Up until yesterday, I didn't know there was such a thing as math museums. I knew of art museums, car museums, train museums, historical museums, etc but never a math museum.

One I found yesterday was the 3DXM Virtual Museum filled with all sorts of mathematical art. It has material on minimal surfaces, famous surfaces, conformal maps, plane curves, fractals, space curves, polyhedral and math art.

Each topic has a variety of drawings for each type along with the mathematics involved. The pictures tend to be in color and appear three dimensional. It isn't until you get to the math art section that things really come together to show math in a different way.

I checked out the work of several artists and their work is absolutely breathtaking. The rose shaped parametric surface, the Kleinian double spiral, iso construction, and Tori reflections are spectacular and if shown out of context would be taken as straight artwork.

This site is the type of place students can go and just explore while having fun checking out the interactive elements.

Other places that are not so much museums in the traditional sense of the word but have interesting things to offer both students and teachers.

1. The Exploratorium in San Francisco offers activities that can be done with your students. One suggestion is creating a math walk locally. The site offers instructions for creating one locally for people who are not sure how to go about setting one up.

2. The Smithsonian has some nice mathematical opportunities available on line. There is a virtual exhibit on Slates, Slide rules, and Software a history of mathematical teaching in America. In addition, there are several videos on the math of prehistoric climate change, or the math involved in the fish population. There are quite a few stories and articles all associated with math. So many items to integrate into class.

3. Finally is the National Museum of Mathematics in New York City does not have much in the way of online materials but it does have a page full of videos made by several different mathematicians. If you want to see what topics are covered by checking out the map of the museum.

Have fun letting students explore some new places filled with mathematics. Let me know what you think. I'd love to hear.

# Thoughts on Teaching Math with technology

## Friday, November 17, 2017

## Thursday, November 16, 2017

### Exploring Math

The idea for this column came from a tweet by Tina Cardone. The idea is to create an ongoing project filled with four elements designed to have students do one activity from each column each quarter. At the end of the quarter, the students report back.

She basically created a choice board with choices to play by doing math art, math comics, or a puzzle, research either a career, a mathematician, or find an article on math to read and summarize. There is an explain section where the student explains a new math topic discovered in the last 100 years, or explains about an ancient math system, or an unsolved math problem. The final section has the student try a math challenge problem, or problems from two other sources.

The play column is in there to show students that math can be fun and is not always answering problem after problem after problem. So if you want to create your own, where would you go to find comics, art, or puzzles that students might enjoy. You could borrow her version or you could personalize it to make it more relevant to your students.

Here are places that offer math comics that are actually rather funny.

1. Comic Math filled with samples and links to various mathematically based comics both popular and lesser known ones.

2. This list has comics based on topic.

3. This is a list of 7 places with great math comics.

Now for the art possibilities:

1. Math art for kids has some great ideas including creating a city scape out of the numbers of pi. There are 21 different suggestions listed here. Even though its for kids, I think some of my high school students would enjoy them so I think I'll try to find time to incorporate them into my class.

2. Smith Curriculum has some great art projects including the Pythagorean Snail based on the Pythagorean Theorem. I should try this in my geometry class since some of the students are more artist than anything else and would rather draw than do school work.

3. To see works check out the Virtual Math Museum with some fantastic art based on mathematics.

This site is filled with mathematical puzzles that students might find interesting.

Check this site out as it lists both male and female mathematicians and their ethnicity. Its quite a list.

My students are not ready for something like this but I like the idea of playing with mathematical art to add another layer to my classroom. Let me know what you think. I'd love to hear.

She basically created a choice board with choices to play by doing math art, math comics, or a puzzle, research either a career, a mathematician, or find an article on math to read and summarize. There is an explain section where the student explains a new math topic discovered in the last 100 years, or explains about an ancient math system, or an unsolved math problem. The final section has the student try a math challenge problem, or problems from two other sources.

The play column is in there to show students that math can be fun and is not always answering problem after problem after problem. So if you want to create your own, where would you go to find comics, art, or puzzles that students might enjoy. You could borrow her version or you could personalize it to make it more relevant to your students.

Here are places that offer math comics that are actually rather funny.

1. Comic Math filled with samples and links to various mathematically based comics both popular and lesser known ones.

2. This list has comics based on topic.

3. This is a list of 7 places with great math comics.

Now for the art possibilities:

1. Math art for kids has some great ideas including creating a city scape out of the numbers of pi. There are 21 different suggestions listed here. Even though its for kids, I think some of my high school students would enjoy them so I think I'll try to find time to incorporate them into my class.

2. Smith Curriculum has some great art projects including the Pythagorean Snail based on the Pythagorean Theorem. I should try this in my geometry class since some of the students are more artist than anything else and would rather draw than do school work.

3. To see works check out the Virtual Math Museum with some fantastic art based on mathematics.

This site is filled with mathematical puzzles that students might find interesting.

Check this site out as it lists both male and female mathematicians and their ethnicity. Its quite a list.

My students are not ready for something like this but I like the idea of playing with mathematical art to add another layer to my classroom. Let me know what you think. I'd love to hear.

## Wednesday, November 15, 2017

### Finding Errors

As I mentioned yesterday, its hard for students to find the error they made if they do not get the correct answer. I've been wondering about techniques I can include to help students learn to find their errors.

It seems that once the student has completed a problem, their mind shuts the door on it and moves on because they are finished with it and don't need to check it.

One suggestion I ran across is to have a poster in the classroom for the top 11 errors made in math calculations hung somewhere in the room so they can check it before they move on.

1. Did not distribute the outside term to both terms inside the parenthesis. This includes not distributing the negative sign with the number.

2. Multiplying by 2 instead of squaring. In other words they multiply by the exponent, instead of applying the power.

3. Adding instead of subtracting or vice versa.

4. Adding instead of multiplying or vice versa.

5. Misplacing or loosing a decimal.

6. Making a rounding error.

7. Forgetting to carry a number or to borrow.

8. Forgetting to change the inequality sign when dividing or multiplying by a negative.

9. Making a mistake when cross multiplying ratios.

10. Making a mistake when adding/subtracting/multiplying/dividing a fraction.

11. Omitting units or incorrectly converting units.

I think I'm going to run this list of common mistakes off and give each student a copy so they can use it to double check their steps. Of course, I'll have to model its use but if I use it regularly, perhaps they will choose to use their list.

It is also suggested that the teacher change the way they identify mistakes for students. Rather than saying "You made a mistake", say "I'm glad you made the mistake, it means you are thinking about the problem and you can learn from it." I tend to let the student know they missed a step when solving it, so go back and check to see if they can tell where they missed the step.

In addition it is good for the teacher to make a mistake, correct it, and let the students know what the mistake was and why they did it. It shows that teachers are not infallible. Teachers are human. Too often students are under the mistaken impression that math teachers are extremely smart, like Einstein. Its important to show them we are human. Make it normal to look at mistakes so they are no longer something to be feared but celebrated.

When a student makes a mistake, it is important to correct it but also to understand why the mistake was made. By correcting the error and knowing why it was made, it gives the student a personal sense of success. Furthermore, the type of the mistake provides an assessment for the teacher. The mistakes let the teacher know, what has not been mastered yet.

In a sense, this is something that should be started in elementary but it isn't always so it is necessary to work with students in high school.

Let me know what you think. I love to hear from my readers. Have a good day.

It seems that once the student has completed a problem, their mind shuts the door on it and moves on because they are finished with it and don't need to check it.

One suggestion I ran across is to have a poster in the classroom for the top 11 errors made in math calculations hung somewhere in the room so they can check it before they move on.

1. Did not distribute the outside term to both terms inside the parenthesis. This includes not distributing the negative sign with the number.

2. Multiplying by 2 instead of squaring. In other words they multiply by the exponent, instead of applying the power.

3. Adding instead of subtracting or vice versa.

4. Adding instead of multiplying or vice versa.

5. Misplacing or loosing a decimal.

6. Making a rounding error.

7. Forgetting to carry a number or to borrow.

8. Forgetting to change the inequality sign when dividing or multiplying by a negative.

9. Making a mistake when cross multiplying ratios.

10. Making a mistake when adding/subtracting/multiplying/dividing a fraction.

11. Omitting units or incorrectly converting units.

I think I'm going to run this list of common mistakes off and give each student a copy so they can use it to double check their steps. Of course, I'll have to model its use but if I use it regularly, perhaps they will choose to use their list.

It is also suggested that the teacher change the way they identify mistakes for students. Rather than saying "You made a mistake", say "I'm glad you made the mistake, it means you are thinking about the problem and you can learn from it." I tend to let the student know they missed a step when solving it, so go back and check to see if they can tell where they missed the step.

In addition it is good for the teacher to make a mistake, correct it, and let the students know what the mistake was and why they did it. It shows that teachers are not infallible. Teachers are human. Too often students are under the mistaken impression that math teachers are extremely smart, like Einstein. Its important to show them we are human. Make it normal to look at mistakes so they are no longer something to be feared but celebrated.

When a student makes a mistake, it is important to correct it but also to understand why the mistake was made. By correcting the error and knowing why it was made, it gives the student a personal sense of success. Furthermore, the type of the mistake provides an assessment for the teacher. The mistakes let the teacher know, what has not been mastered yet.

In a sense, this is something that should be started in elementary but it isn't always so it is necessary to work with students in high school.

Let me know what you think. I love to hear from my readers. Have a good day.

## Tuesday, November 14, 2017

### Right Answer?

Too many of my students entering high school are more concerned with getting the right answer. The ones I've had for a while are moving from right answer to did they do the problem correctly. Yes, they want the right answer but they are beginning to look at the whole process rather than only the solution.

The same group believes as long as they get the correct answer, its fine, even if the calculations contained an error. Their answer is usually "So what! I got it right." Even if the calculations is correct but they messed up on one thing in the process but still got the right answer, they still say the same thing. It doesn't matter if it won't work for any other numbers, they don't care because they got the right answer for the problem.

I think its a mind set they get into in elementary school when the problem is right or wrong. As far as I know, most of the elementary school teachers do not take time to teach students to find their mistakes. They focus on teaching process and getting the correct answer.

I've heard of teachers moving away from numerical grades into using a rubric based grading system so its not like 67 percent but rather they are not quite proficient in the topic. It actually sounds more realistic since it eliminates the "How can I bring my grade up?" question. The last test I gave, many students got upset because I said they could make corrections but the corrections themselves would not raise their test grade. Making corrections for the test only allows the students an opportunity to retake a similar test. They don't want that.

I've spoken with the English teacher who said students only want to write one draft, the final draft, before turning it in. They don't understand that in both math and English, it can take multiple tries to get a finished product that finished and ready to be read.

Because they are so focused on the correct answer, they are unable to take what they just did and do the next problem without asking "What do I do?" or "How do I solve this one." Its as if they are totally separate from the problem. I do get students who manage to put it all together to the point they can do all the problems and can help other students learn it. But too many never reach the point.

Then if you ask them how they got the answer, they give you the "I did the work." answer. I realize their is a move to explain how they got the answer but when I was in school, we were told as long as we showed the math we used to arrive at the answer, that explained it all. If I'd been asked to explain how I got the answer, I would have done it by explaining my work at each step.

Right now, I just work on getting the students to look at the whole problem rather than the answer because its easy to make a calculation error such as 3 x 2 = 5 and get the wrong answer while having completed the process correctly.

Is there an answer? I don't know because by the time they get to me in high school, they are convinced getting the right answer is the only thing math focuses on. Let me know what you think.

The same group believes as long as they get the correct answer, its fine, even if the calculations contained an error. Their answer is usually "So what! I got it right." Even if the calculations is correct but they messed up on one thing in the process but still got the right answer, they still say the same thing. It doesn't matter if it won't work for any other numbers, they don't care because they got the right answer for the problem.

I think its a mind set they get into in elementary school when the problem is right or wrong. As far as I know, most of the elementary school teachers do not take time to teach students to find their mistakes. They focus on teaching process and getting the correct answer.

I've heard of teachers moving away from numerical grades into using a rubric based grading system so its not like 67 percent but rather they are not quite proficient in the topic. It actually sounds more realistic since it eliminates the "How can I bring my grade up?" question. The last test I gave, many students got upset because I said they could make corrections but the corrections themselves would not raise their test grade. Making corrections for the test only allows the students an opportunity to retake a similar test. They don't want that.

I've spoken with the English teacher who said students only want to write one draft, the final draft, before turning it in. They don't understand that in both math and English, it can take multiple tries to get a finished product that finished and ready to be read.

Because they are so focused on the correct answer, they are unable to take what they just did and do the next problem without asking "What do I do?" or "How do I solve this one." Its as if they are totally separate from the problem. I do get students who manage to put it all together to the point they can do all the problems and can help other students learn it. But too many never reach the point.

Then if you ask them how they got the answer, they give you the "I did the work." answer. I realize their is a move to explain how they got the answer but when I was in school, we were told as long as we showed the math we used to arrive at the answer, that explained it all. If I'd been asked to explain how I got the answer, I would have done it by explaining my work at each step.

Right now, I just work on getting the students to look at the whole problem rather than the answer because its easy to make a calculation error such as 3 x 2 = 5 and get the wrong answer while having completed the process correctly.

Is there an answer? I don't know because by the time they get to me in high school, they are convinced getting the right answer is the only thing math focuses on. Let me know what you think.

## Monday, November 13, 2017

### Comparing costs.

Every math textbook seems to have a lesson or two on unit costs. Unfortunately, the prices given, even in new textbooks, do not reflect realistic prices where I live in the bush of Alaska.

A Tino's pizza costs around $6 to $8 each. A pizza you can get at one of the grocery chains runs $21.00 for a $8.00 one on sale in town. If I use the problems in the book, my students think something is off and don't relate to them.

Anything frozen that is shipped to the village has the added cost of air freight. This means I might be able to get a half gallon of ice cream for $5.00 in town but by the time its shipped out here, it costs $11 to $13.00. Quite a mark-up.

Then there is the difference in prices locally. For instance, a can of soda runs about $1.25 at the store but various groups buy a 12 pack at the store then resell the soda at $2.00 each. It is a matter of convenience because they do not have to leave the building, drive to the store, and buy it. I have no idea what a soda sells for in the lower 48 because I don't usually buy it.

So in order to give students a real idea of cost comparisons, I have to use local prices. There are five ways I can have students compare prices.

1. Comparing different sizes of the same item to see if the cost per unit is consistent, or which one is the better buy.

2. Rectangular vs circular pizza by weight, are they the same cost?

3. School prices vs city prices. Our school runs a concession stand that is open during sports activities but not during the day because it sells junk food.

4. Comparing prices between the village and Anchorage so students get a better idea of the markup and cost per unit differences. This is easy to do because two stores in Anchorage do bush orders and have websites listing prices.

5. Compare prices between the two stores in town. The stores are more like convenience stores but sell guns, bullets, freezers, and just about anything else a K-mart might sell. I know that sometimes one store has a better price on eggs, milk, or cheese.

Its hard to teach comparing prices using price/unit cost since the store shelves do not have the tags you would normally find. In a normal store, you might have several different brands or sizes of items but since the stores here are small and limited, they do not carry those tags. This means my students do not have the opportunity to learn to read shelf tags the way most students do.

They do not get to see the tags showing the price/unit cost in different units. I've seen two of the same type of product listed with price/product vs price/ounce which makes it much harder compare. So I have to get creative for this type of comparison in the classroom.

What I do have available is the SpanAlaska Catalogue which allows people to order items in bulk. I can have students look up various items and have them calculate the price/unit cost using the same unit. I could have a friend take pictures, send them so I can show students how the tags normally appear. I have to create the experience for my students so when they go somewhere with the shelf tags, they are educated and capable of using them.

Let me know what you think. I'm interested. Have a great day.

A Tino's pizza costs around $6 to $8 each. A pizza you can get at one of the grocery chains runs $21.00 for a $8.00 one on sale in town. If I use the problems in the book, my students think something is off and don't relate to them.

Anything frozen that is shipped to the village has the added cost of air freight. This means I might be able to get a half gallon of ice cream for $5.00 in town but by the time its shipped out here, it costs $11 to $13.00. Quite a mark-up.

Then there is the difference in prices locally. For instance, a can of soda runs about $1.25 at the store but various groups buy a 12 pack at the store then resell the soda at $2.00 each. It is a matter of convenience because they do not have to leave the building, drive to the store, and buy it. I have no idea what a soda sells for in the lower 48 because I don't usually buy it.

So in order to give students a real idea of cost comparisons, I have to use local prices. There are five ways I can have students compare prices.

1. Comparing different sizes of the same item to see if the cost per unit is consistent, or which one is the better buy.

2. Rectangular vs circular pizza by weight, are they the same cost?

3. School prices vs city prices. Our school runs a concession stand that is open during sports activities but not during the day because it sells junk food.

4. Comparing prices between the village and Anchorage so students get a better idea of the markup and cost per unit differences. This is easy to do because two stores in Anchorage do bush orders and have websites listing prices.

5. Compare prices between the two stores in town. The stores are more like convenience stores but sell guns, bullets, freezers, and just about anything else a K-mart might sell. I know that sometimes one store has a better price on eggs, milk, or cheese.

Its hard to teach comparing prices using price/unit cost since the store shelves do not have the tags you would normally find. In a normal store, you might have several different brands or sizes of items but since the stores here are small and limited, they do not carry those tags. This means my students do not have the opportunity to learn to read shelf tags the way most students do.

They do not get to see the tags showing the price/unit cost in different units. I've seen two of the same type of product listed with price/product vs price/ounce which makes it much harder compare. So I have to get creative for this type of comparison in the classroom.

What I do have available is the SpanAlaska Catalogue which allows people to order items in bulk. I can have students look up various items and have them calculate the price/unit cost using the same unit. I could have a friend take pictures, send them so I can show students how the tags normally appear. I have to create the experience for my students so when they go somewhere with the shelf tags, they are educated and capable of using them.

Let me know what you think. I'm interested. Have a great day.

## Sunday, November 12, 2017

## Saturday, November 11, 2017

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