Category Archives: Math

Why Are Our GIRLS So Resistant To Student-Centered Learning?

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I’ve always had a student-centered approach to teaching: my classrooms have always had tables (or desks formed into pods) and I try to limit my time standing in front of the room lecturing at the board.  My classes often involve student activities/discoveries and conversations.  The term “guide on the side” really resonates with me because I like to think of myself more as a coach than a teacher.  This may be because I actually have no formal training as a teacher.  I went to a small liberal arts college, majored in Math and Sociology and found myself figuring out how to teach as I went as a 22 year-old fresh-out-of-college young woman teaching high school students just five years younger than me.  Looking back, I realize that I was greatly influenced by my incredible department chair who had this same student-centered approach and I naturally followed her lead.  It didn’t feel new or novel and rather just became my style.

Carondelet is the fourth school I’ve taught at, the only all-girls school I’ve taught at, and the only school where I’ve received such resistance to this style of teaching.  I’m used to the usual hazing that happens to new-to-a-school teachers, but I’m surprised that I’m still dealing with fierce resistance to my style of teaching.  Given that the direction of the school is to a more student-cenered, guide-on-the-side style, I wanted to share some of the pushback I’ve received as a warning to all of us for what may be coming.

Here are some of the comments I received on a recent survey of my Algebra 2 with Trigonometry class:

[My previous math] class was very traditional and i was able to learn and do very well in that class, this class is all over the place and confusing and I’m not doing well at all.


[I prefer a traditional math classroom because] It is organized and has a plan this class room is very frustrating and all over the place I juts want to learn the section and do my homework everyday in class


I just feel really frustrated with the way we have been doing testing and learning this year.
I can’t teach myself, I need a teacher teaching me subjects or else I just don’t get it.
I DO NOT mesh well with growthmind. I like to have a sense of what will be on the test
so that I can come into class on test day feeling prepared. I feel that these growth mindset
tests do not reflect my knowledge of the information and I am just feeling very frustrated.



My whole life I have been taught math the traditional way and it is what I am used to and
it is in that format hat I understand the concepts.

I want to be in a different class with a better teacher.

I absolutely do not like the problem solving way of math. I used to love math and this way makes
me hate the subject. I need to be able to be taught a way to do something and for me to practice
what I’ve been taught so I can grow my math skills. Me teaching myself is not helping me at all and
I walk into every test scared and stressed because I don’t feel prepared even though I complete almost
every practice test and it’s really frustrating. I’m concerned that I will not be prepared
for pre-calculus next year.

This class in general makes me very anxious because I do not know what will come next and
I am a person that appreciates predictablity and control.

You may notice that some of the students above refer to a growth mindset, something I’ve been pushing hard in this group.  Also on the survey were two questions:  Where are you on the growth mindset spectrum (1 = fixed mindset, 10 = growth mindset) and what type of classroom do you prefer (1= traditional, 10 = student-centered).  These results were fascinating (p= .0013) showing that there is a significant positive relationship between students who have a growth mindset and prefer a student-centered classroom.



Here are some comments from some of our more willing students:

I like the collaborative [classroom]. It requires me to think more instead of just doing to do.


I like how we are actually making an effort to switch education


[I] much prefer the collaborative teaching style because it helps me learn from my mistakes and think through problems more. it is also more engaging.

I did realize the thoughts that go into problem solving, however actually doing a problem solving question made me feel as if I was getting somewhere. At some point I did not know how to solve it or could only solve it half way. So I think it made me think a lot more than I have in awhile which I liked.


Collaborative classrooms help me to see math in ways that I usually don’t. It doesn’t seem so black and white.


Finally, I want to pose a theory as to why our students GIRLS may be so resistant.  In reflecting on why this resistance is happening here and now, I’m thinking about the other schools that I’ve worked at.  All three were much smaller than Carondelet (class sizes closer to 12-15) and because of their smaller size perhaps more exclusive and competitive.  At those schools there was an expectation that school would be hard, challenge was expected, and although not always enjoyed, this was seldom equated with bad teaching or some form of injustice.  But, I think there is a bigger factor at play:  the fact that this is the first time I’m experiencing this in an all-girls setting.  I think in my previous schools, the boys loved the challenge, the unpredictable nature of our learning and the messiness of the class structure.  And because they loved it (or at least didn’t complain) the girls too got on board.  So, I’m finding myself challenged with motivating these girls by myself, convincing them that I am still teaching them (even if not directly) and they will be better served in college and life by having a class that isn’t neat, tidy, predictable and laid out on a platter for them.

An Algebra Teacher’s Lament

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Here are two questions that were on my recent Algebra 2 with Trigonometry test:  A car dealership offers a 10% discount on a car.

a.  Write a function, f(x) to represent the price of a car after the discount.
b.  If the original price of the car is $20,000, what is the price of the car after the discount?

Could you answer these questions?  Is one easier than the other?

I pose these questions to you because I believe that these are questions that most people with high school math education should be able to answer.  In fact, these were on my test somewhat by accident.  These were lead-up questions to a much more advanced topic, composition of functions, and as such we were not in a unit on percentages.

As my title suggests, so many of my Junior students in this elective upper-level math class were unable to do this correctly.  In fact over half of the students got [a] wrong.  Of those who got [a] correct, only one student wrote an efficient function:  f(x) = .9x.  The other “correct” responses included:  f(x) = x-.1x and f(x) = x-(x/10).   While these are correct they don’t represent a clean, efficient solution.  It’s almost like a student who writes a sentence and you know what they mean but there is incorrect grammar.  It’s the same here.  I know what they meant and I appreciate their reasoning, but as Junior mathematics students I expect the more “grammatically correct” version.

Also interesting, 25% of those who got [a] wrong, were able to do [b] correctly.  Here are two examples:

What does this tell us?
These students can do the problem, but can’t connect it to math language.  Our students can do problems, but they can’t communicate as Mathematicians.

Finally, many of the students who go this wrong wrote something along the lines of f(x) = .1x and calculated the new price of the car to be $2,000.  Here’s an example:

It’s amazing to me how many of them didn’t realize their mistake after seeing that the new car, by their calculation, would be so inexpensive.  And this gives us the other important lesson:  that our students often forget to interpret their answer and apply number sense.  We would hope that there is an intuition as to how big 10% is, and a feeling that 10% off means 90% left and thus that the answer should be close to $20,000.

As sad as this grading experience makes me, I find it somewhat fascinating too.  And, it makes me excited for the changes we’re making in the Math Department, and in particular the way we’re planning to welcome our freshman into our new Math culture that emphasizes communication, understanding and number sense in addition to doing.  Too often, the focus of Math classes is just on the doing, and solving problems, this ability to communicate and make sense of our work gets lost.

Alternative Assessment

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Chapter 7 in our Algebra 1 textbooks is all about exponents and exponent rules.  It is super dry (lots of rote memorization), but critical to their success in future math chapters and classes.  So I decided to try something a little different because I wanted my students to be able to fully understand and explain these exponent rules, not just regurgitate them on a typical test.  As I often do in class, I referenced my favorite Einstein quote, “If you can’t explain it simply, you don’t understand it well enough,” and decided to ask the students to explain the rules to me–via video.  The expectation was that they would explain why the exponent rules work, not just show me that they work.

Not quite sure the best way to do this, I attended Joan’s lunch demo of screencasting (such perfect timing!) and she really helped me get the ball rolling.  We found a great app (ShowMe) to record videos with, but when the time came to submit these videos, there were quite a few issues.  The biggest being that most of these videos are 5-10 minutes long and few platforms have the capacity to store that amount of content.  We are still working through this part of it.

I haven’t made it through all of the submissions yet, but so far they have been amazing.  Allowing the students to prove their understanding in a creative way really helped me see them in a different light … and we don’t even need to get into the brain science that supports using both “sides” of your brain in a math classroom.  The creativity of this one in particular blew me away:

Chapter 7 Alternative Assessment Student Sample
*It is over 7 minutes long so you obviously do not need to watch all of it, but the creativity comes across right away.

Even after all of the logistical snafus, the feedback from the students was very positive:

Almost 70% said that this type of assessment helped them understand these concepts more than if they had taken a traditional test.

A Number Talk Sparks Lots of Question About Student-Centered Learning

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As part of our online course, “How to Learn Math for Teachers” by Jo Boaler, he Math Department is learning about something called Number Talks.  In a number talk, more info here, students are presented with an open-ended problem and are encouraged to think of many ways to solve and many ways to represent their solution (including both numerical and visual representations).  A number talk might start with asking students how to multiply 36 x 5, for example, without a calculator and without pencil/paper (i.e. beyond the procedure traditionally taught).  These talks teach students about the flexibility of numbers, how strategy can be applied to numbers, the connections between numbers and other concepts, and the creative, artistic nature of numbers.   At the same time, it teaches them to expect multiple solutions to problems (i.e. Math is not about getting one right answers) and lets them practice explaining their ideas, methods and solutions.

I really love the idea of number talks and think that even doing a simple problem like 36 x 5 in a high school class has real benefits.  But, I’d rather find a way to change the way I’m teaching so that I use the idea of a number talk to talk about the more advanced topics that we teach in our classes.  And that’s why I was so excited when Lesley sent us a example of such a number talk that she had just played with as part of the Mindset Mathematics Leadership Conference.

It helped that I was just wrapping up a unit on radicals in Algebra 2 with Trigonometry and I was totally hooked on how to solve this visually.   I of course knew how to solve Algebraically/procedurally but this was asking for much more.  Did I really understand what a square root was?  It took me a good hour thinking hard about what a square root really is.  A finally settled on thinking of the square root as the side of one square.  But, even then it took me time to figure out what that meant, and what the expression x+15 meant.  I was thinking, not simply doing.  I was stretching my brain and it was exciting!

I finally came up with this solution and felt really satisfied with the experience:

Because we were just wrapping up this unit in Algebra 2, I decided to pose this problem to them as a number talk.  And, here’s where my failure began.  Because I was at the end of the unit, and a bit behind the other Algebra 2 class, I didn’t feel I could devote class time to actually do the number talk.  And if I’m being totally honest, I doubted that many of my students would have been able to handle it.  Instead, I put it on my board and asked students to think about it and contribute whenever they had an idea.  I told them it would live on my board for a couple of weeks and we’d see what gets filled in.  I had visions of some of my more motivated/math-interested students thinking about this as I did and using their free time to come to my room to make their contribution to my board.

Well, it’s been about two weeks and here’s what my board looks like:

Don’t be fooled.  The pictures you see have nothing to do with the problem.  That’s work by my Geometry students who needed some board space to work on their problems.  Not one student contributed to my number talk.  It’s not their fault.  To really have done this right, I needed to model it for them by using class time.  I chose not to, under pressure to stay on schedule, and perhaps missed out on a really deep Mathematical experience.

This is making me think a lot about much of the innovation we’ve been talking about both in our department and as a school.  In order to be truly student-centered, we as teachers need to be able to go off-schedule, right?  We need to have the flexibility to follow the curiosities of our students.  But, how does this work when we have a Scope & Sequence that dictates how long and which topics to cover?  Isn’t this teacher-centered?  If we are truly student-centered, are we comfortable if some sections of Algebra 2, for example, cover different topics than other sections?  How might this affect our sequential courses?  Or do we do enough re-teaching in our sequential courses that we could accommodate such a student-centered model?  Beyond sequential courses, would this compromise a student’s ability to do well on standardized tests, such as the SAT, if we go deep in one topic and miss another all together?

Sorry, that was a lot of questions but I am confused about how to do this.  Fortunately, our new Algebra 1 program will remove the timing pressure that the Scope & Sequence creates.  Students will self-pace through the material and we’re intentionally building in lots of opportunity for deep thinking activities, such as number talks.  The Scope (the curriculum), however, is still built by us, the teachers.  Might there be a way for us to make the scope more student-centered, so that students determine the concepts they cover?

I’d like to argue that if we focus on deep thinking, we can move away from our current approach of covering concepts and move toward an approach that teachers math strategy/math flexibility so that when they are presented with a topic they’ve never seen (whether on the SAT or in a later math class) they can use their mathematical intuition to figure it out.  After all, all Math concepts can be derived from basic principles.

Growing brains in Algebra Honors

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My Algebra Honors students had a test last Friday on systems of equations in 2 variables.  We had done a lot of work on this chapter and I knew that I didn’t want to assign them more of the same types of problems on our review day Wednesday.  Instead I had the class work in groups of 3 and I gave each group a sheet of paper with two word problems written out.  I told them that they had 50 minutes to solve these two problems using any strategy they wanted.  They just had to justify their answer with math.  I told the class that they are expected to explain their problem to the class toward the end of the period.

What I didn’t tell the class was that the problems involved 3 variables (an Algebra 2 concept) and that we hadn’t learned how to solve systems of equations in 3 variables.  Instead I told them that I believed in them and I knew they would be able to solve them. 
The students got to work solving the problems.  As I walked around the groups I loved seeing all of the different strategies the students used.  One group solved their problems on the white boards and every time they got stuck or realized they made a mistake they would start solving it again without erasing their work.  They labeled each try as a “take” and would refer back to what they did before to help them figure out where they made mistakes.  Finally “take 6” was successful and they were so proud of themselves for figuring it out.  
They labeled their final, correct work as “Holy Ground” and they were so proud of themselves.  
As this group worked I would hear them say things like “we’ve never solved equations with 3 variables before…I’m not sure how to set this up but I know Mrs. Schooler wouldn’t give us a problem we couldn’t solve.”  I also heard a lot of comments about how their brain was growing from the mistakes they made!  The groups for the most part worked without my help.  At times if a group was really stuck I asked them some questions to get them thinking about the problem in a different way 
and that was usually sufficient to get them working again.
As students presented their answers it was fascinating to see how almost every group solved both problems but even better was that not one group solved them the same exact way.  

The students listened to each group present and they would exclaim that they hadn’t thought of solving it that way or telling another group they were impressed at how they approached the problem.  


I surveyed the students after class asking them what they thought of the two problems and what approaches they used when they got stuck.  Here were some of their responses:


“When we got stuck we would look back to see where we went wrong and we listened to each other’s advice.” 
“Whenever we got stuck we would try a new strategy but left the old strategy on the board in case it helped us.”

I also asked the students how they felt after class.  Here’s what they said:


“I felt accomplished and proud of myself because I got to figure out a hard problem without the teacher’s help.  I also felt proud of my group because we worked really hard together to solve it.”
“I felt very challenged but in a good way.”
“I felt like I understood the problems a lot more after hearing how each group solved it.” 

I was so proud of my students.  They were given challenging problems to solve and were successful in solving them.  They made connections to what they knew about systems of equations in 2 variables and applied it to 3 variables.  This is a reminder to me that I need to always remember that my students are capable of so much and that if I am to prepare them to be strong mathematics students I need to give them more problems like this so I can help their brains grow.

Demanding Excellence

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On Friday I gave my AP Statistics students a test that should have been easy.  The test was on hypothesis tests and confidence intervals.  They had to decide which procedure of the two to do for each problem (looking for code words:  “evidence” or “claim” = hypothesis test and “construct” or “estimate” = confidence interval).  For each procedure, we had gone over, in detail (or so I thought), that each requires a 4-step process with each step labeled clearly.

Imagine my dismay on Saturday when I graded their tests and found short one-sentence answers or disorganized work with incomplete steps or steps out of order.  What happened?  Of course I had taken time to make the test, and another hour to grade the test.  I could have just given them the bad grades and called it a day.  In fact, the grades weren’t even that bad.  No one scored below a 60.  But, I simply couldn’t sit with such a deficit.  How could they not know how to do such straightforward problems?  I simply couldn’t move forward.

Instead, first thing Monday morning I told them I wasn’t accepting their tests.  I handed them back without entering them in the grade book.  Instead, we spent Monday going over the answers to the test and they would have a new test on Tuesday.  This means double work for me and I told them that this extra work on my part went into my decision.  Meaning, I think these concepts are so important, I am willing to work double!  The kids who did well of course weren’t happy.  Although as I told them:  if you did well on Friday, you should be able to do just as well on Tuesday.

I’m sharing this experience because I think sometimes we have our “deal breaker” concepts; items that simply must be mastered before we can move on.  This happens all the time in life, right?  If you don’t pass your drivers test, you need to keep testing until you can drive.  I’m not happy that I had to scrap my plans for this week and spend two extra days on this material (something really hard to swallow with an AP curriculum) but I feel it’s importance that as Statistics students they know when and how to do basic inference.  It’s a good lesson for our students too that sometimes what they put in just isn’t good enough.  And they will have to go back and do better until it meets a certain level of excellence.

Creating Innovators–Let’s Play!

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Over the
break I finished the book Creating
Innovators by Tony Wagner.  The
author interviewed many people considered to be “innovators” as well as their
parents, teachers and/or mentors in order to identify trends that help to create
innovative people.
Throughout
the book, the most common trend that developed was the fact that these
innovative people were given time to “play”(by parents or teachers or mentors)
and through that play they developed a passion and through that passion blossomed
into a purpose for their careers and other life goals.  That’s when I realized that we don’t take
enough time to play in a math classroom so I made it my 2nd semester
goal to incorporate more “play” into my classes.  I am trying to do this in three ways:

1.     Number talks
with my algebra classes.  The basic idea is that I put a problem on the
board (i.e. 18 x 5) and the students have to figure it out without any
calculator or paper/pencil.  Once they
think they have the answer they put a “thumbs up” in front of their chest.  When everyone is showing a “thumbs up” I
invite volunteers to explain their thinking while I transcribe it on the board.  For example, one student did 5 x 10 then
added it to 5 x 8.  Another student did
20 x 5 then took away 10 (2 groups of 5). 
We had about 6 different strategies up on the board after this
problem.  The purpose is to show students
that there are many different ways to think of one problem—not just one.  I am very transparent with them, explaining
that my goal is for them to “play” with these numbers and start to see the
flexibility and creativity in math.

2.     Creating
time in my schedule for non-curricular math.  Students need to see how
math (and mathematical thinking/strategies) apply outside of the
classroom.  If I believe this, I need to
prioritize it and create time for it—so that’s what I did.  In planning for this semester, I set aside
nearly every Friday for this purpose.  To
start out we will be working through Khan Academy and Pixar’s collaboration Pixar in a Box, which provides students with
videos and practice activities to see how math, science and technology come
together to create a Pixar movie.  Last
week we learned how Hooke’s Law (physics) was used to animate Merida’s hair in Brave. 
I’m not grading this.  We are just
playing.




3.     Would You Rather?
(Math)  With my financial algebra students I am
trying to incorporate more opportunities for them to defend their thinking with
mathematics.  I am using this to start
class (similar to the number talks with algebra).  Again, I am emphasizing the fact that there
is no one correct way to think about these questions, but they do need to
support their decision with math.  For
example, in the picture below, students cannot just say “I would pick the beach
one because I like the beach better.” Most students select the beach based on
the basic multiplication (they can use calculators for this exercise) but then
some students argue that they would pick the city location because it is more
likely to be full throughout the year).


So my
question is, do you incorporate any aspect of “play” into you classes?  How?  I
need more ideas.  Also, if you are
interested, the best chapters were the ones about “social innovation” and “innovating
learning”.  You do not have to read the
entire book to get value from those chapters.

Debating Systems of Equations Methods

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My Algebra Honors students have been learning how to solve systems of equations.  We learned 3 methods:  graphing, substitution, and elimination.  While each method will always work, there usually is one method that is easier to use based on the two equations given.  

In groups of 3 I assigned my students one method to solve.  I then gave the class a system of equations.  I gave each group a few minutes to discuss why the method they were given was the best method to solve that system, even if it may not be their preferred method.  Each group then shared out with the class why their method was the best.  Groups could then engage in a friendly debate on which method was the best method to solve that specific system.  It was wonderful to hear the reasons students came up with for each method and how they tried to sell their method to the class.  The students got excited and were really advocating for their method.  After we debated which method was best students then had to solve the system of equations by all 3 methods, starting with their method first.  After everyone solved the system of equations we circled back to our debate and asked if anyone’s opinions were changed based on solving the systems.  We were able to repeat this process 4 times with students getting assigned a different method each time. 
This lesson reinforced that no matter what method you use to solve the system of equations you should end up with the same answer.  I hope it also got students to think about the benefits of each method so they can decide which method to use when presented with the equations.  I loved how it got students talking about math as well which is a skill we are working to develop in our department.  

Tall Goals in Geometry

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Similarity is one of my favorite units in Geometry.  I love it because it provides a fun opportunity to get outside and use our knowledge of Geometry to do the seemingly impossible.   Let me ask you this:  Could you tell me the height of this tree using just a mirror and a yard stick?

Guess what?  My students can and did!

It turns out that if two pairs of angles of from two different triangles are congruent, then the triangles are similar.  And similar triangles have proportional sides.  Combine that with our knowledge of how mirrors work, that your ingoing line of sight bounces off the mirror at the same angle, gave us this sketch as we planned in the classroom:

We then headed out in groups to measure various tall objects around campus.  Some great number sense happened as the groups measured and calculated.  “Wait a minute, that tree can’t be 3 ft.  That doesn’t make sense,” I heard one student say.  Upon looking at their paper they had measured their height in inches but other distances in yards.  A quick fix to make sure everything was consistent fixed the problem.

In hindsight, I wish I hadn’t dictated what objects they would measure.  I wanted to give them the freedom to choose their own objects and the space to not be on top of each other.  But, it would have been a fun discussion of measurement error to see why our estimates of the same object differed.  Maybe next year!

Inverse Trigonometric Functions War

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The ability to evaluate inverse trigonometric functions correctly is a fundamental and critical skill that a student needs for Calculus. For whatever reason, many Precalculus students struggle to master this skill and many who do master it do not retain it by the time they take Calculus. I decided to step outside the box and have my students play a game that would force them to make more sense of the range of answers that can be found when evaluating inverse trig functions.
I had read on some math blogs about teachers using the card game War to help students evaluate logarithmic functions and so I thought I could make this work for my situation. War is a card game in which each player gets half of the deck of cards. For each turn, the players each flip a card. The player with the higher valued card wins both cards. The player with the most cards at the end wins the game. I adapted the game with homemade cards that had all the inverse trigonometric functions that use angles found on the unit circle. I prepared a visual aid that diagramed the allowed range of answers for the six functions. I divided the students into groups of three: two players and a referee. I explained the rules and then let them play.

It was slow at first while they got used to the rules and got comfortable with evaluating the functions. I circulated the room, checking in with students and responding to questions. It was very apparent to me just how many students really didn’t understand what they were doing. Group by group, I responded to their questions. As soon as one in the group understood, I would leave that person to explain to the others. Slowly but surely, they started to “get” it.

While I doubt that anyone would claim that this was a particularly fun game, I had many students tell me on their way out how much this helped them gain confidence with these functions. While I was circulating around the room I also witnessed many “aha” moments where students for the first time seemed to grasp the purpose of evaluating these functions. I’m not sure how I will be able to assess how well this extra day and extra activity really served to improve their understanding and retention of the subject, but subjectively I conclude that this was definitely time well spent.