Category Archives: Amanda Jain

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.

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.

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.

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!

Design-Thinking the Final Exam Review Process

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In an effort to put my students in the driver seat of our Geometry Exam review, and to help them see the connections of what we’ve studied this year, I tried a new review format that mimicked much of the design thinking activities I’ve engaged in, in the past. In advance of our week-long review period, I asked my students to make a list of everything we learned this year.  Then, on the first day of review I divided them into four groups and give each group a large board and lots of sticky notes.  They had 5 minutes to get everything they wrote down on the board (one concept per sticky note). The one rule throughout the process was that they could never refer to a chapter or section. They couldn’t say, “Section 4.3,” for example. They had to know what CONCEPT was covered in that chapter and use real math vocabulary as opposed to artificial chapters and sectioning. See video here.

Then all groups rotated.  Each group ended up at another group’s board full of stickies and they were instructed to group and organize them into larger topics, much like we’ve done with our design thinking work this year.  Again they had five minutes. See video here.

 Then they all rotated again. For this round they could add any stickies that were missing and they were also encouraged to make arrows connecting stickies to multiple topics. See video here.

 Finally they rotated again for five minutes with the same instructions. See video here.

 At the end they went back to their original board and digested what was in front of them. See final boards here, here, here and here.
I chose to do this for the following reasons:


  • So often our teaching, reviews and even assessments are organized by chapter.  In an effort to make sure we cover everything, our reviews and tests follow the chapters of the textbook: Two questions from chapter 1, then a few from chapter 2 and so on and so forth.  A predictable set of unrelated problems where a student might be able to (and a teacher definitely could) draw the lines where one chapter ends and another begins. Instead, I wanted these students to see the interconnectedness between the CONCEPTS (not chapters) we had learned.
  • I wanted them to see topics written by their peers that perhaps they didn’t think about.
  • I wanted them to be at the center of the review process.  Instead of me providing the review content, they had to generate it.  
This was a nice activity for this seventh period class.  They were up, active and very engaged in what they were doing.  Much moreso than if I had led the review or provided a review packet to complete (I did eventually do this).  When I asked how they liked this process, they were overall positive but they did say over and over that they wished I had given them the topics they needed to know.  This is not surprising given the cultural change we are trying to make in the Math department:  away from teacher-centered direct instruction and more toward student-centered discovery.  We still have a lot of work to do but this was a fun way to change up the Final Exam review process and continue to move us in that direction.  

Quadrilateral Music Videos

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The Quadrilateral Unit of Geometry is usually my least favorite to teach.  While there are very cool properties of quadrilaterals and their interconnectedness, it is SO MUCH information to teach (not to mention so much information for the students to learn and remember).  Last year, I came up with the idea to have my students make a music video highlighting the various properties of quadrilaterals.  This was inspired by watching videos like these at home with my two young children.

This year, I took it one step further.  I made this Unit a Project-Based-Unit in which I led with the Music Video project and instead of teaching them all of the properties, I gave them large pictures of each quadrilateral.  With many tools available to them (rulers, patty paper, protractors, etc) they were instructed to figure out what was special about each of these shapes through direct measurement and investigation.  They had two weeks of free work periods to find what was special about the shapes, to make lyrics that incorporated what they found for a song of their choosing and to record their music video.

Only two of the six groups finished on time.  This continues to remind me that our students have a really hard time with self-pacing and open, free time.  Fortunately, the groups that didn’t finish on time were still able to produce something by the time the “Share” period was over:  Some had to do a live performance because of technical difficulties, and some had videos that showed amazing potential but that were ultimately disappointing due to time running out.  Two groups asked at the last minute (i.e. 15 minutes before the project was due) if they could have an extension.  To be fair there were no extensions, instead I told them they had to pull something together and “Make it Work!” to reference Tim Gunn.  The girls worked like crazy in those last moments and I could physically see their adrenaline pumping in the form of sweat, labored breaths and rosy cheeks!  While I certainly didn’t want the end of the project to go this way, I think it’s important for them to know that some deadlines are hard and to feel the urgency and teamwork needed to pull things together quickly.

I highlight here two videos that I think were nicely done for two different reasons.  The first is simply just fun to watch.  They had great visuals, choreography, and they clearly had fun with this project.  While their lyrics had some issues, the video matched the spirit I was hoping to see in this project.  The second video shows successful teamwork.  These girls split up the work nicely and put the effort in at home (or even in the car!) to complete the video on time.  Their lyrics were awesome.  While I wish they had a bit more passion and creativity, there is something about the raw nature of this video that just makes me smile.  Finally I share this video that clearly has technical issues.  This group created lyrics to a song by Macklemore and Kesha’s song, Good Old Days.  Their lyrics are awesome and I love that they chose this modern song which combines melody and rap.  If they could have executed this, it would have been amazing.

Finally, the students did have a fairly traditional test in which they could have their lyrics printed to use as a study guide.  Overall, this was a fun project and I’ll definitely do again, but perhaps with more check-ins so that the kids can finish and execute their amazing ideas on time.

How Do We Use Our Long Blocks?

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After a great long block yesterday in my Algebra 2 class followed by an inspiring meeting with my Math Department colleagues, I’m thinking a lot about how we do (and should) use our long blocks.  Yesterday, during our long block we spent 80 minutes on one problem, which is linked here.   I organized the class as follows:

  • 0-20 minutes:  Students worked in groups and were not allowed to ask me (or any other group) questions.  It’s during this time that the students need to show grit, perseverance and confidence in their ability to handle new and challenging tasks, on their own and without me showing them the way.  Here are two videos showing what this looked like yesterday:

  • 20-30 mins:  I opened the class to public questions to me, meaning I would take questions and provide feedback with everyone listening.  The rules were that the questions had to be specific, (i.e. not “I don’t know how to start this”) but not so specific that they gave away the answer (i.e. not “This is what we did.  Is it right?”).  My feedback was strategically reflective during this time.  If they asked, “Do we need to make equations?” I answered, “What do you think?”  “What might the variables be?”  “How many equations might we need?”  “What might they represent?”  
  • 30-40 mins:  Another period of time to work in groups without any help from me.  It’s during this time that they should reflect and think critically about the feedback I’ve given them.
  • 40-80 mins:  I circulate and offer help and feedback to the groups individually until they finish the problem.
In all reality, I could use 1.5-2 hours to complete a problem like this really well.  Toward the end, it becomes a mad dash to complete the problem.  The groups are active, spirited and the adrenaline is running.  To make sure they really understand the problem, I wish I could give them more space to come to an understanding on their own but as the period comes to a close I end up giving more help than I’d like because I want them to have the satisfaction of completing the problem and finishing the task.  
I share this experience because, after our Math discussion yesterday, I’m thinking a lot about how we use these long blocks.  I know that many of my colleagues like to use the long blocks to give tests, and I understand the appeal:  there’s more time for students to work and more time to give a longer test (i.e. more questions/variability in what we ask).  But, I wonder how this reflects our recent discussions about making our math teaching less focused on discrete, right/wrong answer tasks and more focused on larger, open-ended tasks that require critical thinking, innovation, grit and perseverance.  What does this say about our priorities if we devote our longest class period to an assessment?  Could we (should we) commit as a department to devote our long blocks to more open-ended tasks?  
I know the first reason to say “No” to this question is time, and that is a real concern.  The problem that I describe above could have been taught by me in about 15 minutes, if I had used a traditional format of me demonstrating the answer on the board.  Instead I chose to spend more than five times that amount of class time because I believe that teaching skills is a worthwhile investment and if I have to sacrifice some content later, I can live with that.  In other words, at the end of my course, I want my students to have grown in their critical thinking skills, their problem-solving skills, their ability to collaborate and their understanding that math is all about modeling the world around them.  If focusing on this means I don’t get to Conic Sections or Coordinate Geometry, that is a sacrifice I’m willing to make because I believe they are better served by learning and practicing skills than by learning content.  I am aware that not everyone shares this view.  And, I am aware that this becomes really difficult when we start to talk about AP courses, or even sequential courses, where skipping content could have real consequences.  
I wonder how other departments make use of their long blocks.  How many of us only assess in a long block?  How many of us give double lessons in a long block as a way to keep our sections together (or simply as a way to cover all of the content of our courses)?  In what ways can we leverage these extended periods of time to do the 21st century teaching and learning that we keep talking about?

A “Not Yet” Quiz

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I made a mistake today.  I had intended to give a quiz today and while I put it on our chapter schedule, I forgot to put it on Schoology.  It wouldn’t have been fair to give them the quiz, but I still wanted to make sure that they knew the material and have a way to assess that.

Inspired by Carol Dweck’s Ted Talk The Power of Yet, I decided to give a “Not Yet” quiz.  In this quiz everyone was required to get 100%.  Some students would get a 100% on the first try, others would need to retake the quiz (and continue to retake the quiz) again and again until they got a 100%.

The quiz I would have created had I not forgotten to schedule it would have looked like this:

1.  Quadratic function in vertex form:
     a.  Find the vertex
     b.  Sketch a graph by hand

2.  Quadratic function in standard form:
     a.  Find the vertex
     b.  Sketch a graph by hand

I partnered the students and instructed everyone to make a quiz for their partner and pass it to them.  They would take this very short quiz and check that they did it right by graphing it on their graphing calculator to verify their vertex was correct and that the pictures matched.  If they got it right, they were done.  If they made a mistake, they would work with their partner to find the error and they would ask their partner for a new quiz.  Partners were instructed to praise their peers when this happened with messages of “It’s great that you made a mistake!  Your brain just grew!  Let me make you another quiz.”

The kids did this and were so kind and supportive of each other.  One student, who took a few attempts to get her 100% at the end said, “I made it!  I feel so accomplished.”

Awesome.  Best Mistake Ever.

(Note:  I am aware that this will cause some grade inflation.  I won’t do this all the time but I do believe that the kids learned from this experience and got a big confidence boost.  As well as a lesson in the reward of grit and perseverance.)

Bring on the Challenge!

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This post is a bit of an update from a prior post of mine in which my attempt to foster a growth mindset through challenge majorly failed (to say the least).  Fast forward two chapters and my students and I have evolved.  After listening to their concerns, I changed my testing strategy.  I didn’t want to give up on the challenge, but I realized I had perhaps done too much too soon.  As a compromise I gave the students two options for the Chapter test:  [1] Take a standard test with fairly expected problems or [2] Complete an application-style test which consists of one large-scale real-life problem that would be different than anything they had seen but which would use the concepts taught in the chapter.  The carrot?  Option [2] would be completed in groups of 2-3 and have use of any and all resources (internet, books, notes, you name it).  Option [1] would be completed individually with limited resources (calculator and study guide).

Last chapter, when I first implemented this approach, 8 out of my 80+ Algebra 2 students went for the application style test.  Each of those students worked their tails off in the 45 minute period but all ultimately figured it out.  It was a small success!  This chapter, an amazing 23 students opted for the more challenging test, 11 being from my Period 4 class which had originally shown the most resistance to challenge and the idea that to truly grow one must struggle.  Again, it was a 45 minute period filled with sweat, hustle and adrenaline and again they all got to the end and correctly solved the problem.  This is remarkable considering that in this chapter, where we covered systems of equations and inequalities, I took my own risk and gave them a linear programming problem.  While related to systems of inequalities, I didn’t teach this method and these types of problems are complicated even when taught well.  It just goes to show that if we set our expectations high, our girls will rise to the challenge.

I feel like we’ve started a revolution in the Math Department and it’s so exciting.  When students make mistakes I celebrate and tell them, “your brain just grew!”  I regularly hear my students telling each other, “I believe in you!  You’ve got this!  Growth Mindset!”  While they are gently mocking me with these statements, I have to believe that some of these ideas are sinking in.  And now, I see them choosing challenge over predictability.  It’s awesome and exciting to be part of it.

Finally, I leave you with some encouraging feedback from a unit evaluation I gave immediately following today’s test.  Yes, I’m focusing on the positives (and no they weren’t all positive) but the balance of positive to negative comments is shifting and I’m riding that momentum:


I love doing challenging problem, like the performance task.

The group test really helped open my mind on the chapter and I get to view things differently.

Keep challenging us

I am starting to have a better growth mindset and open to challenges.

the performance task was kind of hard, but it made me work harder

Where are the Girls?

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I showed this short video clip from the CBS nightly news to my Algebra 2 classes.  It’s only 3 minutes but if you don’t have time to watch, here’s the synopsis:


Four exceptional Kansas high school students who are smart, politically active and forward-thinking are running for governor of Kansas. The state’s constitution has no age — or other requirements — to run.  


What struck me about this clip was WHO these “smart, politically active and forward-thinking” students were:  all boys.  And so, I decided to show it in my classes as a way to continue to remind our girls that they have been socialized to be perfect while boys have been socialized to be brave.  


Something unexpected happened when I showed the video and the Sociology Major inside me is freaking out about it.  After showing the clip, I asked the students whey they thought I chose to show it to them.  I received many responses such as, “You want us to take chances.” and “You want us to reach for the stars” and “You want us to try new things.”  I pushed them more.  Why else might I have showed this to you?  More and more of the same but not one comment on the fact that they were all boys.  I couldn’t believe it!  They were so socialized to expect boys to take these types of risks, they didn’t even see it.  



When I told them I showed this to them because they were boys, there were many a-ha moments, like when you look at a magic-eye puzzle and the hidden image suddenly appears.  I told them that this clip made me mad, because I believed that girls are just as smart and innovative as boys (if not more so!).  I (again) encouraged them to go big, take risks, and that now is the time when they have a safety net under them to catch them when they fall.  None of those boys is going to win the governorship.  But, each of them will experience incredible growth throughout the process, not to mention a killer college essay.