Author Archives: galenaylor

About galenaylor

Documentation engineer at Facebook. Math tutor. Former aerospace engineer and programmer/analyst.

Testing the “Open Middle” Waters

I have been itching to try an Open Middle math problem.

As a tutor, I am always looking for ways to enrich and deepen the understanding of the students I work with, as opposed to just band-aiding their current math struggles.  I finally got my chance last week.  I chose this 3rd grade question: Draw three rectangles with a perimeter of 20 units.

First, I asked my student to draw one rectangle with a perimeter of 20 units.  Here is his initial attempt.

G_OpenMiddle_Perim 1








Rectangle…check.  Perimeter-is-the-edge…check.  Perimeter = 20…not so much.

We add the side lengths and he sees the total perimeter is much more than 20.

Aha! #1  “Oh, I get it!”  Immediately, he draws a square and proudly labels each side 5 units long.  Yay!  

G_OpenMiddle_Perim 2






Wonderful!  Now:   Draw another rectangle with a perimeter of 20.

Complete puzzlement.

So, we talk about what perimeter means and I ask Is there any other way to have the sides add up to 20?  Aha! #2, and he draws the following “rectangle:”

G_OpenMiddle_Perim 2b






Hmm.  Okay.  Perimeter = 20…check.  Rectangle…not so much.  So I draw his “rectangle” to scale.

G_OpenMiddle_Perim 2c1





Hmmm.  What’s going on here?  We talk about the properties of rectangles and his next attempt is a rectangle, but again the perimeter is not 20.

G_OpenMiddle_Perim 3a






So, I ask him to try again.  He thinks for a bit and then draws:

G_OpenMiddle_Perim 3





Great!  We have two rectangles with a perimeter of 20.  Now draw another one.

Much less thinking time this time:

G_OpenMiddle_Perim 3b





And we get to Aha! #3, at which point he stops drawing, and starts talking:  “Or, the sides can be 2-8-2-8, or 1-9-1-9, or …!)”  Pretty soon he’s listed all of the different configurations/rotations of rectangles (with integer-length sides) and HE GETS IT!

This was more than an “aha!” moment; it was a groundswell of understanding – like a wave of comprehension crashing on the beach:  I get it!  I get it!  I get it!

And it didn’t happen after creating one rectangle, or even after two.  It required him to come up with three different rectangles before he understood.

SO fun to watch.  It literally made my day.  Can’t wait to try more Open Middle problems!


When is a Line Not a Line?

3rd Grade Line Plot

3rd Grade Line Plot (From K12Math Passion Third Grade Written Assessment)

After a long hiatus, I am finally getting back to my assessment project.  The end of the third grade assessment is in sight!  As I work to understand each standard and turn it into an assessment question, I do a fair amount of checking to make sure (1) I’m using appropriate levels of problems and (2) my assumptions are good.  (I don’t always succeed, but that’s another story!)  This check is particularly important with elementary topics because my credential and most of my experience is with secondary math.

Two of my favorite sites for checking my assumptions are IXL Math and Illustrative Mathematics.  They are both very organized and easy to use.  IXL Math has lots of practice problems for each standard and Illustrative Mathematics has amazing tasks and a great interactive graphic showing the domains across grade levels.  (I’ll be using both of these sites over the summer with my 3rd and 4th grade tutoring students.)  There are others, of course, and I also use Google to research topics.  (If  you have a favorite site, please share it in the comments!)

Thank goodness I checked my assumptions about line plots (3.MD.B.4).  I don’t know about you, but when I think of line plots my brain constructs an image of coordinate pairs connected by line segments.  You know, something that has lines in it.  So, I was a bit surprised when I Googled “line plots for 3rd grade” and images of stacked X’s came up on my screen.  (See example above.)

Okaaaay.  Hmm.  Well, the making-a-line-plot standard doesn’t have an image of a line plot.  So, I did a little digging and discovered that, apparently, in the 3rd grade world a line plot looks like stacked X’s.

Don’t get me wrong.  I think the stacked-X’s-as-line-plot is an outstanding stepping stone between bar graphs or pictographs and traditional line plots.  I understand it’s an important bridge, which will greatly help students make the transition.  In fact, I hope teachers facilitate this transition by eventually asking students to draw a dot at the top of each stack of X’s and connect those dots with line segments.

It’s just the name that gets me: line plot.

I realize I’m a very literal person, and third grade was a LONG time ago, but I think even third grade me would be confused.  “Where are the lines?”

When is a line not a line?  When it’s a third grade line plot, of course.  Anyone can see that.

So Much More Than People Think

To educate

Microsoft Office Clip Art, November 2013

The announcement for Mission #7 of Exploring the MathTwitterBlogosphere happened to cross my consciousness this morning.  The mission: to explore a day in the life of a teacher/educator.   (Thank you for including non-teachers!)   It’s Sunday, my big tutoring day, so I resolved to take whatever happens today and write about it.  I wondered if I would have anything to write about.  This turns out not to have been a problem.

This morning, with latte in hand, I continue unpacking my calculus knowledge so I can preteach the next section to one of my tutoring students later today.   I love this concept – can you love a concept? – that as we process knowledge it becomes compressed, and then as teachers we have to go back and unpack our knowledge so we can help students learn.  This model perfectly matches what I feel in my own head as I learn and process new concepts.  BTW my calculus knowledge has been packed for a very long time!

In the middle of all this calculus, I find out my dad was in the hospital last week and didn’t tell me.  (Sigh.)  He blacked out twice recently, and was planning to drive somewhere today.  OMG.  I make him promise not to drive and wish I lived closer to him, or vice versa.

Tutoring session #1 starts out shaky.  It turns out my student is having a personal crisis.  I don’t ask outright, but wait until confided in.  I then spend 10 or 15 minutes responding with my best growth mindset messages:  this happened only one time; you will do better next time; this event does not define you.  Whew!  I’m able to help the student process the event and move on.  This job is so much more than people think it is.  At the end of the session, I strongly urge my student to “step away from the math” for a few hours, and then go back and review each of the problems before submitting the assignment.  Recognizing the benefit of letting time pass so you can tackle a problem with fresh “eyes” is a good life lesson and only one of many I find myself sharing with my students.

Tutoring session #2 starts out great:  I discover my student scored one of the highest grades in the class on the latest test.  You can just see the increase in confidence and a willingness to try more difficult problems instead of sitting back and saying “I don’t get it.”  Gotta keep this positive feeling going!

As I reflect on the day’s educational interactions, I realize (once again) how much the affective side impacts our ability to be effective educators.  This job is so much more than people think it is.  And isn’t that great!

Scary Real Life Math Stories


Or, Why Rates And Proportions Are Important.

It’s Monday morning.  I am having a medical procedure done in which medicine is pumped into my arm through an IV.  My nurse is calm, efficient, and experienced.  She is also training someone new.  No problem: Everyone is new at one time or another.

After the experienced nurse inserts the IV, the two nurses leave me to retrieve the medicine.  This is the gist of the the conversation I hear through the curtain:

The amount of medicine is 100 ml and it needs to be infused over half an hour.  The machine needs the infusion rate to be entered in ml per hour.  What rate should you enter on the machine?


The medicine is 100 ml and it needs to be delivered in half an hour.  What do you enter on the machine?


Think.  We have to infuse the entire 100 ml in only half an hour.  How many ml/hr do we need to enter on the machine?

Oh, 200!



You can be sure I was paying close attention to this conversation.  And even though the trainee eventually got to the right answer, I was still relieved to see the experienced nurse check the machine.

One thing I noticed:  The strategy employed by the experienced nurse was essentially to repeat the question, with very little modification, and to calmly ask the trainee to “think.”

My approach would have been different.  When the trainee first answered “100,” I would have said something like, let’s think about that.  If we enter 100 ml/hr into the machine, and we have 100 ml of medicine, how long will it take the medicine to be completely infused?  I presume the trainee would have said it would take an hour.  (At least I hope so.)  At that point, I would have said, now if we want to infuse the medicine in half that time, do we need a faster rate or a slower rate?  What rate do we need to enter?  

I think this approach would have made a better connection for the trainee and allow her to see how her answer needed to change.  It would also better model the thinking needed to solve the problem.

What would be an even better way to have handled this situation?

MTBoS Mission #1 – Exploring Functions

Welcome!  Here is my contribution to Math Twitter Blog-o-Sphere Mission #1:  What is one of your favorite open-ended/rich problems?  How do you use it in your classroom?

I was looking for a way to open a discussion of functions with one of my tutoring students last summer and found a lesson on the Internet called The Shapes of Algebra, which (at least in one version) included a great Ice Breaker activity.  (See references below.)  This activity asks students to consider five functions – represented in five different ways (verbal, tabular, graphical, function notation, and as an application problem) – and determine which representations belong together.  One example is shown below.

AfAY1 icebreaker

What I Did

After printing the file, cutting apart the different representations and mixing them up, I gave my student the pile of cards with only the briefest of explanations:  Which ones go together?  It was wonderful to watch how engaged he was as he first figured out what he was looking at, and then went to work puzzling out which cards described common situations.  His initial groupings contained several errors, but he corrected most of them on his own as he worked through the set of cards.  To finish the activity, he taped each set of representations to a piece of paper labeled with the name of the function, creating a personal reference for the rest of our lessons.

In addition to creating a introductory experience with different types of functions, this activity generated lots of great conversations about interpreting graphs, identifying slopes and rates of change, as well as the meaning of function notation.  Each of the five functions are completely different (linear, quadratic, reciprocal, absolute value, and exponential) and nicely set the stage for the rest of my planned function lessons.

How I Would Use It

In a classroom, I would definitely use this lesson as a group activity.  I would create “kits” of the cards so that (ideally) each group would have a complete set for a different function along with lots of cards for other functions so they had opportunities to compare and contrast different representations.  I think this activity would spark a great discussion and I would follow it with The Shapes of Algebra explorations using the Function Explorations document I created.  The document both structures the activity and provides a nice way to collect students’ observations about the way the functions change.  Desmos is a great tool for students to use for graphing the functions, but a graphing calculator (or even pencil-and-paper!) will also work.

What would you do differently?


There are several links for The Shapes of Algebra handout (including one for an online version), but I can’t seem to find the link where I found the ice breaker activity.  You can use my link, or Google (including the quotation marks) “Mary owes her mother $7” and you should find the file.



NOTE:  The following is my answer to Question 3.5 in Jo Boaler’s online class: How To Learn Math.  “Which ideas from this paragraph struck you? Write 1-2 paragraphs on why you like it. What does it mean for learners of math?”  The original paragraph by Peter Sims can be found here Daring To Stumble on the Road to Recovery (The New York Times, August 6, 2011, paragraph 16.)

The paragraph written by Peter Sims (NY Times) is both succinct and tremendously thought-provoking.  He seems to have captured, in a very brief space, the key elements supporting intellectual creativity.  I think what most resonates with me, however, is the author’s core belief that we must be unafraid.  In short, we cannot conceive new ideas,  create new inventions, or learn new concepts if we are afraid.  To become unafraid, we need to move out of our comfort zone, physically and mentally, and even entertain wild and crazy ideas.  We need to suspend judgment of a situation so we do not limit the possible solutions and thereby stifle our creativity.  Furthermore, we must allow ourselves not only to be wrong but also to be alone in our beliefs – if we believe strongly enough that we are right.   Finally, we must be courageous and not give up when the task is difficult or even overwhelming.

Many students (and adults) fear math, which creates a wall that greatly impedes learning.  Peter Sims’ paragraph lists several specific characteristics that when developed will help students overcome this fear.  By sharing new research, such as we grow new brain synapses when we learn from our mistakes, we may encourage students to become more comfortable with being wrong.  Ironically, being wrong will allow them to learn better.  Also if we create a classroom environment that withholds judgment of all ideas, students may be less inclined to pre-judge their own ideas as unworthy, which will help them to persevere in working toward a solution.  In addition, this positive classroom environment may embolden students to stand up for their solutions, further deepening and enriching the mathematical discussions.  When we fear something, we are often tempted to give up.  If we can help students to be less afraid of making mistakes or being wrong, they may persist when faced with a difficult problem, which will increase their chances of making and learning from their mistakes.

(Photograph from MIcrosoft Office Clip Art, search string “fear.”)

How Does Math Make You Feel?

Session 1 - Concept Map

Well, I hope math doesn’t make you feel like the Stanford University freshmen (see the concept map, above) who were interviewed for the OpenEdX course entitled How to Learn Math.  You might think anyone who was accepted into Stanford would be good at math (and feel good about their abilities), but not so.  In fact, the professor (Jo Boaler) includes in the first class session an interview with a scientist from the UK who, when awarded a medal by the Queen, still harbored feelings of math inadequacy.

Many thanks to Karl Fisch, for his blog post that pointed me in the direction of this class.  One minute I was reading about Karl’s introductory assignment for his students and parents, and the next thing I knew I was signing up for the class.  This is just up my alley and I am really enjoying it.  (Oh, and did I mention, the class is free!)

The concept map is one of my first assignments.  After listening to the students talk about their school math experiences,  I created the concept map to capture the essence of their responses.  I think it came out pretty well.  I’m interested to see what some of the other 20,000 (!) students came up with and if anyone had a different take.  Back to class!

Desmos Graphing Exploration

Graphing Exploration 1

Graphing Exploration 1

I have been reading many blogs lately, some of which have lead me to Benezet’s experiment,  A Mathematician’s Lament, and reminded me that “Perplexity is the goal of engagement.”  As a result, I have adjusted how I am approaching the super-speedy-but-in-depth review of Algebra One I am doing this summer for one of my tutoring students.  I know, that sounds like an oxymoron and it definitely is.  I had planned a lovely, and still too short, 10 week course based on one of Geoff Krall’s Problem Based Curriculum Maps when I found out my student (I’ll call him Josh) was going to be gone much of the summer at various camps and on vacation.


I’m not sure how quickly Josh and I will cover the material this summer. He’s very bright, and with only one student I should be able to proceed much faster, while still achieving depth of understanding.  On the other hand, he also has many holes in his math fundamentals, which I deal with as they come up and which take time.  Still, I’m optimistic we’ll make it through enough of my planned curriculum that he’ll be reasonably well prepared for next year.  Also, I know Josh’s Geometry teacher reviewed selected algebra topics during the last month of class, and his Algebra II teacher will review concepts as well, so I’m not too panicked.  (Funny.  I can no longer think about Algebra II and Geometry without a wry smile.  If you aren’t also wryly smiling, read A Mathematician’s Lament – be sure to read all the way to page 25.  It’s worth it, and a little depressing at the same time.)

What to do?  My current approach to solving the speedy-but-in-depth challenge is to begin by helping Josh obtain a solid (and dare I hope, intuitive?) grasp of functions and relations, using the lessons outlined in the Algebra I curriculum map.  I also hope to finish the initial linear and quadratic functions sections and still have a couple of days to look over the Algebra II textbook so I can pre-teach some of the concepts.  I know, I know; summer is shorter than I think.

But back to the graphing exploration.  As part of my recent reading frenzy, I came across Fawn Nguyen’s cool Des-man lesson, in which she very creatively uses the Desmos online graphing calculator.  (So fun!)  I plan to work Fawn’s Des-man lesson in later, but for now I created a simple graphing exploration activity I will use to introduce Josh to the unit on functions.  Although my activity doesn’t start with a problem per se, I hope to get him intrigued (if not perplexed) about what makes linear, quadratic, and other functions change before we start a more in-depth conversation.

The activity starts with two graphing explorations, one for a series of linear equations and one for quadratics, and then has a blank template that can be used for as many student explorations as desirable.  In the first two explorations I plan to teach Josh how to use Desmos to enter equations, add a slider for variables, and record observations about the graph of each equation. (Okay, after about 3 minutes he will probably be teaching me how to use Desmos, but I have to start somewhere!)  At the bottom of each page are questions intended to provoke thoughtful reflection on his part (I can dream, can’t I?), or at least remind me of the questions I wanted to ask.

It’s easy to see when Josh is intrigued and engaged (or not), so I will get immediate feedback from a student perspective.  I’ll let you know how it goes.  Meanwhile, if you have feedback, comments, or questions, please leave a comment.  Thanks!

Assessments Update

Demise of Soccer Balls

Okay, this is becoming a little daunting.  I knew the fun pictures of soccer balls and ice cream cones I used for the Kindergarten assessment were going to be replaced by more number-dense problems, but I’ve just finished the third grade Operations and Algebraic Thinking domain and I already have 29 problems.  Yikes!  My philosophy of including each word problem variation may not survive third grade.  But how do I choose?  Will it be the Goldilocks solution: one easy, one hard, and one “just right?”  Or is there a better way?

I just looked back at my 3rd grade assessment to see how many word problems I have and noticed  many of my other problems have multiple parts.  For example:  Write a related multiplication fact for each division expression has 4 division expressions to solve.  So, my assessment is really much longer than 29 problems.  (My count: 11 word problems for 3.OA.3 and 7 for 3.OA.8.)

Oh, yes, and looking back at my notes from a month ago before I broke my foot (but that’s another story), I had already eliminated some of the word problem variations.  Here’s what I wrote: “Naïve to think I could do them all… hard and easy array problems, array versus area problems, measurement versus non-measurement, [and] compare problems.”  And that’s not even changing the location of the unknown in the problems, which makes a huge difference in complexity.


I think I will continue to create the assessments as a complete mapping of the CCSS-M (including all word problem variations) because it is a way for me to more deeply understand the standards.  However, it’s clear my original (and, apparently, naive) idea of creating a series of assessments that would illuminate concepts not mastered in previous math classes, and that would be practical to give students, is in need of some rethinking.

Dinosaur Math

Because I’m home with a broken foot, my daughter happened to overhear part of my latest tutoring session.  What she heard was my student abruptly asking, “Do you know how long ago dinosaurs became extinct?”  (This type of comment is very common during our sessions.)

Later, my daughter, who is 15, told me she was amazed I didn’t immediately shut down the dinosaur conversation.  I explained to her that listening to this student is an important part of teaching him.  Over several months we have built a relationship; we listen to each other.  As a result, I can challenge him to change how he solves problems, or to try something new.  It works.

It was only later I remembered another reason I listen (at least for a while) to these apparently random, off-topic utterances.  It’s because they frequently spring from a math-related source.  Take the dinosaur comment.  He wasn’t asking because he didn’t know how long ago dinosaurs became extinct.  Rather, he wanted to know if I knew because something about it was bugging him.  He wondered:  If 200 years ago scientists discovered that – 65 million years earlier – dinosaurs had become extinct, then why don’t we now say dinosaurs have been extinct for 65 million two hundred years?  (I chose to ignore the when-did-scientists-know-it question and instead focused on the concept of rounding.)

I used a whiteboard and wrote the numbers 65,000,000 and 65,000,100.  (Okay, I finessed the date a little and said, let’s just use 100 years.)  We talked about rounding and the fact that it would take a really long time before it would make sense to say anything other than dinosaurs became extinct 65 million years ago.  He seemed content and I was happy that I had answered what turned out to be a good question.

It was only a day or so later I realized I had missed an opportunity to make the concept more real.

What if, instead, I had said:  How long is 65 million seconds?  Next tutoring session we start with this one!