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?