Posts filed under ‘Technology in the classroom’

Relax… and watch the kids shoot some squid or crash a car

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THREE weeks from the end of the semester I decided to bring back the laptops for my three Algebra 2 classes and see where we would go. We would go further than I could possibly have hoped. And it would be me who would learn the most… the more the teacher pulls back, the further the kids go on their own.

The students would be using Geometry Expressions, a symbolic geometry system that reveals the geometry behind algebra and the the algebra behind geometry¹. To get us going we would be using problems from Geometry Expressions activities for Algebra 2 and Pre-calculus, written by high school math teachers Tim Brown an Jim Wiechmann.

For a start, I photocopied some of Tim’s and Jim’s problems and hints on parametric functions which took the students through the basic steps of using Gx and launched in. I could model the steps using Geometry Expressions on the interactive whiteboard.

First impressions were positive as the first lesson logs record:

pirate_ship “Positive response to the program from students (girls + boys). Most problems around familiarity with the program, but no complaints about impossibility… Several said liked the way it draws straight lines, can see numbers, etc. (p2)”

“Three girl students who normally do nothing working well on task and asking mathematical questions… Higher level of engagement than normal for this class… Girls who are rarely engage in the lesson comment, “it’s fun… we should use it more. (p4)”

But not entirely: “This class (p5) has high rate of deliberate helplessness and unwillingness to figure stuff out. Ditto here, many students get stuck, do little to get unstuck and then blame me. It’s par for the course… Least successful class to date… as expected… Need re-think for next class.” Classic grumpy teacher response. It’s the kids fault! Bad teacher!

And so it continued for several lessons. Some ups and quite a few downs. The lesson logs record:

“The first part of the investigation seems to be useful and keeps most students absorbed. (p2)”

“TAG boy resists experimentation… Explain the aim is to get him to push assignment to one side and to take over under his own initiative and inquiry. Try to explain a method of changing one thing at a time in the equations and spotting/writing the change. Look for pattern. Explain he’s used to being more spoon fed… this will seem slower, but will be more effective. He does not look convinced. (p4)”

“Better than usual focus (still not brilliant). (p5)”

But I’m still nervous. I’m off the curriculum, ignoring the upcoming finals and can see a lot of students being much better at toggling in/out of their web browser as I walk the room (we need internet access for the students to logon and the laptops to work). “Move students on… seems to indicate that need to put in more traditional lesson somewhere to check some links are established. I seem to be more phased by this than the students. (p5)”

Focus on trying to lighten up the atmosphere and take one problem very slow. Enter the powder-puff kicker trying to kick a 33-yard field goal over a 10-foot high goal.

For starters, we looked at investigating how to find one possible path for our powder-puff kicker’s shot. We decided the path would be a parabola, a quadratic.

Then by guess and check we started to use Geometry Expressions to find some possible paths.

❏ First, the parabola opens upwards, which does not fit the problem. Why?
❏ Then how to get it to start at the origin (where we had drawn a powder-puff kicker using the interactive whiteboard ability to draw over the Geometry Expressions projection).
❏ Where might the ball land if the kick had cleared the goal? Would picking a landing spot help us find a parabola and its quadratic function? Why? Is there more than one?
❏ But the highest point of the ball’s flight seemed to be at more than 2,000 feet! What would give us a reasonable curve?

Most of the classes — working in pairs — get curves that show how the ball could fly over the goal.

This gave a good review of quardratics, and particularly how we could use the kick spot and the landing spot for the x-intercepts, the zeros, and find a function in factor form.

But we still don’t know how the speed of the ball at kick-off or the angle of the kick affects the curve.

The lesson logs: “Apart from a few rough edges and reluctance of students to actively participate in much discussion, lesson feels good. Most pairs follow along and are still working using Gx at end of lesson. (p2)”

Well, most of the kids did 70 minutes of work. But, notice, they followed along, they were reluctant to participate. I ran the lesson, not the kids.

Same exercise for the next class, but with the aim: loosen up teacher-man.

“Explain we’re going to be looking at parametric equations, but we’re not likely to get there this lesson, it’ll take two. I like the idea of encouraging students to think that we’re not in a rush to get an answer, that the process is more open-ended… Second time around, much smoother. Also class more actively engaged in discussion back and forth. Keep almost all students actively engaged and on task for all lesson… I found it interesting to teach by finding the curve that fits the problem with guided guess and check approach. More students found the final curve, more quickly and more independently in this lesson.”

And I’m learning from lesson to lesson:

“Spoonfed up to this point, but with input from class,” records the p5 log. “Now decide to leave and not intervene… ‘it doesn’t work’ insists best student in class… ‘we found out what we were doing wrong’… Announce remember we’re trying to do it like mathematicians have done it through history… a sort of guess and check&hellip ‘We got it… awesome’.”

Two steps forward, one step back: “Summarize where we were at… try to get responses. Little comes back… Front of room led lesson, spoon feeding all the way… but attempting to get feedback at every step.” The lesson moves on to open up the ideas of parametric functions.

It depressingly concludes: “As a didactic lesson ok. As a discovery lesson, useless. First lesson of the day always an issue with sleepiness, etc. Difficult to motivate students. (p2)”

One class goes out, the next comes in. Teacher is starting to think:

“Smoother and more straightforward explanation of trig relationship to the problem… More feedback from students and most of lesson achieved in shorter time… Tomorrow will try again, but will ponder on how to move in direction of less didactics and more student discovery. One key might be to slow the process and simply give less… the fear here is that the students simply don’t respond but just sit and talk! The issue-elephant in the room is that students who are used to being spoon-fed, see math education as getting an answer and who are not expected to be engaged on their own behalf are slow to rise to the challenge… My challenge, next year, is to work out how to build a new culture in the classroom specifically aimed at teaching the kids how to learn by themselves for their own sakes.” That’s what I thought I’d been trying to do this year.

Hands off with p5, “encouraging them to explore” using a table to record their results.

“The students in this class (p5) who are normally willing to engage, did so in this lesson. There was some engagement by those who are normally resistant to the lesson. I wandered around the room on several occasions and asked if students needed help and most said they were ok… Unfortunately could not get much for the table of results… Asked one of the more engaged students how it felt just being left to get on with it… ‘it was fun… I knew what to do and you gave us support… it was ok’.”

It’s getting close to the semester’s end and finals week. p2 “sleepy and non-responsive.” We get into a muddle with degrees and radians which throws off the results: “One of those problems that gives students the excuse to stop,” notes the lesson log.

I’m becoming resigned to spending the summer thinking about what went wrong. Last day for our seniors, the sun is out, carnival atmosphere, so try to keep things light. Let’s try to get some pirates to shoot a cannon ball at a giant squid: “What do we do first? What did we do the other day? Draw a silly picture. I want to see lots of pirates and squids… and a bit of math thinking.” A coloring-in lesson to get us through the 80 minutes!

After two weeks, some pretty drawings. “I hit enter and the point just disappeared,” says the most engaged girl in class with apparent irritation. Another frustrated girl: “The point just kept going up and down.”

Next class (p4)… “I want lots of drawings” I say with resignation. After ten minutes I decide to walk the room, for the sake of form, expecting I know not what, but not a lot. Here’s the lesson log (the numbers represent times, TAG = talented and gifted):

09:55 TAG girl trying to reconstruct a Quadratic to plot the curve. Two TAG boys appear to have drawn a giant quid in the computer (at the correct coordinate) plus a pirate ship complete with sails and a cannon (at sea level!). I go up to suggest they now… when they fire off two cannon balls complete with parametric coordinates, both of which directly hit the squid! I point out the cannon balls are traveling at different speeds… also why not put the cannon on the deck of the pirate ship?

10:00 TAG boys have shifted their cannon and can still shoot the squid.

A plethora of pirate ships drawn in Gx now appearing on laptop screens. TAG girls importing pictures of pirate ships off the internet and placing inside Gx.

10:05 TAG girls who plotted quadratic have stopped work… explain the need for them to drive a point using parametric functions as the coords.

10:17 Lots of experimentation now going on… one boy suggests putting less or more powder into the cannon to change the velocity.

TAG girls wrestling with working out the parametic functions in the coordinates… what’s theta?  Why t? “We’ve forgot”

TAG girls(2) have animated Jack Sparrow firing the cannon… alas not so much math

Remind need to find two angles.

10:20 TAG boys firing salvos from a deck full of cannons, with many seemingly many direct hits… “we’re trying to fix them, but now we’re lost” they say laughing!

TAG girls… “we’ve got it animated, but the squid is shooting back! It’s going the wrong way”. Encourage and help to examine the functions… they’ve added the distance 1000m inside the x-function

TAG(2) girls are working on the math… trying to get the cannon ball to come out of the cannon which is on the poop deck along with an animated Jack Sparrow! Discuss how they could do this.

TAG boy (loner) struggling to understand how cos and sin work and why we use them. Brief lesson on trig with some scribbled cannons and right triangles.

09:38 (lesson ends 09:40) some students still working to complete and email to me.

I am just flabbergasted! Unbelievable! I’m bouncing around the ceiling!

At the next lesson (p4): “Class keen to see its work projected from last lesson. Lots of pride. Show a range of the work.”

We move onto plotting moving cars and finding whether or not they will crash.

09:50 TAG boys solve, they don’t collide. So… how do you make theme collide? Change speed car A. What about car B? What if a car starts earlier or later? What if a car takes a different route?

TAG girl takes time to remember the variable that will change is t… we have 2-minute dialogue where discuss everything about a car journey with kids in the back seat moaning till she hit on time is what makes them impatient. She gets the coord change +45*t immediately. Changes car A coord v quick too. She checks in she has it correct… she needs lots of validation, even tho v smart.

TAG who struggled last class, gets it quick this time.

09:57 TAG boys “they crash” Now what do you want us to do? I want you to decide how to change the problem… change time, direction.

10:05 couple of pairs need support to change the coordinate. Need to emphasise idea of time driving x and y. Up till now x has always driven y, now t for time drives x and y

TAG boy (loner) completes last lesson exercise showing the two cannon balls. The target (the eye) on the squid is clearly marked in red, points labelled, explanation included. Meticulous completion of the task, with reluctance to move on till completion. Moves on to new problem. (So, why are tests timed? What are we testing? People work at different speeds to their own different standards)

Pix of cars off internet being put on screen.

10:23 TAG girl creates a car out of points and drives the car.

TAG boys make four cars travel diagonally till all hit pedestrian in the center of the screen.

TAG boy draws on-screen map, has a miss and a collision, carefully color-coded and labelled. Other students build the Starwars battle.

10:28 TAG girl now has two cars built of points that collide… suggest she changes the nature of the crash… full on, glance, 

10:32 TAG girl 2 crashes five cars… boys sneer, they’re going too slow. I can make them go faster responds girl.

Lower ability girl, who is rarely engaged, continues to work with furrowed brow trying to change the coord to get cars to meet. Ask if she’s ok and get told she’s fine.

New boy knows exact speed change. Can run the actual collision in slow-mo back and forth. Suggest he changes time of start of one of the cars. He realizes straight away that he needs to change t in some way… he multiplies it by 1/2… hence making the car travel at half the speed… suggest he needs to think about just making the car start one hour later… he gets (t-1)

10:40 TAG boys getting 8 points to crash into one point… no longer talking cars talking “points” More serious in tone and attitude.

All students achieve basic aims of the investigation with many going much further. Solid engagement in a fun atmosphere for 70 minutes.

TAG girl reports the exercise on her Facebook page… suggest she posts the parametric coordinates. “I will,” she says.

10:47 TAG boys now have 12 cars crashing.

The other classes seem to pick up too. The log for the p2 lesson on cars crashing records:

The concept of miles per hour times t gives a distance not much of a problem, but need a little pushing to go there. Class not prepared to ask itself questions, but will respond to questions asked.

08:45 Model on board to make sure everyone roughly at the same spot. Class engaged + feedback is flowing freely. Working in pairs v cooperatively. Brief discussion about nature of time, no wrist watches any more (kids all show their left wrist), clocks used to tick, the only sound when no cars, etc… Someone asks how did old fashioned clocks work…

Suggest consider changing the problem. How do we ensure they meet? What if one car starts at a different time? What if a car decides to drive a different direction. Talk briefly about how math developed… by changing the problem and asking what if?

09:01 class silent and completely engaged.

J (struggles with math, but tries really hard, concepts v slow to hit home) Shows me how he has got the cars to collide, what he has changed, how he has changed the coordinates + how he has adjusted the speed of the animation so that he can see the collision clearly plus he has put a point on the screen to show the point of collision.

TAG girl raises the fact the car is going in the wrong direction… girl who has not attended class for three months(!) suggests need a negative.

On to the “final”. Ditch the common assessment and go for a Gx-based investigation: a sister/brother is using a hose to water a basket of flowers hanging above brother/sister who is sleeping in the sun underneath the hanging basket. What parametric coordinates get the water into the basket, without drenching the dozing brother/sister… or, drench the sibling and miss the basket? I add: How can you then change the problem?

p2 the first to go. We hit difficulties because I’ve misjudged some of the numbers. But now I’m not the teacher just someone in the classroom so I don’t get phased. Fixing the numbers just becomes part of the problem. The log records:

Despite the difficulties all class sticks with the problem for allotted time of 65 minutes.

Interesting… recall by students is a lot less than teachers would like to imagine. Even TAG students need a lot of reminders even though we’ve been doing this sort of problem for two weeks. Each time they catch on quicker and some can recall bits well, but few can recall all aspects. Taken back to traditional teaching this means that few students probably remembering v little in a sense that includes understanding.

The log for p5 records:

This class not able to complete this investigation without considerable amount of support, however all pairs remained on task for allotted time of 70 minutes with most completing the basic investigation. Two pairs (working together) managed to work out how to generate a stream of drops of water by adjusting the variable t… (t + 0.1).

Although this class has been working on this type of problem for several weeks using Gx, retention is an issue. The big success with this class is that students who adamantly refuse to engage with the subject, (now) do so with enthusiasm and determination when using Gx.

The log for p4:

Students willing to go further. All round good engagement. Even those who struggle, ask questions and move forward slowly.

Pair work seems to be accepted as beneficial, even where several students sit together each with a laptop!

This class v keen to draw a pic of the problem inside Gx, including placing artwork culled from the internet. All pairs manage to get to a solution where the water does not hit the  basket, but drops from above into the basket. Many plot a stream directly at the person asleep under the basket.

Explain that the parametric point represents the path taken by just one drop of water. Recount the example of the park in France where water is fired in short bursts over the pathways into barrels on either side of the pathways… bizaar effect of short isolated strings of water flying along a perfect parabola through the air. Suggest as inspiration of how to take the problem forward… how to fire a stream of drops one after the other… need to adjust t… is it t + 0.1… t – 0.1?

One student picks this up and works hard to get it to work. She does so.

One student asks why we would want to expand the problems once got an answer… explain idea that math developed through this process (heuristics).

Good end to the year.

¹ The development of Geometry Expressions is funded in part by grants from the National Science Foundation. The author sits on the NSF committee monitoring the project. The logs are being used as part of an NSF-funded research project un by Oregon State University looking into issues raised by using technology in math classroom.

July 11, 2009 at 3:25 am Leave a comment

We should be teaching mathematical thinking

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The use of IT in the math classroom raises implications for pedagogy in the math classroom.

This brief discusses the purpose of math education and tests the arguments for thinking more in terms of teaching creative math thinking skills instead of the current practice of teaching a series of mostly algorithmic skills. In particular the brief argues for a new form of assessment reflecting students attempts at math thinking as opposed to an ability to demonstrate facility with applying algorithms.

The brief concludes that such changes could mean more students feel success in math, grow to enjoy their math and attain more useful math skills — math thinking skills.

+++

WHAT’S the point of teaching and learning math?

For some time now, I have increasingly felt that a major issue in math education is that math teachers — plus those that create and administer the structures within which math teachers work — are not clear in giving good reasons to students about why it is so essential to study mathematics, exactly what is it we are all trying to achieve.

Current course structures, strict curriculum, standards, the emphasis on testing… all do not help teachers to reflect on or explain the purpose of what they do. Probably a majority of students, but certainly significant numbers, remain confused&hellp; and uninspired.

Part of the problem arises from the fact that just about all involved — except the students and the parents — are at least reasonably good at math, they get it, they enjoy it, they value it for its own sake, they see its value in the wider world. Few will have experienced the debilitating confusion, demoralization, despair that is the lot of substantial numbers of their students.

They see math education as turning out kids like themselves, mathematicians or one sort or another. This is fine when they encounter kids who get it, enjoy it, value it, etc.

This approach is compounded by political forces, which can include some mathematicians and their associations such as the NCTM, which view math education as little more than a utilitarian function at the service of corporate America.

For this latter group in particular, the maximum of math teaching is the delivery of a numerate workforce. And, it should be acknowledged, that is an aim of math teaching.

However…

There are swathes of young people, children, who do not get it, do not enjoy it and do not see any value in their math education. Yet many of these young people may, indeed, have math thinking abilities. They will certainly need math thinking skills, math understanding, in the fast-changing world in which they will live.

Crudely put, these children are not always well served by mathematician math teachers. Or, are not well served by mathematician math teachers who do not reflect wider on issues of pedagogy or wider (social, political and philosophical) concerns about education in its widest sense.

One dire result is that math education in the public schools is often confined and restricted to training children to be numerate, with little more.

The emphasis on standards, state testing and curricula driven by textbook adoptions, militate against wider reflection. Math teachers simply have too much on their plates to reflect… or they do reflect, and then knuckle down.

Students themselves give the clue to solving the dilemma when they ask — and they always do — “when am I going to use this in real life?”

The frank answer is most are not going to use much of it at all, and certainly not the math that gets tested in multiple-choice computerized testing. One realistic answer to the question is that math education is attempting to teach abstract thinking skills, or problem-solving skills.

But that doesn’t go far enough. It’s not just about what and why students are learning&heliip; but also how they are learning. Indeed, the how can really be the embodiment of the what and the why.

Focussing on how children learn math may be the answer to the student-question. If the experience in the classroom is totally focussed on math thinking — with the student feeling in meaningful control of the progress, mentored Vygotsky-style within the zone of proximal development by adult guidance and peer collaboration — then the teaching pedagogy itself may give real meaning, be itself the explanation of what math education is all about.

And if the assessment supports this approach, helps guide it forward, focussing on helping the student to find their own thinking skills — rather than seeking to reward or punish — the student will not so much be learning math thinking skills but experiencing using math thinking skills.

The how becomes the what and the why.

+++

The how does not include didactics, the pressure of tests and quizzes, points, grades.

The how does include peer collaboration, teacher as mentor, student control… and time for the child to play, think and work out their own solutions.

That can be done without technology. But there is no doubt that technology can really aid the approach, providing the classroom atmosphere and activities allow the child to get lost in their thought, explorations and discussions. Indeed, the creative use of technology almost demands a new approach to pedagogy in the math classroom.

It also demands a new approach to assessment…. still working on this!

July 4, 2009 at 11:50 am 1 comment

Avis foregoes the restroom and expresses some geometry

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AVIS was in my Geometry class this year. Or, occasionally he was in my Geometry class. For a few minutes at a time.

Avis is not disruptive in class, as much as he simply leaves class. He’s frequently found just wandering the halls.

He has numerous accommodations to account for his behavior. Some teachers respond with referrals, but I’ve decided little is achieved by repeated referrals when the aim is learning.

In my class Avis asks — insistently — to go to the bathroom. Again and again. On his return it can be minutes before the demand is repeated. If I explain he has just been, he disappears from the room the moment my attention is elsewhere.

And when he is in class, it is hard for him to sit in one place. Avis doesn’t do much geometry, though he will turn in some work if he can sit next to an accommodating pretty girl willing to help.

Earlier in the year the class had used laptops to do various constructions and investigations using Geometer’s Sketchpad. The exercises had not been as successful as hoped, more likely because of the didactic approach I had adopted, making the investigations quite formal.

But Avis’s attention had been grabbed, and the frequency of his bathroom visits slowed. Avis had spent much of his time playing, using the program to draw pictures. As play is a positive start to learning, this seemed a welcome step.

At the end of the year I decided try again with the laptops, using a different program, Geometry Expressions, and adopting a looser teaching style.

Geometry Expressions ¹ is a geometry-algebra system. Algebraic notation is used to “constrain”, or fix, the drawings. The outcomes can also be expressed as algebra. The focus of the mathematical thinking involved is different to that used in attempting to do traditional straight-edge and compass geometric constructions.

If you constrain a triangle side-angle-side, and then try to add a second triangle using the third, unconstrained, side Gx forces some interesting thinking if you try to impose constraints on the second triangle.

Teachers and academics who have worked with Gx readily acknowledge its power for older or higher-level students — my own AP Calc class produced some lovely work based on animating hypocycloids after their exam.

The question has been could the progam help younger or lower-ability children with their mathematical thinking? Avis was not the student I had most in mind when we tried our first exercises using Geometry Expressions.

Students were invited to draw some triangles and then constrain — fix — side lengths, angles, to create right triangles.

The class then had a variety of problems involving Pythagoras. They were asked to solve the problems using Gx to recreate the problem. One particularly interesting task was to find all the possible sides of a right-triangle if two of the sides are lengths 10 and 15.

Below are extracts from the lesson log written as the the lesson progressed (TAG = “talented and gifted”, RR = “rest room”, AA = Avis):

30 mins in, most students able to change sides a and b for numbers + observe how the formula calculates the answer.

Some students use the program to draw pictures not related to the exercises. (This also happened when students introduced to GSP).

40 minutes in… higher level of engagement with activity/focus better than normal for this class. Students interested in confirming answers in discussion.

TAG-type student complains the exercises can be done more quickly without the computer… explain am training him to use the program with aim of using it as a thinking tool + more difficult problems later. Agrees to give it a second go on a more complex problem.

45 mins in… one of most difficult students (usually can’t focus at all, frequent trips to RR, etc) still engaged and asking questions… and has not visited RR! Call this student AA. AA working with girl BB who does not find subect easy, but who tries.

50 mins in… some signs of disengagement among some students.

60 mins in… TAG student happier… still feels he’s working slower, but is seeing the program can aid his thinking.

Two students raise calculation that gives pi/2. Leads to brief discussion that this is 90 degrees (this class not done radians).

65 mins… engagement now down to about one in three students. This is an improvement for this class. Class also much more quiet than normal.

Allow internet access for last 20 minutes… some students still continue with Gx.

Girl BB + boy AA still v engaged and 70 mins in are now trying to work out how to constrain a side using a radical. Still no visit to the RR by AA!

TAG student comments that Gx good “verification tool”, useful for checking answers you’re not sure of.

Pack away 80 mins in.

AA cleans board, still quiet (!!!) and no visit to RR. Wow! AA smiles genuinely brightly when I congratulate him on not going to the RR.

Two days later the same class works on using Gx to solve problems involving the equations of circles. This time Avis is very much the focus of my attentions.

Below are extracts from the log written as the class progressed:

blitz start to lesson… doors locked, etc. Computers eventually arrive from another classroom.

Students specifically told to work in pairs.

13:30 all students working. Enough computers.

AA + BB work together and ask frequent questions (on task).

Questions from students make it v easy to answer with a question focusing on why?

14:48 Class engaged. One TAG student (girl) asked for help, but figured answer while I worked my way round. Questions focus on basic use of Gx… partic how the animation works. These are easily solved questions and students pick up quick.

Talk in room is almost exclusively on the task.

pair work v successful.

13:54 Deep discussion between two TAG students on equations of circles.

Respond to earlier request from BR for help… “I got it! it’s ok…” Doesn’t look up, stays completely focussed.

LI, BB, AA have discussion about where the “variable”s go… these students never talk about “variables”!

14:03 Five students pulled out of class for OAKS state testing… they’re really upset! Includes LI who is working really hard on task.

14:12 AA + BB want to discuss what are the two things on which they must agree for them both to draw the same circle. BB using words like “segment”. AA has not been to the RR!

Two students working in Gx, but drawing pictures. Only occasionally are students caught on the internet.

Easy to get students moving.

Three TAG students now working on the extension activity.

14:12 BB and AA in deep discussion looking at the calculated equation of a circle.

14:22 AA + BB still on task. HT ok writing out definition of circle with help. BB points to an equation and asks “does this make sense to you?” We spot that she had not replaced r for radius with a specific value.

14:24 Tell students to start to shut down… BB shouts out to AA “quickly, let’s do this one.” And they do.

Sadly, departmental decisions about the need to deliver aspects of the curriculum before the end-of-semester test meant an end to the experiments with Geometry Expressions and a return to textbook-based didactics for Avis’s class. At the following lessons Avis mostly left the classroom, though he did frequently ask whether or not the class would be working on the computers.

But the brief experiment did demonstrate that Geometry Expressions can motivate and aid the mathematical thinking of younger and lower-ability children. This was confirmed by similar reactions by lower-ability students in Algebra 2 classes where it was possible to use the technology over a much longer period.

¹ The development of Geometry Expressions is funded in part by grants from the National Science Foundation. The author sits on the NSF committee monitoring the project.

June 30, 2009 at 4:33 pm Leave a comment


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