Professional Development

Mathematics: the Universal Language?

                Sharon Hoffert discusses important issues when it comes to working with students whose primary language is not English.  Many American teachers have very little training in teaching English Language Learners (ELL) and do not know what to do when they have these students in their classrooms.  Recently, the No Child Left Behind act required that non-native English speakers have to pass the same standardized exams as native English speakers. This means that teachers will have to find innovative ways to teach ELLs because there is essentially, all teachers will have to work with these students at some point in their careers.

                The realities of ELLs is that learning academic English can take up to 4-7 years, even if they can communicate socially in English. So, some methods of teaching ELLs are listed in this article, and from Hoffert’s experience, these methods have greatly helped her students succeed in her classes and on the standardized tests. 

                Whenever possible, assess your students’ current academic level by testing them with adapted exams and letting them use methods of translation. This way, students won’t be put in a remedial math class just because they couldn’t understand the prompts on the exams. This will help instill confidence in the students and give them opportunities to learn and build upon what they already have an understanding of.

                Make sure prompts are written in very concise, clear form, where key vocabulary words are highlighted. Also include diagrams and pictures so that students can visualize concepts and match up words to the visuals.  Put students into cooperative learning groups so that they can practice speaking mathematical language which will help their understanding of the concepts. Also do discovery based lessons so that they can gain confidence in knowing that they discovered a concept on their own.

                Lastly, Hoffert encourages teachers to have faith in their students and understand that these students are very capable and can succeed. 

Students' Real-World Problems

Problem-solving process of a Real-World situation

1.       Read/understand problem: Initially, we had to read the problem together and try to understand what exactly we were needing to find. If we weren’t reading a problem from a textbook, this part of the process would involve us determining that we needed to know how far away the ship was from the shore.

2.       What does this look like? At this point we needed to take what we comprehended from our problem and visualize what we needed to find.

3.       Need to solve for distance, x, and the second side to the ship. After visualizing, we were able to think about different ways that we could solve our problem.

4.       We drew out what we were visualizing to confirm what the problem is and to help us decide which method to utilize.

5.       After visualizing and drawing the problem and analyzing methods we could use, we decided that the Law of Sines would work really well because we had the appropriate measurements and data.

6.       Before we could use the Law of Sines, we needed to find one measurement. We found an angle in the triangle so that we could use it correctly.

7.       Then we used the Law of Sines by writing down the correct measurements at the right parts. Then we computed it.

8.       Then we interpreted that this number was the length of the ship to the shore.

9.       In context of the problem, we evaluated that swimming to the ship is a very bad idea because it was a far distance.

Mathematics and Historical Foundation of Mathematics

A Discussion of a Couple Ancient Numeration Systems

If you were to chose between the Babylonian Number and the Egyptian Number Systems, which would you chose? 

Egyptian:ancientegyptianfacts.com

Babylonian:kreannasandoval.wordpress.com

After some reading, and contemplation I decided that I would favor the Babylonian over Egyptian for the main reason that Babylonian has much less symbols to memorize.  The Egyptian numbers require many symbols for even simple and smaller numbers. The Babylonian system is a more simple way to represent numbers because there aren’t as many symbols, they’re just repeated to represent the bigger numbers.  There’s also a simpler way to represent numbers involving a way to represent a number minus one.  It involves simply placing a certain symbol before the other symbols rather than after.  That is another advantage of the Babylonian system, is that it is read left to right, and if we were required to switch number systems at this time, that would be an easier transition. The Egyptian is read right to left.  The Babylonian system uses a base 60 rather than base 10 like we do in our current system. The Egyptians do not have a base system, so that is part of why they have to create so many more symbols and hieroglyphs.  So although the transition to a base of 60 would be difficult and confusing, I think it’d be better than a transition to a number system with no base system at all.

What is The Rhind Papyrus?

                                                   https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus

The Rhind Papyrus is a long scroll discovered in Thebes, Egypt that gives us insight to what mathematics the ancient Egyptians performed.  From what we can tell, the most mathematics that the Egyptians did was practical problems involving money managing, creating the calendar, measuring and working with the Nile River, and proportioning food.  It is what we now call applied arithmetic. The Egyptian number system is an additive number system. Rather than creating symbols and functions for multiplication, they added the numbers repeated times. Then to divide, they would do multiplication in reverse.  When dealing with fractional numbers, the Egyptians would only use 1 as a numerator, making representations of a fraction much longer.  We think that the Egyptians didn’t ever get around to proving anything, or creating theorems to be proved because of the amount of time and work it would take to complete practical problems. That was the Rhind Papyrus is mostly composed of, practical problems. The Egyptians dipped into algebra by solving some problems dealing with only one variable that were fairly simple.  Their methods of solving these algebraic problems are longer and more complicated than what we do now with everyday algebra, but they are very interesting and I think it’s quite incredible that they were able to come up with this and make it work. For example, one of their methods is the method of double false position. Ultimately, you guess two solutions to the problem, plug them back into the equation and use the guesses and the solutions when the guesses were plugged into the problem to compute the correct solution.   I think it’s really exciting and interesting to learn about the way that these people came up with solutions to their problems. Their methods are different, but ultimately they mean the same things as what we do now to solve problems, and I think that’s pretty darn cool. 

My Favorite Mathematician

Descartes marked the beginning of pulling mathematics back out into the picture after the centuries of dark ages when mathematics were deemed unimportant. Rather than only utilizing mathematics for application such as basic finances and arithmetic, he pursued mathematics in a more pure form. He wanted to be able to logically and mathematically deduce all types of propositions from the most basic principles such as axioms or self-evident truths.  In accordance with that, he wanted to be able to define absolute truths that could not be countered. His ideas greatly impacted and influenced the start of the debate between faith and reason. 

www.edublox.com

He also contributed to the new and coming ideas and theories involving the nature of the universe. He argued against the Aristotelian and Ptolemaic view about the universe. He believed that mathematics and mechanics were the backbone to true and certain laws of nature. His theory about the universe was that the sun is at the center of a vortex and that is how the planets revolve about the sun. Although this theory had major holes and could not be proven, it was influential in increasing the study and discovery of the universe beside theologian beliefs and the theories of the Christian Church.

One of Descartes’ most well-known contributions to modern mathematics is his discovery of the coordinate plane.  This is why the coordinate plane that is most commonly used is called “Cartesian coordinates.”  Descartes certainly wasn’t the only mathematician to really officially formalize Cartesian coordinate geometry, in fact, he only really worked with the first quadrant of the plane (only positive x and y values.) But he was able to begin to piece together the notations and concepts of both algebra and geometry.  This means that he could represent geometrical figures on a Cartesian graph. This way, instead of writing a segment from point A to point B, he could represent this as an algebraic line.  Then with the segment graphed as an algebraic line, he was able to prove theorems easier and be able to measure lengths, distances, areas, etc., of all sorts of shapes in a more algorithmic way that was more accessible to people including those who were not skilled mathematicians. 

Technology

Technology Enhanced Lesson Plan

Task Sheet:

1.    Go to this website http://mste.illinois.edu/activity/regression/ and in the box “Fit Line Angle (Radians)” put .7.  Now, wherever you’d like, click on the graph to create 15 or so points. Notice that a green bar will appear below the graph that says “accuracy.”  See what happens to the accuracy when you create new points on your graph.  Hit the “reset” button to try a new set of points. See if you can get the green bar to 100% accuracy.  Explain  what you have to do in order to get the accuracy as close to 100% as possible.

2.  Now check the button for “Show Best Fit.” Again, select a few points and notice what happens to this line when you create a new point. (This is the blue line)  Explain what happens to this line when you select new points.

3.  Now, please patiently listen while your teacher instructs the class about Correlation Coefficient and Best Fit Line.

4.  Using these listed steps

https://www.mathsisfun.com/data/correlation.html


compute the correlation coefficient for this set of numbers.

            x={1, 2, 3, 4, 5, 6, 7}  y={10, 7, 3, 5, 6, 9, 8}

5. Consider these questions: Was this a tedious process? Was it a long process?  What would we do if we had a larger set of data?  How could we more easily and quickly compute r for a larger set of data? Write down a couple of ideas for the last question and discuss them with your group:

6. Now, open Geogebra on your computer. Under the “View” tab, select “Spreadsheet” and a spreadsheet very similar to excel will open on the right side of the screen. Open http://mste.illinois.edu/regression/species.html and download/open the Excel 5 file of Species data. Copy and paste the Brain Weight column and the Total Sleep column into the Geogebra spreadsheet.

Since one of our Total Sleep cells says “missing” click the number of the row that the cell is on, right click and hit “delete row.”

Now highlight all of your data. On the top left-hand corner, there are four boxes, click on the second box from the left and on that menu select “Two Variable Regression Analysis” Then select “Analyze.”  Be sure to select the button in the top left hand corner that looks like this x.  This will pull up the statistics for your data, including the correlation coefficient listed as r.

Now, based on your correlation coefficient, explain the relationship between the Brain Weights of various animals and the amount of Total Sleep these animals get.

Exceptionality and Accommodation

Helping parents/students with disabilities:

An IEP (Individualized Education Program) meeting is where a written document will be organized to suit your child’s individual needs/related services.  You must have an IEP to receive special education.

 We will be discussing your child’s current academic performance and what we can do to help them learn in a way that’s better suited for them.  We’ll make individualized goals for your child and discuss any related services that need to be provided.  How much time spent in mainstream classes will be discussed, their participation in district/statewide assessments, transition services regarding postsecondary goals related to their training and education in secondary schooling.  The IEP will also keep a record of how the child’s progress is toward their goals.

I, a regular education teacher and another special education teacher will be there.  There will also be someone who can access and knows about the resources and education curriculum that the school has available.  The child may also be there when appropriate.  

Multicultural Education

              As part of learning about Multicultural Education, I signed up for a program on campus called the Global Aggie Partner program. This program is designed to help international students get involved with on or off campus activities and make new friends. I was assigned a partner, his name was Saeed.  We mostly had meals together, lunch and dinner, and had good conversations. We had lunch at Aggie Ice Cream once, and another time he invited my roommate and I over to his apartment for delicious Saudi Arabian food.

                I learned a lot from Saeed.  I learned some things about Saudi Arabian culture, for example, they usually sit on the ground for their meals and have everyone dish out the food from a large revolving plate in the middle of the table.  The dinner that Saeed cooked was a traditional dish and he even used spices and ingredients from Saudi Arabia.  It was a very delicious meal.    I also learned that typically Saudi Arabian families are large, Saeed has four siblings, and that is unusual.  Apparently, seven to ten children in a family is typical in his culture.  We found a lot of connections because I also have a large family.  We discussed how much we like having a lot of siblings, and how much fun it can be to have family gatherings when there’s a lot to talk, joke and laugh about.  We also discussed that it can also be hard sometimes having a large family because sometimes we can fight and argue.  Saeed also told me about how he came to Utah to get an education because the schools were a little cheaper, and then he intends to move back to Saudi Arabia after he graduates and make more money over there as an engineer.  We are both in a calculus three class and we’ve talked a lot about how hard the class can be.  He has an especially hard time because he is still learning English.  As an international student, he had to spend two years learning English before he could start his program.  For only knowing English for two years, he communicates pretty well.  Though when we were having dinner at his apartment, my roommate and another American that was there were talking about school and other things and Saeed and his brother (who is also here, spending his two years learning English,) both told me that they didn’t understand anything my roommate was saying because she was talking too fast.   Something I learned about Saudi Arabian culture is that it is offensive for men to touch a woman’s hair.  Saeed didn’t tell me what it meant, only that it was offensive.  That is also one of the reasons that the women wear the niqab.  Saeed also showed me a large robe-like thing that he wears for warmth.  I can’t remember what it was called, but it was like a very very thick robe that had long sleeves and went all the way to the floor.   We talked about how we wish that everyone could have something like that and wear it everywhere, especially here in frigid Logan!   Saeed also talked about how he isn’t used to the cold and doesn’t really like it.  He was very used to the really warm weather in Saudi Arabia.  We made a connection about that because I grew up for a time in Southern Arizona where I was also used to very warm, dry weather, and then moved to a cold, wet Utah. 

                We also talked about what we like to do for fun.  I told Saeed that in Utah, outdoorsy things are very popular because of the mountains and lakes that are close by.  I told him about rock climbing, hiking, water skiing, four-wheeling, etc.  He told me he doesn’t like heights, and hasn’t really done very many things like that.  In Utah, for fun he likes to hang out with friends, try out new recipes or restaurants and go to parties and activities.   In Saudi Arabia he said it’s about the same, he really likes their social gatherings and big family gatherings. 

             Spending time with Saeed gave me insight to how I should teach when I have students of other languages and cultures. I will need to be patient and understanding when students have a difficult time keeping up with the other students and make accommodations for them to continue learning and understanding even though there may be language or cultural barriers. 

Assessment

Sample of various Mini-Experiments (Test prompts)

EXAM 1  Name:_________________________

Take Home Portion:

1.  The mean and standard deviation of the set {49, 23, 25, 24, 28, 28, 24, 26, 28, 30} are 28.5 and 7.15 respectively.  Please give brief explanations for each answer.  (8 points)

·         What would the mean be if we added 7 to each number in the set?

·         Would the standard deviation change if we added 7 to each number in the set?

·         What would happen to the mean and standard deviation if we were to remove 49 from the list of numbers?

·         What if we were to include the number 32 in the set of numbers? What would happen to the mean and standard deviation then?

2. Researchers interested in determining if there is a relationship between death anxiety and religiosity conducted the following study.  Subjects completed a death anxiety scale (high score=high anxiety) and also completed a checklist designed to measure an individual’s degree of religiosity (belief in a particular religion, regular attendance at religious services, number of times per week they regularly pray, etc.) (high score=greater religiosity) (http://www2.webster.edu/~woolflm/correlation.html including table) (5 points)

Compute the Correlation Coefficient by hand. (Show your work, there is more space on the next page)

Based on this coefficient, is it safe to assume that being religious causes less death anxiety? Explain.

 

3.  Does the table below reflect gender bias towards pets? Explain. (Hint: rewrite the table in percentages) (5 points)

Preferred Pet	Dog	Cat	Other	Total
Boys	10	5	9	24
Girls	8	7	1	16
Totals	18	12	10	40

In Class:

4. Entry to a certain University is determined by a national test.  The scores on this test are normally distributed with a mean of 500 and a standard deviation of 100.  Tom wants to be admitted to this university and he know that he must score better than at least 79% of the students who took the test.  Tom takes the test and scores 585.  Will he be admitted to this university?  (6 points) http://www.analyzemath.com/statistics/normal_distribution.html

5.  Leonardo da Vinci theorized that if you put your arms out to the side and measured from the fingertip of one hand to the fingertip of the other, this “wingspan” distance would approximately equal your height.  A group of fourth grade students measured their height and wingspan and found:

Ave Height: 49.5 in, SD height: 1.8 in

Ave Wingspan: 48.9 in, SD wingspan: 2.1 in

Predict the wingspan of a randomly chosen fourth-grader who is 52 inches tall.  Explain what your answer means. (6 points)

6.  Compare the two different distributions A and B.  (6 points)

  

a) Describe the relationship between the mean and mode of A and B

b) Which of the two is skewed, and is it positive or negative?

c) What summary statistic determines how wide or spread out the distribution is? How does this statistic change what the curve will look like?

Motivation, Engagement, and Classroom Management

An Evaluation of Cooperating Teacher for Clinical Observations
November 12, 2015

Period 1: 8^th grade Honors Math, Daren Lentz

During this class, Lentz played an “Extra credit game” with the students. The game involved basic arithmetic where three dice would be rolled and then the team would have to create a combination of those numbers using operations such as add, subtract, multiply, divide, square, etc.. I’m not entirely sure what goal was trying to be achieved by this game besides helping students improve their mental arithmetic. After the game, Lentz went over one example with his students. He let the students work on the question for a minute or two, but finished it with a large class discussion to make sure the students understood it correctly.

The last fifteen minutes of the class, the students worked on homework either with their neighbors or individually.

Period 2: 8^th grade Regular Math, Daren Lentz

Lentz also started this class with an “extra credit game.” This game involved rolling a die as well and filling in the spaces of a tic-tac-toe board with the numbers that were rolled. Different combinations of numbers had different values of points.  I think the objective of this game also was to help students work on mental arithmetic.  Then for the rest of the class period, Lentz had the students working on a work-sheet that they had been working on previously as groups or individually.

Comments:

It’s hard to define one specific type of learning activity that Lentz used in class. It seemed to be that multiple types were integrated into one class period. The great advantage about that is that the advantages of the different types of learning activities used are all present.  For example, in the honors class, the students spent some time in a large class question-discussion session, and the advantage was that they could get questions answered definitely by the teacher and make sure they are on the right track to solving the problems they’re working on.  Then when they move to the cooperative-learning session where they worked on the homework together in groups, they have the opportunity to discuss the problems with their peers and improve their understanding by explaining and verbalizing what they are doing.

In the Regular class, there wasn’t really a question-discussion session, since the game wasn’t really a discussion. But it was mostly a cooperative-learning session as they worked on the worksheet with their peers.  I think this was advantageous because students could work on problems at their own pace and explain things to each other.  Also, at this point, there were several preservice teachers in the classroom who could walk around and help students with specific problems. I also think it was advantageous because students were engaged in their work.  From what I could tell, the students stayed on-task pretty consistently with only minor distractions. This is much harder to accomplish with a large group presentation.

In general, I would say Lentz’s classes were mostly cooperative-learning sessions, and I have always found it to be very advantageous to do this type of learning activity for the reasons I have described already.

I think the disadvantage to this though, is that sometimes students still aren’t exactly sure what they are supposed to be doing and leads them to off-task behaviors. Also, if all of us preservice teachers hadn’t been there, it would’ve been harder for Lentz to be able to answer all of the questions that students had as they worked problems.  Whereas in a large group presentation, he could have addressed some of those questions in class so that all of the students could get help on the question. 

Another disadvantage could be that students would not learn the basic ideas of concepts because they would not have been explained to them (which is what I would assume would happen in a large-group presentation.) But, from what I have observed, Lentz uses different learning-activities almost every day, and he incorporates large-group presentations and question-discussion sessions depending on what goal he has for the students that day.  So ultimately, I think the best thing to do, would be to define your objective and determine the best learning-activity or combination of learning-activities to achieve that objective.  

Alternate Activity:

8^th grade Honors:

Instead of playing the extra credit game, I would engage my students in a cooperative-group learning activity as follows:

First thirty minutes of class: Put students into groups of four (or three if needed) and have them work on this task sheet

After students finish the group-work task sheet, I would initiate a full class discussion to see what people discovered. Be sure to lead students to the idea that they can break down an irregular prism into multiple regular prisms to find the volume.

After only 5 or so minutes of discussion, I’d let the students work on their homework either individually or in with peers. Their homework has more examples of this same material. At this point I would be sure to wander and answer students’ questions.

  

Cognition, Instructional Strategies, and Planning

Sample Course Syllabus and Parent's Newsletter
**************************************
Course Syllabus for Secondary Math I

Sunnyville High School

Room 123

Instructors: Heidi Clawson, Elise Bollwinkel, Arielle Meredith

Required Text: Core Connections Int 1

Course Description: This course is designed for students to engage in learning new mathematics material and build upon previous concepts they have learned. General concepts that will be covered are: Number and Quantity, Algebra, Functions, Modeling, Geometry, Statistics and Probability. Students will learn how to solve problems involving these concepts and engage in critical evaluation of how these concepts work.

Course Units and Goals

1. Relationships Between Quantities
1. Reason quantitatively and use units to solve problems
2. Interpret the Structure of Expressions
3. Create Equations That Describe Numbers or Relationships

2. Unit 2 Linear and Exponential Relationships

1. Represent and Solve Equations and Inequalities Graphically
2. Understand the Concept of a Function and Use Function Notation
3. Interpret Functions that Arise in Applications in Terms of a Context
4. Analyze Functions Using Different Representations
5. Build a Function that Models a Relationship Between Two Quantities
6. Build New Functions From Existing Functions
7. Construct and Compare Linear, Quadratic, and Exponential Models and Solve Problems
8. Interpret Expressions for Functions in Terms of the Situations They Model

3. Unit 3 Reasoning with Equations

1. Understand Solving Equations as a Process of Reasoning and Explain the Reasoning
2. Solve Equations and Inequalities in One Variable
3. Solve Systems of Equations

4. Unit 4 Descriptive Statistics

1. Summarize, Represent, and Interpret Data on a Single Count or Measurement Variable
2. Summarize, Represent, and Interpret Data on Two Categorical and Quantitative Variables
3. Interpret Linear Models

5. Unit 5 Congruence, Proof, and Constructions

1. Experiment with Transformations in The Plane
2. Understand Congruence in Terms of Rigid Motions
3. Understand Congruence in Terms of Rigid Motions (Honors)
4. Make Geometric Constructions

6. Unit 6 Connecting Algebra and Geometry Through Coordinates

1. Use Coordinates to Prove Simple Geometric Theorems Algebraically

Assignments and Grade Determination

Assignments: You will be assigned one assignment per day that will be due the next day in class. Assignments will only be checked for completion.  Before turning in your assignment, be sure to check for these things:

-Name, period, and HW #___ in the top right hand corner

-You have ripped out your notebook page on the perforated line

-Your problems are listed in order and adequate space is given between

each question.

-Final solutions are circled or highlighted.

Bell Quizzes: Every day at the very start of class you will do a bell quiz that consists of one of your homework problems (so you’ll want to finish all of your homework problems.)

Exams: You will take a Unit exam after we have completed each unit. These exams will consist of questions similar to your homework questions, but not exactly the same.

Semester Comprehensive Tests: At the end of each semester you will take a comprehensive test as to prepare you for the standardized test at the end of the year.

Cheating and Discipline Policy

Plagiarism and cheating are not tolerated at our school.

Parents’ Newsletter for Secondary Math I

The Instructors

Heidi Clawson: “I completed the Mathematics and Statistics Composite Education program at Utah State University. During my time at Utah State University, I had the opportunity to get experience teaching and tutoring for lower-level college mathematics courses.Teaching has always been a passion for me, and although Mathematics weren’t always my passion, I have learned to love mathematics and statistics through my deeper learning and understanding of it.  I am excited to teach your children math and statistics and I hope to instill a love for learning and mathematics in them.”

Contact Information for this Instructor: heidiclawson17@gmail.com

Elise Bollwinkel: “I attended Utah State University where I completed my Mathematics and Statistics Education degree. Throughout my education, I had experiences teaching and tutoring students of various ages in mathematics. I have found great satisfaction from seeing students grow in their understanding of mathematics, engage in critical questioning, and increase their own love for learning. I hope that when you or your child have questions concerning this class that you may feel comfortable approaching me. It is my hope that your child will have a rewarding experience this year in Secondary Math I.”

Contact Information for this Instructor: a.e.russell@aggiemail.usu.edu

Arielle Meredith: “I graduated from Utah State University with a degree in Physical Education and Mathematics Education. I have tutored many students in mathematics in the athletic department. Math has always been a passion of mine, solving problems and gaining success is something I loved. I love teaching and seeing students find that same success in mathematics.  I hope to teach your children through my love of mathematics and watch them succeed throughout the year.”

Contact Information for this Instructor: acmeredith@aggiemail.usu.edu

Resources: We encourage our students, if at all possible to use the given resources for deeper learning and understanding of the material. They can also be used for additional examples and practice problems. These resources are also helpful if you’d like to understand more about what your child is learning during class.

• Khan Academy: https://www.khanacademy.org/  This site has helpful explanation videos and practice problems
• Just Math Tutorials: patrickjmt.com, More helpful explanation videos
• Desmos Graphing Calculator: https://www.desmos.com/ An online graphing calculator that also has examples and explanations of the different types of graphs.
• WolframAlpha http://www.wolframalpha.com/
• After school tutoring programs
• After school help from the instructors

Course Expectations

• In Class: We expect our students to behave with respect to their instructors and their peers. Every day in class, students need to come prepared with a pencil, eraser, and notebook. Students will be marked tardy if they are not sitting in their desk prepared for class and working on the Bell Quiz when the bell rings. Students who disobey classroom standards and policies will be written up and disciplined by our Principal, Dr. SoandSo.
• Home: Any homework assignments that students don’t finish during class-time will need to be finished at home or after school. As stated above, students have many resources to which they may turn to for help if they are having a difficult time. We expect them to finish and hand in assignments at the beginning of the following class.

This Week

During this first week of the Term, we will be discussing the Course Syllabus, and explaining to the students how the classroom will be managed.  We as instructors look forward to getting to know each student. We expect to gain an understanding of their current mathematical knowledge so that we can teach in the most effective way possible and build upon the concepts that they have already learned. By the end of the week we will begin our first unit of Mathematics Curricula.