Number Sense and Math Reasoning Inventory

I have two children of my own. Often, they end up becoming guinea pigs when it comes to math instruction (or just about anything). As a secondary teacher, I feel qualified to teach integration, quadratic functions, or any other topic beyond Algebra 1. When my son brought home his first set of flash cards from school, I was at a loss. The only tool up my sleeve was memorization by repetition. Needless to say, we spent many hours drilling and crying (ok, I was the one crying). My younger son watched and didn’t speak, but he was absorbing. Little ones are like sponges and sometimes that is not good.

A few days after the flashcards disappeared, my then 6-year-old said, “Mom, I know why 3 x 5 is 15.” I was curious. His response shocked me. He explained that if he counted by three’s 5 times, he would end at 15. He also told me that he could do it the other way and count by five’s 3 times. Of course, I knew this, but in my pursuit to push my older son to memorize faster, I skipped the understanding. What my 6-year-old was developing was number sense.

One of the complaints I most often hear from high school teachers is that students do not have number sense. I’m not sure how you measure number sense or how to teach it, but you know it when you see it and you definitely know when it is lacking. I need students to understand that 3/4 – 1/4 is 1/2 without having to perform an algorithm. Sometimes I think we become too engrained in teaching algorithms without understanding. My hope is that the new common core assessments will force teachers to fix this.

Recently a colleague introduced me to a website, Math Reasoning Inventory, that encourages formative assessment for understanding and building number sense. Maybe some of you already use it and can offer feedback. I have not seen anything like it in mathematics before and am excited about how this could change elementary mathematics education. The site encourages teachers to conduct interviews with the students on mathematics problems. The focus is not only the correct answer, but how a student achieved that answer. For example, if a student is asked to calculate 7000 – 70, using a standard algorithm would be considered an inappropriate strategy. I have to say I love this! Students forget algorithms and cute tricks, but true number sense and understanding will always work. I encourage you to look at the website and provide feedback below. I would love to have students at the secondary level who had been evaluated with this method. 

Number sense is hard to define and harder to assess, but I believe that the Common Core will help us achieve what is so necessary for mathematical success.

A Calculus Wedding

We had a wedding in Calculus.  I gave the students a copy of an invitation the day before, asking them to come dressed in black and white to observe the occasion.  I purchased a cake for the event and provided a bouquet of flowers for the bride.  On the day of the wedding, students arrived in suits and dresses.  I was really surprised with the effort the students put into this event.  I thought they would think it was cheesy, but I guess they know me by now and expect nothing less.  One student went as far as to  borrow a wedding dress from the theater department.

I allowed the students to vote for a female student to represent Deriva, the Derivative and a male student to represent Integroom, the Integral.  Each person was allowed to select an attendant.  I played the wedding march and directed the ceremony, officially wedding the Integral and the Derivative using The Fundamental Theorem of Calculus.  The wedding was all the buzz of the school and several students in Pre Calculus came to ask me about it.

 This is my first year teaching Calculus and AP Calculus.  I think it is important for the class that we establish an identity in the school so other students see the importance of being in such a rigorous class.  My goal next semester is to order t-shirts for us.  Does anybody else have any good ideas for setting the AP Calculus class apart?

Roller Coaster Fun

The STEM academy at my school is starting a Science Olympiad team this year.  To recruit members, the engineering teacher and I collaborated for a roller coaster project during class.  We took our students to the auditorium and put them in groups of three or four.  We gave them a handout explaining the details of the activity.  The students were asked to create a mock roller coaster using the supplies they were given.  They were told the roller coaster would be evaluated using the following equation:

The students had to first discuss how each of the variables in the equation would affect the final score.  The goal was to achieve the highest score, while creating an aesthetically pleasing roller coaster.  After the student finished the roller coaster, they were asked to create their own equations that would give them a higher score.  They had to defend why they weighted each item as they did and why they put it in the numerator or denominator. 

We supplied each group with a piece of foam tube for track, four notecards, four straws, a styrofoam cup, and a roll of tape, and a razor blade.  The students were allowed to use items in the room as supports, but not as actual parts of the roller coaster.  The passengers were marbles of varying weights.  The students had a great time and it was amazing to see the differences in  each roller coaster.  I definitely see expansion ideas for this project.  Next time, the engineering teacher and I will create a store for the supplies and make the students purchase their supplies with a limited budget.

Senior Farewell

In the past, I’ve always been so tired at the end if the school year and ready for summer break. I survive graduation and move on to the next year. This year is different. Due to scheduling and personnel changes, I have taught this senior class as sophomores, juniors, and seniors. I’ve never looped with a class before, so this was a new experience. It was great!

I wasn’t sure I would like teaching the same students for three years, but I built relationships that are priceless. Sure, there were moments when my nerves were stretched and I’m sure my cheesy jokes and sarcastic attitude lost its charm by year three,  but I know these kids.

I’ve watched this group of students mature from 15 year old, scared teenagers to young men and women ready to conquer the world. I know their parents, siblings, extracurricular activities and dreams. We have shared struggles, gains, and losses. Many of them know my children and have shared in the lives of my own boys growing up.

This is a difficult year to let go and move on with school. I will miss these kids. I have poured three years of my life into them.

I know next year I will have a new group of students and they will be just as special. This is one part of teaching no one talks about and no program can prepare you for. Teachers invest so much of their hearts, lives, and passion into these students and then they leave. As a mom of two boys, I can only imagine the emotion I will feel when they leave home. I hope my boys have teachers that invest in their lives.

Many students will stay in touch. I love Facebook for this reason. I will watch from a distance as they enter college and embark on their exciting futures. (I am a little jealous. I loved college.) Next year, I will start the cycle again. I’m sure it has to get easier.

A Kindred Spirit

I love the blogging world of teachers. I find kindred spirits here. I find people who are passionate about students and learning. I find teachers who are not just passionate about their subject, but about their pedagogy. I find people who want moe than the latest craze in education and don’t want to use the buzz words of the moment, but want to create real change.

I also find this among several of my colleagues. I’m blessed to teach and a school with several people who don’t consider my passion for education an automatic geek label. (Ok, maybe they do, but they hide it well.) I’m blessed to have teachers that want to sit down and struggle with the challenges in education and have a desire to look for solutions. I know not all schools have this and not all teachers can find this.

My husband has a job, and it is a great job, but at 5:30 he comes home and doesn’t think about his job until 8 am the next morning. He doesn’t read blogs about how to be a better claims adjuster or tweet his friends about the most cost-effective manner to fix a vehicle, so sometimes he doesn’t get me. He chalks it up to my overachiever personality. He doesn’t understand why I spend hours every night analyzing student data and why I get excited when I see correlations. He doesn’t understand why my free time is spent reading blogs about math or tweeting fellow teachers. I tell him I don’t have a job, but a calling. I am called to teach and I want to do it to the best of my ability. (Ok, so maybe he is called to handle claims, but I doubt it.)

So, thank you to all the great bloggers out there who inspire me. Thank you to my fellow teachers at my school who push me to be better. I have found kindred spirits and it encourages me to continue my calling.

Worlds Apart

I have a foreign exchange student in my Pre Calculus class this year.  She is a wonderful young lady from Sweden, Sandra.  I think I am learning more from her than she is learning from me.  She attends an IB school in her country.  After attending a recent math competition in our state, she informed me that her math club travels to China to compete with other IB schools.  I think sometimes in education, we are so focused on what is happening in our backyard that we forget to look at the amazing things other countries are doing to educate their youth.

Sandra’s brother does not attend her IB school.  When the students in Sweden complete ‘middle school,’ they choose a higher learning center based upon their interests.  He chose to attend a technical high school.  All Sweden citizens can also attend college for free.  Sandra knows that she must perform at a high level to maintain her enrollment at the IB school.  She knew the expectations when she arrived and she will meet them.

I think the most surprising thing I have learned from Sandra is that while mathematics is a universal language, how we teach it is not.  We just started our unit on Trigonometry.  When I introduced the unit circle based upon special right triangles, Sandra informed me that she had never seen the unit circle.  She knows trig and has already completed a Pre Calculus course in Sweden, but they do not use the unit circle.  My initial thought was, “How do you know the Sine of pi/6 if you don’t use the circle?”  One of our Calculus students thought that it would be easier to make the transition to graphing trig functions without the unit circle.  I guess I see his point.  I love the Unit Circle so much, so I’m not ready to abandon it.

I know that Sweden sent us their best when they sent Sandra.  I’m so excited to see what else this young lady can teach me.  Who knows, maybe someday when my own children are grown I will explore the possibility of being a foreign exchange teacher.  Sounds like fun!

Sentence Frames

LaQuinta High School uses sentence frames school wide.  They presented at the Model Schools Conference in Nashville this year.  They focused on choosing one strategy to help all of their students succeed across the curriculum.  Sentence Frames provided an opportunity for all students to practice literacy across all disciplines.  I did some research on sentence frames and found a very limited Wiki.  All of the sentence frames I discovered were pretty generic and not exactly what I was looking to use in my classroom.

So, I started my own list of sentence frames (more like paragraph frames).  I’m going to try using them this year in my Algebra 1 class.  I am fortunate to have a co-teacher in this classroom who will help me develop these.  I am going to use them as starters in the class before I introduce the lesson.  Here is an example:

Algebra 1 Sentence Frames_002

My objective is to have the students copy the paragraph and fill in the blanks.  I will then allow time for the students to work with a partner (Think/Pair/Share) to explain how and why they placed the words in the appropriate blanks.  We will then discuss the correct answers as a class.

I’m trying this for several reasons:

1.  I’m tired of putting math problems on the board as starter problems and having students sitting there doing nothing because they “can’t do the math,” even though we went over it the previous day.  Everybody can put words in blanks.

2.  I want the students to use the correct math vocabulary.  I’m tired of them saying things like, “that house thing over the number.”  I’m hoping that if they write it, then they will remember it.

3.  This is a co-taught class, so I want the students to experience success.  Many students come into this class hating math or thinking they are bad at math.  I want to provide opportunities for students to feel a sense of understanding and accomplishment, even if it is just putting words in the correct blanks.

4.  Most of my students don’t know how or when to take notes.  These starters will then serve as our notes for the day.  We will work sample problems under the paragraph that represent the standard of the day.

At the end of class, I will put the slide up on the projector again and review the paragraph with the students.  I want to reinforce what we learned that day and make sure we close our topic and not just end the class with “work these 10 problems.” 

LaQuinta High had great success with this strategy.  They presented several examples from math, science, and history.  Here are my creations so far:  Algebra 1 Sentence Frames.  It’s a work in progress.

Class Size Matters

Gov. Bill Haslam downplayed concerns today that budget cuts would hurt schools by increasing class sizes.  “Most studies have shown that teacher class size is not as direct a relationship to achievement as people have thought in the past, that having a great teacher with 25 students is better than having a mediocre teacher with 18 students, OK?,” Haslam said today.   I agree that it is much better for a student to be in a crowded classroom with a great teacher than in a classroom with 15 students, if that teacher is mediocre at best.  I do believe that the quality of the teacher has a huge impact on learning.  This raises several questions, including, how does the state determine who is a great teacher?  How is it measured?  The current answer includes test scores.  That is a blog post for another time…

Another concern, and a more pressing one for me, is class size.  In Tennessee, we currently average 30 students per class in grades 7 – 12, but by law, we can have 35 in a classroom.  In my personal experience, I normally have 27 – 35 students.  It is really difficult to teach a class containing 35 students.  I explain it to my students this way:  Our class periods are 90 minutes long.  There are 35 of you.  How much of my individual time do you get per day?  (Math problem…)  That’s right, about 2.7 minutes per day.  This means we must learn as a group.  Normally, this would not be a problem.  My Pre Calculus students handle this very well.  It does make me sad that there are several in the class that I do not get to know very well.  I don’t have time to listen to personal stories from each of them or talk about their crazy weekends or last night’s game.  Don’t get me wrong, I try, but it is really hard.  The quiet students tend to fall through the cracks when there are 35.  My relationship suffers with my honor students, but I don’t feel like the learning does to a significant degree.  Let’s be honest, most of these students would learn if you placed them in front of a computer with random math videos and then asked them to answer 10 questions in a row correct.  My role in the room could be optional. 

My standard level classes are another story.  With the adoption of new standards and the gaps in learning that are a result of implementing them all at one time, a class of 35 is just too many.  I have several students that can not learn in a lecture environment and when I put them in small groups, 7 small groups is too much for one classroom.  I prefer not to lecture.  I want students to learn and discover math on their own and not just copy or mimic steps or algorithms I demonstrate on the SmartBoard.  For learning to be authentic and memorable, students must have the ability to direct their own learning with input from the instructor.  This is very difficult to do with a crowded classroom.

With all the pressure placed on teachers to have high-test scores, anything that can be done to ease our burden is not only welcomed, it is necessary.  Yes, Governor Haslam, I can teach 35 students at one time.  Yes, I can do my best to get them to pass their standardized test.  No, I can not build a meaningful relationship with all of them, encouraging  them to value education and continue their learning beyond the classroom.  I can not discover their individual interests and goals and tailor my instruction to motivate them.  Believe me, I will try, but I will fail.  Instead of believing random studies (I can find my own study) believe the teachers:  Class size matters.

The Day of the Tornadoes April 27, 2011

I’m sure that many of you who follow me on Twitter already know about the devastation that reached our county yesterday.  We were warned, but I don’t think anyone really knew the destruction that was coming.  My family and home were spared.  We are truly blessed.  Other than a night spent in the basement and some minor roof damage, we are fine.  My school system and county were not so fortunate.  The photos and stories are beginning to come in via Facebook and texts.  I know of several of our students who have lost homes.  We lost two elementary schools that will not reopen this year.  We won’t have school the rest of this week.  They are meeting today to plan when to reopen. 

As I sit and try to plan the last few weeks of school, I wonder:  How do you continue?  When I face my students who have experienced tragedy, how and why do I teach them Parametric Equations?  I only have a week left with my Seniors before graduation and two weeks with my Juniors and Sophomores.  I don’t think I can cram in the last few weeks of topics in any way that will make them care.  I know other schools have suffered tragedies.  How do those teachers return to classes and some standard of normal?

NCTM Review #NCTM11

I had the pleasure of attending the NCTM conference in Indianapolis this year.  This was my first time attending any NCTM sponsored event.  Overall, I had a good time.  I wanted to dedicate this blog post to some of the highlights of the event.

• NCTM Android App:  NCTM released an application for cell phones that had the complete schedule and allowed the user to build their own schedule.  This was amazing.  It is nice to see conference utilizing the technology that is available.  I enjoyed not having to carry the large book around.  The application also notified the user when a session was canceled.  There was a twitter feed on the app as well as a map of the conference center. 
• Loring Coes did an excellent job presenting on how to make movies in the mathematics classroom.  He addressed the benefits of LoggerPro and Fathom as software packages that make building graphs and gathering data from videos very accessible.  I was sad to hear the Flip Camera is no longer going to be manufactured.  I am going to have to try to get my hands on some before they are no longer made.  Some ideas that I walked away with were tossing a hacky sack to create a parabolic curve and spinning a hacky sack on a string to create a sinusoidal curve.  Fathom was nice in that it allowed the user to build an equation to match the video and create sliders for the coefficients to discover the equation.  I think this is beneficial to the students over just running a regression in LoggerPro and having no idea how the software created the curve of best fit.
• Angie Morgan and Gordon Wells from Ohio Valley University had a good session on Quantitative Reasoning or Quantitative Literacy.  For years the push has been reading and writing across the curriculum.  It is nice to see a push towards Numeracy across the curriculum.  They talked about getting other disciplines in our school to utilize their disciplines and bring mathematics into their classroom.  This can be as simple as analyzing data in science class and comparing local and regional global warming and looking for instances of ‘cherry picked’ data.  I know personally, I have been looking for opportunities to team up with our history department.  The study of government lends itself to incorporating numeracy.  They cited the Mathematical Association of America and their work on Mathematical Literacy. 
• Sherrie Wisdom conducted a session on Applied Physics in the Mathematics Classroom.  I did walk away with a good activity for my Geometry Students that could be modified for different levels of students.  She suggested that we have students trace their shoes on graph paper and find the surface area of the bottom of the shoe.  They could do this by counting squares or grouping the drawing into Geometric shapes that they know.  Students then use the equation Pressure = Weight / Area to find the amount of pressure.  The students need to take their weight and divide it by the area of two shoes.  If you have students do this with multiple pairs of shoes, they can then determine which pair of shoes should be the most comfortable.  Hopefully, students will discover that as the shoes surface are increases (slippers) the pressure decreases (as compared to heels).
• My final session was conducted by Leigh Nataro from Moravian Academy and addressed the use of Facebook with your students.  She created a closed group in Facebook and had her students join.  Facebook now allows you to upload documents and photos.  This took the place of her class blog.  She used it in her geometry class by posting a picture of a geometric figure and requiring each student to post a comment about the shape.  Students could not duplicate comments.  This means that if a student logged on later, they had to read all of the other posts to make sure they did not duplicate a previous comment.  She did allow students without a Facebook account to submit their comments on paper directly to her and she posted the comment for them.  She encouraged teachers to ‘like’ their students comments and posts.  She also recommend that teachers refer to posts in class discussions.  He entire session was based upon the idea that “You need to go where your students are.  You’ll get more traffic when you are in their neighborhood.” 

Overall, my first experience with NCTM was beneficial.  A few modifications that need to be made, in my opinion, involved the use of technology.  Each presenter is required to provide handouts.  At most of the sessions I attended, the speaker ran out of handouts.  The presenter posted their email address and told people to email them if they wanted a copy of the handout of presentation.  My colleague who presented was bombarded with emails the night after her talk and had to respond and upload her documents to each individual.  I think that NCTM should provide a website with a link for each speaker.  The speaker can then upload their documents directly to the site and people can go on their own to download what they want.  It still amazes me, the number of teachers and speakers who do not have websites or places to post their work.  I guess I’m spoiled being in the blog world and having access to excellent resources and the opinions of my fellow teachers.  I would love to see someone from the blog world present next year about blogging in education.  (Hint to my fellow bloggers…)