Category Archives: Algebra 1

Individual Interviews vs. Typical Test

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What I did:

1.  I collaborated with my students to develop a rubric for their understanding of the Chapter 10 content.  The left side lists the four main skills from this chapter and the top shows the criteria necessary to earn each score.

2.  I created a schedule (in a shared Google Sheet) for students to sign up for their 10-minute appointment window.  Most were during class time, but some were before school or during a lunch/break/X-block.

3.  I created a sheet (for myself) with sample questions from each section so that I had questions ready to ask during the interview.  This also allowed me to ask different students different questions.  Students also used this sheet when they assessed themselves at the end because I broke the questions in to sections that matched the topics on the rubric.
4.  I interviewed each student for 7 minutes (with a 3 minute buffer).  During that time, I asked them questions about each topic (not just “Solve this…” but “Why did you do that step?”), students assessed themselves on the rubric, then I assessed them (as they were walking back to get the next student).
Why I did it: 

A few chapters ago, I asked students to make me a video (similar to a Khan Academy video) explaining the concepts we learned in that chapter (exponential expressions).  They did this in place of a typical written test.  The feedback was very positive and since then their ability to simplify exponential expressions has really impressed me.  Whereas a lot of math concepts tend to get forgotten after the chapter ends, these concepts seemed to stick.



PROS of the interview process: 

  1. I watched students hold each other to a higher standard when they were explaining and reviewing the concepts.
  2. The students said that they studied harder/better knowing they would have to explain themselves.
  3. Students were way more involved in the assessment process.
  4. I was finished grading as soon as the interviews were finished–there was nothing to take home!

CONS of the interview process:

  1. I had to limit myself to 7 minutes per student which limited the number of questions I could ask each student.
  2. The timing also limited how long we could spend on any given section.  If a student didn’t understand the concept, I would eventually have to move them along to the next section.
  3. I do not have the same amount of evidence to support my score of the student as I would have on a typical test.  Since the majority of the “work” for this assessment was a conversation (not every problem was worked out) I do not have a lot of evidence to base my score of the student on.  I wrote little notes and the students worked through some problems on paper (and they assessed themselves), but I definitely do not have written evidence to support every score.

Next steps:

I am going to continue with an “alternative assessment” for our last chapter, but my students and I are redesigning it with some changes.  Here are our ideas so far:
  1. Students pair up and assess each other.  This will allow students to have more time in their “interview” since multiple interviews will be going on at one time.
  2. Students will have the opportunity to re-explain a concept for a higher score on the rubric.

Do you have any ideas for us to consider?

Shooting for the Stars

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On the first day back from Christmas break I decided to jump start the math brains of my Algebra 1 with Math Lab students.  I gave them the following problem:
You have 10 fewer quarters than dimes and 5 fewer nickels than quarters.  The total value of the coins is $4.75.  How many of each coin do you have?
I didn’t tell them until we were almost finished that this is an Algebra 2 Trig problem.  To solve it you need to write and solve a system of three equations with three variables.
I gave them about 10 minutes to work with their group without any help from me.  They could use any method – I even brought in some coins for those that need to “see” it.  You can see some of their efforts on the papers below.  A few solved it by “guess and check.”

After 10 minutes, I started prompting them with the following:
            Which do you have more of: quarters or dimes, nickels or quarters?
            Write an equation that relates the number of quarters and the
                    number of dimes.
            Do the same for nickels and quarters.
            Define variables for the number of each of the coins you have.
            What is the value of one quarter?  What is the value of all of the
                    quarters you have?
            Do the same for dimes and nickels.
            Write an equation that represents the value of all the coins you have.
We ended up finishing the problem together on the board.  And I think most of the students understood the steps and why it worked.
When we finished, I asked them why they thought I chose this problem for today.  They said:
            To get our brains going again.
            To show us that struggle is ok.
            To have us make mistakes and get the synapses firing.
            To encourage us to stick with a really hard problem and not give up.
Yay!  Growth mindset is sinking in!
Notice I didn’t tell you the answer.  Can you figure it out?  I believe in you!  You can do this!

Redesigning Algebra

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The math department has been rethinking our math classes and
how we can personalize the learning for our students.  Inspired by what we’ve
learned from Jo Boaler’s How to Learn Math forTeachers online course, we’ve decided to remove tracking
in our Algebra classes.  We want to show
students that mathematics is creative and open ended.  It’s
about identifying problems and working toward finding a solution.  It’s
collaborative learning, real life problem solving, and sharing this information
with others.  We’ve had the opportunity to spend time in a small group discussing
what we want Algebra (with the intention of expanding out to other classes) to
look like next year.  Our work has just
begun but I thought it might be helpful to others to share our progress so far.
We really want to encourage our students to develop a growth
mindset.  We believe that all students
can do math and we want them to believe this about themselves as well.  
“When I asked over 800 teacher leaders in the
US recently which educational
practices develop and maintain fixed mindset
ideas in students the number one reason given was ability grouping. I agree, it
is hard to give a stronger fixed mindset message to students than by putting
them into groups and telling them they have a certain ability.  In a
recent study Romero (2013) found that significantly more students developed
growth mindsets after they were placed into high track groups. Students who
develop fixed mindsets will often do anything they can to maintain the idea
that they are ‘smart’ which can make them vulnerable to unproductive learning
behaviors and the avoidance of challenging work or higher-level math courses.
 It is extremely important that schools communicate growth mindset
messages to students, and don’t limit students’ achievement by giving fixed
mindset messages through grouping and other practices. This is important for
equity, it is important for students of all levels, and it may be the key to
unlocking the potential of millions of students in mathematics.”

We’ve decided that our Algebra classes
will all be of mixed abilities.  Instead
of having Algebra with Math Lab, Algebra, and Algebra Honors, we will mix in all
of the students together.  We will have 3
sections of Algebra during one period with 4 teachers team teaching these
students.  The curriculum will be broken
up into units.  We are moving away from
chapters and sections and instead we will be presenting the concepts in an
interconnected way so students see how one concept relates to another (we are
thinking of having the girls create their own concept maps and continue to fill
them in as we progress through the curriculum). 
Each unit students will self-pace through the following items:

  • Presentation of concept – this can be through a video or a discovery activity
  • Practice Problems – this is where students demonstrate mastery on the basic idea of the
    concept (can they solve an equation in one variable, for example)
  • Topic Challenge – these
    are application problems where students work collaboratively in small groups to
    apply the concept they learned.  Many
    problems will be presented as an open ended idea/problem and students will
    develop the plan to solve it.
  • Unit Challenge – this is
    a larger project that will connect all or most of the concepts from the
    unit.  This will also be completed
    collaboratively.  Some unit challenges
    will be worked on throughout the unit while others will be completed at the end
    of the unit.  Here is where students can
    really see how mathematics is used, how it connects big ideas, and how it
    applies across disciplines.  During the
    topic and unit challenges students will be working collaboratively to solve
    these challenges with the teacher acting as a coach to help guide them along
    the process.
  • Unit Assessment –
    Students demonstrate mastery on these short, application based
    assessments. 
  • Honors Challenge – If a
    student wishes to have the honors designation on their transcript at the end of
    the year then they need to complete this challenge.  These are what we like to call the     “wicked problems” where students really need to
    persist and make connections in the data and apply it to concepts they may not
    have initially been taught.      In addition,
    students will have a peer tutoring requirement and need to create instructional
    videos for their peers.     

We are envisioning using 3 classrooms where the students will move
fluidly among each one based on the needs of that class period (or a portion of
that class).  The 4 teachers will be
moving among each room too each day (which means we will need to touch base at
the end of every day to determine who will be facilitating each room the next
day).
Room 1: “Traditional,”
direct instruction
Topic challenges
Practice problems, collaboration
Room 2:  Teacher coaching/guiding, student-centered learning
Unit challenge
Honors wicked problem
Room 3:  Teacher monitoring, no direct instruction
Assessment
Practice problems for mastery
We have outlined the units for the Algebra curriculum and are now
beginning to determine which concepts will go under each unit and create the
assignments.  There is still a lot of
work left to do but we are excited to continue moving forward!

Scatter Plots

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I was really excited to teach my Algebra Honors students about scatter plots and lines of best fit today.  I have been looking forward to this lesson because it allows me to teach statistics, which I love, to my students but also because we were going to use for the first time the HP Prime graphing calculator app on their iPads.  I knew I wanted to make this lesson interactive working with data they collected that would be interesting to them.  I found a PowerPoint that had 20 pictures of celebrities and students were supposed to guess their ages.  The students loved seeing pictures of Justin Bieber, Kim Kardashian, Nick Jonas, etc. and guessing their ages.  It definitely grabbed their interest!  I was feeling really confident about this lesson.  I then gave students the actual ages of the celebrities.  Students now had two rows of data and it was time to have some fun with the HP Prime.

Mistake #1:  I had thought that the HP Prime app was automatically downloaded on all freshmen iPads but I never checked with the students prior to class if this actually was the case.  Some students had it but many did not.  I thought this was easily fixed, the app was free, the students could just download it.  For the majority of students this worked but there were 3 students that were unable to download any apps. They unfortunately had to just watch on with a partner.

Mistake #2:  The HP Prime calculator is very different from the TI-84 calculator and there is a learning curve.  I had budgeted some time to play around with the basic functions of the calculator before we started the activity.  I was thinking of the first time I used the HP Prime at a conference last summer and how I was intimidated to push buttons randomly so I waited for the instructor to walk me through step by step initially.  What I didn’t anticipate was students feeling much more comfortable with new technology and have them start pushing all the buttons and not paying much attention to my tutorial.  This resulted in many students asking me how to do the same things over and over.

Mistake #3:  Once I got everyone back on track and stressed the importance of staying with me while we entered data we started creating our scatter plot.  Students did great entering their two columns of data but as soon as I showed them how to graph the points they immediately got excited about the touch screen of the graph and started going off in a bunch of different directions.  This resulted in a similar replay of mistake #2.  It was fine at first that they were playing around with zooming in on their window but when it became time to sketch a graph with their finger many instinctively hit OK which then saved their line – even if it was a line they didn’t feel represented their data well.  Rather than ask for help they continued to draw multiple lines which I had to walk around and delete.

While we did finish the lesson and I think I demonstrated to my students that the HP Prime calculator is superior to the TI-84, the lesson did not go at all how I envisioned or wanted.  In hindsight I think it might have worked out better if I gave students written step-by-step instructions for how to enter the data and do the functions working in their groups.  I then could have circulated around the room to see how things were going and troubleshoot if needed.  The girls are way more comfortable with technology than I gave them credit for and I think they could have figured out what I wanted them to do working together.  We then could have had a class discussion on the different lines of fit each student created and discussed the similarities of them.  Overall I would do things differently but I am glad that we approached scatter plots this way rather than just completing problems in the textbook in a more traditional sense.

Finding Slope

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So I thought it would be really clever to insert a picture of Nemo here (Finding Slope – Finding Nemo), but I doubt it’s in Creative Commons and didn’t want Joan upset with me!

Last week I realized that my Algebra 1 students weren’t understanding slope as “rise over run” and therefore couldn’t find the slope of a line graphed on the coordinate plane.  So, armed with 50 cent brightly colored rulers purchased at a dollar store, I sent them out to find slope around campus.

Their first instinct was to make the ruler the slanted/diagonal line.  So I had to explain (and they had to demonstrate!) that the vertical (up/down) is the rise and the horizontal (left/right) is the run.  I then had them measure and calculate the slope, and document it with a photo.  After about twenty minutes of wandering around campus, we came back to the classroom and shared photos.

Their calculation weren’t always correct, but that wasn’t the point.  I wanted them to be able to visualize and have a “hands-on” experience of slope.

Maybe they’ll look at the Inner Court stairs differently now!