Notes from Multiplication Models Class Week 1

[Steve Thomas]

Maria, showed what she had setup at http://mathfuture.wikispaces.com/MultiplicationModels as a starting point for cataloging and documenting our work.

During the session we looked at: Area Models for Multiplication and Division at learner.org

Good Points:

• the use of shading to distinguish units
• One thing we did not look at during the session, but is a nice feature is that when you click on solution, is that you can roll over parts of the area model or the "algorithm" to see the connection. See examples below:
• 
  
• Also they had a note that "This is one solution to the problem. The same manipulatives can be arranged in different positions to fill the same area".  This works towards addressing one of our questions: "How can we get learners to think about different models to solve the same problem?"
• Also reviewed the video after class and one nice use of this model is to show an area model like the one below (created in Etoys) and ask "How can this picture help you figure out the product of 12 x 13?"
• Go ahead class discuss.

Areas for Improvement: (Note: These apply to most on-line manipulatives except those created in Etoys and there are a set from Oxford I believe that allow you to "save" the models. They actually save an XML file representation of a particular state, which you can load.  In Etoys I find it simpler to copy and modify, then save the project. Then simply open the modified project)

• You could add more problems
• You could allow teachers and students to create/model their own problems by allowing them to specify the area. I have a model in Etoys already, will tweak it some and post it.

One other idea I have tried is to give a learner 1-100 block, 5-10 blocks and 6 one blocks and say, make a rectangle.

The video that goes with this manipulative also points out where a model fails.  We should document where the models do not work.

Questions for evaluating manipulatives:

1. Can a teacher modify it?
2. Can a student modify it?
3. Can you save a modified version for later re-use/exploration?
1. We noted that if you design the manipulative properly, it can be parametric, to facilitate easier modifications.  The modifications may not even have to know the implementation language (flash, Etoys, ...) if designed well enough.
4. How can you share objects?
1. Share objects from my manipulative with someone else
2. Have multiple kids play with the same manipulative.
3. Share a screen shot/image of the model
4. We looked at how Wolfram Alpha does this by "saving" what you entered in the "search box" (the yellow box where you enter your question) as a parameter in the URL string, so you can share the URL and the same question/answer should be displayed (unless they updated the answer/search engine and you get a different result).
5. Wolfram Alpha also has widgets to let you copy text/image/pdf's.
6. 
   
5. How easy is it for a learner to use the manipulative to communicate their understanding? One way is to take static "pictures" of the manipulative to put in a presentation/portfolio.  Another would be able to put a "screencast" of the steps the child took as they explain what they are doing.
1. Screen Capture and Screen Casting could be used to solve this, but I think part of the issue is how easy is it for the student to do this and embed it in their presentation/report/homework.  If it is too hard or requires the purchase of software that makes it less likely to be used.

We then looked at: NLVM Algebra tiles, while playing with it, it was not obvious how to rotate the X*Y block, but that could simply be shown and kids should pick it up quickly.

I then showed a model I have been working on in Etoys slide show here

The advantage of this (warning endorsement from Author, be suspicious, very suspicious) is that it is easily modifiable, and teachers and kids can use their own images and create their own problems. We did a real-time creation of a fractions in the array model as well.

We discussed the Pan Balances - Shapes from Illuminations.  This is a great example of good manipulative.

One thing we really liked was that it was a "puzzle me" rather than show me.  The only limitation was that there was only one set of problems and teachers/kids can't create their own. We discussed a similar project I had created which allows teachers and kids to modify the weights and images.  Kids can draw or import their own weights and set their values as well. It does not yet have the guessing part which is a big plus.

We also compared Pan Balances - Shapes to Pan Balances Numbers.  While the Shapes version is excellent and engaging the Pan Balance version is one of those manipulatives/games a kid only plays with when forced to by their parents and teachers.  

We then discussed the problem of how do you go from the manipulative, which can be very useful for kids getting the concept, to taking the next step.  Maria mentioned the /(Pirie-Kieren model) image making, property noticing, but the next step formalizing is missing.  Here is a paper she co-wrote

We discussed some other software we have used and liked (www.poissonrouge.com, okay while I am confident I could probably find one multiplication model on the site, it is so beautiful and well done you should visit and just start clicking and playing, its a treat).

We decided another test is: Would a kid play it, if someone did not make them?

And how would you test that?

Got a little into game mechanics and there use for good and I'll, discussed the dangers of kids distracting themselves to death (sorry I see no value in Call of Duty, and just heard some kid on an NPR story about the Sony Playstation outage, that he enjoyed playing Call of Duty with others over the interent because "He really liked killing real people" and it wasn't fun playing the standard game.  Nuf said.

Natural Math showed up #2 on a search of Multiplication Models!!!

Got into a discussion of how important (or not) it is to learn your multiplication facts and it would be good to combine some of the methods of Anki with some games like TimesAttack.

We talked about how we can get kids to think about different methods/models for solving a problem. Not just to know the different methods but how and when to use them and when NOT to use them. This will be part of our discussion next week.

Stephen

  If you give a kid an answer, you solve a problem for a day.

  If you teach a kid how to solve a problem, you prepare her for life.

[Maria Droujkova]

Hello,

We are creating rich and free descriptions of models, to see what categories gel together. This is somewhat similar to how "Music Genome Project" created its categories for Pandora Radio: http://en.wikipedia.org/wiki/Music_Genome_Project

The recording of the meeting Steve summarized is here: https://sas.elluminate.com/p.jnlp?psid=2011-04-27.1731.M.B41FC4AB0A3565F8BCAE1B600D6595.vcr&sid=2008350

On Sat, Apr 30, 2011 at 11:08 PM, Steve Thomas <stho...@gosargon.com> wrote:

Maria, showed what she had setup at http://mathfuture.wikispaces.com/MultiplicationModels as a starting point for cataloging and documenting our work.

During the session we looked at: Area Models for Multiplication and Division at learner.org

Good Points:

• the use of shading to distinguish units
• One thing we did not look at during the session, but is a nice feature is that when you click on solution, is that you can roll over parts of the area model or the "algorithm" to see the connection. See examples below:
• 
  

I am listing "dynamic links between multiple representations" as feature of models defining their quality. This is something computers do better than most physical manipulatives, by the way. 

• Also they had a note that "This is one solution to the problem. The same manipulatives can be arranged in different positions to fill the same area".  This works towards addressing one of our questions: "How can we get learners to think about different models to solve the same problem?"
• Also reviewed the video after class and one nice use of this model is to show an area model like the one below (created in Etoys) and ask "How can this picture help you figure out the product of 12 x 13?"
• Go ahead class discuss.

When I work with kids, I ask the series of questions:
- Find 12
- Find 13
- Find multiplication
- What is the light square? Where is it in the equation?
- What are blue bars? Where are they in the equation?
- What are dark little squares? Where are they in the equation?

All questions are about correspondences. How can the questions be captured in the software?!!
 

Areas for Improvement: (Note: These apply to most on-line manipulatives except those created in Etoys and there are a set from Oxford I believe that allow you to "save" the models. They actually save an XML file representation of a particular state, which you can load.  In Etoys I find it simpler to copy and modify, then save the project. Then simply open the modified project)

• You could add more problems
• You could allow teachers and students to create/model their own problems by allowing them to specify the area. I have a model in Etoys already, will tweak it some and post it.

One other idea I have tried is to give a learner 1-100 block, 5-10 blocks and 6 one blocks and say, make a rectangle.

The video that goes with this manipulative also points out where a model fails.  We should document where the models do not work.

Probably along several limitations:

• What types of math entities are limited (e.g. whole numbers only, positive numbers only)
• Disability access (e.g. visual models representable/not for the blind)
• Development levels and prerequisites (toddler-kid-researcher-etc level)

 

Questions for evaluating manipulatives:

1. Can a teacher modify it?
2. Can a student modify it?
3. Can you save a modified version for later re-use/exploration?
1. We noted that if you design the manipulative properly, it can be parametric, to facilitate easier modifications.  The modifications may not even have to know the implementation language (flash, Etoys, ...) if designed well enough.
4. How can you share objects?
1. Share objects from my manipulative with someone else
2. Have multiple kids play with the same manipulative.
3. Share a screen shot/image of the model
4. We looked at how Wolfram Alpha does this by "saving" what you entered in the "search box" (the yellow box where you enter your question) as a parameter in the URL string, so you can share the URL and the same question/answer should be displayed (unless they updated the answer/search engine and you get a different result).
5. Wolfram Alpha also has widgets to let you copy text/image/pdf's.
6. 
   
5. How easy is it for a learner to use the manipulative to communicate their understanding? One way is to take static "pictures" of the manipulative to put in a presentation/portfolio.  Another would be able to put a "screencast" of the steps the child took as they explain what they are doing.
1. Screen Capture and Screen Casting could be used to solve this, but I think part of the issue is how easy is it for the student to do this and embed it in their presentation/report/homework.  If it is too hard or requires the purchase of software that makes it less likely to be used.

6. Quest or roleplay?

1. Are there open-ended tasks that come with the manipulative?
2. Are there closed-ended tasks that come with the manipulative?
3. Are tasks organized into chains by difficulty or conceptual dependencies?
   

 

We then looked at: NLVM Algebra tiles, while playing with it, it was not obvious how to rotate the X*Y block, but that could simply be shown and kids should pick it up quickly.

I then showed a model I have been working on in Etoys slide show here

The advantage of this (warning endorsement from Author, be suspicious, very suspicious) is that it is easily modifiable, and teachers and kids can use their own images and create their own problems. We did a real-time creation of a fractions in the array model as well.

This "cow array" interactive brings to mind another tag or description: "Good source of math circle problems."

You can simply ask kids to arrange counters in arrays, and a series of questions suggests itself:
- What can be sides of arrays?
- Which numbers work with a particular width?
- Which numbers have "many" and "few" possibilities?

We discussed the Pan Balances - Shapes from Illuminations.  This is a great example of good manipulative.

One thing we really liked was that it was a "puzzle me" rather than show me.  The only limitation was that there was only one set of problems and teachers/kids can't create their own. We discussed a similar project I had created which allows teachers and kids to modify the weights and images.  Kids can draw or import their own weights and set their values as well. It does not yet have the guessing part which is a big plus.

We also compared Pan Balances - Shapes to Pan Balances Numbers.  While the Shapes version is excellent and engaging the Pan Balance version is one of those manipulatives/games a kid only plays with when forced to by their parents and teachers.  

What game mechanics would go with Pan Balance Numbers? Maybe kids do like it in some form and we just don't realize it?
 

We then discussed the problem of how do you go from the manipulative, which can be very useful for kids getting the concept, to taking the next step.  Maria mentioned the /(Pirie-Kieren model) image making, property noticing, but the next step formalizing is missing.  Here is a paper she co-wrote

We discussed some other software we have used and liked (www.poissonrouge.com, okay while I am confident I could probably find one multiplication model on the site, it is so beautiful and well done you should visit and just start clicking and playing, its a treat).

We decided another test is: Would a kid play it, if someone did not make them?

And how would you test that?

Got a little into game mechanics and there use for good and I'll, discussed the dangers of kids distracting themselves to death (sorry I see no value in Call of Duty, and just heard some kid on an NPR story about the Sony Playstation outage, that he enjoyed playing Call of Duty with others over the interent because "He really liked killing real people" and it wasn't fun playing the standard game.  Nuf said.

Natural Math showed up #2 on a search of Multiplication Models!!!

Got into a discussion of how important (or not) it is to learn your multiplication facts and it would be good to combine some of the methods of Anki with some games like TimesAttack.

There are two separate needs that have to be addressed by drill software:
- Sophisticated memory tracking. Serve me facts I am about to forget to reinforce them. Do not serve me facts I remember well. Anki is excellent for it. http://ankisrs.net/
- Amusing, beautiful or funny delivery. TimesAttack is decent at this (for some kids), but words like "underhanded" and "sugar-coating" come to mind. Delivery has to be honest, too.

If these two needs can be served by one package, the maker will get rich quick. So, whoever read this far can start on it :-)

Cheers,
Maria Droujkova

We talked about how we can get kids to think about different methods/models for solving a problem. Not just to know the different methods but how and when to use them and when NOT to use them. This will be part of our discussion next week.

Stephen

  If you give a kid an answer, you solve a problem for a day.

  If you teach a kid how to solve a problem, you prepare her for life.

--