# Thoughts on Teaching Math with technology

## Wednesday, March 22, 2017

### Time Zones

I just spent about 8.5 hours flying from Philadelphia Pennsylvania to Anchorage, Alaska a span of four times zones. It is exhausting because of having left so early in the morning and arriving at my destination mid afternoon.

Its interesting that the world is divided into a minimum of 24 time zones based on the idea that each time zone is 15 degrees from the next time zone or about an hour apart but in reality it does not quite work that way. There is the GMT line or Greenwich Mean Time, the International Date Line which have added a couple of extra time zones to everything. Then there are a some places such Singapore or North Korea as which only have 30 or 45 minutes in the time zone.

One large country, China, only has one time zone. China has operated on Beijing Time or Chinese Standard Time since 1949 when the communists took over. A question asking about travel time in China would require no additional time zone calculations but if you flew to Singapore, you'd have to keep in mind the 30 minute time zone.

There are several sites which provide some very good time change problems complete with explanations and very real problems. For instance, Space Math by NASA takes time to explain the time zones in the United States but its problems deal with what time astronomers need to be ready to watch a solar flare. I've actually done some calculations like that to determine if I could watch a solar event.

Berkley has a nice set of problems which take this a step further by involving more countries after having students practice finding times when going from one time zone to another. The questions require students to calculate differences between Central Australia and Alaska or Universal Time and California.

I was unable to find problems in which the traveler began in Germany and ended in one of the countries with a 30 or 45 minute time zone. I think it would be cool to have students create a a trip through certain countries with information on time zones, take off and landing times for a realistic activity.

Let me know what you think. I'd love to hear from you.

## Tuesday, March 21, 2017

### 9 Common misconceptions.

While researching yesterday's topic, I stumbled across a list of mathematical misconceptions some of which I've had students happily share.

I'm sure you'll recognize some or all of the misconceptions listed below. I'm also sure some will make you smile at the memory of a teacher telling you that exact thing in elementary school.

I know, I heard them myself. So here is the list.

1. Three digit numbers are always bigger than two digit numbers. This rule comes about because when they first learn numbers, they are only exposed to whole numbers. In that case, this rule is correct but once decimals are thrown into the learning, it no longer applies. 3.24 is not bigger than 6.2.

2. When you multiply two numbers together, the result is always larger than either of the original but that is only true with whole numbers. Once students begin using fractions or decimals, this may not be true. one example is 1/2 times 1/6. The result, 1/12, is smaller than either one.

3. Often students think the fraction with the larger number in the denominator means its larger such as in 1/4 and 1/8. They sometimes think 1/8 is larger than 1/4 because 8 is larger than 4. I think this has to do with 8 is larger than 4 normally with what they've been taught so when the context changes their understanding does not.

4. Most students see two dimensional shapes in only one orientation such as a triangle with the base always at the bottom part of the shape rather than placing it at the top with the vertex pointing downward or off to the side. Teachers need to change the orientation so students do not get in the habit of seeing it one way.

5. In squares the diagonal appears to be almost the same length as the sides and students may assume they are the same.

6. When multiplying by 10, simply add a zero. This works for a whole number but not for a decimal number. You could add a zero but it does not help you to remember to change the position of the decimal.

7. Ratios where students get used to comparing one object to another such as two carrots to three peppers rather than looking at two carrots to five vegetables. When the situation comes up where they need to set up a part to a whole, they often have trouble.

8. Students often confuse perimeter to area because they count squares for both of them without understanding the whole square inside the shape is counted for area while they are only counting one side of the square for the perimeter.

9. Students often have difficulty determining the scale used by the measuring item. Not all scares are divided into 10's. Many students do not count the markings to figure that out, they assume its always going to be 10.

I understand why students are taught many of these rules when they are in elementary school but it does a disservice teaching these are "rules". Students need to to quit learning "rules" which only apply to a narrow population of numbers. Hopefully, teachers will quit doing these so students are more open to learning new situations.

Let me know what you think. I'd love to hear.

I'm sure you'll recognize some or all of the misconceptions listed below. I'm also sure some will make you smile at the memory of a teacher telling you that exact thing in elementary school.

I know, I heard them myself. So here is the list.

1. Three digit numbers are always bigger than two digit numbers. This rule comes about because when they first learn numbers, they are only exposed to whole numbers. In that case, this rule is correct but once decimals are thrown into the learning, it no longer applies. 3.24 is not bigger than 6.2.

2. When you multiply two numbers together, the result is always larger than either of the original but that is only true with whole numbers. Once students begin using fractions or decimals, this may not be true. one example is 1/2 times 1/6. The result, 1/12, is smaller than either one.

3. Often students think the fraction with the larger number in the denominator means its larger such as in 1/4 and 1/8. They sometimes think 1/8 is larger than 1/4 because 8 is larger than 4. I think this has to do with 8 is larger than 4 normally with what they've been taught so when the context changes their understanding does not.

4. Most students see two dimensional shapes in only one orientation such as a triangle with the base always at the bottom part of the shape rather than placing it at the top with the vertex pointing downward or off to the side. Teachers need to change the orientation so students do not get in the habit of seeing it one way.

5. In squares the diagonal appears to be almost the same length as the sides and students may assume they are the same.

6. When multiplying by 10, simply add a zero. This works for a whole number but not for a decimal number. You could add a zero but it does not help you to remember to change the position of the decimal.

7. Ratios where students get used to comparing one object to another such as two carrots to three peppers rather than looking at two carrots to five vegetables. When the situation comes up where they need to set up a part to a whole, they often have trouble.

8. Students often confuse perimeter to area because they count squares for both of them without understanding the whole square inside the shape is counted for area while they are only counting one side of the square for the perimeter.

9. Students often have difficulty determining the scale used by the measuring item. Not all scares are divided into 10's. Many students do not count the markings to figure that out, they assume its always going to be 10.

I understand why students are taught many of these rules when they are in elementary school but it does a disservice teaching these are "rules". Students need to to quit learning "rules" which only apply to a narrow population of numbers. Hopefully, teachers will quit doing these so students are more open to learning new situations.

Let me know what you think. I'd love to hear.

## Monday, March 20, 2017

### HIgher Education.

I had to go to a family gathering over the weekend. I spoke with one who is currently working on his PhD in Chemical Engineering. He is working with polymers in an interesting way.

He and I discussed the skills he needs in his line of work. It came out he doesn't bother keeping track of certain chemical interactions because he looks it up anytime.

We also discussed when he needs to do any type of data analysis, he has programs to complete the analysis. He does not worry about remembering various formulas.

He stated, it is more important for him to know how to use these programs and interpret the results than it is to remember how to do it by hand. I found that interesting because the school system is still way behind this belief.

It does emphasis the idea that math provides answers and its important for students to interpret the results they get from their calculations. I don't do this enough. I teach students how to solve equations but I do not take the extra time to ask them to interpret their answers in terms of the problem.

I have them solve one and two step equations but I do not take the time to discuss what the answer might represent. When I studied math in high school, it was only important to solve equations, not understand anything about the meaning of the results. That was not important.

According to current thinking, it is important for students to be able to create mathematical observations about their solutions. They need to connect the mathematics with the situation. One facet of a mathematically proficient student is their ability to interpret results in context of the situation and reflecting if their solution makes sense.

They are able to take this reflection and make changes to create a model which is closer to what it should be. This is a real life process. The young man, talked about using the results of the data he's collected to determine what the next step should be. He adjusts factors, tests, and recalculates.

Having students work on performance tasks which require them to examine their works to find tune their assumptions is important and used more in life than having a problem done once and accepting the answer is done so you have nothing more left.

Yes, it sounds like a science experiment but if you are creating a mathematical model of a situation, it often takes several tries to get the right equation. It seldom happens immediately with the first try. Its important to create a situation to help students create models which take several tries to get right.

Let me know what you think. I'd love to hear.

He and I discussed the skills he needs in his line of work. It came out he doesn't bother keeping track of certain chemical interactions because he looks it up anytime.

We also discussed when he needs to do any type of data analysis, he has programs to complete the analysis. He does not worry about remembering various formulas.

He stated, it is more important for him to know how to use these programs and interpret the results than it is to remember how to do it by hand. I found that interesting because the school system is still way behind this belief.

It does emphasis the idea that math provides answers and its important for students to interpret the results they get from their calculations. I don't do this enough. I teach students how to solve equations but I do not take the extra time to ask them to interpret their answers in terms of the problem.

I have them solve one and two step equations but I do not take the time to discuss what the answer might represent. When I studied math in high school, it was only important to solve equations, not understand anything about the meaning of the results. That was not important.

According to current thinking, it is important for students to be able to create mathematical observations about their solutions. They need to connect the mathematics with the situation. One facet of a mathematically proficient student is their ability to interpret results in context of the situation and reflecting if their solution makes sense.

They are able to take this reflection and make changes to create a model which is closer to what it should be. This is a real life process. The young man, talked about using the results of the data he's collected to determine what the next step should be. He adjusts factors, tests, and recalculates.

Having students work on performance tasks which require them to examine their works to find tune their assumptions is important and used more in life than having a problem done once and accepting the answer is done so you have nothing more left.

Yes, it sounds like a science experiment but if you are creating a mathematical model of a situation, it often takes several tries to get the right equation. It seldom happens immediately with the first try. Its important to create a situation to help students create models which take several tries to get right.

Let me know what you think. I'd love to hear.

## Sunday, March 19, 2017

## Saturday, March 18, 2017

## Friday, March 17, 2017

### Formulas

I assume most of you have to teach students to rewrite literal formulas in isolation because that is one of the standards we are required to teach.

What I don't understand is why that is necessary. It makes more sense for most students to connect the literal formulas to real life use.

Lets face it, r*t=d and I = V/R are just a collection of letters to most people until its put in context. We use literal equations all the time but not without values.

Most of us select the literal formula for the appropriate situation, substitute values to find the answer for the missing value. I don't know of anyone who rewrites literal equations just for the fun of it. Is it really important to rewrite I = V/R to I*R = V? Isn't it more important to have students substitute values before solving?

I don't think of rewriting the equation, I think of solving the equation with variables. There are now calculators out there where you type in the values and the answer pops out without doing the calculations.

Why is this considered an important skill? Why do we make students rewrite the literal equation in all its ways rather than focusing on showing you are solving a one step equation. If we expect students to be good in mathematics, we need to provide more connections and more real life applications of what we are teaching.

I'm not even sure why this particular skill is still in the standards. I wonder if it is there due to people who have a fond memory of doing this in school. I thought it was a waste when I took math in high school and we are still making students learn this even though they can just find the missing value.

Is this necessary? I don't think so. I think its time to get rid of this particular standard and focus on more important things.

Let me know what you think. I'm in transit till Tuesday.

What I don't understand is why that is necessary. It makes more sense for most students to connect the literal formulas to real life use.

Lets face it, r*t=d and I = V/R are just a collection of letters to most people until its put in context. We use literal equations all the time but not without values.

Most of us select the literal formula for the appropriate situation, substitute values to find the answer for the missing value. I don't know of anyone who rewrites literal equations just for the fun of it. Is it really important to rewrite I = V/R to I*R = V? Isn't it more important to have students substitute values before solving?

I don't think of rewriting the equation, I think of solving the equation with variables. There are now calculators out there where you type in the values and the answer pops out without doing the calculations.

Why is this considered an important skill? Why do we make students rewrite the literal equation in all its ways rather than focusing on showing you are solving a one step equation. If we expect students to be good in mathematics, we need to provide more connections and more real life applications of what we are teaching.

I'm not even sure why this particular skill is still in the standards. I wonder if it is there due to people who have a fond memory of doing this in school. I thought it was a waste when I took math in high school and we are still making students learn this even though they can just find the missing value.

Is this necessary? I don't think so. I think its time to get rid of this particular standard and focus on more important things.

Let me know what you think. I'm in transit till Tuesday.

## Thursday, March 16, 2017

### Animation.

Sorry, about yesterday but my plane was 2.5 hours early and I had to rush out before finishing it. So you get it today.

One of the math classes I'm teaching this semester is an animation class based on Khan Academy's Pixar in a Box.

I say based because I'm using the online material provided with the videos and the practice activities but I added in a more in depth math component.

As we work through the package, I include instruction on the actual math associated with the lesson.

When they did the section on animation, I integrated more instruction on linear interpolation. For the character modeling, students learned about all the different ways you can find weighted averages in real life.

Right now, the students started simulation of the hair. Specifically, the hair of the lead character in Frozen. I've never seen the movie but my students have. They've learned her hair was simulated by looking at the workings of springs because springs bounce the same way as naturally curly hair works.

So as part of the lesson, they are learning to use Hooke's law. Someone one, I know, asked why I was teaching science in my math class. I pointed out Hooke's law is a mathematical equation which students can learn to solve for force, the constant, or distance.

Its interesting the separation of subjects is found even among teachers. You can't solve a formula or equation without doing math. Math and science go hand in hand. I just introduced rewriting the formula to find distance or the constant but its a real life application of math.

I really like exploring the math in more depth so students see exactly what the math is that is lightly touched on. This prepares students for completing the second part of the topic which focuses more on the math but doesn't always teach the details.

Many of the students in my animation class struggle with math or are under motivated. This class gives them a reason, a real reason, to learn mathematics. They love playing with the animation activities but accept they have to learn the math.

I'll keep you posted. I am happy to use Pixar in a Box because its all set and ready to go. The only issue I have is that some days, the internet slows to a crawl. A total crawl and only half the kids can be on at any one time.

Have a good day and let me know what you think.

One of the math classes I'm teaching this semester is an animation class based on Khan Academy's Pixar in a Box.

I say based because I'm using the online material provided with the videos and the practice activities but I added in a more in depth math component.

As we work through the package, I include instruction on the actual math associated with the lesson.

When they did the section on animation, I integrated more instruction on linear interpolation. For the character modeling, students learned about all the different ways you can find weighted averages in real life.

Right now, the students started simulation of the hair. Specifically, the hair of the lead character in Frozen. I've never seen the movie but my students have. They've learned her hair was simulated by looking at the workings of springs because springs bounce the same way as naturally curly hair works.

So as part of the lesson, they are learning to use Hooke's law. Someone one, I know, asked why I was teaching science in my math class. I pointed out Hooke's law is a mathematical equation which students can learn to solve for force, the constant, or distance.

Its interesting the separation of subjects is found even among teachers. You can't solve a formula or equation without doing math. Math and science go hand in hand. I just introduced rewriting the formula to find distance or the constant but its a real life application of math.

I really like exploring the math in more depth so students see exactly what the math is that is lightly touched on. This prepares students for completing the second part of the topic which focuses more on the math but doesn't always teach the details.

Many of the students in my animation class struggle with math or are under motivated. This class gives them a reason, a real reason, to learn mathematics. They love playing with the animation activities but accept they have to learn the math.

I'll keep you posted. I am happy to use Pixar in a Box because its all set and ready to go. The only issue I have is that some days, the internet slows to a crawl. A total crawl and only half the kids can be on at any one time.

Have a good day and let me know what you think.

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