Is Candy Math A Thing Of The Past?

Finding a dollar on the sidewalk growing up meant a trip to the one-stop pharmacy/convenience/self-care/candy store adjacent to the local grocery store. The candy aisle was the place I truly learned how to add & subtract decimals in my head. 

A small box of Lemon Heads is $.35, so that leaves me with $.65.

Jolly Ranchers are $.05 each, I'll grab three of those (two watermelon and one fireball), and now I have $.50.

Uh oh. A Fun-Dip is $.60! I don't have enough. I'll put the Lemon Heads back, and grab the Fun-Dip. That leaves me with (a quick look toward the ceiling while I calculate) $.25 to spend. 

Laffy Taffys are $.05 each, so I'll grab five of those (three banana and two grape).

A great day!

Today I feel like the old man yelling, "Get off my lawn!" at the youngsters in the neighborhood. Instead I'm telling students they don't understand the excitement and joy of Candy Math! 

With the rise of technology, using payment options other than cash has also increased dramatically. First credit card companies made it easier to pay for your gas at the pump with a swipe of the card. I still remember Jerry Seinfeld's commercial for American Express when this technology first hit the petrol stations. Then that technology made it's way into the stores while shopping for groceries, or at the mall. Then banks offered Debit Cards to be used the same way, and that way you won't have to pat credit card interest. 

Eventually chip readers were embedded so a person didn't even need to swipe their card. They could just hold it up to the machine. Now with pay apps like Venmo, Apple Pay, and PayPal, the need for cash is nearly nonexistent.

Gone are the days of adults carrying coins in their pockets. Missing are the days when kids could search through couch cushions to find change that may have fallen out of said pockets. Cars were designed with cup holders and change holders, but not anymore. No longer can kids sneak a few quarters out of the family car's coin pile to buy themselves candy at the store. 

For years as a middle school math teacher I could use quarters as a way to discuss the fraction 1/4, or the decimal .25, and ask students, "How many quarters are in a dollar?" And the room would fill with every student voice, "Four!"  In recent years, only a few students answer under an almost muted tone just in case they are wrong.

See a need, meet the need. 

In order to create a learning experience with coins, I needed coins. And during the pandemic, their has been a cash/coin shortage, so getting enough coins for a class activity was a challenge. I visited several grocery stores buying rolls of coins until I had enough for each group in class to have: 5 quarters, 10 dimes, 9 nickels and 30 pennies. $3 in total.

Thanks for checking my math in your head just now. :)

Asking students to use a specific number of coins to equal a specific quantity, and then asking for the fraction and decimal math problems to prove the amount led students on a path filled with a teacher's favorite sounds, "Ooohh, Ahha." "That makes so much more sense." Some students used addition problems, while others multiplied, and some used both. The noise level grew with excitement as their learning increased. 

As technology improves parts of our lives, we need to recognize where some learning might be lost. How can we recreate that learning in a lasting way? How can adapt our instruction to the ever changing world? 

I know one thing. When my 3 and 5 year olds are ready for chores and an allowance, I'm paying them in coins!

6th Grade Block Classes: A New Challenge

Switching from Junior High to Middle School creates many new experiences for both students and staff. The leadership and maturity of the 9th graders are gone, and is replaced with the sweetness and neediness of 6th graders. The comfort of having two grade levels of students that "get what our school is like" is gone and replaced with having no one know what our school is like, because we are reinventing ourselves. Teachers who have always done things this way, can no longer always do things this way, because it doesn't fit with the new model.

Part of our new model, six period days were replaced with block classes on Monday, Tuesday, Thursday and Friday. Some training was given on how to teach in a block, but it was generic and lacked purpose. No experts were brought in, and planning time was scarce as we also needed time to discuss all of the other changes coming down the pipeline. I've read excerpts from books and articles online, but without someone that has expertise to ask questions, the tips for success become catchphrases with little meaning.

Included in this transition, teaching a new grade level. I'm a teacher that utilizes investigations, group-worthy tasks, and explorations of concepts daily. So when I read the term "standard algorithm" so many times in the 6th grade standards, my soul takes a beating. I'm sure both can be accomplished, but I, for the first time since the start of my career, don't know where to start.

How do I solidify previous understandings of place-value without just showing them? How do I convince students that long division of decimals on paper is relevant and important when they all have a calculator in their pocket? Multiplication tables, flash cards and timed tests in grades 3, 4 and 5 have not helped 50% of my students learn these facts, so what do I do when the lack of facts hinder progress?

Perseverance. That what this year is all about for me. Seek knowledge from experts, and persevere.

I'm a 14-year veteran, 1st year teacher.

Student Led Corrections

Thought inspiring administrators are imperative to successful teachers and schools.  Administrators that can pose a question about your teaching without judgement or evaluation, and get you to genuinely reflect on your teaching are the reasons why my pedagogy has improved over the years.

One instance dramatically changed my correction process on assignments.  Whether the class decided that they want to correct every problem on an assignment, or a select few, I used to be the one at the board showing the work and explaining the math.  I used formative assessments during this process to check for understanding, but it was usually me at the board.  Then my principal asked me during a post-observation meeting, "I know you can do math, but do the kids really know if they can do math when you are at the board?"  In pondering my response, I thought about my formative assessments.

I was able to figure out which students had understanding and which did not, but were the students getting the same feedback for themselves?  Do they need validation from me? Do they need validation from each other that they understand?  Were they learning that mistakes are good and help us learn, or were they merely fixing their work?  How could I get less of me, more of them, still be safe to make mistakes and maintain a learning focus?

Student led corrections are a staple of my classroom now.  At the beginning of the year, it takes a lot of training and practice, but after 5-6 weeks, the learning and engagement level during corrections is well worth the time and effort.  Here how it works:

1) Select the problems to be corrected.  This can be entire assignment if it's not too long, or can be a few intentionally chosen problems picked by you or the students.

2) Assign the problems to groups of students in the room.  I have eight small groups of four students in my room, so if I were assigning 8 math problems, each group would be assigned one.  They will send one student to the board for that problem. The next day the student that goes to the board must rotate around the group and one student cannot correct at the board for a second time, until everyone has taken their turn.

3) Give them one minute to discuss the work and explanation in their group before sending their representative to the board.  I have four largish whiteboards in the front of class.  The first four students can be writing their work on the board at the same time, then when #1 finishes the correction process, #5 comes up and starts their work while #2 is explaining their problem. This rotation of problems keep the class moving forward and we do not have to wait for work to get on the board.

4) When student #1 has their work on the board, they grab the microphone (our district has classroom speakers that has really improved listening) and explain the steps and mathematical reasoning of their problem.  Expectations for the explanation will vary from teacher to teacher.

5) Once the student at the board has finished, every single student should raise their hand to be called on by the student at the board.  If called on, they have only 3 possible things to say:

• "I agree."  By saying this, the student called on is claiming that the work on the board matches what is on their own paper, or that the work may be slightly different, but mathematically is equivalent. They agree with the work and explanation.
• "I disagree because on line __, I think ____"  The phrasing here is important.  The student is disagreeing with the work and says so by saying the line of work they see an issue. Disagreeing with work is safer than disagreeing with a person.  It feels less like an attack on the student. This also gets students to focus on the steps, math reasoning and accuracy more than just the answer.
• "I did not complete my assignment."  Students will take ownership of not doing their work and realize they still need to participate in the correction process in order to have a better understanding.  These students will not be writing anything down during this time, as the hope is that they do it on their own eventually.

Having three options of what to say makes this a safe process for the student at the board and safe for the student being called on as well.  Instead of just saying, "I got ___."  They can speak about the work.  All math teachers that I have spoken to do not like the "I got..." game that students like to say, and this definitely helps alleviate that issue. 

6) Once the student at the board has three "I agree" statements in a row, they can return to their seat as long as no one has a question for them.

The repetitive nature of this correction process also means I do not need to assign as many practice problems because students see and hear the math so many times.  At the end of the process, the students that were at the board feel good that other students agreed with their work.  They view mistakes on the board as an opportunity for the class to learn.  And at the end of the process, I know who can do the math and more importantly, the students know for themselves. 

Give Me Five!

No, not a High-Five.

How do you start a unit on Systems of Linear Equations, or Simultaneous Equations (depending on how your curriculum or school refers to this concept)?  Do you front-load the vocabulary?  Do you start with the visual solution, i.e. graph?  Or do you start with whatever example is in the book for that chapter?  I start with a Give Me Five!

Students have a good understanding of what it means to be tardy to class, and running laps in Phys. Ed.  I attach a problem to that understanding and ask a simple question.

Tardies have been a growing problem in P.E. lately, so the teachers have created their own system to try and quell the problem.  Mr. Crunch usually starts class with 2 laps around the gym as a warm-up, but now he will be adding an additional lap for every tardy student.  Mrs. Yogger usually starts with 5 laps as a warm-up, but now will add 1/2 lap for every tardy student.

Which class would you rather be in this semester?

It seems simple enough, but as students start to discuss in their groups, they come up with a lot of, "Well, it depends..."  Argumentation among students quickly ensues.   Once the groups are fully entrenched in their response to the question, I tell them they must back their response up with a Give Me Five.

Students must justify, prove or back up their answer using five representations of the problem and their answer.  Writing an equation for each teacher happens quickly.  Then the groups decide if they make two tables or one table and while making that decision they are debating the efficiency of the choices.  The graph spurs conversations on how far to go on the x and y axis, and then what scale to use to best represent the problem and solution.  The picture is a challenge for some students, but if they can draw a picture that makes sense, they definitely understand the math.  Then the written explanation makes connections among the representations and deepens the learning.

By the time the students finish the five representations, each student knows and understands that the point (6,8) is the most important point in their response.  They can clearly explain that in the table, the point (6,8) is where the number of tardy students and laps are the same, therefore the points before and after that depend on if you like running more or running less.  Students can clearly demonstrate understanding that finding the point (6,8) on the graph allows them to show where students would run more or run less.  There are so many teachable moments throughout this activity and you'll want to stop class use them all, but I suggest letting a lot of them go.  Allow for mistakes. Allow students to make tables that stop at five tardies. Allow students to make two graphs even though one is what you expect.  Respond to the mistakes later with respect and a learning focus. It pays off throughout the unit.

At this point students have an understanding of solving systems of linear equations using a table and graph. You may want to practice with different equations for a day or two.  Eventually students will ask, "Is their an easier way to solve for this point than building a table or graph every time?"

Yes! Now there is a desire to learn!

Red, Yellow, Green

While attending a PD session years ago I saw a video in which a math teacher had students using Red, Yellow and Green cards on their table tops to indicate how they felt about the day's learning target as a formative assessment.  Each day most students would start with a red card on the top right corner of their desk, and as the lesson progressed, students would take cards from their binder and switch them to yellow to indicate they were gaining understanding, but not quite ready to run alone with the content.  When a student felt like they understood it enough to help someone else, they would place their green card on their desk indicating they were good to go.

The simplicity of the colored cards was intriguing but I had so many questions.  Are the students being honest, or do they change their cards so they are not the only red in the room? Are middle school students organized enough to keep these cards with them for the year? What do I gain from seeing a lot of yellows?  How does this impact the students' math self-esteem?  A student's math self-esteem can be one of the biggest barriers to success in my experience.  I try to create a WE learning community in my class. Our goal is not that YOU do well, it's that WE do well.  This takes a little pressure off of individuals and helps to create a safe place when asking for help.

Attempting to use the simplicity of Red, Yellow, Green but also address my questions, I first tried using the idea with colored pencils and a unit review.  My 7th grade class was approaching the end of a unit on Linear Relationships and we were gearing up for the final assessment.  Two days before the assessment I passed out the unit review and asked the students to have red, yellow and green colored pencils on their desks. I also asked that they not speak to each other and not to begin working on any of the problems.  I told them that before we begin working, we are going to have some private think time about each problem. I asked them to follow along as I read each problem aloud.

After reading the first problem I asked them to color over the problem number with the red, yellow or green colored pencil with these thoughts in mind.  If you color the number ________ that means if you see a problem like this, or similar to this on the assessment:

Green: You are going to dominate this problem. If your grade were based on this one problem, you'd have an A. You are saying, "I can demonstrate understanding of this problem with math work and a written explanation. I am good to go!"

Yellow: You have a pretty good idea of how to attempt this problem. Sometimes you do well on these problems, but sometimes you don't do well.  You will need to slow down and really think to make sure you are not missing something. You are saying, "I see this problem as a challenge but I am willing to take on that challenge."

Red: You are not yet ready to take this challenge on and would like to have more conversation, practice, instruction or think-time.  You are saying, "I will be able to take this problem on, but I need some help before I can rise to this challenge."

I repeated these descriptions for each of the first five problems that I read aloud.  Once they got the idea, I let them finish coloring each problem number.  Then I asked anyone that has at least one red or yellow to raise their hands.  This gets just about every hand in the air, and students see that even the "A-Kids" have red and yellow.  It's important for the students with a lot of red and yellow to see that everyone struggles from time to time.

Next I asked students who colored #1 green to raise his or her hands and keep them up. Then I asked students who colored #1 as red or yellow to find a green person and have a conversation.  Students began teaching students while I floated around the room collecting data which problems had a significant amount of reds and yellows. I also began jumping in to help the "greens."  We continued this process through all of the review problems.  Students were using the colored pencils to change their colors after each conversation.  Reds became yellow, yellow to green and some greens found out they were really yellow.  After the dust cleared, students knew exactly what they understood and could demonstrate, and they made a plan for the next two days to get themselves where they needed.  I used the red and yellow data to plan for the next day, and the learning was more purposeful and targeted than just passing out the review and saying we got two days to be ready.

Since that first time, I have adjusted the process slightly from year to year, and class to class.  I have used my group roles (see early post on group roles) to guide groups through red, yellow, green conversations, and I have had students make posters of what they feel is demonstrating red, yellow, green level of understanding.  The students will ask me for a red, yellow, green activities when they feel like the levels of learning in the room are out of balance because they understand that until everyone is green, we have not met our goal that WE will do well. Each time I try something new, the basic idea stays the same, but it's the subtle differences that keep the students digging my red, yellow, green chili.

Hope Can Raise More Than Grades

Many years ago in my first 9th grade Algebra 1 class I had several students that entered the school year having always struggled in math, had low math confidence and were lacking the skills required to be successful in Algebra.  I started math help sessions in my classroom after school for anyone that wanted to attend.  Some students were "forced" to attend by their parents, but eventually they saw the benefit.  During this school year it started as Tutor Tuesdays, then Want Help Wednesdays and finally Math Mondays.  The expansion to three days a week was to supply enough help for the demand coming from my students.

We used this time to supplement Algebra with the concepts they were lacking from previous math classes.  Numeracy, fractions, decimals, percents, integers and basic math properties.  The road was rough at first, but as we drew pictures and developed a better understanding of how numbers worked in relation to each other, tremendous gains were made. Students who once failed an assessment on two-step linear equations, were solving systems of equations with ease.  I was curious as to how well these students would do on assessments from earlier in the year, but first semester grades were "final" and we were approaching the end of third quarter.

I asked my school registrar about the possibility of changing someone's first semester grade because I no longer felt it was representative of their knowledge.  I was told it was not possible from her point of view, but that the Principal can override the rules.  So I was off to plead my case. Luckily I work for a  Principal that is all about whatever is best for student learning, and keeps kids on track to graduate. I began reassessing these students and everyone one of them increased their semester grade by at least two letter grades.

Since then I have used a Recovery Contract in which students can recover their first semester grade if they are willing to relearn concepts and reassess.  The offer is made to anyone that earned a D or F first semester. Of course I do not inform my students of this contract until late in first semester because I do not want students thinking they can slack off and then just recover their grade later.  And I have been amazed that students from year to year do not talk about the contract and thus incoming students do not have a clue until the time comes.  No one has ever declined the offer either.

Using the contract the past two years have been particularly inspiring for me and my students.  One particular student this year has made all of the extra hours of tutoring, designing new review materials, new assessments and grading worth it!  She transferred to my school because of the progress her cousin made in my class (on contract) last year.  Her childhood has been rougher than most students.  She now lives with her cousin's family and English is not her first language.

When she transferred to my 9th grade Algebra class in October, we spent three weeks learning integers.  She did not understand the concept of that positives and negatives cancel each other out.  We drew pictures, number lines, used mini poker chips and when I finally put some cash money on the table, she finally started to get it.  From there we worked on combining like terms, distributive property, equivalent expressions and two-step solves.  While the rest of class was working on systems of linear inequalities, she and I worked on ratios/proportions.  She finished first semester with 26% on Alg 1 assessments.  Today is the end of third quarter and she has now earned an 86% B for first semester, and is up to a B+ for second semester.

The hope of recovery has led to a completely to new student. A student that needed help with her math on a daily basis is now helping other students.  Her complete lack of math confidence is now climbing and is having a positive impact in her science and tech classes.  A student who saw a gloomy future, is looking towards an open and bright future.  Hope.

Problems for Problems

The #slowmathchat discussion on Twitter got me thinking about what I assign for homework, why I assign it, how it has changed over my career and where do I think it is going?  I started as most teachers do at the beginning of their career.  After correcting the previous day's assignment, which consisted of me at the overhead (dating myself) demonstrating what was expected on each problem, I would then give a brief lesson on the day's new topic. After the lesson I would write the day's assignment on the board at the mid-point of the class period.  Whatever did not get finished in class, was to be completed as homework.

I relied on my sense of humor, engaging personality and excitement for learning to keep my students interested each day.  While I still use those aspects to my advantage, they are not the cornerstones to engagement in my class. Lessons have evolved to inquiry-based activities and investigations. Learning comes from student talk and not teacher talk, and thus I needed to change the homework assignments. They became hand-selected purposeful problems to practice previously learned concepts and not to finish the day's learning.

Usually on Monday and Tuesday we investigate new concept(s) in class and I assign specific practice from the concept(s) we investigated on Thursday and Friday.  Wednesday I start to assign problems we investigated on Monday and Tuesday.  The homework assignments typically are "naked math" and we work on "real life" in class.  Naked math are straight up math problems, no words.  This continues with each set of problems scaffolded in difficulty to the class's progress with a concept.  Once the skills have been mastered, I will assign "real life" problems for homework.

Depending on the concept(s), the typical assignment is 4-8 problems.  I find that 10-15 minutes of hand-selected purposeful practice improves skills, improves confidence, reduces frustration in students and provides access for all learners.