Warmup

 

If a coconut tree begins producing 52 coconuts per year, 18 years after it being planted and it continues the average production till it is 80 years old, how many coconuts did it produce total?

Warm-up

 

If the average coconut gives 1.5 cups of coconut water, how many coconuts do you need for one gallon?

African American Mathematicians part 2

 

Today, I’ll be sharing more information on African American Mathematicians both male and female. Each and everyone of these mathematicians made an impact on their field and on history.  For instance, Martha Euphremia Lofton Haynes gained the title of the first african american woman to receive a Phd in mathematics from the Catholic University of America in 1943. Although christened Martha, she preferred using Euphemia.  Her father was a prominent dentist in Washington D.C and her mother although a stay at home mother, was actively involved in the Catholic Church.

After graduating from high school in 1909, she obtained her B.S. in mathematics four years later.  In 1917 she married Harold Haynes who eventually became superintendent of the african american school district in Washington, D.C.  Eventually she went to the University of Chicago to get her Master’s degree in Education with a significant number of classes in mathematics.  She went on to obtain her Phd in Mathematics in 1943.  She spent 47 years teaching in Washington D.C and taught everything from first grade to high school.  In addition to being the first African American woman to receive a Phd in mathematics, she was also the first woman to chair the District of Columbia School Board.  

During her lifetime, she also established the department of mathematics at Mines teacher college where she taught mathematics.  In addition, she taught mathematics at the District of Columbia Teachers College and occasionally taught at Howard University.  When she died, she left $700,000 to the Catholic University to support a Chair of Education and provide monies for a student loan program for the School of Education.

Then there was Walter Richard Talbet who was born in 1909 in Pittsburgh, Pennsylvania.  After graduating from high school, Walter attended the University of Pittsburgh in 1927 where he began studying physics.  Unfortunately, the depression hit so in order to afford college, Walter worked several part time jobs.  This meant he could not continue studying physics so he changed his major to mathematics.

He received his Bachelor of Science in mathematics in 1931 and continued studying at the University of Pittsburgh where he received his Phd in 1934.  Upon graduation, he accepted a staff position at Lincoln University in Jefferson City, Missouri.  He began as an assistant professor of mathematics but over time became a full professor, then Dean of Men and finally the head of the mathematics department in 1944.

He remained at Lincoln University until 1963 when he accepted the position as head of the mathematics department at Morgan State University till he retired and passed away the day after Christmas in 1977.  It is said that Dr. Talbot improved the ability of African Americans to study mathematics.  

As you can see both people went on to great careers in education.  Let me know what you think, I’d love to hear.  Have a great day.

Tropical Geometry

Monday, we looked at tropical arithmetic and algebra but that is only half of it.  The other half is tropical geometry which takes questions on Algebraic varieties and transforms these questions so they apply to polyhedral complexes.  Tropical geometry developed as an offshoot to Tropical Algebra due to questions that arose in computer science.  In addition, this topic is often referred to as tropical algebraic geometry.

There are three approaches to tropical geometry.  First is the synthetic approach, and is considered algebraic geometry applied to a tropical semifield.  It led to finding tropical theorems equivalent to those applied to algebraic curves and their associated geometry.  The second is the valuation theoretical approach which looks at tropical versions as shadows for the algebraic ones.  The final approach is the degeneration theoretical approach which is related to the degeneration theoretical approach for algebraic varieties.

Tropical geometry looks at things differently such as when polynomials which may be continuous in algebra are turned into piecewise functions. In fact, tropical geometry makes it easier to do polyhedral geometry because algebraic geometry doesn’t explain polyhedrals as well. It has been found that many of the invariant varieties are preserved when using tropical geometry.

Furthermore, tropical geometry works well with enumerative geometry, especially Mikhalken’s work.  Tropical geometry has developed its own formulas to determine the number of rational curves based on its degree. Furthermore, tropical geometry looks at solution spaces.  

If you look at this site, you can read four lectures by Dianne Maclagan given back in 2008/2009 to students taking a class in this topic.  The four lectures cover an introduction, fundamental theorems, more examples and explanations, finishing off with enumerative geometry.  Although it is only 67 pages long, it provides some fascinating information.

If you notice, tropical math encompasses both algebra and geometry to work together to form a new way of explaining things.  Let me know what you think, I’d love to hear.  Have a great day.

Tropical Math - Arithmetic/Algebra

 

I just heard this term the other day and the first thing my mind did was to picture numbers lying on the beach, sipping drinks, while sitting under umbrellas. Yes, I know it's not that but there are certain visuals we associate with certain words.  

Tropical math actually covers two different mathematical  sub groups - Algebra and Geometry.  It began developing around the beginning of this century and the adjective "tropical" came from several French mathematicians including Jean-Eric Pin.  

It is based on the "tropical semi-ring" which uses a set of real numbers with the additional element of infinity.  The tropical sum is defined as the sum of their minimums while the product is actually their sum. An example of a tropical sum for 4 and 8 is 4 because 4 is the minimum of both numbers while the product is 12 because 4 + 8 = 12. The notation used for tropical sum is a circle with a cross in the middle and tropical product is indicated by a circle with a dot inside it.  

What makes this even more interesting is that these two operations are actually commutative and can have the distributive property applied to them.  Furthermore, infinity is considered a natural element of addition and zero is a natural element for multiplication.  Unfortunately, subtraction within tropical arithmetic is a bit harder because there is no value for the phrase "10 - 3" thus it is best to stick with addition and multiplication.

As far as polynomials go, tropical monomials are actually linear functions with integer coefficients and a tropical polynomial is actually defined as "a finite linear functions with integer coefficients". Tropical polynomials are also continuous, composed of a piecewise of linear functions and it is concave. In addition, the Fundamental Theorem of Algebra apply to tropical linear functions.

Furthermore, curves in tropical algebra is shown in a hypersurface composed of all "roots" of the polynomials.  This can be extended to polynomials in two variables in which the curve is contained in the plane of real numbers squared with both bounded and unbounded edges.  In addition, the slopes are rational and if the sum of all vectors is taken, the result is zero.

When graphing such polynomials, one sees three half-rays heading off in three different directions. The degree of the polynomial determines the number of half-rays, vertices, bounded and unbounded edges but they half-rays tend to head off to the north, east, and southwestern directions.  This leads to the idea of linear spaces in which solving linear equations requires a person to determine the intersections of a certain number of hyperplanes.  

This is a brief look at tropical arithmetic and algebra but there is also the field on tropical geometry.  I'll be providing a short look at tropical geometry on Wednesday.  Let me know what you think, I'd love to hear. If you want to learn more about this check the internet for some really interesting reading.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If there about 540 peanuts are used to make 12 ounces of peanut butter, how many peanuts are needed for 32 ounces of peanut butter.

Warm-up.

 If one pound of pecans yields 4 cups of pecans, how many pounds of pecans will you need for 128 pies if one pie uses 1 3/4 cups of pecans?

African American Mathematicians Part 1

 Since the month of February is Black History Month, I’d like to take time to focus on several African American Mathematicians.  The movie “Hidden Figures” introduced us to Katherine Johnson who worked at NASA but was not recognized as a mathematician. Most of us are only familiar with the Europeans such as Descartes or the ancients like Pythagoras. 

If you look at many African American Mathematical figures, you will notice that  many of them contributed to the growth of mathematics.  For instance, Elbert Frank Cox influenced the mathematical program at Harvard.  He was born in Evansville, Indiana in 1895.  He was able to attend college and secured his Baccalaureate from the University of Indiana in 1917 with a major in mathematics.  After serving in World War I, he spent time teaching before returning to school to obtain a Masters and finally his Ph.D in Mathematics from Cornell University in 1925.

He is considered the first African American to be granted a Ph.D in Mathematics in both the United States and the World.  After graduating from Cornell, he spent four years working at West Virginia State University before moving to Howard University. At Howard University, he worked there until 1965 when he graduated.  He is said to be responsible for setting up the program which was later used by Ph.D students when Howard started granting Ph.D’s in 1975.  He also chaired the mathematics department from 1957 to 1961.  

Then there was Marjorie Lee Browne who was born in Memphis, Tennessee in 1914.  She began college during the depression but using a combination of work, scholarships, and loans, she was able to attend Howard University. She graduated with her Baccalaureate in Mathematics in 1935. Upon graduation, she accepted a teaching position in New Orleans but left  after one year to pursue her Masters at the University of Michigan at Ann Arbor.  She received her Masters in 1939 while becoming one of the first African American women to receive this advanced degree.  

She accepted a job working as one of the teaching staff at Wiley College in Marshall, Texas after graduation but she worked during the summers on her Ph.D at the University of Michigan.  She ended up accepting a teaching fellowship at the University of Michigan where she completed her Ph.D around 1950.  In 1949, she accepted a position with North Carolina Central College where she worked for 25 years and she was the only one on staff with a Ph.D.  She is credited with obtaining enough grant monies for North Carolina Central College to purchase it’s first computer and to provide grant monies to outstanding mathematics students.  Furthermore, she spent her summers instructing secondary school teachers.  She passed away in 1969 of a heart attack at the age of 65.

I’ll be looking at more outstanding African American mathematicians later in the month.  This is just a start to show they are out there and made tremendous contributions.  Let me know what you think, I’d love to hear.  Have a great day.

Textbooks Over Time

I ran across a couple of articles dealing with how math textbooks have changed in the 100 years between 1900 and 2000.  Back in 2010, someone took a look and analysed the content in 141 math textbooks for elementary students that were published between 1900 and 2000.  They discovered that up until the 1960’s, most of the material (85%) covered focused on basic math but by 2000 the percentage dropped to 64%.  In the meantime, textbooks began teaching more on topics like advanced Algebra or geometry.  

In addition, many of the early textbooks were written by teachers teachers and supervisors rather than real mathematicians.  This meant many authors of mathematical textbooks didn’t even hold a Bachelors in Mathematics.  Many people who qualified as teachers in the 1930’s only attended a “Normal School” for two years so they didn’t have a strong background in the subject and most students were not expected to take four years of math in high school.

Often, these authors thought they could do a better job of creating a textbook than the one they used so they’d write one and at the same time, they’d earn a bit of extra money.  Even in the 1950’s prospective university students were not expected to have taken a lot of math but then the 1960’s hit.

In the 1960’s,  there came the push to have students learn more math and science so they could think more like mathematicians and scientists.  This became especially true when the Soviet Union managed to launch the first man into space.  At first, math was taught based more on inquiry and by the 1980’s there was a move away from rote memorization towards the standards used in today’s schools.

With the push towards having students focus on conceptual learning and rich mathematical ideas, teachers have been using various activities more which don’t always support these topics. In addition, the writing of textbooks has moved from teachers and supervisors creating their own version, to publishing companies creating a series of textbooks written by mathematicians and educators who have a solid understanding of learning.

Furthermore, analysis indicates that prior to adopting the common core standards, most standards were an accumulation of beliefs on what they should know so the curriculum was mostly set up to minimally look at so many different topics and was referred to as “scope and sequence”.  The change has certain topics appearing across several grade levels moving from an introduction into a deeper use.

If you look at universities and colleges, you’ll find that they also underwent a change.  Prior to 1900, many math classes were taught by professors as that one elective they needed to teach outside of their area of expertise much like coaches teaching math in high school.  In addition, many mathematics professors taught classes in subjects outside of their field like art, or science.  Over time, mathematics came into it’s own with departments, journals, and real textbooks.  

So textbooks have changed over time to what we see today. If you get your hands on a textbook from the past, it has little explanation but with mathematical proofs to explain the topic.  I’ve got a few from the 1940’s and 50’s for physics which can be difficult to follow. Right now, it is known that things need to change but it takes so long for textbooks to change.  Textbooks only change if school districts make the demand or states want it.  Let me know what you think, I’d love to hear.

Happy President’s Day

 

Happy Valentine’s Day

 

Warm up.

 

If 250 million roses are sold this year for Valentine’s Day and 69 % are red, how many red roses are sold?

Bees As Mathematicians

 

The other day, I discovered a series of articles where they state that bees are mathematicians.  Interesting thought so I read several to see why they made the connection.  Honeybees make these beautiful combs where each cell is hexagonal shaped.  Think of it! A creature who can make something that is hexagonal without a ruler, compass, or written plan!

In fact, mathematicians have been making conjectures and hypotheses about honeybee’s ability since 36 BC.  The first known conjecture from 36 BC was the honeybee conjecture in which it was speculated that hexagonal shapes are the most efficient way to fill an area. This wasn’t proven until the 1990’s. 

People claim bees are mathematicians for several reasons. First they can construct hexagonal shaped cells for their honey and hive without the use of instruments.  Next, they seem to understand why the hexagon is the best shape for the cells.  They also appear to have a grasp of the concept of zero.  Finally, they can solve simple problems and even other types of problems such as the “Traveling Salesman” one.

In addition, at the age of two weeks, bees are capable of changing sugar into a waxy substance used to build the hexagonal cells which have a more economical shape because they use the least amount of wax to construct.  This shape has the least amount of surface area while providing the maximum amount of space. Furthermore, there is no wasted space between cells like one would have if they used circles instead.

Simple experiments have been carried out to test how well honeybees can perform simple math.  In one, honey bees were asked to fly through  a maze which contained a variety of shapes either in blue or yellow.  When the bee came to a color they had to make a decision of which way to go.  If they made all the right choices, they were rewarded with a treat of sugar water.  It took 100 bees between 4 and 7 hours to figure out the maze and associate their decisions with simple math tasks.

They are able to communicate geographical information using a specific type of dance.  There are indications that bees can also remember human faces.  Scientists would like to figure out how bees are able to perform complex tasks with a limited number of neurons in order to apply the information to machine learning.  The answer to this questions, will help AI learning improve by leaps and bounds.

I find it fascinating to learn how well bee’s do math.  Let me know what you think, I’d love to hear.  Have a great day.

Financial Algebra

 Last year, several teachers from my district attended a mathematics conference and one of the vendors offered something that I’d never seen before.  As you know, most high schools offer the traditional Algebra I, Geometry, and Algebra II but this vendor offered a financial algebra course which could be used to replace the traditional class.

This is a class that offers many of the same topics but shows how they are applied in finance.  It might show students how advanced algebra is used in things such as discretionary spending, banking, home or auto ownership, business, or retirement.  Furthermore, financial algebra also offers a better opportunity to integrate modeling with the topics.

I looked at one textbook from National Geographic which had chapters on the stock market, learning to model a business, banking and services associated with them, consumer types of credit, employment and taxes, and planning for retirement along with several other topics.   In other words, the class gives students a more in depth understanding of financial topics for personal and business settings.   It takes things a step further than the usual consumer math classes and gives them a better mathematical exposure to finance.

This particular class is a good one to add to the current offerings because it offers a more relatable context for most students who see Algebra as totally unrelated to their lives.  A good financial algebra class will cover topics like linear equations, fractions, decimals, percents, moving averages, exponential functions and exponential growth and decay, limits, recursive thinking, piecewise functions, expected value, parabolas and quadratic formulas, scatter plots and correlation, natural logs and lns, rational expressions, square roots and so many more topics found in a traditional Algebra II class.

If done properly, students will only need to complete the Algebra I class in order to be prepared and a good financial algebra class will reinforce what they’ve learned while introducing them to more complex topics.  

I like the idea of financial algebra because it will answer the question “When will I ever use this?”  I think it will help students to see when and where the material is used in the real world and in their lives.  This is important because it gives context to the math they are learning in school. 

In addition, such a class provides students who struggle with traditional algebra 2 classes a nice alternative and is good for students who do not plan to major in one of the hard sciences.  It offers more advanced algebra than the traditional consumer math classes which are more arithmetic based.  

So schools should offer financial algebra in addition to the more traditional algebra II classes so more students have their needs met while preparing them for the future.  Let me know what you think, I’d love to hear.  Have a great day.

Data Science In High School

 

Things are changing. When I went to college many years ago, one wanted to have enough math in high school so you could go straight into calculus if you were majoring in the hard sciences.  In fact, if you could manage to take calculus in high school, even better but the requirements for what is needed in the real world has changed.

People are now thinking that a different approach is necessary for advancing in the hard science fields because data and statistics are being used more and more rather than calculus.  In fact, the collection and interpretation of data is being used to make predictions, interpret the events of the world, or explain the world.  

In fact, most data science has become a major player in the world and many are predicting that all high schools should offer a class in it.  Although sciences traditionally have used data and statistics over the years, other professions such as economics, politics, and education are relying more heavily on it.  Jo Boaler is recommending that schools provide instruction to students to help them develop data literacy.

Unfortunately, traditional pathways in modern education still have students taking a sequence leading to calculus. Furthermore, most schools still do not offer stand alone data science or statistics courses.  Statistics is often incorporated into Algebra I and II classes such as in my district.  This means students only get a surface knowledge of the material rather than a more in depth understanding.

It is hoped by offering additional classes in data analysis and statistics in high school, students will be better prepared for college and the workforce. If this cannot be done, it has been recommended that classes offer students the opportunity to work with large sets of data.  It is something students need to do before they graduate from high school.

In fact, if students take data science classes in high school, it opens up a chance to participate in the up and coming career of data scientist.  A data scientist sorts through data and provides the information necessary for various industries to make informed decisions. At the moment, there are more jobs available than candidates to fill them.

So if we want our students to understand more of the world around them while preparing them for a career or at least offer the tools needed to help them succeed, we need to begin offering data science classes in high school.  Unfortunately, education tends to be a bit slow in changing to meet the needs of a rapidly changing world.  Let me know what you think, I’d love to hear.  Have a great day.

Warm-up

If there are 33 dried apricots per pound, approximately how many apricots are there when you have 258 pounds?

Warm-up

If it takes four pounds of grapes to produce one pound of raisins, how many pounds of raisins will you have if you start with 2688 pounds of grapes? 

Japanese Multiplication

One of my students showed me something new.  She’d seen it on Youtube and chose to share it with me.  It was interesting. It’s called the Japanese multiplication method also known as Chinese multiplication, or the stick method.  Apparently, it's been around a while. I've used it and it reminds me a bit of the box method without the box.

It uses parallel lines that run diagonally and when the line cross, they form intersections and it is the intersections that provide the multiplication. I've done a problem showing each step so you can see how it works. I'm using the problem 24 x 13.

For 24, I've got two parallel lines in green with a space and then four more that are dotted. 13 is represented by one red line and then 3 more a bit a way. It is easy to see the 24 and 13.

The line are supposed to cross at 90 degree angles or be perpendicular. Usually the lines are running along the y = x or y = -x pathways.

The next step is to place dots on the intersections so they are so much easier to see.

These dots are what provides us with the product for each step of the multiplication.

The intersections at the middle left represent the hundreds, the group at the top and bottom represent the tens, and the group to the right represent the ones.

So based on the last photo, we have 200 from the left, 60 plus 40 from the middle and 12 from the bottom. Each of these represents one number from the standard multiplication algorithm.  When you follow the standard multiplication algorithm you would go 3 x 4 = 12, 3 x 20 = 60, 10 x 4 = 40 and 10 x 20 = 200 so when you add all the numbers up, you get 312 using either method.

There are problems when trying to use this method with larger numbers.  For instance, if you multiply 8 x 9 you'll have 8 lines crossing 9 which gives you 72 intersections to count.  Or if you so something like 599 x 798, you end up with lots of lines which can be quite confusing but when children are learning to multiply, they tend to use small numbers so this system works well.

I am glad to know about this as it gives me one more method I can use with students who struggle with multiplication.  In fact, I have a couple of students who do not know their multiplication tables so I want to show them this method.  It might help them so they are not as reliant on a calculator. Let me know what you think, I'd love to hear.  Have a great day.

Is John Hattie Incorrect?

I don't remember where I stumbled across this but it was about John Hattie and his work. As you known, he's written several books which contained conclusions based on synthesized data on how to improve learning. My district uses his information out of his "Visible Learning books to talk to us about what we should be doing. In fact, I set my personal goal based off of something I found in his "Visible Learning in Mathematics.

There are quite a few articles out there questioning the methodology used. One claim is that Hattie conducted a meta-meta-analysis of over 1200 meta-analysis studies that synthesized over 50,000 studies which covered all sorts of topics from interventions to pay performance teachers. These studies had sample sizes from one to several hundred, or carried out with everything from great parameters to bad. It was also claimed that any biases contained in these meta-analysis studies into his final work.

In addition, several people argued that since many of the original studies had sample sizes of one or lab studies using paid volunteers and other issues, the results may not be valid. Supposedly, Hattie synthesized all the results and applied a metric of confidence so they could utilize the date.

At least one statistician claimed the method of calculating averages and statistical deviation used by Hattie and his people shows a lack of sophistication. Many questioned whether this method of calculating averages and statistical deviation can provide enough data for accurate conclusions. It has also been said that he does not use proper baseline comparisons and has also used comparisons of other factors in correction. Furthermore, they allege that Hattie didn't really understand most of the normal methodology used which lead to incorrect conclusions.

Another person raised the issue about using a multitude of studies that do not share the same methodologies, implementations, or having students who are in the same socio-economic group. These differences can make comparing results even more problematic. Others have asked if conducting a meta-meta-analysis of meta-analysis studies is the best way to obtain information since one is synthesizing information that has already been synthesized.

I don't know if his conclusions are correct or incorrect. For me, my district follows the results published in "Visible Learning", and I have to accept that they want me to use the information. Let me know what you think about this topic. I'd love to hear. Have a great day.

The Problem With Remainders

I am returning to the topic of division, specifically remainders because so many of my students do not seem to understand what it represents.  I’ve observed many students who took a remainder and made it the decimal portion of the final answer.  For instance, if they have 24/5, they do the division and instead of 4 4/5 or 4.8, they put 4.4 because the remainder is four. I have no idea where that misunderstanding comes from.

I’ve tried to research why this happens but have not been successful as most every article only discusses the process of division and how difficult it is.  It is hard finding articles discussing why students have issues with writing down the correct form of the remainder. I wonder if they do not connect the remainder with a fraction and  a decimal equivalent. I'm not sure they even understand the concept of a remainder . 

Most articles state that the process of long division is extremely difficult and students struggle to learn the multi-step process but few address misunderstandings associated with remainders. I found one author who indicated that higher order thinking skills are needed to interpret the type of remainder. The remainder requires students to determine its context in order to figure out the type of remainder that is needed.

There are four types of remainders that students will run into when doing division.  The first type of remainder is the one you leave alone as a fraction such as in 25/4 gives 6 and 1 left over as the answer.  The numerator tells you how many were left over out of the groups.  In other words, the part of a whole.  This is the case where my students tend to write 6.1.  The second type of problem looks at only the remainder so the problem might tell you that you have 25 quarters and wants to know how many you have left over once you turn the quarters into dollars.  In this case, the answer is one quarter. 

The third type of remainder is one that is either rounded up or down depending on the circumstances.  For instance, if your can of paint will cover 300 square feet with two layers and you need to cover 1100 square feet with a two coats, how many cans should you buy.  If you do it with a calculator, you’ll end up with 3.666666667 or 3 ⅔ cans.  In this case, we need to round up because we cannot buy ⅔ of a can.

The final type of remainder is the sharing remainder where you want a fractional answer such as you are sharing 25 cookies among 4 people, how many cookies will each person have?  It is 6 ¼.  The answer tells us that each person is going to get 6 ¼ cookies.

Based on my own experiences teaching, students have issues with remainders because they have not fully learned how to use long division in addition to not knowing how to interpret remainders.  Furthermore, they are also weak on connecting remainders with fractions and changing those fractions into decimals.  Let me know what you think, I’d love to hear.  Have a great day.