Wednesday, August 24, 2016


John Stevens' tweets (above) got me thinking and primed for day one back.   This summer has made me think a great deal about what is important to teach our children.   So far none of the truly important had anything to do with content. A brief list:
1. Understanding and acceptance of all people
2. Respect for authority
3. How to highlight and rectify injustice
4. Respect oneself
5. How to debate an issue without resorting to character defamation
6. How to value someone else's argument even if you disagree
7. How to compromise
8. Critically interpret information and data

As I write this list, I can't help but be reminded of Robert Fulgram's All I Really Need To Know I Learned In Kindergarten.   In it, he creates a short list of everything people need to know to make the world's a better place. As a parent, I've tried hard to instill these characteristics in my child.  There's always room to improve.

Just a reminder to myself that as we enter the school each day, these are the priorities.  Curriculum should be supplied to support the goals of raising critical, involved, and educated citizens.  We also desperately need more role models that can highlight these virtues.

Wednesday, May 4, 2016


There's an app on my phone that is a game of memory with 16 cards.  Each cars has a point value.  When you pair, the points go in your bank which can then be redeemed for free items.  Pretty sweet deal!
You can miss four pairs.  On the fifth miss, the game is over.  Being a super nerd, I'm working on figuring out both the probability of clearing the board (matching all 8 pairs before having 5 misses) and the expected value of a play.
The 5 plays accumulate over time.  Last night at dinner I introduced the game to a colleague, who started with 4 plays.  I advised him to wait for the fifth play but grew played any way.  He lost. No points.    It got me thinking of the big difference between 4 and 5 plays, as well as the difference between 5 and 6 plays.  At 4 plays, you are guaranteed nothing, at 5 plays you are guaranteed only a single pair and at 6 you are guaranteed to clear the board.  Someone put some serious math into these probabilities!  Well done app maker!

Tuesday, May 3, 2016

Standard Algorithm for Subtraction

Kay is working on money in her math class.  She had homework that required her to compute two people's money based on the coins they have, decide who has more and by how much.

She incorrectly calculated the amount of the difference for the two people.  I asked her how she decided who had more money, since there were some markings on the paper.   Using the comparison strategy, she explained, she was able to show that one person had an extra dime and penny.  But there was a mistake that she was able to catch.  One of the coins in her comparison was a nickel, not a dime.  She then totaled the two and came up with 36 and 42 cents.  She came up with an answer of 16 initially.

Standard response, "How did you get that?"  She thought a moment, then said she did it wrong and it should be 6.

"How did you get that?"

"I know that 6 plus 4 is 10, and another 2 makes 42."  Content she settled down and moved on.  30 seconds later...

"I could have done that so much easier I had just remembered my doubles facts...6 plus 6 is 12!"

Love this kid!  She's making sense of problem, developing and evaluating strategies, and coming up with alternate paths to solutions.  She is not, however, using the standard algorithm.

When is the right time for this introduction?  She had a good conceptual grasp.  Looking back to my post, she can use models to explain her thinking.  She uses strategies that make sense to her.  I'm torn on deciding if standard algorithm well become just one more tool for her or become a confusing abstract tool.  How do you know 5 right time for the algorithm?

Sunday, May 1, 2016

Kay 35 + 7

So Kay (6) became intrigued about the calendar.  She started reading dates as we completed our monthly ritual of changing the dates.  She told me that being able to count by 7's would be a good skill for keeping track of the date.  She then went over to the calendar, paused a minute and started reading off, "7, 14, 21, 28..."  She paused as she calculated the next number.  "35".  She stopped.  To egg her on I told her, "If you add another seven, you get my favorite number."  (Hitchhikers' Guide anyone?)  She asked what it was to which I replied, "I'm not just going to tell you".
She came up with the correct answer of 42, but instead of confirming, I asked her how she got that.  Matter-of-factly she stated, "I used my Make 10 facts".  I assumed she did what I did, which is break the 7 into 5 + 2 and add the 35 and 5 to make 40.  Just to be sure, I asked her what fact she used.  She told me "7 + 3 is 10".  I was totally confused.
She wanted to show me on a calculator, but since one was not handy, she ran over to where we keep our base 10 blocks.  "I can show you".  It was hard to contain my excitement!  What ensued, was a pretty amazing explanation, I think.

It was about more than just the addition.  Kay showed me a completely new way to think about decomposition.  I never would have thought to take from the larger number to make my 10.  Here I am, a math teacher in my 11th year, with a Bachelor in Math and a Masters in Math Ed, learning about addition from a 6 year old.  I was truly and genuinely interested in her method, especially since she's my kid, but she was oh so eager to share it with me.  It took her a lot of processing time, and she made a few mistakes.  Once the ball was rolling, my job was to listen.  As always though, whenever she shares her method, I try to share a different one, just for comparisons sake.

So how can we replicate this in the classroom?  How can we make sure that students have a place to voice their thoughts?  How can we give students enough time to make sense and explore their sense making?  How can we be genuinely curious about the voice of each of our students on a daily basis?

Monday, April 18, 2016

Connect 4 With My Kid

Inspired by Joe Schwartz's Connect 4 using a multiplication grid, I tried it out on my unsuspecting 6-year-old.

A little background.  K's understanding of multiplication is in the frame of repeated addition.  She can count by 2, 5 and 10.

The beginning of the game was quick.  K was able to calculate multiples of ten quickly, even commenting "Oh, this is easy".  When she got to her first entry in the 9 row, she interpreted the multiplication as 9 groups of whatever number is at the top of the slot.  So she saw 9x4 as 4+4+... 9 times.  She used this strategy for most of row of 9s, but when she did 9x7, she suddenly switched strategies and added 9 to 54.  I asked her about the strategy and she said each spot is 9 more than the one before it.  As the game progressed, she used a similar strategy vertically, recognizing that each column had its own progression.

Obviously, I was very pleased, as she was both enjoying the game and working on some important foundation arithmetic.  Her strategies revealed a lot about how she interpreted the game.  She noticed patterns and created her own meanings.  I was also really intrigued by her choosing to consider the multiplication m*n as m groups of size n, but then proceed in rows by adding a group of m.

Overall, the game is a win for math, but I learned that my kid needs to work on her Connect 4 strategy...

Tuesday, March 29, 2016

Number Headbands

My 1st grader just brought home a test.  On it was a question about playing Number Headbands, a twist on the popular kids' game.  The game is played like this:

Kids are broken up into groups of 3.  A deck of cards with numbers on it is shuffled.  One person is the referee and the other two each take a card without looking at it.  The referee looks at both, must add them, then gives the total to the two participants.  It got a little tough asking a six-year-old what happens next.  I assumed it was a competition between the other players to get the answer, but I think a time could be ascribed to the round to allow for more participation, then both players can give their answers.  Kids switch roles and repeat as often as time allows.

In hearing my daughter describe the game, I thought about the benefits: engagement, students checking each other's work, and the relationship between inverse operations.  How can this structure be used in other classes?

I'm thinking about how this might look in intermediate grades, middle school and beyond.  Introducing negative numbers, decimals, fractions, and other operations can help extend this engaging practice technique.  Looking at algebra, it can help students with multiplying and factoring polynomials, or composing functions.  A two player version could help students write inverses.

Wednesday, January 27, 2016

Percents and Number Lines

So I was monitoring this conversation on Twitter the other day:

I was going to join in but realized that my thoughts required WAY more than 140 characters.  There were so many great points made by members of the #MTBoS (as always).  My initial thought was "of course a percent is a number, so it can be placed on a number line", but then I began to doubt myself following some points made by @letsplaymath.
It took a while but I think I clarified it all for myself.  The question is can we put a percent on a number line.  I'm sticking with my initial YES.  Here's why: When we create a closed number line (not an open number line used by @Mr_Harris_Math to teach arithmetic strategies), we MUST put at least 2 numbers on the number line.  This inherently defines a unit on the number line.  This unit is, in a sense, the interval.  It is the distance covered by 1.

All the other numbers on the number line are defined by this unit.  The number 42 is 42 of these units lined up away from zero.  A percent is a special type of fraction where the denominator is 100.  Putting 85% on a number line means taking that unit distance, dividing it into 100 equal sub-units and traveling 85 of these sub-units away from zero.  It is both a number and a ratio.  When we have a closed number line, the unit distance (interval if you will) is inherent to the line as soon as two numbers are placed on it.  Any ratio or percent you'd like to plot on this line are then defined in terms of this unit.  A percent just defines that we will be traveling in increments that are 1/100th of the defined unit.

There was an argument that we can't put 80% on the number line because it is relative to another number.  Is it 80% of 20?  Is it 80% of 1?  When we place 80% on the number line, we are placing it relative to 1 unit on the number line.  This is the key to the argument that we CAN indeed put percentages (and any other ratio) on the number line.  It is implied that we will be placing them on the number line relative to the distance of 1 unit.

Wednesday, January 20, 2016

Intro to the Coordinate Plane

I always had a handful of students that continually reverse coordinates or plot points incorrectly because they forget that a negative sign indicates going down or left.  So I designed a lesson that I was hoping would help students understand the importance of conventions in the idea of an ordered pair.

I told students that they would be on a mini-treasure hunt.  We blindfolded a student and sent them out of the room.  The rest of the class was charged with giving the blindfolded student directions to the "treasure".  For my class, I used a dollar bill and put it in the ceiling which conveniently had square drop tiles.  The class had to give all of the directions to the blindfolded student BEFORE that student was allowed to move.  

They brainstormed and came up with a plan to tell the student to walk 5 steps forward and 3 steps right.  Immediately they began to see some issues with the directions.  The blindfolded student took much smaller steps than the student that had measured the 5 and 3 steps.  The student was way off.  We had a discussion to decide where things may have gone off track.  After some more brainstorming, students decided to use the tiles on the floor as a way to track distance.  This required me to change the blindfold for a "You're only allowed to look straight down" direction to the treasure seeker.  This process allowed students to understand the need for a consistent interval when plotting on the coordinate plane.

We repeated a few times with success (I kept the dollar bill though!).  After two successes, we did the same, but I went into the hall and had the student come in through another door.  The directions the class gave were assuming we were going to use the same door as the last few times.  The class was upset with me because I was being "unfair", but it opened the discussion that if we were to give directions ahead of time, we needed to know WHERE to start.  Again leading to the understanding of why we always plot points starting at the origin!

The last twist was that I put a challenge to the students to give directions in as few words as possible.  They quickly reduced directions to something like 2 left, 5 forward.  Then I said, no words, all numbers.  Having familiarity with a number line, students were able to arrive at a negative for one direction and a positive for the opposite, but it took some serious prodding for them to get to the point where they understood that they needed to come to a consensus about WHICH number would come first.  This was exactly what I was looking for in terms of their coming to grips with the coordinate plane.  

Friday, January 8, 2016

Equation Golf

We were talking about PD experiences the other day and it inevitably turned to "What was the best PD you've ever been to?"  Having never attended a TMC (and unable to go this summer...sad face) I reflected upon one of the more useful district supported PD days.  We met a group of math teachers from another high school in the district.  Prior to the meeting we were asked to write out a description of our best.  It could be a lesson, a technique, an explanation.  I honestly don't remember what I brought to the table, but it wasn't nearly as good as the stuff I got out of the day.

Below are quick descriptions of two review games that have been great of great use in the classroom for a tweak on an "quality, basic" problem set.

Equation Golf:  Have a problem set of 15 problems that you want the whole class to do?  Give students a goal, or a par.  You want to get all of the correct answers to the problem set by calling on 18 people or less.  (Numbers used are purely for illustrative purposes, you can have 20 problems and call on 26 people, just make the goal reasonably attainable).  Have students work on the problem set for the desired amount of time.  Individually, partners, groups, your call.  At the end of the time, tell students that they must pay attention to play golf.  Pick a student at random (Popsicle sticks, random # generator) and that student picks whichever problem they would like to answer.  You verify or reject the answer.  New person.  Each person counts as a "stroke".  Goal is to make par.

This helps work on listening skills as well.  I always tell the students that they must pay attention to which answers have been given AND marked as correct.  The students must announce the problem number first, then give their answer.  This way, if a correct answer has been given for #3, and a student says "I'd like to answer #3, the answer is..." I can cut them off, let them know that that question has been answered correctly already and that counts as a stroke.

In the past, I had given an entire class grade for this activity.  Usually a relatively small grade that didn't really affect the overall grade in the class, but enough to keep them interested.  As I tried to move towards more standards based grading, I might make the change to allowing a student that answers correctly to use that as evidence of learning for a particular objective.  This would have the added bonus of helping students to identify their own areas of need, work on that with students in the class during the class work portion, and trying to answer more difficult questions based on their individual needs.

Bluff:  Take a similar problem set as Equation Golf but divide the class into two teams.  Give one team a problem to complete (the other team should simultaneously complete the problem).  On Team 1, after a certain time, ask anyone that feels that they have the correct answer to stand up.  Someone from Team 2 picks a Team 1 member to answer.  If it is correct, Team 1 gets a point for each person standing.  If it is incorrect, Team 2 can steal the points.  Then the next question goes to Team 2.

This game can be a bit touchy so you have to have a responsible group of kids that will be willing to put themselves out there.  Class culture is extremely important to this game.  I have seen times where students will continually pick on one person, so it is up to the teacher to ensure that the game stays positive.  In cases like this, the teacher doing it was awesome (disclaimer, it was not me...though I am pretty awesome!) and when this situation arose, he turned it and made the kid a champ when he kept getting the questions correct.

A bit of strategy goes into the game as well.  A team cannot call on the same person 2 turns in a row, so the last person called should always stand.  I don't share this with kids, but it is fun to watch them figure it out.  And a student can stand even if they do not think they have the right answer in an effort to increase the point value for their team (hence the name Bluff).