More History With Math.

This past Saturday, I checked for new and interesting math apps for my iPad and I came across one put out by the Midway museum in San Diego California.  I was surprised to find something on the Midway but the authors of the app decided to combine some history with math to make it a bit more interesting. 

The Midway Museum STEAM app has combined information on the aircraft carrier with mathematical problems.  Admittedly there is only one math question per section, this is one of the first apps I've seen to combine history with math.

The app covers 6 areas of the ship.  It looks at the enlisted berth where men sleep in triple decker beds, the foc'sle where the anchor is kept, the galley, the amount of food needed to feed the crew while they are underway, helicopters and the flight deck. 

Each section either has a short video to watch or an augmented reality exploration as a way for students to learn more about things.  After they've checked the video or AR activity, students are given a situation and one math question to answer.  If they get it right, they are told great job but if they miss it, they are told they didn't come up with the proper answer and to try again.  

From a math teacher's perspective, I think I'd ask students to provide their thinking or work as they answer each question.  I would also find additional information to create some sort of sheet to go with this.  For instance, when a student finishes watching the video on the enlisted they are told there are 180 enlisted men and asked how many will occupy a top bunk?  I'd want to know the actual number of enlisted men at various points during it's time of being a part of the navy.

For the information on the anchor, I'd want to know more about the anchor itself such as it's weight, measurements, etc so I could ask additional questions including if we could fit an anchor inside the classroom because most students have no idea how big it is.  We can give them weights but they don't relate to 30,000 pounds but if we said it weighed the same as 5 cars, that is something they can see.

I think it is important to add a bit more math to what the app provides to give students a better idea of how math is found throughout the whole ship but also include activities where they have to relate things to their real lives so they create a connection.  The galley talks about taking 9 pounds of flour to make enough pancakes to feed 100 people and asks how many pounds will be needed to feed 4000 people. I see taking this a couple of steps farther by having students figure out how many pounds of flour will be needed to feed 4000 people pancakes once a week for a month.  Then ask them how many 50 pound bags of flour would need to be ordered for one month, two months, or three months.  Once they have an answer, find out the 3 dimensional measurements of a 50 pound sack of flour and have students calculate the volume the sack is and then ask how much space will the sacks take for a three month voyage.  

By extending the basic questions, it gives students a better feel for what the ship is like.  It could be made into a project where the student is in charge of ordering supplies for a one month voyage.  So in addition to the amount of supplies, students can calculate the total cost of the order.  A real life based project.  Let me know what you think, I'd love to hear.  Have a great day.

Visualizing Combining Like Terms

We have been told to provide visual representations for everything we teach in math.  Unfortunately, the visual of us use tends to be pointing out the squared or cubed associated with the variable.  We assume students know the difference between them but if that is the only way they've seen the terms, they wouldn't know how to visualize them. Without a visualization, they won't see the differences between a squared or a plain x or a constant.  

I took a class this semester on teaching math either remotely or via a hybrid model.  In it she said graphs work as visualizations but what do you use to show the differences between x^2 and x's visually.  I had to think about it but realized I could have students use either algebra tiles or jam board to help them visualize why you cannot combine the two. 

I looked at Jamboard and it is possible to create the large squares, rectangles, or small squares in different colors so they have one color for positive and another for negative.  It would require setting up ahead of time and making copies for each student.  

Jamboard allows students to move pieces around so they can group the like terms and then count the final total for each group.  This can be done by students via distance learning as easily as ones in class.

On the other hand, an easier solution is to use an Algebra Tiles app or online version so students can make as many x^2 as they need in two different colors so they can easily identify positive and negative terms.  They can make as many of each as is needed and move them around  to work out problems.

In addition, this type of visualization can help connect the dots on why the terms in the second part has to change signs for each term following the subtraction sign.  It supports the explanation of subtracting from the original.  

I realize that both of the suggest apps rely on having a mobile device and possibly the internet but what about students who only have a phone with limited data or no internet access?  How do we allow them to do the same exploration as those who have the devices.  I don't have Algebra Tiles as part of my classroom supplies and if I did, I don't think I'd be allowed to loan them out.  Fortunately, there are templets available on the internet.  These can be used to copy tiles onto colored paper to make a physically based set.  There are templets here or here.

Since many teachers are both teaching virtually and have students who do not have internet or computer access, we may have to provide some sort of manipulative with instructions so that we meet the needs of all our students.  This version might require us to send home Algebra Tiles, a mat for them to work on and directions both written and visual.  If a student has a computer at home without internet, the teacher can send a thumb drive home with the video.  If the student does not have a computer, one can create a series of photos, print them out, and send them home.

I have to create two packets worth of work for the first two weeks after Christmas break and I have to figure out how to create the support materials since at least half my students do not have internet access so I'm figuring out how to do it.  Let me know what you think, I'd love to hear.  Have a great day.

Pearl Harbor + Math.

 

On December 7, 1941, the Japanese bombed Pearl Harbor, sinking the Arizona and damaging several more ships causing so many to die.  In addition to Pearl Harbor, a couple of other military places were hit, in addition to quite a few civilians.  This is one of those topics that can be used to create a cross curricular unit connecting history and math.

Many years ago, I helped create a cross curricular unit with math, science, history, and social studies about Pearl Harbor.  Although math can be one of the harder  topics to find this type of activity.

I created three units for students to work through associated with the bombing of Pearl Harbor and the battle of Dunkirk.  I admit that it took a lot of research on my part to find all the information but once I had the units finished, it was worth it.

I'll start with my unit on the battle of Dunkirk in which a flotilla of mismatched ships and boats managed to move over 300,000 people from the continent to the United Kingdom within a short time.  I researched the length of the route, the number of people moved each day, and the type of ships and boats used to evacuate the military after I showed them a video clip on the event.  Students looked at maps, numbers, to see the size of the operation.

For the mathematical part, I asked students how many boats they could scrounge from around the village, and then how many total if we included the next closest village.  In the process they had to figure out how many people could be moved per load.  I had them assume one round trip per hour.  They had to calculate the number of hours it would take people to move the same number of people and then convert the hours into days, and weeks. At the end of this activity, students were amazed at the results and impressed.

I also created a unit on Japanese mini subs, the type that snuck into Pearl Harbor during the attack.  I had to research to find the type of submarine that carried the mini sub from Japan to Hawaii, it's size, the speed of the "mother" submarine and mini subs, etc.  Students used the speed of the mini sub to calculate how long before the bombing they had to leave the mother sub to get to Pearl Harbor, the number of cubic feet the pilots had to fit in as they travelled, and the speed of the mother submarine to go from Japan to Hawaii to place the mini subs in position.  

I used this exercise as a way for students to understand what the Japanese commanders had to consider as they planned the attack.  I admit, they probably used subs closer to Hawaii but I wanted them to see what went into planning something of this size and distance. 

The final activity had to do with the balloon bombs the Japanese released into the air.  Some of these travelled all the way from Japan all the way the states like Oregon, or Washington.  Research is a wonderful thing because it allowed me to find the size of the balloons, the amount of sand used as ballast, the distances from where the balloons were launched to where they landed, the speed of the jet stream and the amount of bomb materials included.  

I included a map of the Pacific Ocean for students to mark down the places bombs landed and draw lines from the place of launch in Japan to the bombs.  They marked down the distances for each one.  Once they had all of this done, they needed to calculate how many hours it would take as a minimum for the balloons to travel to each place.  In addition, they needed to calculate how much paper was needed to create the balloons which required them to calculate the surface area of a sphere and use the results.  They also needed to calculate the total amount of sand needed for all the balloons.  Again it showed students what the Japanese had to calculate before they could even launch the balloons.

I could just as easily have researched the trajectories involved in guns firing from on board a ship, or used trig to determine how thick the steel should be on a ship to prevent ammo from penetrating, the approach of a plane to an aircraft carrier or angle of take off from the deck of a ship.  So many possibilities.  I admit, it will take quite a bit of research to find the information but it is worth it because it makes some of these events more real.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If a dolphin normally swims at 6 miles per hour but has a top speed of 20 miles per hour, what is the percent increase from normal speed to top speed?

Warm-up

 

Give your age without using years.  For instance, I am one quarter, one dime, one nickel and three pennies.  My friend will tell you her age is equal to four 12 foot ladders.  How old are you?

Near and Far Transfer..

I love researching new topics because I’m often lead to new ideas and thoughts.  During my research on the two types of knowledge, I ran across some new information on transference of learning.  Information that helps me understand why students often time have difficulty in taking the information they learned in one situation and transferring it to another situation.

I just discovered that transfer learning can be divided into two types.  The first is near transfer which involves skills and knowledge which are applied the same way every time while the second, far transfer, is the ability to transfer those skills and knowledge to other situations.  An example of these would be for near, the students are taught to calculate percentages and can apply them to any problem in the book. This is an example of near transfer because students can apply what they've learned to the same type of problems.  To be considered far transfer, students would have to be able to go to the store and calculate the percentage taken off a jacket based on the original price and the new price.  

This is where so many math teachers get frustrated.  They see students can do the problems and can pass a test but when they are asked to use the same knowledge in a more real world situation or to a situation that is not identical to the context of what they learned, students can't and we wonder what happened.  Now we know.  They've mastered the material as a near transfer but not a far transfer.

One thing that's been noticed is that real world application is often times more complex than the problems students have solved in the classroom.  I've heard it said that real life is so much messier because the answers are not as neat and tidy as most problems students experience in class.  Consequently, it is possible that the problems taught in class are too simplistic and this can make it much more difficult for students to perform far transfer.

Fortunately, there are some things teachers can do to help students learn to apply knowledge via far transfer rather than remaining in near transfer.  The first is to engage students in working with real world applications of what they've learned.  Unfortunately, this can be much harder in Math because many teachers, including myself, do not always know how certain topics and concepts are used in real life.  Next, it is important to help connect what students are currently learning with what they have learned in the past.  For instance, we teach students to solve equations and inequalities the same way so as we teach inequalities, we can talk about solving regular equations.

 

If students are given extensive practice for routine skills such as addition, subtraction, multiplication, and division so students can perform them quickly and accurately.  Being fluent and comfortable, also contributes to a better near transfer because students can focus on the concept rather than struggling with the arithmetic.  As far as assigning problems for a specific skill or topic, it is important to assign a variety of problems because the more varied contexts students see, the easier it is for them to transfer the knowledge from near to far learning.  This means that instead of having all problems with the equal sign on the right, make sure it is appears on the right side.

Furthermore, it is important to for the instructor to point out the underlying principles are used in different situations so students see how to use them in a variety of situations.  This makes it easier for students to utilize far transfer because they've been learning to recognize the underlying principals and then apply them.  Finally, students need to be taught to reflect on their own thinking so they can improve their learning.  

It has been suggested that all learning goals be written in two parts.  The first part covers the procedural objective which is the part where students learn the steps such as in learning to solve two step equations.  The second is the declarative part which involve the conceptional knowledge or how to apply it to a variety of situations or contexts.  The second part is so much harder because I've never thought of doing it.  Let me know what you think, I'd love to hear. 

Inert vs Generative Knowledge.

When we teach math, we usually seem to want to teach procedures without taking time to focus on having them see how everything relates.  The other day, when I read up on curriculum, I ran across a reference to inert and generative knowledge.  I honestly don't  remember hearing about either.  Knowledge was knowledge.

Briefly inert knowledge is knowledge that is not used while generative knowledge is used to solve a problem.  Remember back in elementary when you had to memorize all the state capitals and never did anything more with those?  That is inert knowledge but if you used the information to write to the governor of each state, then it becomes generative knowledge.

There are three types of inert knowledge.  The first is that the knowledge is there but not accessed while the second states there is a problem with the structure of the knowledge.  The knowledge is in a form that cannot be applied and the final is there is an issue with the situational usage of the knowledge.

In math, we often teach students process used to solve  various types of problems and we give lots of practice problems but we do not provide situations that require students to apply the processes to solve problems.  In fact, most real world problems found in textbooks are neat and only require students to apply the math learned in the section.  

Generative knowledge is often referred to as generative learning.  In generative learning, it is believed a student is not going to learn the material as well as when they are able to construct meaning by generating relationships between what is learned and it's usage. In other words, they are generating understanding.  If teachers do know help students generate their own understanding for each new topic or section, then they will know how to use it to solve problems, otherwise each topic or section will be treated as in isolated skill that cannot be applied to problems unless the student is taught to explicitly apply it.

This may explain why students seem to know what they are doing but are unable to apply it when they take the state test, or a district test.  This is because it remains as inert knowledge rather than being moved to generative knowledge.  Fortunately, there are ways to help students to this.  One way is to ask questions of students that make them look at similarities and differences between processes or topics.  Another way would be create discourse that encourages discussion, debating, or generalization.

In addition, it is important to create situations where students can apply the knowledge they are learning so it is no longer inert.  For instance, when I teach solving one and two step equations, I take time to show how the same equation written in a general form can be graphed as a line and how the solution for x and the answer (y) is a point on the line.  In other words take the problem 2x + 3 = 7.  We solve the equation to find x = 2. I teach that x = 2 and y = 7 and that point is on the general line 2x + 3 = y.  

I try to relate topics or concepts to things students have had before but I need to provide more activities and discussions to help students create their own understanding rather than trying to do it for them. It is hard sometimes because they arrive in high school not having had a lot of experience creating their own meaning.   Let me know what you think, I'd love to hear.  Have a great day.