Gender Bias in Mathematics

My latest issue of Mathematics Teacher arrived in the mail a few days ago.  One of the first articles, "Staring Down Stereotypes," gave several examples of stereotypes about mathematics:

• Men are better at mathematics than women.
• Mathematics requires only logic, not intuition.
• Mathematics is not creative.
• There is a best way to do a mathematics problem.
• Mathematicians do problems quickly, in their heads.
• Mathematics requires a good memory.
• Some people have a "math mind," and some don't.

The article also mentions Harvey Mudd professor Rachel Levy and her blog "Grandma Got STEM," which confronts the stereotype implicit in the direction to write an explanation to a math problem so simple that your grandmother could understand it.  The blog is a tribute to grandmothers in STEM fields and includes biographical information as well as many historical pictures.

The blog has a picture of the women from Bletchy Circle, including a link to an interview of Joanna Chorley, one of the women who worked there.  It also has a great collection of links including this interview with Eleanor Kolchin in the Huffington Post.  She was a programmer in 1946 and shared what it was like to work for IBM at the time.  She had to get married secretly because she would have been fired by company policy at the time.

The blog is inspiring, and proves that talent in mathematics and science is not confined to the young, or to men.  My own grandmother used to say to my two sisters and me, "Don't settle for a job serving a man - like a secretary.  Be an engineer or a doctor or a lawyer."  Her message was meant to convey that we could do anything we wanted.  My older sister is a computer programmer, I am a math teacher, and my younger sister went into the IT field.  Thanks grandma!

What I am Learning about Differentiating for Equity

I do not have a lot of diversity in my college classrooms.  This semester, I have 88 students total, and 6 are students of color.  I have never been challenged to differentiate instruction for equity by my division or even by the administration.  I am a novice in the area of differentiating, especially in relation to diversity or equity.  I have been learning a lot this past week by reading through recommendations made by school districts, entire states, and individual teachers.  Most of what I have read is written to the K-12 world, but could extend into the college classroom.

Mark Ellis argues in Chapter 1 of the book Mathematics for Every Student: Responding to Diversity, Grades 6-8 that "differentiation is to vary one's actions as a teacher to meet the needs to students."  (p.10).  He argues that the focus is on changing instructional practices, "moving beyond the standard, or normative, habits that characterize mathematics teaching (e.g., teacher-led lecture and demonstration followed by individual student's work on rote procedures)." (p.10)  He argues that the alternative understanding of differentiation in which the curriculum is changed in response to students' abilities should be avoided, especially in "school mathematics, with its history of providing inequitable access to content on the basis of perceived ability." (p.10)

I read another succinct way of describing differentiated instruction in a document by the Metropolitan Center for Urban Education called Culturally Responsive Differentiated Instructional Strategies: "Through differentiated instruction, students will get to the same place, but take different paths." (p. 2)  So culturally responsive differentiation is not watering down the curriculum for some students, or even individualized instruction.  "Differentiated instruction suggests teachers look at “zones” in which students cluster so they can offer three or four routes to a goal on a given day." (Culturally Responsive Differentiated Instructional Strategies, p. 3)

I also came across a Research Brief titled "Increasing the Achievement of African American Males" which was a report from the Department of Research, Evaluation, and Assessment in Virginia Beach City Public Schools.  The report detailed factors that contribute to the achievement gap of African American male students, and suggested strategies aimed at improving their achievement.  The report stated on pages 5 and 6:

Culturally responsive instruction pertains to classroom practices that draw meaningfully on the culture, languages, and experiences that students bring to the classrooms in order to increase engagement and academic achievement for students of color (Au, 2006; Ladson-Billings, 1995).  A few of the most common tenets of a culturally responsive pedagogy included (1) acknowledging that valuable knowledge resides in students’ home languages and cultures and that this home knowledge is not always valued in the schools; (2) pursing academic success for students of color; (3) creating valuable connections between students’ home and school experiences; and (4) fostering social justice (Dutro, Kazemi, Balf, & Lin, 2008).

One way in which I can implement these tenets is to spend the time getting to know my students' native language and culture.  I each lunch on campus everyday and I value that time to connect with other professors and staff members on campus.  However, I can also take the time to connect with students of diverse backgrounds.  I recently had lunch with an international student who was born in China but grew up in Italy due to her parents' jobs.  She explained some of the differences between Eastern and Western culture that she has encountered in her education.  She helped me understand some of the challenges that international students face.  In order to provide effective culturally responsive instruction, I need to become more familiar with the experiences and backgrounds of my students, and this will require spending time with them outside of the classroom.

The report also listed characteristics of effective African American teachers on page 10:

In his work, (Howard, 2001) assessed three major culturally responsive strategies used by highly qualified African American teachers who taught mainly African American students. Holistic, culturally communicative, and skill-building were the three strategies discussed in his study. In essence, the report emphasized that culturally relevant teachers are personally warm toward and respectful of, as well as academically demanding of, all students.

Holistic strategies recognize that not only the intellectual, but also the social and moral capabilities of students need to be developed.  This involves teaching students "character building, honesty, responsibility, respect, cooperation, sympathy to others, and behaving in ways that are consistent with the social norms of the classroom and society." (p.10)  At a Christian institution, these holistic strategies fit into the goal of developing the whole person, not just the mind.  I usually only have students for one semester in my classes, but once I am finished with my master's degree and have more time to connect with students, there are mentoring opportunities available.  In fact, there are mentoring opportunities available at my son's middle school (grades 7 & 8) and also with an organization called Urban Transformation Ministries that my church is connected to in which I can help to encourage students of color.

Culturally communicative strategies take into account the language patterns of students.  The report gave an example of this on page 10:

According to one teacher, students “talk all day in class and don’t get focused, but if you give them leadership roles of leading a group or leading a class discussion, they are very successful” (Howard, 2001, pg. 190).  

The report mentioned that teachers took into account that many African American students prefer oral rather than written expression, and gave an example of having students listen to a story before writing about the story.  I need to provide opportunities for students to express their mathematical knowledge orally, both in class and on assessments.

Skill-building strategies recognize that all students have the ability to think, or are "smart", but that some students have developed more skill.  This aligns with the ideas from the Mindset book we talked about earlier in the semester.  These ideas can help counter the cultural myths that some students just can't do math.  I can talk about these ideas at the beginning of the semester, and refer to them whenever I encounter the fixed mindset in students.  

Another related quote from Mark Ellis in Mathematics for Every Student: Responding to Diversity, Grades 6-8 about working with a 7th grade student named Alonso:

Although I realized that he lacked proficiency with many mathematical concepts and skills, I did not equate this lack with Alonso’s having a low ability to do mathematics.

I think that sometimes it is difficult to see students who have lower math skills when entering our class as equally capable of doing math, but I believe that it is critical in addressing the inequities that are present in our culture.

I am still at the beginning steps of differentiating in culturally responsive way.  I can make progress toward better understanding the students in my classes who have a cultural background different than my own, affirming their ability to learn mathematics, and providing alternate paths to achieve success.

5 Practices for Orchestrating Productive Mathematics Discussions

I teach three sections of College Math every semester - usually one right after the other on MWFs.  I usually have about 100 students in total.  While teaching the same class three times a day can sometimes be challenging (I always hope that my most talkative class is last), it has given me momentum to improve the class each semester.   Usually there are five sections of College Math each semester and I collaborate with the two adjunct professors who also teach the course.  Last year these colleagues and I started experimenting with the flipped class.  Our motivation was the desire to have more class time for discussion and application of the knowledge that students were learning.

One example is the lesson on measures of central tendency and measures of variation - mean, median, mode, range, five-number summaries, and standard deviation.  We usually had to spend a significant portion of time reviewing how to calculate these measures and ran out of time to analyze when to use them or what they reveal about sets of data.  So I created a video explaining how to calculate these measures and assigned watching it and filling out guided notes as homework due before class.  During class we spend time discussing when to use each measure and how to interpret what the measures are saying, especially in comparing sets of data.

I realized pretty early on that I needed to learn how to structure the flipped class time differently and that I easily slid back into a lecture mindset on those days.  I have not experienced a math class as a student where discussions are the central format, nor did I learn how to facilitate mathematical discussions in my teacher preparation program in college.  Since I was able to choose the topic for a project in my current grad. school class, learning about how to do this better seemed like the obvious choice.

I just finished reading a book by Margaret S. Smith and Mary Kay Stein called 5 Practices for Orchestrating Productive Mathematics Discussions, recommended to me by my professor.  I learned so much from this relatively short book.  Here are some quotes:

"In mathematics classrooms, high-quality discussions support student learning of mathematics by helping students learn how to communicate their ideas, making students' thinking public so it can be guided in mathematically sound directions, and encouraging students to evaluate their own and each others's mathematical ideas." (p. 1)

"Teachers often think of lessons in relation to the activities that they will ask students to do rather than what students will come to know and understand about mathematics as a result of having engaged in the lesson." (p. 15)

"I must consider the mathematics in relation to the children and the children in relation to the mathematics." (p. 50)

"What students learn is intertwined with how they learn it." (p. 61)

"Almost all good classroom discussions begin in the same way: by inviting a student to share how he or she solved a particular problem.  After the initial student response, however, classroom discussions diverge--separating into the relatively rare fruitful ones and the much more frequent unproductive show-and-tells.  It turns out that issuing an invitation to students to reveal their thinking is a relatively easy thing to learn how to do.  And most students will comply......Typically, however, teachers treat all presentations as equally good, they ask few questions of the students, and they do not connect different student presentations with one another or the disciplinary ideas under investigation." (p.69)

This book lays out five practices designed to remedy this situation:

1. Anticipating - considering ahead of time how students might approach a mathematical task, both correctly and incorrectly, and how those relate to the things that a teacher wants students to learn
2. Monitoring - paying close attention to students are they are working on a mathematical task and keeping track of strategies used as well as asking questions of students to help guide their thinking and keep them engaged
3. Selecting - picking particular students to share their work in order to get certain mathematics into the open for examination by the class
4. Sequencing - purposeful ordering of the student work being presented so as to maximize the chances of achieving the goals for the lesson
5. Connecting - drawing connections between the student work that is presented and the key ideas of the lesson

A necessary foundation  for these practices to be successful is setting clear goals and selecting tasks that are conducive to discussion.  The goal of the lesson needs to clearly identify what students should know and understand about mathematics and consequently will guide the selection of appropriate tasks.  The tasks need to be of sufficient depth to stimulate and provide opportunity for discussion.

I also appreciated the chapter on asking good questions and keeping students accountable for their understanding of the material.  Here is a list of questions asked by teachers in various classroom scenario presented in the book:
"What does this mean?"
"Why did you do it that way?"
"Can you explain more to me about that?"
"Does anybody want to add to that?"
"How do you know that?"
"Do those two things mean the same thing?"
"Does everyone agree with this?"
"Is there another way?"
"Can you repeat what [insert student's name] did to solve this?"
"How can we know for sure?"
"Can you always do that?"

The value in these practices is to provide a structure that will help transform class discussions from a "show-and-tell" format to an intentional interweaving of student work with the mathematical goals of the lesson.

The next step is to implement these five practices with a lesson in the College Math class that I teach, involving the adjuncts along the way.  Collaboration will improve the process.  Let the project begin!

Which is the Better Buy?

I am always on the lookout for good questions to ask my college math classes.  One of the questions that catches students' attention is this:

Which is the better buy:  33% off the price of an item or 33% more quantity of that item for the same price?

At first students guess (neither deal tends to draw significantly more votes than the other initially) and then I ask them to try running the deals with some numbers.  It takes some students awhile to figure out that they need to choose both a price and a quantity of an item to compare these deals.  Students work in groups and the numbers that they choose are interesting.  Some groups choose an item that they have bought for which they know the price and quantity.   Others make up numbers that don't have a connection to real life.  Other groups choose the numbers with simplicity in mind (like $1 for 1 gallon).

The next hurdle for some students is figuring out how to compare the deals.  Comparing unit prices can be a good way of finding which is the best deal, but not all students come into the class remembering how to do that.  Comparing unit prices (price per ounce, for example) is the most common strategy that students employ, but when I taught this lesson last week, one group used unit quantities (ounces per dollar, for example).  They puzzled for awhile about whether the higher unit quantity or lower unit quantity was the better deal.

After we come to a consensus that 33% off the price is the better deal, I ask them if it is always the better deal.  Since each group of students usually chooses different prices and different quantities and they all calculate that 33% off the price is the better deal, some students conclude that it is always the best deal.  A short discussion about whether many examples prove that the idea is correct is followed by me challenging students to think of how they could prove that the discounted price is always better than more quantity for the same price.  A few students will realize that they need to create a variable for price and a variable for quantity and do the calculations with the variables rather than their particular price and particular quantity.

This lesson prompted me to determine whether it is always better to get the same quantity at a discounted price for any discount amount - not just 33%.  Here is the way I figure that out:

Question:
Is it better to get ¹⁄_x off the price of an item or ¹⁄_x more quantity of that item for the same price? (where x > 1)

Solution:
Let p = price and q = quantity.

First consider getting ¹⁄_x off the price of an item for the same quantity:
New price = p - (¹⁄_x) p = p - (^p⁄_x) = ^{(px - p)}⁄_x = ^{[p(x - 1)]}⁄_x
Unit Price (price per quantity) = (^{[p(x - 1)]}⁄_x) ÷ q = (^p⁄_q )(^{(x - 1)}⁄_x)

Next consider getting ¹⁄_x more quantity of an item for the same price:
New quantity = q + (¹⁄_x) q = q + (^q⁄_x) = ^{(qx + q)}⁄_x = ^{[q(x + 1)]}⁄_x

Unit Price (price per quantity) = p ÷ (^{[q(x + 1)]}⁄_x) = (^p⁄_q )(^x⁄_(x+1))

How should these unit prices be compared?  The lower unit price is the better deal.

So we are comparing (^p⁄_q )(^{(x - 1)}⁄_x) to (^p⁄_q )(^x⁄_(x+1))

Since each expression contains (^p⁄_q ), we can simplify and just compare (^{(x - 1)}⁄_x) to  (^x⁄_(x+1)).

Getting a common denominator results in comparing (^{[x2 - 1]}⁄_[x(x+1)]) to  (^x2⁄_[x(x+1)]).

Since both fractions have the same denominator, the smaller fraction will be the one with the smaller numerator.

So now we can compare (x² - 1) to x².

(x² - 1) is less than x² for all x.

Therefore, getting  ¹⁄_x off the price of an item is always a better deal than getting ¹⁄_x more quantity of an item for the same price!

Promoting Mathematical Discourse

I want to facilitate and promote better discourse among my students.  About 40% of the lessons in the core math class that I teach are flipped.  For me, the biggest challenge in the flipped classroom is developing valuable tasks for students that will extend and build on the content presented in the video watched by students before class.  The quality of the group discussion and problem-solving that takes place in class are critical for students to benefit from the flipped structure.  I found the list of resources below on an NCTM website about promoting mathematical discourse.  Time to get reading!

• Breyfogle, M. L., & Herbel, B. A. (2004, April). Focusing on students’ mathematical thinking. Mathematics Teacher, 97, 244-247.
• Choppin, J. M. (2007, November). Teacher orchestrated classroom arguments.Mathematics Teacher, 101, 306-310.
• Fernsten, L. A. (2007, November). A writing workshop in mathematics: Community practice of content discourse. Mathematics Teacher, 101, 273-278.
• Himmelberger, K. S., & Schwartz, D. L. (2007, November). It’s a home run: Using mathematical discourse to support the learning of statistics. Mathematics Teacher, 101, 250-255.
• Kitchen, R. S. (2004, January). Challenges associated with developing discursive classrooms in high-poverty, rural schools. Mathematics Teacher, 97, 28-31.
• Koellner-Clark, K., Stollings, L. L., & Hoover, S. A. (2002, December). Socratic seminars for mathematics. Mathematics Teacher, 95, 682-687.
• Kotsopoulos, D. (2007, November). Mathematics discourse: It’s like hearing a foreign language. Mathematics Teacher, 101, 301-305.
• Manouchehri, A. (2007, November). Inquiry-Discourse mathematics instruction.Mathematics Teacher, 101, 290-300.
• Manouchehri, A., & Lapp, D. A. (2003, November). Unveiling student understanding: The role of questioning in instruction. Mathematics Teacher, 96, 562-566.
• Manouchehri, A., & St. John, D. (2006, April). From classroom discussion to group discourse.  Mathematics Teacher, 99, 544-551.
• Mason, R. T., & McFeetors, J. (2002, October). Interactive writing in mathematics class: Getting started. Mathematics Teacher, 95, 532-536.
• McIntosh, M. E., & Draper, R. J. (2001, October). Using learning logs in mathematics: Writing to learn. Mathematics Teacher, 94, 554-557.
• Stein, M. K. (2001, October). Mathematical argumentation: Putting umph into classroom discussions. Mathematics Teaching in the Middle School, 7, 110-112.
• St. John, D., & Manouchehri, A. (2006, April). From classroom discussions to group discourse. Mathematics Teacher, 99, 544-551.
• Truxaw, M. P., & DeFranco, T. C. (2007, November). Lessons from Mr. Larson: An inductive model of teaching for orchestrating discourse. Mathematics Teacher, 101, 268-272.

Student Difficulties with Graphs

I teach College Math, and we start out the semester with a lesson on functions.  One of the things we ask students to do is look at the graph of a function and decide if it is possible or impossible.  This task is surprisingly difficult for some students.  I got the activity from Shodor, a nonprofit organization serving students and educators by providing materials and instruction relating to computational science, at http://www.shodor.org/interactivate/activities/PossibleOrNot/.  Here is the first graph that students discuss:

Some students immediately discern that this graph implies traveling backwards in time and is impossible, but others see the graph as a path an object is traveling and think that it is possible.  Some students have trouble interpreting the first two linear segments as representing an object traveling away from something (distance is increasing) and then traveling back towards something (distance is decreasing).

The other graph that students struggle with is this one:

Many students resist the conclusion that this graph is impossible, even after I ask them to interpret the points lying on the vertical segments on the graph.  Many students persist in seeing this graph as a path in two dimensions, as if the grid represented a blue print of the floor of a room.

I recently read an article about student difficulties with using and reading graphs ("Examining Students' Reluctance to Use Graphs" by Frances Van Dyke and Alexander White in Mathematics Teacher, Sept. 2004, p. 110).  One of the types of difficulties discussed in the article relates to the Possible/Impossible activity that I use.  Students seem to do well with interpreting graphs of value as a function of time, but struggle with distance as a function of time:

I showed these two graphs to 12 students, some in a Trigonometry class and some in a College Math class.  Every single one got the correct answer for the first question, but only 4 got the correct answer for the second one (every other student chose 'd').  I also asked my 13-year-old son to look at them.  He got the first one right, and also chose letter 'd' for the second one.  I asked him to explain his answer, and he said that the problem didn't state that Chris stood at a distance from the statue for any amount of time (basically he was arguing that the problem didn't explicitly say that any time passed).  It is possible that students may have chosen a different answer if the problem stated that Chris stood at a distance from the statue for a few minutes, but I suspect that some students would still choose letter 'd'.  To improve, students need activities where they see/experience motion happening through time and then create the graph.  This will help strengthen their skill in interpreting these types of graphs.  Dan Meyer (in conjunction with BuzzMath) has collected several videos for this purpose (http://graphingstories.com/1Pp).  These videos are a great way for students to process and represent the relationship between distance and time.

The article also discussed the difficulty that students have with understanding the connection between a graph and an equation.  I have seen evidence of this in my students, even the ones at the top level of achievement.  An example from the article is a problem that attempts to determine whether students understand the connection between the equation of a line, the graph of the line, and a point on the line being a solution to the equation:

I gave this to a handful of students in my Trigonometry class, a group that is fairly accomplished in mathematics (certainly not remedial), and many of them did not choose the correct answer.  I was amazed at students' confusion with the wording of the question.  One student even substituted x=12 and y=1 into the equation and got 29, but still chose letter "b" as the answer!  I also showed the problem to my 15-year-old son, who is an advanced math student, and he responded that he didn't understand what a solution to the given equation looked like.  I suspect that when asked to find a solution to an equation, he is used to having an equation with only one variable in it.  He is currently in Algebra 2 in high school.  Out of the 12 students that I asked, only 4 got the correct answer.

In the College Algebra course that I teach, the unit on Systems of Equations offers an opportunity for students to practice describing solution sets that contain more than one point (when there are a certain number of variables in the system, but not enough linearly independent equations to generate a unique solution).  Many students struggle with how to express the solution set - both the idea that there are an infinite number of solutions, but also with the notation.  In the future, I could introduce students to these concepts in the earlier unit on linear equations by presenting the type of problem above and discussing how to express the solution set to the equation 3x-7y=29.

Another type of question that my College Algebra students struggle with is the type below (also from the article cited above):

Another question that I ask my students about this type of graph is to find when f(x) is greater than g(x) and also when g(x) is greater than f(x).  Students struggle with how to give the answer and usually want to define the intervals in terms of the y-coordinates on the graph (the range) rather than in terms of the input of the function (the domain).

Students need more practice in understanding how the graph of an equation relates to the set of solutions to the equation.  Students need to better understand that solving equations are finding the conditions that make it true.  Too often finding a solution means finding one number that makes the equation true, and students don't get enough practice with finding solution sets that contain more than one discrete value.  Students also need more practice in relating a graph to function notation.  Many students struggle with function notation, and I think that using graphs when teaching function notation would be a help to students.  Finally, students need to practice creating and interpreting graphs of distance versus time.

My Teaching: Resources and Challenges

I teach at Cornerstone University, a private liberal arts college in Grand Rapids.  The classes I teach include PreAlgebra, Algebra, College Math, College Algebra, and Trigonometry.   I enjoy teaching at the college level.  The students are beginning a new phase of their lives by becoming more independent, learning to think for themselves, and striking out away from home.  I don't have much interaction with parents other than the occasional helicopter parent of a new freshman student.  Many students appreciate the opportunity to develop relationships with their professors and will seek out advice about more than just class schedules.  There is an emphasis at the school to view the role of being a faculty member as more than just a classroom teacher.  I have developed and maintained significant relationships with some of my students, and even had the opportunity to extend hospitality to a former student who needed a place to stay for a few months after graduating.

The remedial courses (PreAlgebra and Algebra) are small, with fewer than ten students.  This allows for a lot of individual attention.  I believe that many students in those courses have undiagnosed learning disabilities.  The cost of being tested is usually prohibitive for most students, so if they weren't diagnosed in high school, they likely won't discover the disability.  One of the major challenges for students in the remedial courses is the mindset of failure.  Many students in those courses just expect to do poorly because of the many years of struggle.  A much larger aspect of teaching the remedial courses in comparison to the other courses is mentoring, encouraging, and motivating students.  This can be both extremely rewarding but also challenging.

The core courses include College Math, College Algebra, and Trigonometry.  If a student doesn't have a math or science major or minor, they will take College Math, which is probably the last math class (other than perhaps statistics) that they will ever take.  Class sizes are larger than I would prefer (usually 30 to 32 students).  Some students lack the motivation to think mathematically in deep and significant ways.  They see the course as a check mark in their list of required courses, something to endure.  Others really enjoy the class, which has a quantitative reasoning focus, and appreciate seeing math in applied contexts.  I appreciate the latitude that I have to experiment with alternative teaching methods.  We have a lot of freedom make yearly switches in textbooks if desired.  We have "flipped" about one-third of the lessons and use Pearson's MyMathLab online course management system which includes online homework and quizzes.  All of our students have university-issued laptops, which is both an asset for and distraction from learning.  I find myself needing to address appropriate use of technology in the classroom with my students every semester as well as call out the students who persist in using their laptops during class for activities other than the lesson at hand.  Missing too many classes is a challenge for some students who for the first time are without the structure of a parent ensuring their compulsory attendance.  Students are practicing making wise choices with their increased freedom, something many first-time freshman struggle to do.

College Algebra and Trigonometry are taken by students who have a science major or an elementary education math minor.  These classes are taught in a more traditional way than College Math due to the reality that they are prerequisites for other courses whose instructors expect their students to have a specific set of math skills.  The students are generally motivated and for the most part have the prior knowledge that they need due to the math placement test in place (students have to achieve minimum scores in arithmetic and algebra in order to take the class).  Class sizes for College Algebra range from 15 to 25 students and usually less than 10 for Trigonometry.  Most of the same resources for teaching mentioned above also apply to these courses like the flexibility to make choices about technology and textbooks.  The main difference is in the type of student who takes the course.