In Search of A Puffin

images Riding over the waves on a chilly summer morning off the coast of Maine, my eyes scanned the horizon in search of a puffin.    A child of the desert, the wonders of the ocean filled me with awe.    The captain of the boat kept up a running commentary about the seabirds native to the area, those we encountered and those we might see.  I was in search of a puffin.    With every new species that we encountered, there was a moment of excitement and then a little sigh of disappointment.   It was interesting, but not quite what I was hoping to see.   We saw whales and rare seabirds that day, seabirds that are in fact more rare than puffins.   Alas, no puffins, though.

What did I take away from that day off the coast of Maine?  A wonderful experience discovering the wonder of the world in which I live, a nasty case of bronchitis, and the discovery that sometimes the things we aren’t looking for are the most important things that we see.

I was reminded of that discovery this week.   I had recently purchased Block By Block , thinking it might be an interesting addition to my math class.   Block by Block is a puzzle with 60 different building challenges.   I wasn’t quite sure how I was going to use it – maybe during our work with nets, surface area, and volume; maybe as a brain break activity; maybe as something else entirely.   I started out by giving it to my student aides to explore.     Seeing how they engage with something in play sometimes gives me insight into how it might work with my students (who are a couple of years younger).

img_2082-e1489367482156.jpgAs I glanced over while teaching my 2nd period   class, something caught my eye.   I looked closer to see if I was really seeing what I thought I saw.   I was pretty sure that I was seeing something I hadn’t expected.   Every time I looked over for the 50 minute period, my suspicion was confirmed.   So, I watched again during my 5th period class.   Two different student aides, but the same thing happened again.   Do you see it?

In both cases, I had two student aides – one male and one female.    In both cases, the student aides are highly developed mathematical thinkers.    In both cases, I gave them puzzle without any really direction, just a “try this out and see what you think”.   In both cases, the same thing happened.    In both cases, the male student aide reached out and started building.    In both cases, the female student aide was more passive, offering comments or suggestions, maybe taking a block and moving it around in her  hand, but letting the male student aide continue building.   This struck me for many reasons, not the least of which was that I have known these students for three years and these young women are not passive in an academic setting.     It also was an echo of what I had seen in many engineering labs during my college experience.   Young men jumping in and young women (not all, but many of them) letting them.

I tried an experiment.   The next day, the male student aide arrived first and jumped right into playing with the puzzle.   I sent him on an errand.   When the female student aide arrived, she saw the puzzle out on his desk.   She reached over  and pulled all the pieces onto her own desk and started building.    By the time he got back, she was deeply engrossed in the puzzle.   I watched to see what would happen next.   The previous day’s events were repeated, but this time the roles were reversed

As the week progressed, I found myself pondering what this little unplanned experiment implied for my instructional practice and for my female students.

  • When presented with an unstructured task, without clearly defined roles, why did the boys take such a dominant role and why did the girls let them?    Was it a matter of personality or was it a matter of social norms?    What role did experience play in the equation (the boys both play with puzzles frequently, the girls less so)?      Was intensity of interest a factor (the boys love puzzles)?   Why did the girl dive into the task more intently when the boy was not present?
  • What does it mean for my girls futures in STEM when they accept a more passive role?   The 10,000 hour rule for becoming expert at something implies that my girls are going to have a harder and harder time maintaining pace with their male counterparts on hands-on tasks if they are relegated to such a passive role (for whatever reason that it  happens).
  • What instructional decisions do I need to be making to ensure that my girls have equity?   While this experiment wasn’t intended as instructional, what does it tell me about the instructional decisions I should be making?   Do I need to formulate single gender groups all of the time?     Is it enough to just ensure that each member of each group has clearly defined tasks/roles and to ensure that those roles rotate?  While I use cooperative learning structures often, do I need to use them more often when engaging mixed gender groups in STEM activities?

I set out hoping to see how my students would engage with a puzzle.   Instead, I got to see how they interact with each other can have a profound impact on each student’s outcome. The decisions I make can do more or less to ensure that they each have full access to the best outcomes.   Just as was true when I was in search of a puffin, the things I saw were more significant than the things I started out hoping to see.

Seeing the Big Picture – Comparing and Contrasting Different Forms of Linear Equations

In order to review how to transform between different forms of linear equations on their upcoming quiz, I  revisited the concept using a foldable comparing and contrasting the different processes needed to move from one form to another.    In this foldable, I wanted my students to see the big picture.   As they worked down a column, they focused on the process for a single type of transition.    As they looked across a row, I wanted them to focus on the big ideas of each type of transition and how they are the same and how they are different.   I am hoping that this idea of seeing both the big picture and the details into the foldable will help them to make sense of the different processes.    Marzano’s research has shown this use of comparing and contrasting produces significant gains in student outcomes.   When I have used it with other concepts, my students have said that it really helps them.   This is true for all of my students, but is especially true for my students who are more conceptual in their thinking and who have difficulty with sequential thought processes.


You can download the file for the comparing-contrasting-changing-forms-of-equations foldable by clicking on the highlighted text.

Math for Small Children and Others – MTBosBlogsplosion Week 3

Math should be fun.    This is true for big kids, but is especially true for small ones.   One of my favorite blogs,  Design in Play, turns math for young children into a thing of beauty filled with play.    The author beautifully marries design into play.    She has posts on gumdrop engineering, spin art with  snap circuits, paper bridges, and architecture body building that all embed engineering into projects for young (and older children).   Her Snow Crystal Geometry post teaches children how to use a compass and a protractor as they create stunning paper snowflakes.    This blog is a must-read for anyone who has young children in their life.

Since I think math should be fun, I have to say that I love Sarah Carter’s blog.   This summer, I read her post on Function Auctions and thought that it sounded like so much fun that I spent the next two days making a Proportional Relationship Auction ,some proportional relationship anecdotal records and a proportional relationship card set that I can use7 different ways. Sarah’s blog makes you feel like you have this friendly, creative teacher down the hall.

In addition to being fun, math should be meaningful.   Students should be making sense and teachers should be thinking deeply to ensure that is happening.    The desire to think more deeply about my work and about my students’ work is part of why I have chosen to be part of MTBoS.   My students deserve the best that I can give them and Mark regularly poses questions that make me think.   His post  on exit cards was the first time I really considered whether my exit cards were addressing the spectrum of the things that I want to know about my students.   As I read his post, I realized I was doing a fairly good job of incorporating conceptual and procedural assessments, I really needed to incorporate more metacognition into my assessments.  I appreciate that his posts don’t allow me to be complacent.

Overheard in Math Class

Students were working on solving a system of equations using a graph.  The problem context involved adults and students who had purchased tickets to a performance.   They knew the ticket price for each and the total amount of income.   They also knew the total number of tickets sold.    Students had to write the system of equations and then solve it graphically.

Student 1:   Should the graph be continuous?

Student 2:   No, you can’t have half a person.

Student 3:   Can it go into the negatives?

Student 1:  No, you can’t have negative people.

Student 4:   Yes you can.   Not everyone is an optimist.

Different Forms of Linear Equations Card Set with Seven Variations of Use

In order for my students to practice changing forms of linear equations, I created a set of cards that they can use multiple ways.   For each equation, there are four cards:   a version of the equation in y=mx+b form, a version of the equation in x=kx+d form, a version of the equation in standard form, and a scaled up version of the standard form.


I love card sets because I can use them so many ways.   I can use them when I first teach the lesson one way.   I can then use them again a different way for additional practice or as a quick review leading up to a quiz or a test.   I can also use them as part of a differentiated instructional piece of a lesson to work with students who still haven’t mastered the concept.   Finally, I can bring them back out in a month in yet another way as a quick refresher.   (In Make It Stick, the author articulates the power of revisiting a concept after some time has lapsed in order to “make it stick”).

Option 1 – Card Sort

Students work in table groups or in pairs to sort the cards.    They need to match the four different forms of the same equation.

Option 2 – Rummy

Students use the cards to play Rummy.  Students play in pairs or triads.   Each player is dealt six cards.   A single card is face up and the remaining cards are face down in a draw pile.   The player to the left of the dealer begins play.   He or she takes the top card in the draw pile or takes the top face-up card in the discard pile.  The player may lay down any matched sets he or she has face up.  A player may add any one or two cards from his or her hand to a matched set already laying face up on the table.   If he or she does not play any matched sets, he or she must play a card in the discard pile (it can’t be the same card he or she drew from the face up discard pile).  Matches consist of cards that are different forms of the same equation.    The first player to play all of his or her cards is the winner.

Option 3 – Odd One Out

Remove one equation card from the set.    This leaves one equation with only 3 equations.   Since the game play involves making matched pairs, the remaining equation  is the “Odd One Out”    Students play in groups of 3 or 4.   The cards are dealt out.    Players look at their cards and lay down their matches (in pairs).    When it is a player’s turn, he or she draws a card from another player and tries to make matches with the cards in his or her hand.   At the end of the game, the player with the Odd One Out card loses the game.    Of the remaining players, the one with the most matches is the winner.  (This is essentially Old Maid with equations.)

Option 4 – Concentration

Students work with a partner or play individually.   All the cards are laid face down.   The player turns over two cards.   If the two cards represent the same equation, it is a match and the player keeps them.   If they do not represent the same equation, the player turns them face down again.   If two players are playing, the next player takes a turn   The player with the most matches at the end of the game is the winner.

Option 5 – Spoons

You need a set of teaspoons in addition to the cards.   You should have one fewer spoon than the number of players.   Put the spoons in small circle in the middle of the table.   Deal four cards to each player   Each player tries to make four of a kind.   The dealer takes a card off the top of the deck, removes one of his/her cards and passes it facedown to the left.   Each player discards to the person on his/her left.   The last player discards into the trash pile.   This continues until someone gets four of kind and takes a spoon from the center.   Once the player with four of a kind takes a spoon, anyone can take a spoon.   The player without a spoon gets a letter in the word SPOON.    When a player has spelled SPOON, he or she is out of the game.  If the cards run out, reshuffle the trash pile and continue play.


I didn’t have enough spoons on hand, so we played spoons with forks.    Spoons took a far amount of time today, but the students had fun and liked the game.  This option is probably best for a review activity rather than on the first day of instruction.

Option 6 – QuizQuizTrade

Each student is given a card of a particular form.   I direct them to change it into a different form (e.g., they are given a card in standard form and I tell them to change it to slope intercept form).   They solve the problem and then do a Quiz Quiz Trade.  Quiz Quiz Trade is a cooperative learning structure.   The first partner quizzes the second partner to turn his/her equation into a different form.    If he or she struggles, the first partner can give him or her a hint.   If he or she continues to struggle, the first partner can give a second tip.  If he or she continues to struggle, the first partner can tell the second partner how to solve the problem (Tip Tip Tell).   Partners then reverse the process and repeat.   Finally, they trade cards and find new partners.   I have students continue for whatever amount of time I want to allocate to the task.

Option 7 – Concept Attainment Cards

When I make the card activity, I use a different color of card stock for each set.   With this activity, I give each group a set of cards that is all the same color for the desired concept (all the pink cards are in slope intercept form) and a different color for the non-examples (purple cards with equations in standard form or in x=kx+d form).   Students then have to look at the examples and non-examples in order to define the concept illustrated by the examples.

Option 8 – Give One Get One

Students line up in two lines, facing each other.   I give each student in one line a card with an equation in a specified form of an equation (e.g., everyone in the first line has a card in y=mx+b form).   I give each student in the second line a card with an equation in a different form (e.g., everyone in the second line has a card with an equation in x=ky+d form).   I then tell everyone to transform their equation into another form (e.g., standard form).   When they have done so, they step forward into the space between the lines.   They trade cards and solve the other card, discuss their solutions, check each other’s work, and then step back into the lines.    When all the partners are done, one line passes their card down and the other line passes it up.   The second line shifts up by one (the person at the head of the line goes to the end of it and everyone moves forward) and the first line shifts down by one.   Now everyone has a new problem and a new partner.   I repeat the process for whatever amount of time I want to allocate to the task.

You can download the  rewriting-equations-card-sort  file by clicking on the highlighted text.


Changing Forms of Linear Equations INB pages

My seventh grade classes started this semester with a unit solving systems of equations and inequalities.    While they have a good grasp of linear equations in slope intercept form, I am less confident in what some of them will do when presented with equations in other forms.    Since so much of our work with systems of equations will depend on the ability to work with linear equations in many forms, I am planning to spend some time early in the unit ensuring that everyone can successfully transform equations between various forms.

Today, they worked through a Connected Math lab exploring how to change between forms.   I really like the way Connected Math introduces this topic.   Students are presented with four pieces of student work in which someone has rewritten an equation in a different form that they must analyze.  Students have to determine which pieces of student work are correct, justify the steps taken for those that are correct, and identify the errors for those that are incorrect.   Each of the pieces of student work tackles the task in a different way, so students have to really think about things a little more deeply.   The lesson follows up with a task in which they move equations in slope intercept form into standard form and equations in standard form into slope intercept form.

Tomorrow, I am going to spend a second day on the concept before moving onto the next lesson in the investigation.   I want to take a little more time to solidify the concept.  I I plan to use these Interactive Notebook pages with my students to practice moving between different forms of equations

You can download the   changing-forms files by clicking on the highlighted text.

I will also have them practice with a game.     I will write about that tomorrow after I see how it goes.


A Matter of Belief

What???   That was my first reaction to Sam Shah’s headline revealing this week’s prompt for the MTBosBlogsplosion.   I had absolutely no idea what he was talking about.   Please bear with me on this.   I’m an engineer.   My undergraduate degree requirements were so tightly defined by the university that I could only take three courses in humanities and there were some pretty tight restrictions on what they could be.   That translated to one sociology course and two philosophy courses.   I only took the philosophy courses because I couldn’t take what I really wanted and word on the street was that they were easy As.   My graduate degree in engineering had no humanities.   So, soft skills?  I don’t think I could possibly face a more difficult prompt.    After a week of serious thought, I decided my only hope was to tackle this sideways instead of head on.

Earlier this week, I contacted a parent because I wanted to nominate her daughter for a STEM camp.   I received this in return.

“Thank you for seeing this talent in …. . She’s not been very confident in her abilities to the point that she came home once several years ago saying that she wasn’t good in math… I could not believe my ears and I am glad she persevered and she was so fortunate to have you as her teacher.”

I find the fact that this young woman ever could have thought that she was not good at math absolutely stunning and I’m not really sure what I did to change that.   I didn’t do anything special.   There were no long pep talks.  There was no special “cheerleading”.   There was just normal math class.

After some thought, I think there might be a few things in normal math class that make a little bit of difference for her (and for other girls as well).

  1. I genuinely believe that she is good at math.    (I believe this of all my students.    Everyone can be good at math)   I don’t know if this mattered, but I find that having someone believe in me can be very powerful.
  2. Because I believe she is good at math, there are no softball questions.   I ask her hard questions and she rises to them every time.    That doesn’t mean she gets every one of them right, but she thinks deeply about every one of them and has learned that she can think deeply.
  3. I am lucky enough to use a really good, research-based curriculum (Connected Math).  It’s not a perfect curriculum (nothing is), but it is designed to allow students to construct meaning.   Students leave with a fairly good conceptual understanding of the math rather than a bunch of rules that they blindly follow.   Having the chance to make sense of something means she gets to make it her own, to know that she knows it, to believe in her abilities.
  4. I use Kagan Cooperative Learning Structures a lot.    These structures provide this beautiful interdependence that helps everyone rise.   Students start out working independently (so they have to figure things out, not just ride on someone else’s coat tails).   Then, they work with one or more classmate (depending on the structure), discussing their work and coming to a consensus (the mathematical discourse is so much richer since I started using these structures).   They have a stake in each others’ success because they don’t know who will speak for the group.   I think this interdependence is good for all students, but girls seem to really thrive on it.
  5. I make time for regular review (5-10 minutes every day) of previously learned concepts.   This delayed recall helps students to retain concepts that they’ve learned.   I also use this time to differentiate instruction.   Sometimes students are assigned to work on a station based on their mastery level.   Sometimes students are paired so that someone with mastery is working with someone who is still working on mastery (providing an opportunity for some extra kid to kid discussion that helps to scaffold the concept).   The groupings are ever changing so every student knows the reality that everyone is good at some things and everyone has some things on which they still need to work.   I think knowing that no one is good at everything helps kids see that not being perfect at everything does not equate to not being good at math.
  6. Both of the years that I have had this young woman in my math class, she has been in a class dominated by girls.    It is just a strange fluke in the way that schedules were run, but both years I have had a period in which 3/4 of the class was girls and a period in which all of the class was boys .or in which there was only one girl.    (I try to get a schedule change for the girl when that happens so that she is not alone).   There is a lot of research out there that shows that girls in math classes do better when at least 40% of the population is female because they don’t have to expend energy fighting a stereotype.   My anecdotal evidence is that both years that I have had these girl-dominant classes, the girls have grown mathematically in degrees that I have never seen in any of my other classes.   It’s not that these classes are filled with perfect, compliant students.   It’s more that these classes are filled with strong, smart girls who interact differently than they do when there aren’t so many girls.  .