Happy New Years Eve

 Happy New Years Eve.

Math For New Years

It is time for the New Year and most schools are not in session but these activities would be good as a way to let the students ease into the school year after being off for a couple of weeks.  So I found a couple activities that focus on New Year's itself.  

First off is one from Yummy Math on the Times Square ball, the one that is dropped at midnight in New York City.  The ball is actually a geodesic icosahedron covered in Waterford crystal triangles.  The activity allows students to examine the ball in more detail. It includes a three page worksheet to do and has a video students can watch to see how a truncated polygon happens. 

There is also an activity from Yummy Math that looks at the year of the rat from January 2022.  Although it is a lunar new year, it can be used to show that different groups may have their year begin at a different point and explains more about Chinese culture.  In this activity, students learn more about the Chinese calendar and how it works.  Students have the opportunity to figure out which New Year it is for 2023 to 2028 or so. They learn the patterns used and so much more.  At the end, they get to answer questions about the calendar and the New Year based on their work.

Another site has several worksheets which are great for the New Year.  Some of the worksheets listed on the page are free and some you need to be a member for.  I'm only reviewing the free ones. For instance, one asks students which discount is better when you buy fireworks or party supplies.  This worksheet is made for younger students so I would ask students to show via mathematics that their answer is correct.  This would add an element of communications. 

The other activity for this site has students practicing identifying reflection by placing the dot in a new location after it was reflected over the x-axis or y-axis.  Both worksheets have answer sheets available so students can check their own work. 

Finally is a short video on the the math of time zones and how they work in regard to celebrating the New Year.  If you watch television, you'll start celebrating with Times Square and continue each hour until you reach your region's celebration. So you have several options to explore various aspects of the New Year in your math class.  Let me know what you think, I'd love to hear.  Have a great day.

Winning At The Gift Stealing Game.

 

I am going to apologize for not having my usual entry this past Monday but I had been traveling and didn't have the ability to access the internet.  I took a semester long job in a beautiful place in Alaska and I'm finally getting back to normal.  Now, on to today.  We are going to look at the best way to win at the gift stealing games.  

One mathematician, decided to use mathematical modeling to determine how one can best "win" the game. The game is played where everyone brings a wrapped gift, puts it in a pile and each person in order has the choice of selecting a gift from the pile and opening it or selecting a gift that someone else opened so you know what you are getting.  If a person has their gift taken they may choose a new one or take an already opened one. Now if you are the first person, you can only choose an unopened gift and if you are the last person, you know what everyone has gotten so you can make a better guess.

The mathematician used agent based modeling which is used for everything from companies bidding in the electricity market to the way the human immune system works.  The person decided to have 16 people with 16 gifts for the model. In addition, all opened gifts were assigned a value of 1 to 10 based on how desirable they were.  The likely hood of the opened gift being taken increased if it was rated at above a 5.  In my family, if the gift was some sort of alcohol, it was automatically at the top of the list for being taken.  

Furthermore, it is known that there are multiple variations in which people play depending on the family.  The fairest rules are the ones that allow a gift to be taken multiple times during each turn, if a person holds the same gift  three times, it cannot be taken, and after the last person has chosen their gift, the first person is allowed to take a gift.  

It is possible to utilize the three rounds and the gift becomes locked rule if you utilize two co-conspirators.  Say you take friend ones gift, friend one takes friend two's gift, and friend two takes your gift  so each of your friends have the gift three times and can keep you.  You can go after another gift so they get what they want.

It is important to steal gifts in this game because the model indicated that participants are 75% less happy with their gift if they don't steal any gifts and steal something even if you don't want anything.  Think about stealing a gift that someone wants so that when it is stolen from you later in the game, you have a wider choice of games to choose from.  So now you know how to mathematically use the rules to help you get the gift you want.  Let me know what you think  

Happy Boxing Day

Happy Boxing Day to all. 

Seasons Greetings

 

Happy Christmas Eve

Happy Christmas Eve  

Bad Weather

Weather and internet issues. Should be back later.

 

Chat Stations

 

When I was researching Hexagonal Thinking, I came across a discussion protocol called "Chat Stations". This is a type of learning center but one with strict protocols for having students talk to each other about a topic.It enhances small group discussions.  In addition, this is a type of cooperative learning tool and it is one I'd never heard of before. 

Chat stations require very little prep but are designed to encourage conversation among students rather than just having them sit around a table and it encourages movement between discussions.

A chat station usually has a discussion prompt that goes with a photo, picture, or object associated with the concept or topic being taught. In math, the prompt might be asking them to analyze the problem to determine what possible equations of with the picture, talking about everything they know on a topic, deciding which answer is wrong and explaining why, list the sequence of steps needed to solve a problem but they don't have to actually solve it, which equation is needed from the reference card to solve the equation, etc.   Each group needs a piece of paper that is divided up so it has enough individual cells to match the number of stations. This is where students will record their thoughts and answers associated with the prompt for the station.  

As far as groups go, each group should have no more than between four and five students but the number depends on how many students are in the class and the number of stations available. In addition, each group should only be at a station for three to five minutes because any longer and the conversation will turn to something else.  You want them engaged so this short time is perfect. Once the time is up, have the students move to the next station, chat, and move one.  

When students have rotated through all the stations, it is time to call everyone back together to discuss their findings from each station.  When asking the groups, choose a different group to ask them for their results  for each station.  This group starts it off but you can call on all the groups to contribute.  The teacher writes down any insightful thoughts or comments from the students. In fact, these chat stations encourage whole class discussions because all the students have explored the topic in smaller less threatening situations.  

Often times, chat stations are used when students are starting a new concept or topic.  It is a way to trigger prior knowledge so they  have something to refer to. Chat stations can also be used as a lesson opener, a way to summarize the lesson, or as a way of integrating writing into the math classroom, and as a way to communicate mathematically. 

When students are at the chat stations, the teacher can go around, monitor the discussions, do a quick assessment to see what they do or don't understand, help if a group needs a bit of a nudge, while encouraging student independence.  So if you have never used chat stations before, give them a try, especially since they break up the normal day, let students move around, and have a chance to talk.  Let me know what you think, I'd love to hear.  

Hexagonal Thinking For Project Based Learning And More

It has been suggested that when we have students work on project based learning activities, we help them by introducing them to something called hexagonal thinking.  Hexagonal thinking is an activity specifically designed to help students learn to think critically, make novel connections, increase discussion while providing evidence to support their reasoning.  This is done by visually connecting ideas that have been written on paper or using a hexagon either digitally or on paper.
Picture this if you will.  You have a bunch of empty six sided figures or hexagons.  You write a name or idea on each hexagon and then assemble the hexagons so ones that are related or connected in some way touch upon each other.  The nice thing about this is that every person and every group will assemble the cards in a different way.  Hopefully students will question some of the connections and discuss which connections work better.  

In addition hexagonal thinking helps promote a more rigorous project in project based learning activities.  There really are three levels of rigor when thinking about PBL’s.  The first is composed of simple I know skills or ideas which is the lowest level.  The second encompasses a more deeper level where the student is able  to relate the skills or ideas.  The last level is the transfer level, so students are able to apply the skills or ideas to other contexts and this is where we want our students to end up.
Within the six steps of hexagonal thinking, these three levels are applied and students move deeper in their understanding.  Each step is designed to help students go from the entry event which helps build the need for surface and deeper learning.  The first step is the entry event or launch.  This is where the context or contexts are created for the students.  The entry event might be listening to a podcast, watching a situational video but it needs to initiate the intellectual engagement.

The second step is for surface exploration.  This is where hexagonal notes are passed out and students are asked to write down key terms, the context and content which shows relationships.  Context refers to the situations where the content applies and content is the specific knowledge or skills to be applied.  If students are not sure where to start, supply them with a word bank to use.
The third step is where students move on to a deep level connections.  This step is done using smaller steps or parts.  Step one is to have students discuss the connections between the words they have on the hexagonal notes.  They can also write down the connections but they need to talk about connections between content, between context, and between both.  Once they’ve done this, they are then asked to compare and contrast their results with the results of another group.  

Step four is designed to help encourage transferring knowledge by asking students to think of other contents and contexts associated with this project.  This is where they might create analogies, compare contexts across groups, speculate on how their configurations might change over time, and draft any questions they have yet to solve.  In step five, students share their thinking from the previous step to help identify the driving questions, learning goals, and success criteria.  One way to do this is by using structured protocols such as chat stations.  

In the final step, the class decides which questions they will work on to answer. It is here that students can write down what they know and what they need to know, what steps they will take to answer the driving questions, and how they know they are successful. Although this is designed to help students get started doing their projects, I can see where it could be used to help students learn how math is used in the everyday world.  For instance, slope is found on road signs, roofs, ramps, and so much more.  This is a way for students to make the connections on their own.  Let me know what you think, I’d love to hear.  Have a great day.

Warm-up

 You and your family harvested 2789 kumquats. If there are 35 kumquats per pound, how many pounds did you harvest?

Warm-up

 If a kumquat tree grows 18 inches a year and reaches a height of 15 feet, how long does it take the tree to reach its full height.

Writing Better Math Journaling Prompts,

 I have tried using writing prompts in the past with less than spectacular results. It might be that I have never learned to write a proper prompt and have had to rely on prompts I found in books or on the internet. A lot of prompts I found didn’t seem right but I used them anyway because they were available. The other day, I came across something that explains how to write better prompts. Ones that will help students think more deeply and not just regurgitate the lesson.

It has been suggested that teachers move towards prompts that encourage multiple solution paths and away from ones that have students simply retell everything contained in the lesson, or just rewrite the steps needed to solve a problem. Prompts should appear in a variety of formats that encourage more writing, more reasoning, and even debates or arguments. The prompts need to be carefully constructed. 

Unfortunately, this not happen in the math classroom as much as it should due to state testing and mandated curriculum and pacing requirements. What is nice, is that with a bit of adjustment, it is possible to create opportunities to engage students in higher order thinking, writing, and conversation. One can make small changes to already existing material to make them more effective. 

Instead of writing the prompt to tell students how to do it, the language needs to be changed so the student has to do more work.  For instance, rather than writing "Use a drawing to show how to add two fractions together", change the wording so it is more like "Describe an efficient way to add two fractions." This makes it more open ended which requires students to do more thinking.  

They have to decide what way to use to show how to add fractions.  This could be done by using a number line, a drawing, or even photos of manipulatives.  This leads to higher level thinking and it provides more opportunities for multiple solution paths. If there is a fear students won't write much, it is easy to rewrite the prompt so the student is asked to provide two to three different ways of solving the problem and have them include their solutions with an explanation of why all the ways are accurate.

Another problem with many writing prompts is they ask students to explain how they solved it.  This is often interpreted by students to mean they need to write down the steps they used to solve the problem.  Instead of asking students to calculate the volume of a box of cereal and explain their answer, ask the student to explain the meaning of each term in the volume formula to a friend.  Think of using phrases like "Explain to your friend how you solved......" or "Describe the meaning of........" or "Explain the patterns you found...........".

These types of prompts allow the students to communicate their understanding of a mathematical. concept in the way that they answer the question. For instance, if they use more mathematical terminology correctly rather than general English vocabulary, it shows a higher level of understanding.  If the written answers are constantly  vague, then one should talk to the individual to determine how much they understand.

Finally, look at prompts that ask students to defend the validity of their answer or friends answers to a specific problem.  This allows students to see there are multiple pathways to get to a solution, see the common errors that occur, and help them become mathematical writers and thinkers.  If you have a prompt which asks if the work of a student is correct, rephrase it as two students are debating their answers.  One has one answer, the other another answer so students have to write a text, an email, or answer to both to explain who has the correct answer and why it is correct.  

When prompts include the phrase "How do you know" offers the opportunity for students to explain how students approached solving a problem, provides an opportunity to practice mathematical writing, and argument while practicing higher levels of critical thinking.  Let me know what you think, I'd love to hear. Have a great day.  

One Way To Encourage Note Taking Via Assessment.

 

In math, we are always trying to encourage students to take notes and them use them.  Many teachers even have students copy down notes so they have them but then need a way to have them look at the notes later.  One teacher found a way to do this. Although she teaches government classes, the technique works as well in Math.

The way this works is that students take the test twice.  First they take the test without using their notes and then they take the test a second time with access to their notes.  The two scores are averaged to provide their final score.

When notes are taken by hand, it improves retention of material and improves understanding. I realize students can have a difficult time staying organized if they take notes on sheets of paper, I usually have students take notes in either a composition or spiral bound notebook.  They write in the notebooks and their notes are there.  The other advantage to this is it allows the instructor to spend time teaching students how to take notes since few learn how to do it.  It is an important skill for college or training.

If you use a system like Schoology or Power School, you can modify it so it allows students the chance to take the test twice and average the scores. When they take the test first without using the notes, they must rely on what they have learned through classwork, notes, and studying.  In order to get a higher grade, they need to do well on the first test due to averaging the test scores. When they retake the test, they retake the exact same test using their notes.  

One thing to remember is that students need to be taught how to take notes.  We need to show them a method to use so they have a way of organizing their thoughts.  I've use Cornel notes in the past because it allows for additional comments and learning. One way to teach students note taking is to give a lecture with examples of good note taking technique. Take time to grade student notes while providing feedback on their notes.  In addition, give open note quizzes so students get practice looking at and using their notes.

As the teacher, you decide when and how often to do this type of assessment.  This could be used on the major tests rather than every assignment, quiz, or assessment.  It is acceptable to tell students they will be able to use notes on tests but not on which tests.  You would never use the double testing method for any pretests.  It works best on material that is much more complex such as algebraic fractions.

You can include short answer questions which ask what the next step is, or why would you do this at this point, or something similar. If students answer these correctly on the first test, let them skip them on the retest because they know them but they do need to retake any calculations problems. No matter what type of questions you use, you need to look at the length of time you have in class.  When you have them take the test twice in a row, that takes time so you may end up with two days being taken up.  In addition, the length of the test needs to be set so students have time to really work the problems.  I had a professor in college who felt that if he could do it in 10 minutes, we could do it in the regular class period.

You need to always grade both tests because some assessment software does not always read the way the answer is inputed.  Once students have taken both tests and you've graded them, you can let them know what their final score is.  On Friday, I'll explore the question of how to get your students more independent so you are not doing all the thinking for them.  Let me know what you think, I'd love to hear. 

7 Ways To Differentiate Math Instruction.

 In today's semi-post Covid landscape, we have students of differing abilities and we need a way to meet the needs of all students.  This means we need to think more about differentiating instruction so none of our students feel left out. Differentiating math means we think about changing up the way we do things, think outside the box, and not get into the habit of doing the same thing every single day.

Differentiated math instruction is when an instructor uses a collection of techniques, strategies, or adaptions used to reach a diverse group of students so math is made accessible to everyone. When one differentiates a math lesson, one is providing a variety of entry points or exit points designed to support student thinking.  This makes math accessible to all students and no one feels left out.

One way is to set up a series of math centers or stations that students can work their way through.  Centers might include watching a video, reading an article, solving a word problem, or doing an activity. Once the teacher has given the whole class lesson, students break up and work their way through the centers, spending 10 to 15 minutes at each stop.  Math centers are great because they help facilitate independence, small group learning, and gives the teacher some time to provide additional support to struggling learners. It is important for the teacher to customize groups and centers to math the needs of students better.

Next, look at using activity or task cards that allow students to decide what they do and the choice gives students more power.  Activity cards might be math problems, tasks, or questions and the material should span several lessons while offering students the opportunity to work individually, in small groups, or with a partner.  

Another possibility is through the use of choice or menu boards since they provide students with a way of them making decisions about their learning.  What ever type of choice board you choose to use, they have to focus on specific learning needs, interests, and skills.  Use of choice or menu boards tends to increase student ownership because they can pace themselves and decide how they will engage with the information, and show what they've learned.

In addition, look at having students fill out math journals.  When students write about math, they can reflect about their learning while having the chance to practice English, especially in written form.  This is a great way for ELL students to practice their language skills while giving all students the opportunity to practice communication.  Students can summarize key points, answer open ended questions, connect math with everyday life, or write about what they find most challenging.  To make the entry point good for all students, do not set a minimum amount they must write.  Give them the choice to write as little or as much as they want or even let them draw their ideas.  

A slightly different idea here is to set up learning contracts.  One way to do this is to ask students to reflect on their learning, set learning which includes what skills they need to learn, or which skills they want to improve, or the areas they want to explore.  This is one way to set up personalized learning plans and these can be done at the beginning of the year and have students revisit these on a regular basis throughout the year.

Don't forget to use math games in class because they are fun, motivational and help students to deepen their mathematical thinking and reasoning.  When using a game in class, make sure the games learning objective matches up with the mathematical objective.  Always have a variety of games to use so you can change them out. Some games might involve the whole class while others allow each student to play individually.  It all depends on what is needed.

Finally include digital practice for math. Look for websites and apps that are specifically designed to reinforce the current material being studied.  Make sure the apps or website is not timed so students do not develop anxiety and are more likely to learn the material.  Using these type of digital materials can increase the level of fun and participation without discouraging them as much.  

When it comes time to implement these strategies, you don't have to do them all at once.  Think about introducing one at a time so they get used to doing them but these are all ways to differentiate math instruction. Let me know what you think, I'd love to hear. 

Warm-up

 Once the flow reaches the Saddle Road, it will cover it.  So if the lava continues flowing at 21 feet per hour and the road is 10 feet wide including the entire paved road, what percent of an hour will it take to cover the road?

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.