Turning A BasketballTournament Into A Learning Experience

  Its time for the annual three day basketball tournament.  The tournament begins today and ends Saturday and involves 7 schools bringing both male and female teams.

Its kind of hard to teach classes when our school is on the court so rather than fight the kids to get them into class, I took them to watch the games but they still had an assignment.

Their assignment was to select a player on one team and mark down all the baskets, attempts, steals, etc that player made.  They had to collect data for each game. Basically, they'd follow that player whenever that team played. Of course, they also had to monitor someone on our team because of school loyalty.

At the end of each match, I collect the forms and hold on to them till Monday.  Every game, students received a new form so by the end of the event, I had a stack of data for the students to analyze.  Come Monday, students are given all the data they collected to analyze.

I expect them to calculate Field Goal Attempts, Field Goals Made, Rebounds, Steals, Assists, Turnovers, Free Throw Attempts, Free Throws Made, Fouls, and Technicals for every player they followed.  Once all these stats are calculated, I ask students to enter it into a shared google document that all students can check.

Once all the stats have been collected and entered, it is time for students to decide who they want for their perfect team.  Students  have to explain why they chose each player, why they are assigned the position they have and justify it using the data from the shared documents to support their choices.

Some years, I pair students up and other years, I ask students to do it individually but either way, at the end, students are expected to share their ideal team with others in class and they have explain their choices.  This adds a speaking component to everything because my students hate speaking in public and avoid it at all costs.  If they are too shy to do it in front of the class, I'll let them make a short video I can play in front of the others.

This is my version of Fantasy Basketball because it uses real players, real stats, a chance for them to apply mathematical principals to a game they play year round. In the summer, they play on wooden decks and in the winter, they play at lunch or at open gym.

Let me know what you think, I"d love to know.

2 Ways to Improve Mathematical Discourse.

I think about half to three-fourths of my students are classified as ELL which means their language development is below a certain level and they are not considered fluent.  This often leads to some really interesting discussions with words like "stuff" and "thingy".

Fortunately, there are some great ways to improve mathematical discourse.  The thing to remember is that mathematical discourse is not just discussion but also diagrams, symbols, words, technology, and models to convey and defend their ideas.

One suggested way is instead of the standard telling them via a lecture is to create something much more interactive than the old question and answer sessions where the teacher conveys information via questions or statements.  Unfortunately, only a few students tend to be involved in providing answers but there is technology out there that can encourage more discourse, especially for students who are shy.

1.  Google Slides offer students the opportunity to share  their knowledge in a safe way so students who might not normally participate will.  Some of the ways google slides can be used to encourage mathematical discourse is by having students create visual representations of problems, explain how the did a problem, or how they visualized a problem.  If students are given the ability to comment on other slides, they can ask questions, post observations, or see others solved the same problem. 

Furthermore, slides allow students to post their thoughts on open ended math questions, or they could post solutions to any "Which one doesn't belong" activity or post their solution to a three-act task.  Each activity requires students to communicate their thinking which is so important.

2. Flip Grid - is a wonderful way for students to record themselves while explaining a problem.  If they don't like showing their faces, they can wear a mask while making the video.  Flip Grid allows students to explain how they worked their way through a problem, share ways they've found math in the real world, share their answers to open ended math problems, or discuss a diagram they made of solving a problem.

Flip Grid can also be used by students to explain their answer from a "Which one does not belong" activity or a three act task or any rich math activity.  They can record themselves explaining their answer in detail and if they want, they could include visual displays of their thinking. 

I'll share additional ways of improving mathematical discourse another day.  These are two easy ways to begin getting more students involved in expressing their thinking.  Let me know what you think, I'd love to hear.

Solar Geometry

Solar Geometry refers to the position of the sun on any day in relation to the earth.  It is used when finding sunrise and sunset, determining where to place solar panels on a house, best direction to build a house for maximum solar exposure so it doesn't get too hot in the summer.

First off, the location of the sun is represented by two coordinates, much like a point on a graph.  Instead of using x and y, they use the solar azimuth which refers to the clockwise movement between the sun and the cardinal direction of true north and solar altitude or elevation which is the angle of the sun from the horizon.

Then there is solar declination which is the angle of the sun's ray as it is extended to the center of the earth and the equatorial plane.  If you look at a graph of the solar declination over the period of a year, it resembles a sine curve between 23.5 degrees and -23.5 degrees.  At the Vernal and Autumnal Equinox, the declination is zero while at the Summer Equinox is + 23.5 and the at the Winter Equinox it is -23.5 degrees.

There is also solar noon defined as when the rays of the sun are hit perpendicular to the planet based on the longitude and noon occurs at exactly the same time everywhere along the longitude.  These are some of the basic terms. 

Penn State has produced a 68 page activity guide covering everything from longitude to latitude, to earth's orbit around the sun but towards the end, it provides information on solar position.  This section includes the three steps needed to calculate solar time using standard time, an equation of time for the position and date, and the longitude correction. The longitude correction is expressed in either degrees or radians based on preferred use. All the equations needed for students to calculate the solar time for their location or any location around the world.

Once the solar time is calculated, the activity has students calculate the solar declination for a specific location. Now a student has everything needed to calculate the sun's position in the sky.  This information is important if someone is planning to install solar energy.

This site has some great diagrams complete with all the mathematical equations for solar geometry. The diagrams provided show how the angle of the sun changes the amount and direction of light which is used to create a sun's path for a specific location which is used when planning and installing solar energy.

Finally, Teaching Engineering has several lessons which could be used to reinforce the above math.  One lesson has students learning more about solar angles and tracking systems, determining the amount of solar energy available at a specific place and time, learn about the maximum power point of a photovoltaic cell, and ways to increase the amount of solar energy hitting a cell. 

Let me know what you think, I'd love to hear.  Have a great day.

Geometry and Sunrise/Sunset.

The other day, I was rewatching NCIS and Jethro Gibbs made a comment about "Which sunrise?  There are three.".  I had to go check that out because I'd never learned that.

The three based on the position of the geometric center of the sun in reference to the horizon. Geometric center is the exact wording used by scientific writers.

There are two twilight times every day.  In the morning, we refer to it as dawn and in the evening it's called dusk.  If a location is far enough north or south, it may not experience all three twilights every day.  It just depends.

First is civil twilight which occurs between 6 degrees below the horizon and above.  Its the last one, the lightest one and only the strongest stars are still visible.  This is the time when Venus is still visible.  In the morning, it is from 6 degrees below and up while at night it lasts until the sun gets past the 6 degrees below the horizon.  Civil twilight is used by many countries for laws on aviation, hunting, etc.

Second is the nautical twilight, lasting between 12 and 6 degrees below the horizon.  It gets it name from history because sailors used this time to get bearings for their navigation.  During this period, there is enough light being produced to see the horizon, so it was light enough for sailors to use the horizon when determining their headings, etc. On a good day, it is possible to see ground based objects but not with great detail.  The United States military uses this definition when they are planning tactical operations

Finally is the astronomical twilight defined as going from 18 to 12 degrees below the horizon.  It is the darkest time and is most interesting to astronomers.  It is also the darkest of the three with no light in the sky.  Most people call this night because the lights have not yet begun to show

I found the equations used to calculate sunrise and sunset times locally in a paper on Solar Geometry  Notice the equations use trig to calculate dawn and dusk.  The times are based on a 24 hour clock with noon being 12:00

The first tangent term represents the latitude of the specific location where the local sunrise is being calculated while the second tangent is the Sun's declination angle for that day.

Tomorrow, I'll share more solar geometry with you.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Warm-up

Applying Retrieval Practice in Math

 Sometimes, we read about these cool practices but we run into problems figuring out how to implement them.  Fortunately, Retrieval Practice has the information we need in the form of free downloads, etc.

Retrievalpractice.org is a website run by Pooja K. Agarwal to provide current research and ideas on applying this to the classroom.

They have several free ebooks to help people learn to implement these strategies including one on math.

The math volume is "Interleaved Mathematics Practice" or ways for students to learn what they really need to know.  This 13 page book discusses the standard way most textbooks arrange patterns and why it gives a wrong impression before explaining how to change things so as to make practice better for learning. It also provides evidence for this method and explains how to create the proper interleaved practice.  The thing I like best about this book is they also provide cravats for when it may not work and why you need a certain type of practice.

There is also a book on using retrieval practice to improve learning.  The 12 page book defines retrieval practice, explains why it's so good, and the many ways it improves student knowledge.  It provides information on implementation, materials, and provides an FAQ.  In addition, there is a check list to follow as retrieval practice is implemented.

The third 12 page book is dedicated to using spaced retrieval practice to increase student learning.  It defines what spaced retrieval practice is and why it works before explaining how to implement it into your classroom in a beneficial way.  Furthermore, the authors take time to discuss potential problems and answer frequently asked questions.

The final 12 page book explores the connection between retrieval practice and transferring knowledge.  They go into detail on what transference of knowledge is and the different forms it may be take.  They also discuss how powerful connecting transference, retrieval, and feedback can be for student learning.  Furthermore, they look at both effective and not so effective methods of transfer and retrieval practice.  In addition, they provide a check list to help teachers implement this effectively.

As if that isn't enough, they provide warm-ups, key phrases, research, and all sorts of other things.  Yes, I've got them all downloaded so I can reread them to properly understand everything.  Check out the site, let me know what you think, I'd love to hear. Have a great weekend.

Peer Tutoring.

My principal told me this past Monday that I needed to change the way I write things in my lesson plans from "Students will work together on "  to "Students will teach each other" because he wants it to be that on my lesson plans.  So I'm thinking of stating "Using peer tutoring, students will teach each other" since its actually more accurate.

Peer tutoring is defined as students working in pairs to help each other learn or practice the current mathematical task.  It has also been around in some form since 1795 when it was first promoted in Scotland and over the past 30 to 40 years, it has become quite popular for pairing up students of mixed abilities.

It has been suggested that when pairing students up, they should be of different abilities and students should switch who does the tutoring regularly because as a person explains the material, they are extending their own understanding and learning. So one student is the expert while the other is the novice. 

In addition to extending their understanding, students often undergo higher academic achievement, better relations with other students, increased motivation, and improved social and personal interactions.  Furthermore, teachers have more time to provide individualized instruction as needed while students are getting individualized instruction when the teacher is not available and it cuts down on misbehavior.

Other benefits include  the fact that the interaction between students promotes learning, reinforces individual learning by instructing others, active learning is promoted by direct instruction between students, students are more comfortable with each other, promotes mathematical discussion between students,  and it is the best way to promote mastery.

One thing about peer tutoring is that it is best to provide training for all students so they know what to do and what the teacher's expectations.  Without those, the expert students may do the work for the other student or allow them to copy.  Furthermore, by training all students, it makes it easier for students to switch from tutor to the one being tutored so both have a chance to show their knowledge. 

I have peer tutoring in my classroom and I've seen students who do very little actually gain confidence and begin turning in more and more work until they are succeeding.  Let me know what you think, I'd love to hear.  Have a great day.

Dice and Card Games

  Computers and digital devices are fun but sometimes its necessary to change pace in class and go back to the old fashioned hands on cards or dice.  I use them sometimes as a back-up to what my students are learning.

One day to give my low performing group a chance to practice multiplication, I grabbed four dice.  I rolled two at a time to make two 2 digit numbers that my students then multiplied.  It might be 56 x 61 or 23 x 64.  Since I used dice, the numbers were as random as they could be with dice.

Another time, I grabbed two dice, each in a different color so one die represented positive and one negative.  I rolled the dice and used the numbers to complete the (x + ?) (x + ?) which students then multiplied.  After a couple of times through, I gave each student a set of dice so they could make their own binomial terms to multiply.

Students can play multiplication war using one deck of cards per two students.  An ace is worth 11, Jack 12, Queen 13, and King 14.  In addition black cards are positive numbers while red cards are negative numbers.  Deal the deck out so each player has 26 cards.   Each person flips over the top two cards and the person with the higher product gets all four cards.  In case of a tie, the cards are set aside until the next hand and the winner of that hand gets all the cards including those set aside.  The person with all the cards wins the game.

There is also exponent war which uses one deck of cards per two students.  All face cards are wroth 10 while the ace is worth either 1 or 10 but it has to be decided before the game starts. The cards are dealt even between the two players so each person has 26 cards.  The game begins the same as multiplication war with the first two cards being turned over but in this case the first card is the base and the second is the exponent.  The player with the higher product takes all four cards. If the products are the same, the cards are set aside and the winner of the next hand takes the four for that round and the ones set aside.  The winner is the one at the end who has all the cards.

To promote thinking, have the students play hit the target in groups of 2 to 5 players per deck of cards.  The players choose a target number and one student draws 5 cards from the deck.  The cards are laid face up and the players have to use addition, subtraction, multiplication, division, and order of operation to use those 5 numbers to get to the target number.  Since students have to follow order of operations, they are allowed to use parenthesis to group numbers.  So if the target number is 22 and the person turned over a 5,5,6,2,A and the Ace is worth 1, the players have to arrange the numbers to equal 20 but the numbers can be moved around to accomplish it.  So you might do 5 x 5 - (6x1)/2 but that is not the only possibility.

I'll share more games in the future but these are a good start.  They don't take much to play and are easy to learn while letting students practice skills in a fun way.  Let me know what you think, I'd love to hear.

Tangram in Geometry

  The other day, I described using Tangram in my geometry class to find convex and concave polygons.  I wondered what learning opportunities Tangram offered in Geometry.  Why are they good to use?

Tangram help develop geometric knowledge, reasoning, and improves geometric imagination.  One reason geometric imagination is important is it helps sense the shapes, their position, changes in size, both in the plane and in space.

These are some of the activities that can be used with Tangram.

1.  If all 7 pieces are used to create a square and the square is one inch by one inch, what is the area of each piece.  Show or explain how you arrived at your answer.  This activity requires mathematical reasoning and thinking.

2.  A different way of looking at activity one is to assign a ratio to each part, for instance the largest triangles might be worth 1/4th of the shape each, etc so that once they've got fractions worked out for each shape, they can then figure out the actual area for each piece based on the size of the original square.

3.  This is a great exercise when looking at the specific quadrilaterals as an introduction because once the shapes are made, students can use the resulting piece to determine the characteristics of each one.
Have students use all 7 pieces to create
          a.) a quadrilateral with four equal angles and equal sides that are opposite of each other.
          b.) a quadrilateral with two sets of parallel sides and no right angles.
          c. ) a quadrilateral with one set of parallel sides and no right angles.
          d.) a quadrilateral with four right angles and four equal sides.

4.  Students start with the parallelogram piece and are asked to recreate it using other pieces in the Tangram.  (It uses one square and two small triangles.)  They are then asked to find the formula for the area of a Tangram which they do by rearranging the pieces into a rectangle.

5. Activity three also allows students to create a formula to find the formula for the area and the perimeter based on the shapes they create.  Once they have the formulas, the teacher can give them a worksheet to apply the formulas to the shapes shown on them.

I believe for geometry that when the students work through activities such as this one, they are more likely to understand the concept and remember it.  I told my students to keep their Tangram sets for future activities, so we'll see if they did. Let me know what you think, I'd love to hear.  Have a great day.

3 Ways For Students to Increase Their Ability To Crique Their Own Work

  It is important to teach students to check their work so they can find errors before turning it in.  Many times students do not want to take the time because they don't know how to spot the errors.  Sometimes, its because they don't like math and wanted to be done.  Sometimes, none of their previous teachers expected them to find and correct work.

For the more advanced students, I take a highlighter and mark exactly where the error took place.  They are expected to make corrections before giving the work back to me.  At this point, I enter the better grade.  I feel the point is for the student to learn, not be penalized for stupid errors and those are the type of errors I see most often.  Errors like 3^2 is 6, or reading the y value instead of the x value.

This works only if the students are willing to look back at their work and if they know how to "see" the error.  Unfortunately, this may not work as well with students who have little faith in their abilities.  So one way to help students is to teach them how to look for errors, how to critique their own work.  Here are three ways to help improve their ability to find mistakes and get past the "I did it and turned it in.  I'm done."

1.  It is suggested that as the teacher, you make mistakes more often.  Ask students to identify what you did wrong, or see if they can determine how you arrived at a certain answer, or even ask if the answer makes sense.  The last one helps the teacher see if they have developed a number sense which can be important in deciding if the answer makes sense.

These questions help students become better at critiquing their own work.  It extends their focus from only the answer to the process so they look at the whole thing, rather than just the end.  It also creates a basis for mathematical discussion. In addition, it is the teacher who made the mistake, making it easier for the students to comment because they do not feel as if they are being spotlighted.

2.  Use problems which require multiple steps.  When the teacher has collected the problems, share several with the students but with no names.  Some problems selected have the correct answer, some do not, and a few might be done incorrectly with the correct answer.  The teacher projects these problems and asks questions at each step of the problem.

The questions might ask students why this person added, or what were they thinking when they did this.  Other questions might ask if the student could have found the solution in fewer steps, or where did the student go wrong in solving the problem and why?

3.  Ask students why.  Ask them to explain why they did something because it requires them to construct a proper explanation or justification of why they did it.  This type of questioning leads to better arguments and reasons because math is more than just right answers.  Its also about communications.

These activities help students learn to pay attention to detail rather than operate on automatic pilot.  They learn to analyze and critique both their own work and others so as to boost their ability to do the work and defend their choice of ways to find solutions.

Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Warm-up

Building Persistance.

  One of the hardest things to help my students develop is mathematical persistence.  Many of my students have gotten in the habit of deciding something is too hard before trying it but I had them do an activity that most everyone managed to finish.  Yes there was quite a bit of squealing, throwing things on the floor, but eventually they plowed through.  I don't think anyone flatly refused to do it.

The first step in the process was for students to create their own set of tangram.  This required them to measure an 8 inch grid with 2 inch subdivisions.  For many students, it took two to three tries because they couldn't use a ruler properly but everyone made themselves a set.

Next, each student received a worksheet with 12 problems, each requiring a different number of pieces to create a different polygon.  About half of the problems required students create a convex shape while the rest of the problems created a concave shape.  At the beginning, many of the students confused convex and concave but by the end, they knew the difference.

Many times when a student could not figure it out within two or three tries, they ask for help.  I'd decided before I made the assignment, I was not going to help them. I suggested they work with another student to see if they could help each other rather than relying on me.   In general this cut down on the help requests. 

Sometimes, I'd arrive to check a student's solution to the problem only to hear their partner point out the shape was convex instead of the requested concave.  The one who called would look at me to see if their partner was correct and I'd have to confirm the statement.  Other times, the student got the convex and concave correct but messed up on the number of sides.

I'd hear screams of "I quit" followed by pieces flying off the table only to be picked up a couple minutes later and the student would be back at work.  At one point, one student over heard  another team crying they'd finally gotten number 11 done.  So he went over and asked how they did it because he'd been trying to find an answer for 15 or 20 minutes.  He asked them how they did it but they refused to share. He got even on the next problem because he'd figured out how to do it and wouldn't share the answer with them.

After four days, about two-thirds of the students have finished the assignment.  Several came in after school to spend time trying to finish it while others will try to finish it today.  This is one of the first times, I've had students push through something they felt was "too hard" so I'm happy.  I think I'm making progress.

Let me know what you think, I"d love to hear.  Have a great weekend.

Ridiculous

We had something happen at school that shows me how the administration does not think things through properly.  One of our science teachers applied for and was accepted to attend the upcoming Response to Intervention conference beginning next week.  Just yesterday morning, she discovered they'd replaced her with an English teacher because science teachers do not teach reading.

When I heard this, I realized that much of what I said yesterday, applies to science too.  In science they also clarify their thinking and understanding while learning to communicate information to others. In addition, they have to infer, predict, conclude, compare and contrast, cause and effect.

Furthermore, in science, mathematics is interwoven through the course and students need to understand and apply common mathematical formulas to various scientific situations. 

Most scientific experiments require students to analyze and interpret data before evaluating it to see if it supports their prediction or if they have to adjust the hypothesis.

Its amazing how many people have the idea that reading and writing belongs only to English thus the English teacher is the one who teaches those two subjects, yet reading and writing appear in math and science.

Reading and writing across the curriculum is especially important when working with students who are anywhere between 3 and 7 grades below grade level.  It is also important to teach students to read specialized material in context and to have them use specialized vocabulary in their writing.  As most people who have taken a regular English course, one usually reads fictional accounts and sometimes biographies or books based on actual events but they do not get the chance to apply the same techniques to non-fiction specialized material.

So the math and science classrooms are the places students get the opportunity to practice their reading and writing  skills in a different context while experiencing cross curricular applications.  I admit, I was extremely disappointed to hear that reason for switching the English teacher for the science teacher, especially as she needs the training as much as anyone else.

Yes, I get a bit upset when administration seems to fall into the same mindset that most students fall into with their "But this is not English" misunderstanding.  It reinforces the idea we do not use reading and writing in math or science. 

Let me know what you think, I'd love to hear.  Have a great day.

Why Is It Important to Integrate Writing into Math.

  I've discussed ways to integrate writing into the math classroom but I'm not sure I've taken time to explain why its important to have both reading and writing integrated into the math classroom.

The fact is that reading, writing, and math are all related but too many students see them as separate.  My students all tell me that reading and writing belong in English and not Math. 

I keep telling them they are wrong because if they can explain something using words, it means they truly understand the material.  So what does research say about this whole topic. The two basic components of a lesson are the hands-on component whose job is to stimulate curiosity, engage interest, and build prior knowledge before asking a student to read or write on the topic.

It is well known that the amount of prior knowledge is the best indicator of a students ability to make inferences from the text.  In addition, the more they gain in prior knowledge, the more they can comprehend and the more they learn from the text.  The prior knowledge is what they access when reading or writing on a topic.

The second part is the reading or writing which helps the student interpret, analyze, and share their mathematically based ideas in addition to being able to evaluate sources to determine how valid the information is.  It has been noted that quite a few skills needed to do this are the same as needed in reading.  These skills include, predicting, inferring, sharing, identifying cause and effect, or comparing and contrasting facets of a topic.

Since there is an interconnection between reading and math, it is easy to design lessons that integrate the two topics in such a way as to capitalize on the skills needed in both.  Furthermore, by including writing in math lessons, it helps students clarify their understanding while improving their communication skills.

One goal is to engage students in their learning so they are more likely to ask questions, learn new vocabulary, and express their thoughts in writing.  One way to do this is to use a variety of books such as trade books, texts, even fiction to hook student interest.  There are tons of picture books out there with a mathematical theme such as Sir Circumference books  but there are more adult books out there with mathematical themes or have sections dealing with math.  There is a science fiction book whose name I do not recall but the basic premise is if you are an athlete competing in this competition, you have to compete in both sports and math.

Perhaps one day, we'll get everyone past the separation of Math and English and begin to see the topics as intertwined.  Let me know what you think, I'd love to hear.  Have a great day.

Anatomy of a Lesson

The new principal is giving very detailed information on exactly what he wants to the point, he even covered the anatomy of a lesson, something I haven't seen since my teacher training days.

I have no idea why he chose the version he did but it was one that broke things down as far as time but he did.  The packet he gave us also listed activities for each stage but no details on how to use them in math and some seemed rather inappropriate.

The one he chose stated the lesson should be 20% devoted to assessment and activating prior knowledge or students show they know and understand the objective of the lesson.  This might include a warm-up, play give one - get one, or other activity.  In order to have my students een check out the objective, I write it on the board as an "I can" statement and have them write it in their notebooks so it is there.

Then another 20% should be devoted to teacher input which is where the teacher provides direct instruction, differentiates instruction, scaffolds instruction, graphic organizer, or other type of instruction.  This is the part where student manipulate the information given into some form.

Following this 45% of the lesson is where the students actually manipulate the material provided by the teacher through think-pair-share, gallery walks, written or verbal responses, etc. It is well known that students need to a way to process the information and practice so they develop better understanding.

The final 15% is when the teacher identifies student success or where they show they know the material through the use of tests, quizzes, exit tickets, etc.  This is not what I learned when I was back in teachers training.

I was taught you have some sort of bell ringer or warm-up for the first five to ten minutes so students immediately started working while the teacher greets students, takes roll, does lunch count and any other house keeping chores.  The next stage depended on where you were in the process.  It might be an exploratory activity to activate prior knowledge while helping to connect to the new topic.  It might be a lecture or something similar followed by guided practice before letting them practice individually. At some point, it would be time to assess if they learned it.

So now I'm trying to reconcile the principals idea of what a lesson should be with what I've learned over the years. I hope to reach a point where I'm able to balance what I know works with what he wants.

Let me know what you think, I'd love to hear.  Have a great day.

The Relationship Between Systems Of Equations and Matrices

I don't know about any other teachers but I still teach systems of equations separately from matrices because I learned them as two separate entities. It wasn't until years later I learned about their connection.

Even now, I have to make a conscious decision to connect the two topics in Algebra II or I end up treating them as unrelated.  So this year, after I finish teaching solving systems of two equations by graphing, substitution, and elimination, I plan to slip in enough matrices for them to solve systems of equations with two and three equations.

First step is to have students become proficient or at least reasonable at solving systems using both substitution and elimination because both skills are used while working with matrix.  Instead of moving on to systems of inequalities, I'm going to have them learn the basics of matrix math.

From here its a hop, skip, and, jump to learning more about matrices and learn enough to easily solve a system of three equations.  This will be like building a good building after laying a strong foundation.  In a sense its like long division versus synthetic division, one method can take more time to do than the other.

So when is using matrices better than using systems of equations to solve a problem better or is there?  Well I spoke with a computer programmer and asked him.  His answer was an emphatic matrices because those are easier to program than systems of equations.  In addition, matrices are used in computer graphics any time there you see light going through water.  This is because the science used in optics to account for reflection and refraction is another use of matrices.

Furthermore, matrix math is used in calculating the electrical properties of a circuit, volts, amps, resistance, and other aspects.  Matrix math is also used in electrical engineering to give approximations of more complicated equations. 

In some circumstances matrices represents data while in others it represents equations. Some IT companies use data matrix to track all sorts of user information, carry out search requests, and help manage databases.  Furthermore, geologists use matrix to carry out seismic surveys, and plotting graphs.  In other fields, matrices allows calculations for optimization problems.  In robotics matrices are used to direct its movement. 

This is important because students like to know how to relate the current topic to the real world.  If they do not see a reason for learning the topic, they are less likely to want to learn it.  So know, I have the beginnings of the answer to "Why do I need to learn this?" and perhaps they will be more willing to learn it.

Warm-up

Warm-up

Helping students learn.

Many of my students arrive in high school without the necessary study skills needed to succeed.  I don't know why that happens but once they arrive, they have to learn. 

One of the first thing I did was to supply composition books and two pocket folders so they are more organized.  With the notebook, they can keep all their notes in one place.  I've tried three ring binders in the past but students usually destroyed the binders by December. 

Once they have the notebooks, I teach them  Cornell notes for most of the notes they take but I also teach the outline form of notes because the outline is often better for taking notes from a book or for writing process type notes.  After one semester, my students are getting quite good at finding the information in their notes when before they'd loose papers, mix them up, or forget to place them in order when they used binders.  

The next step, was to get them to look back at their notes so when I give them a test, or we play jeopardy, I allow them to look at their notes because every time they review the notes, they learn.  If I don't do that, they refuse to even glance at their notes.  In fact, the other day, I had students copy down sentences with blanks for them to fill in as they re-watched a video.  Then we played a Kahoot game with the same type of sentences with blanks.  As soon as they saw the first question with the blanks, the kids pulled out their notebooks to quickly check their notes on the topic.

I also have to have them learn to discuss mathematical topics. They have arrived in high school missing the skills needed to express their thoughts and ideas mathematically.  One way to do that is by using those activities created by Desmos where students are paired up and they each choose a picture of a graph or whatever and have to ask questions to help eliminate the ones it isn't.  Kind of like battleship.

This is the step I'm at right now for students this year.  I am just about to begin the idea of highlighting the point a mistake occurs, handing it back without the grade and having them make corrections before I actually assign a grade for both daily work and tests.  One of the hardest things is getting students to check their work.  Most times, when I ask them to check work, they glance at it and turn it in so I'm hoping this will help them really "see" their work.

Next week, I'm hoping to find some concrete methods to address the study skills needed to do well in math.  Let me know what you think, I'd love to hear.  Have a great weekend.

Finally Figuring Out How To Scaffold Properly.

 Up until recently, our school district has not had any decent software with adaptive capabilities.  I've been trying to figure out where the holes in their learning is which has made it really really hard to differentiate or scaffold learning.

Our benchmarking program only gives general lists of what students need so I've had to go looking for something that is free that will test students, give me more detailed information.

So far, I've found two programs which will provide me with the detailed information I need for one class.  I had students take a pretest for measurement and data in that class and I was surprised at the results.  Two out of ten students performed at a fifth grade level, while most were down at either a first or second grade level.  These are ninth grade students!

The program sent me individual results for each student down to the point of telling me exactly which standard they began at after the assessment.  I can use this information to create packets for each child.  When I say packets, it might include a video, a game, or a website along with some worksheets.  This way for half the period, students can work on closing the holes in their foundational knowledge while the other half of the period, we will have a regular lesson with direct instruction for topics I need to cover.

In addition, the program also has some adaptive lessons so students take a pretest and they can work through the program at their level.  The unfortunate thing is since its their free version, I cannot assign as many lessons as I'd like to so I'm waiting to see if that is a per day limit or a per topic limit.  In the meantime, I signed up for another free program which appears to allow me to assign more lessons using the adaptive features.

I figure that between the two programs I'll get the necessary information I need so I can have them work on the computer to close knowledge gaps and also provide other activities both digital and hard copy so they get the scaffolding they need.

Its been very hard to get the data I've needed because my district has been quite slow at providing the type of software I've needed to provide the detailed information.  They bought something several years ago but never got it installed so it was worthless.  Eventually, they purchased MAP but they didn't buy the extra features so I couldn't get the indepth analysis for each student.  Instead, I got a kind of generic listing but none of it told me much.

I'll keep you posted on how this works out.  By later today, I should find out if they are cancelling the in-service scheduled for Friday.  If it is, I'll let you know what I'm going to plan to cover class so I don't have to teach much and so students will get to practice the current topic. 

Have a great day.

Another Look At Grading.

I am taking another look at grading because of yesterday's column. After thinking about it a lot, I realized that I had a few college professors who offered options to students or had a different way of grading that offered the student more.

In one class, the professor made a list of the minimum required work needed to get a "C", what else was needed to get a "B", and what one could do in addition to the "B" work to get an "A"

In another class, the professor who was European, made appointments with each one of us at test time.  He'd have us come to his office where he gave us a problem or two to do. He'd watch us work it, ask questions, before assigning a grade of A which meant you did a fantastic job, a B for an above average grade or a C because you did an OK job.  Otherwise it was an F because you didn't even try.

I keep thinking my students have gotten to high school with the idea of they should only do work if it counts for a grade and once they get the grade, there is no reason to look back at mistakes or make corrections.  Once my students get their papers back, they throw them out so I've been trying to figure out a way for them to look back, make corrections so they learn the material better.

I stumbled across something that looks like it has the potential to require students to revisit their work until it reaches a certain standard and its based on the idea of highlighting mistakes.  Its a Grade - Revise - Replace idea where you "grade" the assignment but only highlight the mistakes without putting a grade on the paper.  The student goes through to check out all work with anything highlighted until they've "corrected" the mistakes and turn it in.  The teacher checks over all "corrections" before assigning a final grade which is the one that gets recorded in the grade book.

I like this particular idea because in the end, the students get the numerical grades my district requires for their records but I've had a chance to get them to redo work and make corrections to improve their understanding of the topic instead of listening to their "Its done, I don't need to do anything more." comments. 

In addition, it gives my students who freak out over anything less than an A a chance to get the corrections made before the lower grade appears in Power School.  Furthermore, if I ask them to write a few words on what the mistake was before they make the correction, it helps them verbalize their mistakes which I hope will make it easier to find their own.

I think I'll use this new grading idea in my class filled with students who struggle with completing assignments to turn in.  If I assign fewer problems overall, it means they have fewer to do at first but if they go over the same problems, it will take less time and they should be able to finish the corrections.

I'd love to get feedback on this idea.  Do you think it would work?  Do you have ideas for making it better?  Please let me know what you think, I really want to hear.  Have a great day.

Highlighting Mistakes on Tests.

I've decided to try something new this semester based on a cool idea I stumbled across on Twitter.  One reason, I wanted to try it is simply that my students do not like finding where they made errors,  This method, works on helping students find errors and making corrections without the teacher doing the work.

This idea comes from Leah Alcala on the Teaching Channel. She noted that if she graded tests, put the scores on the top and then returned them to students, most students looked at the score and put it away rather than checking for errors.

In response she started using a very different grading methodology.  Once all the tests are turned in, she sits down to grade tests but instead of grading them, she highlights the exact place the student made a mistake but nothing more. 

Some of these mistakes are what she calls flow through mistakes where the mistake was made but the overall process was totally correct but that small error caused the answer to be wrong.  An example might be instead of 3 x 2 = 6, they say 3 x 2 = 5 and they use the 5 rather than the 6 making the final answer incorrect.  In other words, the problem was correct up to that point.

She also marks things like students dropping negative signs, multiply instead of doing exponents properly, etc.   This is all on the first go through.  On the second time through, she goes back and looks for patterns in the mistakes to see if the student makes the same type of mistakes or the mistakes are all over the map. Assign points based on the type of mistakes made such as if the student makes the same type of mistake over and over, they should get more points than the one who makes a bunch of different types of mistakes.

Write the grade in the grade book but not on their papers so when the papers are returned, they have to look at the highlighted areas rather than the grade.  They can check Power School for their grades the next day. She has students look over their papers to determine if they understand the mistake they made.  Students are allowed to retake the test as many times as they need.  In addition,  she uses the information from the second time through the test to find things to talk about in class and to use to create mathematical discussion. 

One variation, I plan to use is to have students make corrections on a separate piece of paper based on the highlighted spot.  If they can figure out their mistake and correct it without my help, they can learn the material better.  I want to give them that chance before I post scored in Power School, otherwise I love what she does.

Let me know what you think, I'd love to hear.  Have a great day.
 

Assessment

 Sorry there were no warm-ups posted over the weekend but I got stuck in a town for a few days waiting for the weather to clear up enough to fly home.  I managed it but the next day the weather got bad again.  So while waiting for the weather to clear, I wasn't able to do much.

Now on to today's information.  When looking at assessments, its important to decide what the assessment is for.  Lets look into the four different types of assessment designed to improve learning.

Assessment should make learning better while supporting good instructional technique.  In other words, instruction and assessment should support each other.

First off, assessment can help the student determine what is more important to learn.  This means, students should be assessed on specific topics. For instance if they are learning about greatest common factors, they should be assessed on that topic.  If they have been learning to use greatest common factors to simplify fractions, that is what they should be assessed on.

This type of assessment is not only teacher driven, it can be used by students to help them assess themselves.  Furthermore, it has been discovered that learning is really restructuring prior knowledge rather than adding to it so assessments can be developed to help improve learning.

Second, students must be able to construct new knowledge from previously learned materials and the proper assessment can help with that.  One way to do that is through assessment tasks which are similar to learning tasks with data analysis, making connections, and making contrasts.  Large groups are often optimal for learning and assessments designed for whole group must reflect that.  Group work is a very real life situation because many businesses have groups find solutions to problems.

Third, if students have been learning mathematics using tools such as graphing calculators they need to be assessed while using them, otherwise the final picture is incomplete.  Furthermore, many of the standardized tests require students to use certain tools so if they do not know how to use these tools, they are at a disadvantage.

Finally, students need to be assessed on more than the usual automated tasks.  They need to be assessed on how well they've connected with the current topic, how well they can see how things connect even with superficial differences, the ability to use the appropriate process, and their own understanding of the material.

Tomorrow we'll look at a different idea on grading tests which I recently read about.  Let me know what you think, I'd love to hear.  Have a great day.

Low Floor, High Ceiling Tasks.

While researching entry tickets, I stumbled across something called Low Floor High Ceiling Tasks.  By definition low floor, high ceiling tasks are tasks that can be done by everyone but can be extended to higher levels.    In other words, everyone can start at the same level but students can easily take these tasks to higher levels if the beginning one is too easy.

In order to plan such a task, there are several things to keep in mind as you look for one or you write it.

1.  What is the learning goal of the task?

2. How can you tell if the students are successful?

3. What knowledge or skills did the task address?

4. What stumbling blocks might arise during the task and what questions can you as the teacher use to help get the student moving.

5. How can the task be adapted for students with special needs?

6. Do you have all the supplies you need for this task?

The great thing about these types of activities is they focus on showing students what they know, not what they don't know.  It automatically provides differentiation to all students.  For confident students, they can explore the idea in greater detail while the students who still struggle are able to consolidate their thinking.  In addition, it promotes discussion among students as they share a common activity.  It helps create a positive atmosphere and mirrors real life math.

An example of this type of problem is as follows:

Folding Papers.
1.  Make a square out of a rectangular piece of paper.
2.  Make another square that is 1/4th the area of the first.  Convince yourself it is a square and its area is a quarter of the original.
3. Make a triangle that has 1/4th of the area of the first.  Convince yourself it is a triangle with one fourth of the area of the original.
4.  Make another triangle that is not congruent to the triangle in step three with 1/4 the area of the original.  Convince yourself it is a triangle with a quarter of the area of the original.
5. Make another square that is one half the original square.  Convince yourself it is a square with half of the area.

I provided a link for the activity so you can check it out.  This type of activity could be turned into a gallery walk to share the reasoning of students with each other.  This type of activity can also encourage students to be less afraid of math and being to blossom.

Let me know what you think, I'd love to hear.  Have a great day.

Entry Tickets

I've read about exit tickets, even used them in class but I hadn't hear about entry tickets until quite recently.

Entry tickets are another way to have students mentally warm up rather than using a bell ringer or a standard warm-up.

Warm-ups can be used to review material or be used to determine the amount of prior knowledge for a new topic.  Students are given a specific amount of time to finish the entry slip.

Collect the slips and quickly spot check a few to see what students responded but set them aside to analyze later in the day.  The information can be used to plan the lesson for the next day.

The nice thing about entry tasks is they do not have to consist of practice problems.  The entry activity may require students to compare answers from homework, discuss a problem from yesterday's work, what's the next step in the process, find the mistake and correct it, or it could be as simple as asking them what they didn't understand.

When creating a new entry ticket, decide the purpose.:

1.  Are you introducing a new topic by connecting it with their lives such as slope can related to skate boarders.

2. Are you using it to bridge topics between yesterday's lesson and today's lesson.

3. Are you using it to check for understanding.

4. Are you using it to review concepts, key ideas, material from the week.

These are much better than spending the first part of class going over problems.

Let me know what you think, I'd love to hear. Have a great day.

Math Practice in a Different Way.

Every so often, I love creating activities for my students to play but I try to make it so everyone can participate.

In one game, I create cards similar to the one you see on the left.  Each card has an equation and an answer but the answer is not the one to the problem.  It is the answer to another problem.

You pass out cards to the students and give them a few minutes to solve the problem on their card.  Once everyone has a chance to find their answers, select a student to read out their problem.  The student explains how to solve the problem before giving out the answer.  Whoever has the answer, reads out their problem, explains how to solve it, asks for questions before giving the answer.

By having the student explain how to solve the problem, it helps clarify their understanding and often they find mistakes in their own math.  It also allows others to ask questions if they don't understand something or if they believe the person solving the problem made a mistake.  This continues until everyone has completed their problem and its back to the first person.

A second activity I enjoy making is pairs math.  I divide the class into groups of two and give each person a different worksheet but each problem has the same answer.  This way they each have to solve their own problems but if they get the same answer, they know they did it correctly.  If they don't get the same answer, I ask them to check the other person's work for a mistake because there is a good chance that one of them made a mistake.

A third activity is one where I place folded papers around the room.  The papers can have the equations or they can have QR codes, it all depends on what you want.  This is actually a variation on the first one but you place the equation on the front and the answer on the second page.  Again the answer on the second page is not the answer to the equation on the front page.

Students spread out around the room, choosing one to start with.  They write down the equation and work out the answer on the answer sheet while circling the answer.  They then find the answer a different paper and work the problem on the front.  If they can't find the answer, it usually means they made a mistake.

The third game is nice because it gets them moving around the room so they get exercise and it makes it easier for them to focus, especially for students who are on the hyper side.

Let me know what you think, I'd love to hear.  Have a great day.

Happy New Years