Mathematical sophistication is when the teacher becomes comfortable in the art of teaching mathematics. This is perfectly good example of having flow in allowing the student to create, test and make conjectures about something on YouTube. Mathematics is not about a textbook. It is about real life application and enjoyment. In this lesson the student makes and test conjectures about mathematical objects and structures. It is also about bringing math alive by creating models, using logical arguments.

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This exercise has a very nice go-easy approach. Students slowly build up information about a subject by doing a series of problems based on real life situations that get increasingly more sophisticated and complex. This should be very good for minimizing math anxiety. The only suggestion I can think of is the possibility of adding some activities for kids with different learning styles.

For example:

1. Role playing the actual scenario, where someone is hailing a cab and needs to figure out which company to use, or have two cab drivers try to persuade the same student that their company’s rates are the best for his needs.

2. Art/Graphic Design. Give students who are interested the opportunity to diagram or draw some of the cab ride scenarios.

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I liked the progression of questions the students have to answer and the way they led to understanding the mathematics of the situation. Perhaps it would be possible to insert a few questions that would prompt students to ponder and conjecture about mathematical rules along the way.

For instance, before question 6, insert a question like: Do you think there is a rule for deciding which company is best to use in certain situations? In every situation? If there is a rule, what might it be?

]]>However, most people can’t pay highly focused attention to oral presentations for longer than 5-7 minutes. People who are successful lecture listeners take microbreaks, switch from listening to note-taking, and otherwise manage their own attention within longer captivating stories. Most TED lectures are under 10 minutes. Most YouTube videos are around 5 minutes. PechaKucha presentations are 400 seconds (20 slides, 20 seconds each). These seem to be the lengths of presentations large groups gravitate to, naturally, when they can choose their rhythms.

Dani Novak, who will be leading the GeoGebra-NA Network Series of events this Fall, proposed an NPR-like format where people prepare media pieces ahead of time, with shorter ones incorporated into the live event (for a more prepared feel, I may add) and longer ones available online. This may capture the best of both worlds.

]]>That was an excellent webinar. You hit on some of the highlights. However, there were a couple of items that were staggering. Don, “The Mathman” Cohen has earned such a place of trust with his students that he is now teaching their children. And in one case, he mentioned that he is teaching a third generation of children from the same family. Regardless of the subtlety with which Don guides his young learners, I don’t think there is a laurel leaf quite so powerful as bringing one’s children to him because you loved him so.

The map also has depth and breadth. He, along with his former and current students, capture the two essential elements of calculus. And he relates them to infinite series and infinite sequences. I was priveleged to participate.

]]>I found the same Mindmap page! I thought it did a great job of putting more context into the rather dry and abstract levels of thinking required for each level of understanding. The Mindmap did a good job of placing the model within the structure of the way that learning occurs.

I agree also that applying the van Hiele’s mode to a group of students would not be useful. Since every student may have a different level of understanding, finding a common ground for the group would require a lot of work. Using it when assessing and then modifying curricula for individual students can provide a good method for moving them along the levels.

Finally, I don’t agree that a single person has to remain at the same level inside the model. I don’t have any evidence to support this. However, I trust my gut feeling that some students have a deeper understanding of some concepts than others. And as a result, they could be on different levels.

Do you think that the levels apply directly to specific concepts in geomerty, like congruence, similarity, correspondence, symmetry and the like? Or do the levels generalize the overall understanding of all principles together? The model is very interesting and I think provides a great source of debate. And from this point of view, I think it is very useful.

]]>I agree that the lesson utilizes the lower spectrum of Bloom’s Taxonomy and van Hieles’ model. From that standpoint, I would certainly use the lesson as an introduction. I start many of my lessons from “first principles”. In that way, I don’t skip right past the students in the class. It may be a bit boring for the students that have a bette grasp. However, I find that working the basics for them is a great help too.

Perhaps a place to begin with this lesson would be to inject some algebra into the lesson. Also, talking about how the exercises are graded and any rubric would be helpful for other instructors too.

]]>I liked this problem, because it made me take a step back. At first, I thought that the expressions should be the same for each of the hypotenuses. Then I recalled they have to be equal, not necessarily the same. And then it was a breeze to discover that x=3 units.

Since there was no units given, they can be inches, feet, miles, yards or whatever. I would dare say that making a glider using a 60 sq inches of fabric would be problematic for a person of my considerable wieght!

Perhaps providing the basic unit would be one tiny improvement.

Also, should you approach the problem from the position of how much the project is worth, and an idea of how you plan to evaluate the solution?

It was a fun problem!

]]>These sites are very interesting forums. I like the ongoing investigation, and the varying level of sophistication. It is not brainiac math whizs showing off their recondite vocabulary and symbology. Although, there was one person with a rather opaque rune as their signature! 🙂

I would like to use this sort of ongoing online community forum as part of my lesson plan. I think it would provide a great connection for kids to find each other, and to build relationships via social networking. That is an area that I am learning to embrace.

Also, I think bringing the solutiosn they found it the community back to class and posting them in a dedicated place would also provide great visibility and a low tech connection to a high tech environment.

Nice find!

]]>I’m getting started reviewing the work on your site. I love the picture of little man! The time flies so quickly. I remember my son being that size. Now, he’s kicking the ball and starting to ask about the potty! How about a new pic?

In your 5 key issues, when you say appreciate, you mean like/enjoy? Or do you mean have an understanding? Either way is good, because I want students to enjoy math, and replace fear with fun. Yet, I realize so many won’t! So, I usually opt for the appreciation meaning an understanding.

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