to a position 4 mm right and 1 mm above its previous location on that screen.
Assume that both ships continue to move at a constant speed on their respective linear courses. Using graphs and equations, find out if the two ship will collide.

Why do I like this question?

  • Students can understand it and it is fun.
  • They can graph it on paper or using a graphing program.
  • It involves finding the equation of a line through 2 points (twice) – good reinforcement.

Why do I think it is a “good question”?

  1. The graph looks like every 2×2 system of linear equations they have solved in Algebra 1.
    • It looks like the boats collide at the intersection point (see below).
    • It seems like all they need to do is solve the system and be done.
    • … until you say “Where is time on the graph?”.
  2. The student can build a animated simulator that “shows time” – easily!
  3. The kids can make the boats collide – what fun!.
    • They can move the starting points until they get the boats to collide.
    • They can also adapt the simulator so that they can change the slopes and get the boats to collide. Directions here.
    • My thanks to David Cox for seeing this!
  4. You can get all kinds of mathematics out of them.
    • You can get them to calculate when each of the boats reaches the intersection point in the original question.
    • You can get them to check the math on their “colliding simulator” to see if the boats really do collide, where and when.
    • You can ask them about a 3D graph and what this would look like when the boats don’t collide and when they do.

To animate, click on the play button at bottom left of graph.

To animate manually, right-click on slider and deselect “Animation on”. Then, click and drag the point on the slider.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Source: I found this question asked on answers.yahoo.com. My webpage for this question is: mathcasts.org/mtwiki/Gq/BoatCollide

12 September 2008

Race Car Activity – Exploring Slope and Intercepts in the Real World

Filed under: applets,education,ICT,math — Tags: — admin @ 10:23 pm

Click and drag the slider points to adjust the cars speeds and positions. Then use the animation buttons.
(The animation buttons may no longer work because there are multiple animated pages on this blog. If they don’t, please go to the webpage – they will work there.)

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Here is the webpage: http://mathcasts.org/mtwiki/Activity/CarRace
My thanks to Jon Ingram for showing me how to do this!   See how! (after September 15)

30 August 2008

GeoGebra on a Blog

Filed under: education,ICT,math — Tags: — admin @ 8:18 am

Successfully embedding a geogebra applet!

You should be able to click and drag the blue point. Please let me know if you have any problems!

Geogebra applet (enable Java to see it).

My thanks to Jon Ingram for showing me how to do this!
To see how to do this, go to: http://mathcasts.org/mtwiki/GgbHelp/Embed
(after September 1)

16 March 2008

Vindicated – How sweet the taste of these grapes :)

Filed under: education,ICT,math — admin @ 3:53 am

In today’s New York Times – I saw a report: Report Urges Changes in Teaching Math

The report: The Final Report of the National Mathematics Advisory Panel

And we had just finished our paper the day before that made so many of the same points (also has some concrete ideas for solutions): ICT in Math Education – Small Interactivities & Specific Goals

How sweet the taste – after so many sour grapes this month!

————

Someone asked me what I meant about lighting the fire and filling the bucket in mathematics education …

In the US, math education has swung completely towards a “lighting the fire mentality” . For example, one doesn’t need to know the multiplication tables by heart. (I really didn’t think anybody could believe this until – no kidding – a university professor of ICT actually said this outloud at a conference: “It is enough to know that 6×7 is somewhere between 30-something and …”. By this time I was laughing so hard I didn’t hear the rest of the sentence.)

On the other hand, here where I live in Europe, the emphasis is on “filling the bucket”. For example, a child must memorize a formula for the volume of a cut-off pyramid. That is, not just memorize the formula for the volume of a pyramid, but also a formula that (a) you will never, ever need and (b) makes our students dummies. Why? Because the top of a pyramid is just another pyramid. So just use the one formula twice and subtract. Let’s just say to these kids – “no thinking permitted”.

The result is the same – neither child can “make the coffee”, i.e. do math at any level of competence AND both hate math.

One interesting feature is - both schools reward lots of “partial credit” on tests. How many bosses do you know reward “partially done jobs”?

And – if you think this problem is only in math:

My child here in Europe – had to know that in the population census in our country in 1935 there were 532,381 people – that is, an exact number for a definitively inexact figure.

On the other hand, in my US education, I am still unsure of even the relative positions in time of Shakespeare, the protestant revolution, the renaissance, Napoleon, the middle ages, Columbus, Walt Whitman, the US Civil War, etc.

2 March 2008

It’s about making coffee – Light the fire & Fill the bucket

Filed under: education,ICT,math — admin @ 1:09 am

I was just looking at the byline on an email I received from a “Professor of Education”.

It said – It’s about lighting a fire, not filling a bucket.

No it’s not.

“It” isn’t about either of these things separately.

“It” is about being able to make the coffee.

29 February 2008

Sour Grapes or Sour Grants: Judging the Judges

Filed under: education,ICT,SL — admin @ 1:50 pm

There was a contest. (It has “winners” – so that makes it a contest.) It was called DML – Digital Media and Learning Competition or HASTAC initiative, and it was supported by MacArthur Foundation (which does many cool things – I checked!) It was supposedly for innovators who have new ideas to improve the education of our young people. Money for the little people with big ideas.

But that wasn’t so. What they really wanted was to brag that they had over 1000 applications for <20 grants and then change the rules and give the grants to their cronies in higher education. Not grants for innovators – just typical university grants. Say it up front. Don’t lie.

Let’s look at what they said they wanted.

As early research begins to illuminate changes occurring in young people who are “growing up digital,” the time is ripe to translate this research into concrete designs for new kinds of learning environments. …

Innovation Awards support pioneers who are exploring new digital models of learning that build upon and enhance the informal, networked, and collaborative styles today, especially but not only among youth. These projects demonstrate new modes of learning in many environments.

Now look below at the judges qualifications*. We don’t need names – this information alone would have been enough to tell us the story.

Look carefully. There are no teachers* - incredible isn’t it. You want experts? Where are the “teachers of the year”? The experts in the field – the ones actually working with kids and teens. NOT ONE! Look again. There is no one from math or economics or physical sciences. Jeez-louis, how is this possible? Why don’t we just stop teaching this stuff in our schools if it isn’t important.

Look again. There are 7 judges who develop and design computer games. That is, 7 out of 35. Unlike so many of our young people, I am able to do math so I will translate: 20% of the judges are gamers. Look again. There are 23 judges who are either university professors or directly related to higher education. Again I will translate (okay, I did it with a calculator): 65% of the judges are in higher education. The other 15% – OMG as we say in SL.

Now it is reasonably easy to deduce what they are after – others need not apply. NP. Just tell the truth.

And then, the unmitigated gall to blog jovially about the 980 who didn’t get a grant applying again – like our time, our dreams and our real innovations for educating youth are games. Absolutely disgusting.

* The initial 35 judges. I won’t even bother with the finalists judges.
** I myself am a university professor and teacher, but you know EXACTLY what I mean! (BTW: not the PI on the proposal.)

The Judges:

University Professors

1.Professor of African American Studies, American Studies and Sociology
2.Professor of Anthropology
3.Professor of Asian American Studies
4.Professor of Behavioral Sciences
5.Professor of Cinematic Arts
6.Professor of Communication
7.Professor of Composition and Computer Music
8.Professor of Early Adolescent Literacy
9.Professor of Education
10.Professor of Film and Media Studies
11.Professor of English
12.Professor of English
13.Professor of Literacy
14.Professor of Media Studies
15.Professor of Political Science
16.Professor of Sociology
17.Professor of Teaching and Learning
18.Professor at Center for Science, Technology and Society
19.Professor at School of Information (social aspects of building and using digital archives)

Directly connected to higher education:

20.Director of CADE, University of Illinois at Chicago
21.Senior Advisor, Ithaka
22.Project Manager, Ithaka (quote from Ithaka website: “Ithaka promotes innovation in higher education”)
23.Executive Director, Coalition for Networked Information (quote from CNI website: dedicated to … scholarly communication … intellectual productivity

Gamers

24.Creative Development Editor of Fantasy Flight Games
25.Freelance Game Designer and Writer
26.Game Designer and Writer
27.Founder Storytron – computer games
28.Online Operations, Lucasfilms Ltd.
29.Freelance Writer and Game Developer
30.Game Developer and Designer

Other

31.Founder and CEO, Viewpoints Network (quote from website: “be the envy of your fellow beauty junkies”)
32.Co-Founder, Fundable Group, Inc. (motto: We Help You Collect Money Online)
33.Assistant Managing Editor, Yahoo! News
34.Program Director, Social Science Research Council
35.Program Director, Social Science Research Council (quote from SSRC website: “to advance understanding of critical social issues”)

2 January 2008

Facts+Skills+Thinking: Applets and the Vitruvian Learner

Filed under: applets,education,ICT,math — Tags: , , , , — admin @ 12:44 am

We want our learners to know facts, to have skills and to be able to think – a perfectly proportioned Vitruvian Learner.

I am preparing a new course on ICT* in Education. Most of the literature on this is depressing – no kidding. Everyone is still so enthusiastic about the future of ICT in education, but most admit that current ICT integration has brought no significant breakthroughs in student learning. ARGH!

At the same time, I was just designing a set of math applets** for measuring angles with a protractor (link). This turns out to be an exercise for which there are many, many, many such programs available. ARGH!

Of course, I think my applets are different – they will bring the breakthroughs. But why should they? For the moment, let’s assume that magically my applets are loved and are being used in every classroom in the land …

What about these applets might bring a breakthrough in student learning?

To be effective:

The applet must be a perfect proportion of facts, skills and logical thinking processes both required to use and acquired from use and

• this required perfect proportion must apply to a large number of learners.

• this acquired perfect proportion must apply to a large number of curriculum.

One easy conclusion – the smaller and more focused the application – the better chance it has of meeting these requirements. Hence, applets and not mega applications probably are more practical.

Even more essential is communication with teachers because a teacher will not use ICT unless it can be used within the curriculum, i.e. within the framework of “this is what we know so far and this is our goal in this lesson”.

So – just from the point of view of bringing a breakthrough in learning – and before designing an applet, we must ask.

• What are the facts one needs to know to use and what facts will be learned?

• What are the skills one needs to know to use and what skills will be learned?

• What types of logical thinking process are required? What thinking processes will be learned?

Then: Do these answers fit a broad spectrum of learners and curriculum?

Yes? On to the next checklist …

* ICT: Information and Communication Technology, sometimes just IT.

** applet: A little application frequently run from a browser window with limited features that requires limited memory resources.

27 December 2007

More than “What kinda map ya want?”

Filed under: education,ICT — admin @ 1:49 am

We all know the story about the kid who couldn’t find Iraq on any map, but when asked to find Iraq on the Internet, the kid said “What kinda map ya want?”. Oh cool – our kids are so smart. But is this the end of the story?

  • Is it enough for our kids to be able to say “What kinda map ya want?” ?

I think not.

The right idea: As teachers, parents and citizens we complain our kids can’t find Iraq on the map. Now, we are told that this is the wrong quiz question for this generation and – in fact – the wrong quiz question for any generation. I can find countries on a map; I can drag out my 35 year-old NG atlas and probably give you alot of information from it without even looking. But I will have to be very, very lucky if I can answer a real question with that information.

  • Eureka! What we really need is all kinds of maps at our finger-tips so that we can find the right map that answers the question.

The (not-quite-whole) story: Just yesterday my son needed to use the formula for the chemical process of a car battery. He could not understand the explanation in his textbook. It took us 5 minutes to find it online; we found several references so that he was able to confirm the formula (different from the incorrect/incomplete one in his textbook). Then he looked through the different explanations, printed out the one he could best understand, reread it, annotated it and started successfully solving the problems.

  • Eureka! What we really need is a new education. Our kids will be able to quickly find all kinds of maps.

The (not-quite-right) conclusion:

Our kids can find any map – hence they are learning the right stuff.

Bad deduction – just the bad kind of mathematics I hate. No thinking involved!

The real story: Notice the “it took us 5 minutes to find it online”. There is a whole story here (and many wasted hours and frustration) before we got to this point and it was me – the old lady teacher, with no knowledge at all about chemistry – that was able to get the right stuff and show him how to use it.

Why? Because whatever and however I got it and whether it is library book or internet based -

  • My education taught me the thinking-steps to use to get me with confidence from question to answer.

Neither question: “Where on the map?” and “What kinda map ya want?” is the right kind of education.

  • What we really want is to hear our kids say: “Yes! Here’s the map I need!”

These blogs – my SL2RL journey – is to explore specific, step-by-step, ready-to-use, free stuff using technology that can help get our children get to this “Yes!”.

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