Warm-up

 In Hawaii, lava is 1.9 miles away from the Saddle road.  It is flowing at 21 feet per hour, how long will it be before the lava meets the road?

Using Math To Make Christmas Easier..

Most of us used the age old method for placing decorations on your Christmas tree.  Look at the tree and put it where you see some space.  Make sure you don't clump the tinsel and hope the top can hold your angel or star.  

A professor of Information Theory at the University of Bristol did some interesting research on the statistics associated with Christmas.  Professor Johnson spent most of the pandemic helping explain the various Covid statistics to the population.  As things have slowed down, he decided to explore the statistics associated with Christmas.

He looked at the statistics on decorating your tree, stacking tree decorations, wrapping presents, to selecting favorite chocolates, and of course Santa. He explored the math of everything from what happens if a person miscalculates the amount of time needed to defrost or cook a turkey to figuring out how to seat everyone at the table for Christmas dinner.  

In regard to decorating the tree, most people try to create a random pattern of decorations so there are no two of the same color right next to each other.  Humans are not really good at randomness so we cannot create truly random patterns.  So, let's say you have 100 ornaments to hang on 100 branches, then if you "randomly" place the ornaments on the branches, you'll end up with all the decorations placed on about a two thirds of the branches. This means about 37 branches will be bare and other branches might have up to four ornaments on them. 

Using Maclaurins inequality, they've found that the best shaped box to use to save money and wrapping paper is a cube because cubes have the smallest volume. In addition, the most popular flavor of chocolate at Christmas time is chocolate orange.  If you want to save money on wrapping it, don't buy it in the box with the individually wrapped orange, buy a regular shaped chocolate bar in that flavor.

According to this same mathematician, the 12 days of Christmas song represents the numbers in Pascals triangle.  On the first day, you get one partridge in a pear tree.  On the second day, you get two turtle doves and the partridge in a pear tree.  This means you got 1 on the first day, 3 on the second day because 1 + 2 =3.  On the third day, it would be a total of 6 since 1 + 2 + 3 = 6.  If you do this for every day, you see Pascals triangle with 1, 3, 6, 10, etc.  At the end of the 12 days, you will have received 364 presents in total.

It is well known that those glass ornaments break so easily.  About 400 years ago, someone decided that the best way to store them was in a hexagonal shapes in layers so each bauble touches 6 others in one layer.  The next layer is set so the baubles are over the openings of the lower layer.  It wasn't until 1998 that someone was able to prove this is correct. 

Finally, let's look at those boxes of chocolate where there are a few flavors that are not that popular.  If you say the box has 30 total pieces, where 24 have the preferred flavors and 6 do not, then you can calculate the possibility of getting a less desirable flavor is 6/30 = 1/5 = 20 percent.  This is assuming people eat the flavor they chose but what if they return the nasty flavor and exchange it for a preferred flavor. That changes the statistics and raises the numbers so you are more likely to get a nasty one toward the end of the chocolates left in the box.  Let me know what you think, I'd love to hear.  Have a great day.

Implementing Project Based Learning In The Classroom

 

Last time, we looked at Project Based Learning in general and today, we'll be looking at how to implement it in the classroom. This can be difficult, especially if the students have never done anything like this before.  One cannot just start doing it full blast because we need to teach students how to do projects.  We can create the best learning experience but if they don't know how to do it, they will get frustrated and no one learns  anything.

1. Begin with small focused steps.  Select a few goals you want the students to work on throughout the year.  Focus on helping students work towards mastering the goals while concentrating on growth.  One might only look at minimal scope and sequence, revisit a previous project, and take time to get feedback from students.

2. Look at the project as if you are a student.  Think about the questions they might have in regard to the project.  Have some easy to understand and use resources available to get them started and to help them make sense of the project. At the beginning, help get them going by providing the steps they will need to navigate their way through the project.  All projects require that students practice a variety of skills such as researching, summarizing, problem solving, working in groups, learning to determine if the source is valid, and so much more.  

3. Plan to introduce students to the project and process over several days.  If there is someone at school who is experienced with doing project based learning, get their help, otherwise look for help from an outside source.  Introduce students to the project using some sort of entry or launch event.  This sets the tone of excitement and a need to get it done. Once you have their attention, clarify by setting clear expectations, clarifying the purpose of the project and explaining why they are doing it, taking time to talk about how they will do it (interviewing, research, etc), and what options do they have for the final project.

4. Generate a list of possible project ideas or head to places that already have projects available such as PBLWorks. If you want to look at possible topics, think about climate change, meeting a design challenge, exploring a question like "Is violence ever justified?", conducting an investigation, or taking a position on a controversial topic.  In math, topics might include having students working for the NSA as code breakers in which they have to break a code in order to determine the who, when, and where of a terrorist attack, working on making connections between geometry and real life such as roof pitch being the same as slope, or which house shape has the most room, or building a house for a spider where they have to determine the size of the house, the size and positions of doors, etc.

5. Think about how this experience will be assessed and make sure students understand what criteria they need to meet to be successful.  In addition, do not put too many checkpoints in the time line but let them know that a final product is expected as part of the final assessment. 

6.  Sometimes trying to do everything including the PBL within the 60 minute class period can be difficult.  It is suggested the instructor focus on one or two objectives and arrange them to model inquiry and and design thinking.  Offer instruction on process or concept for a day or two and then give students a day or two to work on the project after the project launch. Do not have PBL time separate from the rest of the subjects so students see a connection. 

This is a framework for actually implementing Project Based Learning in your classroom.  Let me know what you think, I'd love to hear.  Have a great day. 

Project Based Learning Basics.

One way to help students learn the material better is to integrate projects into their learning but it can be hard, especially when the administration wants everyone to follow the pacing guide or at least cover the same material over the same period of time.  

Project based learning is a teaching method that has students using the math they learning doing projects based on real world situations.  Usually, students are expected to work on these projects over a specific period of time from a week to the whole semester.  It depends on the needs of the students.

If you've never done any project based learning, it is recommended that the initial units be short so everyone gets used to doing them.  A period of no more than three weeks is suggested for a successful project.  One important thing is that there are several different types of projects in project based learning.  One type of project is the one used at the end of a unit so students can practice applying what they learned while another project is the unit itself.  

When designing the project, there are seven things a person should keep in mind.  First, there should be an essential question or problem that needs to be solved with just the right amount of challenge.  Second, all students should be engaged in a process of asking questions, looking for resources to use, and putting together all the information. Third, it needs to be real world task, context, and speaks to the students about something in their lives. Fourth, students need to have choice so their voices are heard. This allows them to work in their way, express themselves, and be creative.  Fifth, there needs to be a reflection component where they discuss their learning, how effective they managed the research and the whole project, and obstacles they encountered. Sixth, students learn to take criticism and apply it to their projects to make them better.  Finally, they need to share their finished work with the public.  These are the most important things to keep in mind when planning the project.

Then there are seven teaching practices to apply when planning the project.  First is to either plan or adapt a project based on student context, plan it from start to finish, and include some student choice. Second, know the standards the project meets and make sure it key knowledge and understanding from the math class. Third, promote student independence, growth, team spirit, learning to produce quality work, and include open ended inquiry. Fourth, work with students to organize all tasks and schedules, set deadlines, find legitimate resources, use those resources, create the products, and learn to share them with the public. Fifth, use a variety of lessons, tools, and instructional strategies to scaffold student learning. Sixth, use both formative and summative assessments for knowledge, skills, and understanding.  Assessments should  include both peer and self reviews of individual and team work.  Finally, it is important for teachers to monitor the class to decide when students need skill building, redirection, encouragement, and when to celebrate.

This can be a lot to think about when starting to do your first project with your students.  The Buck Institute for Education has a site MyPBLWorks which has so many resources. They offer a free account so it's easy to sign up for. There are articles on how to build a PBL culture from the start, a template for a letter home to parents, and projects one can use.  At this time, they have 19 math based projects available for grades K to 12. These projects cover everything from creating financial plans, to voting, to reducing impact on the environment, and so many more.  If you are interested in exploring this site and lessons, click on the MyPBLWorks and explore.  Let me know what you think, I'd love to hear.  In the next column, I'm hoping to share more on project based learning. 

Warm-up

 If a single hazelnut tree produces an average 24 pounds of nuts each year and there are 108 trees per acre, how many pounds of nuts are harvested if you have 9 acres of hazelnut trees you are growing?

Warm-up

 

If a hazelnut tree is 12 inches tall when it is planted in your yard and it grows 15 inches every year, how long till it reaches its height of 17 feet?

5 Ways To Improve Math Instruction.

 I suspect many of you are like me in that you are always looking out for ways to improve your instruction technique since not everything works well with your students and what works seems to change from year to year.  These suggestions are made by teachers who have used them rather than the administration trying to find another magic program to boost test results.

1.  It is important to teach vocabulary in context.  Many elementary teachers teach students that less than means subtraction but depending on context, it could mean an inequality rather than subtraction.  Even then it might mean subtracting the first number from the second such as 3 less than x where it is x - 3, not 3 - x. 

It is suggested that when working word problems, divide students into groups of two. Have the stronger reader, read the word problem and then have the other child summarize the problem. Next both students use the summary to decide what operations and steps are needed to solve the word problem. The summaries help students understand the problem and helps them avoid trying to use all the numbers in some way.

2. Use the Concrete Representational Abstract method of teaching which has teachers making sure students have the concepts that are prerequisites to the new concept. The teacher begins teaching students the concept using some form of concrete representation such as manipulatives. Then the teacher moves from the manipulatives to a representational form such as a drawing to represent the concept.  Finally, the teacher moves to the symbolic stage using symbols such as numbers, operations, etc to show the same concept so the teacher takes the student from manipulatives to mathematical symbols.

3. Don't avoid some sort of project based learning because the project helps students deepen their conceptual knowledge of various mathematical topics.  Furthermore, it gives students a chance to see how the math is actually applied in a real life situation.  Up until I attended a presentation on piecewise functions, I'd never seen a real life situation so I didn't know that a piecewise could be used to track the price of a postage stamp.  

4. Find a way to make the math culturally relevant. This might mean helping students see a connection with their community so they develop a personal connection.   I work in Alaska and there was a program done up at the University of Fairbanks where they took Native math knowledge and created lessons from that.  They worked with elders and others to create the lessons and I use them in my classroom since many of my students can relate to the material. 

5. Always break the material down into smaller chunks to make it easier for students to learn.  I know that some of those pacing charts just don't work with most of the students I teach because they shut down if I try to present too much at once.  Research shows, students learn the material better if if it chunked rather than done all at once.

These are just five easy ways to improve your math instruction.  They don't take much but are easy.  On Monday, I'll talk about project based learning ideas to integrate into your math class.  Let me know what you think, I'd love to hear.  Have a great day.