Saturday, October 29, 2011

Red Pen Reflection

I have learned many things from Rick Lavoie. One of the most important is to remember that I never struggled in school, so I cannot know how it feels to constantly have my work returned covered with glaring red marks. Recently, when faced with returning just such a math test for corrections, I decided to try something else. I had watched the student accurately complete four math activities online and noticed that he only saw one problem at a time. Thinking that a page of problems might be what was overwhelming him, I wrote each incorrect problem on an index card. The next morning, I asked him to do some math review with the cards, letting him choose the order. He completed them all correctly. It got me to wondering how he would have done on the original test if the problems had been given to him one at a time. Perhaps presentation, rather than content, causes some students' failure to demonstrate their knowledge. Are we setting them up to fail?

Tuesday, July 19, 2011

Was the Bus Empty When it Left the Terminal

Image from

While gathering math ideas for the sisters I'm tutoring, I came across a problem that has a problem. I decided to find out if they could see it too. Here it is:

For the Good Cheer food drive, your class collected 320 cans of food. Each family gets 40 cans of food. How many cartons will you need to pack them?

I'm sure the intended answer is 8 - the number of families who would get food. However, the question asked how many cartons you need. Both girls knew the problem; they needed to know how many cans fit in a carton. We talked about vague, ambiguous, or just plain bad word problems. I told them not to be afraid to speak up when they come across such problems. 

We moved on - 
The bus left the terminal and picked up 5 people at the first stop. At the second stop, 2 people got off and 4 people got on. At the third stop, 3 people got off and 5 got on. How many passengers were now on the bus. 

Instantly, the older girl asked, "Was the bus empty when it left the terminal?" I grinned and congratulated her for getting the message. No test was needed to assess her learning.

That morning, all three of us had a great time playing math!

Monday, July 11, 2011

My Heart is Happy

Flickr photo by fauxto_dig
This is a great day! One more student is beginning to replace the I can'ts with I cans when it comes to math. I played math for an hour this morning with two new students, 5th and 8th grade sisters. Prior to today, the one word that described how the older one felt about math is FRUSTRATED. We played with addition, multiplication, and fractions. Both girls are bright, articulate, and eager to understand what math is all about. Their mom didn't want to overwhelm them, so our first session was just an hour. We made plans to get together again on Thursday, so she asked the 8th grader if two hours would be too much. The young lady replied, "I could do this all day!" I had told her when we met that I help kids understand math because it hurts my heart that so many kids feel defeated by it. Today I responded to her upbeat attitude and enthusiasm by saying, "My heart is so happy!" What a great day! I can't wait for Thursday.

Monday, May 30, 2011

There's a Pattern to It

Last month, while an associate sub, I walked out of a middle school math class thinking, "Wow! That was fun!" The students spent the first part of the period using the Percent Of activity on this website - - to hone their estimation skills and increase their number sense. Some of those with higher scores explained their thought processes to others. They all enjoyed trying to improve their outcomes, and most succeeded.

Previously, the class had worked to determine which of three restaurant patrons was the most generous tipper, so the next activity expanded on that. The teacher projected a chart with several dollar amounts across the top and percentages (1%, 5%, 10%, 15%, 20%) down the side. Students worked in pairs to complete the chart. Listening to their conversations was great fun. Most pairs were diligently calculating each result, but I started to hear a few talk about the relationships between the percentages. The most exciting moment for me was when one boy looked up at me with a grin and exclaimed, "There's a pattern to it, isn't there?"

I applaud the teacher for this example of math being taught and learned the way it was meant to be - with a combination of cooperation, collaboration, communication, creativity, and critical thinking.  

Thursday, May 26, 2011

The Power of Positive Thinking

Since March, I have had the opportunity to chat several times with the 5th grader from the homework group. (See From Can't to Can) Each time she mentions something positive about math class. I subbed in that class yesterday while the students worked on the final test. Before they got started, I told them that any can'ts in their heads would keep them from thinking and asked her to tell them what they needed to believe. She grinned and said, "I think I can!" After school today, she and I worked on some strategies to help her learn and remember some of the addition and multiplication facts that give her trouble. Although I am a volunteer after school, I was richly rewarded when she said, "Ever since I've been working with you, I am really liking math." She now believes in herself, and it has made a big difference in her attitude and her accomplishments.

Tonight I rediscovered a poem I had forgotten about and am making a copy for her - "It Couldn't Be Done" by Edgar Guest.

Saturday, March 12, 2011

Thanks for Asking

Lately, I've had several experiences with the one kid in a class who will speak up and ask the question most of the others are wondering about. I love it!

While subbing in the computer lab, one 2nd grader came up to tell me that the teacher lets them have the lights turned off on Fridays. Having discovered in the past that there may be disagreements about such things, I asked the class how many would object if I turned the lights off. About half the hands shot up. Then I heard a small voice say, "What does object mean?" I explained that it means you wouldn't like it, and all of the hands quickly went down.

Near the end of another 2nd grade day, an intercom message told one little boy that his dad would pick him up tonight. Later, I noticed that he was looking confused and not getting ready to go. Finally he said, "How long is it until tonight?" I knew the secretary meant after school, but he did not. This got me to thinking about how hard kids must work every day to figure out the meaning of things they hear. I told my 91-year-old mom, who struggles with hearing, that it must be similar to how she pieces together conversations when she doesn't quite hear all of the words.

Three cheers for the kids who will ask for an explanation!

Wednesday, March 2, 2011

From Can't to Can

Some of my most rewarding experiences this year have taken place after school, while volunteering with the middle school homework group. This week was no exception. I enjoyed teaching K-4 music, although I sing off-key. Then I headed next door to see who needed help. I found a 5th grade girl struggling with a math test review. She was really down; the can'ts in her head were shutting down all possibility of real thinking. I smiled, encouraged, explained, asked questions, and listened. She was working with fractions, so I gave her one of my dry erase cards I made to help her rename and/or reduce fractions. (See Equivalent Fractions) I watched her eyes slowly light up, and we had a great hour of math. She left saying, "I can!" We met again the next day and she dug in to finish the review with a positive attitude. She will take the test today, and my fingers are crossed that she will stay calm and think things through.

Sunday, February 27, 2011

My Expanding Digital Footprint

Recently, I read a blog post requesting submissions for the next issue of Math Teachers at Play, so I decided to submit my equivalent fractions post. It was accepted and published -

While searching my own digital footprint at, I came across a mention of that submission at

It is exciting to see how my digital footprint quickly extends beyond the original source of my posts. It is also a constant reminder to think carefully about what I make available to the world.

Sunday, February 20, 2011

Successful Cure not Symptom Relief

While responding to another post regarding reward systems, I thought about a paper I wrote for my Educational Psychology and Human Development course in 2007. My experiences in classrooms since that time have greatly reinforced the beliefs I expressed in that paper. What do you think?

Saturday, February 12, 2011

Does Your Answer Make Sense

The more time I spend subbing in classrooms, the more I am discovering that I had some pretty amazing elementary math teachers in the 1960s. They taught us the whys and the hows. They also taught us to check the reasonableness of each answer. In fact, I remember learning to look at a problem to determine about what the answer would be before solving it.

I think the focus of math instruction in some classrooms is more on accurate calculation and less about understanding numbers, the procedures, and the processes. Last week, I tried using logic to help a 4th grader understand why his answer was not correct. The problem: 258 - 199. His answer: 159. He knew that 199 was only 1 less than 200 and that 258 - 200 = 58, yet he could not understand how that affected his answer. I've also tried to help students think logically about multiplication facts. If they know that 5 x 8 = 40, they should realize that 3 x 8 can't be 36.

While working on equivalent fractions with middle schoolers, they told me that to find equivalent fractions you do the same thing to the top as you do to the bottom. I asked them why I could do that without changing the value of the fraction. Not one of them knew, so I taught them what I had been taught about equivalent fractions. I could see the lightbulbs go on. They understood that when you multiply both the numerator and denominator of a fraction by the same number, you are really multiplying by 1, so the resulting fraction is equivalent to the original fraction. In my own class, I would take the time to write 4/9 x 4/4 = 16/36, and let them talk about it to see if they would discover for themselves that 4/4 = 1.

After watching Conrad Wolfram's TED talk, I believe he is 100% correct - "Stop teaching calculating, start teaching maths."

Computers can do the calculations. Students need to be able to ask the right question, turn it into a math problem, and check the reasonableness of the computer's answer. If we must use multiple choice assessments, perhaps we could check for understanding by posing questions like this:

I have a garden that is almost 38 feet by just less than 32 feet. About how much planting area do I have?
a) 120 ft    b) 1,200 sq ft    c) 120 sq ft    d) 1,200 ft

Tuesday, February 8, 2011

I Just Can't Say No

I have thought about writing on this topic, bathroom breaks, for a long time, and I am finally going to do it. As a sub, I come across all kinds of classroom rules and consequences regarding student use of the bathroom. I really try to follow all the rules of the classrooms I'm in, but on this matter I just can't say no.  I am aware that the request may be just an excuse to get up and move around, but I understand because I have needed to get up and move around or clear my head with a short walk many times a day during my 30+ years of office work. When I needed a minute to sort out my thoughts, I got up to file a few papers or get a cup of coffee.

I know I can't have a constant stream of students in and out while I'm teaching a lesson. However, if they are all asking to go during that time, I'd better spruce up my delivery in a hurry. When they are working on their own or in a group, I think they should be allowed to make a quick trip to the bathroom. With very few limitations, I think deciding when to use the bathroom should be one of the choices students have in school.

In a video I recently watched, Rick Lavoie detailed some bathroom use rules and consequences he has come across. He mentioned one that kept a whole class in for recess if too many students left class to use the bathroom. Another incident was about a special needs boy who asked to go about the same time every day. When his teacher finally asked him why he didn't go with the rest of the class before recess, he responded by telling her that the adults didn't protect him during that time and how the group of boys always ganged up on him. Rick ended by pleading, "Let my people go!" And I do.

Competitors or Partners

I subbed in PE today and observed something quite amazing. We played Cornhole (bean bag toss) with two people to each pair of boards. I told them we were playing to 21 - 3 points for going in the hole and 1 point if it stayed on the board. Fourth graders were first, and I noticed some pairs were adding the scores of both players to see how high they could go. The next group of fourth graders did the same. They cheered as much for their opponent as they did for themselves. I decided to tell the rest of the classes they could score either way. A large majority of third graders, second graders, and kindergartners chose to go for one big number. They didn't even compare numbers with other pairs, and they all had a fantastic time.

It was a wonderful example of kids opting out, on their own, of a win-lose game and playing just for the fun of it. I set up a competition and they turned it into a partnership. Score one for those who believe cooperation should trump competition in school.

Monday, January 31, 2011

Equivalent Fractions

Almost a half century ago, my teacher taught a math lesson that I still remember today. We were learning how to add and subtract fractions, and she explained how to make equivalent fractions in order to have a common denominator. She started by asking us to multiply 1/2 by 1, 3/4 by 1, 5/6 by 1, etc. Then she wrote several names for 1 on the board - 2/2, 6/6, 10/10, etc. She explained that we could multiply the numerator and denominator by the same number without changing the value of the fraction because we were really multiplying by 1. I realize now that she was teaching us both the how and why in math, and I think that is why I still remember the lesson and have been able to manipulate fractions ever since.

Whenever I help students learn to make equivalent fractions, I always try to replicate my teacher's lesson. This week, I came across a great visual in a Wikiversity article and decided to make a laminated version on 4 x 6 cards, so students could use them with dry erase markers. I think this will help them understand what they are doing and why they are doing it.

February 2011 Last week I started making the cards reversible by changing from multiplication to division, so they can also be used to reduce fractions to lowest terms by dividing by a form of one. Students have had very positive responses to these cards.

Sunday, January 23, 2011

Adding Meaning to Mean

I had the opportunity to observe a 6th grade math teacher add relevance and meaning to the study of mean, median, mode, and range. She compiled formative assessment data from her four math sections to determine specific concepts to focus on prior to the summative assessment. She used the results to design further instruction, but she also presented and discussed the results with the classes, which helped them better understand the meaning and uses of what they were learning. It also allowed them to analyze the data. The assessment contained twelve questions - three for each concept. The data was organized by listing the percentage of students who correctly answered each question under the concept heading. The class discussed their interpretations of what they saw and were able to accurately infer which questions in each section resulted in the lower scores. For example, they decided that the question involving median that had the most incorrect answers involved an even quantity of numbers in the list. Following the lesson, the teacher displayed business websites containing statistical data and talked about how they use statistics to make business decisions.

This was one great way to answer the most often asked question in math classes - "When am I ever going to use this stuff outside of school?"

Tuesday, January 18, 2011

Playing with Words and Numbers

I got to teach 2nd grade today and enjoyed watching one of the Word Journey groups type the words into Wordle - - and print them out. They loved it and enjoyed looking at those created by their classmates. Practicing spelling words can be fun.

Their feedback during my math lesson also gave me something to smile about. We were working on double digit addition. I love helping kids use what they know to figure out what they don't, so I used think-alouds to talk through the relationships between numbers. When I needed to add 7+6, I said that I know that 6+6=12, and 7 is one more than 6, so 7+6 must be one more than 12. Pretty soon I had the class talking through the problems with me. I felt good about the lesson and even better when one girl said, "I like the way you teach math", and several classmates agreed. Before becoming a teacher, I enjoyed playing with numbers on the job for 30 years. It is really exciting when I see kids enjoying playing with numbers too.

Saturday, January 15, 2011

Tell Me and I Forget ... Involve Me and I Understand

As a 3rd grade associate this week, I watched a veteran teacher guide students to an understanding of the use of a coordinate grid. Prior to this lesson, they had used a map with letters on the horizontal axis and numbers on the vertical axis to locate cities. This time numbers were on both axes. The class discussed if this would cause any problems. About half thought it might. One student plotted the point (3,2). The teacher asked if anyone had a different location in mind. Another student placed a point in another location. The class discussed the problem. Next, the teacher explained the rule that solves the problem and results in each set of coordinates identifying only one point. I thoroughly enjoyed watching the students discuss their ideas with their shoulder partners and with the whole class. This lesson is an excellent example of involving kids in their learning.

Thinking About Mummies

After an extended weekend due to two snow days, I was back in the classroom. One of my favorite activities this year is volunteering with the homework help group at the middle school. The one-to-one aspect allows me to help students really understand the why, so they can consistently implement the how, especially in math. This week, I also had the opportunity to observe how peer discussion can get the brain's wheels spinning.

I was sitting with a 5th and 6th grader, and the 6th grader was working with the BBC's Mummy Maker at Discovery Kids - . He commented about using a hook to stir up the brains, and the 5th grader was hooked as well. The next step was to remove vital organs. The instructions said that one must remain to help the person reach the afterlife. A clue revealed that the conscience lived in that organ. The boys expressed their thoughts in conversation. The 5th grader said he thought it was the heart because it is the source of feelings about right and wrong. The 6th grader added that it might be the stomach because of the feeling you get in your stomach when you do something wrong. He opted to try the stomach and discovered heart was the correct answer. I was again reminded of the importance of letting students learn by sharing thoughts with peers.

Wednesday, January 5, 2011

It Doesn't Get Any Better Than This!

     I don't think I've felt this good since I found out from a group of third graders that I rock the world. This week I have had three amazing days in classrooms. The first two were in 7th grade math. On Monday, I got the impression that some of the students couldn't do the math because they didn't believe they could, so Tuesday morning I wrote one of my favorite sayings on the board. If you think you can't, you're right! I told them that the first step in accomplishing anything is to believe you can do it; if you don't believe in yourself, you will fail. I told them I believed I could become a distance runner at age 40, and I did. I believed I could take my first college class at age 50 and become a teacher, and I did. I believed I could run a half-marathon at age 55, and I did. I urged them to look at things that challenged them and say, "I think I can!" I don't know the exact words I heard from one girl in the back because I was so overwhelmed by the meaning. She said something to the effect that starting today she was going to have a whole new attitude about life and she was adopting my saying as her motto. Wow! I have had a lot of positive responses to my Can Do spiel but never one that touched me so profoundly.
     Today I got a call to teach in a 3rd grade class where I have subbed before. Before school started, I walked out of the classroom and down the hall. I said hello to a boy I passed, and he responded, "Hello, Mrs. Downing." He continued down the hall, and I thought I heard my name again, so I turned around. He was excitedly telling his classmates, "We get to have Mrs. Downing today!" I knew right then it would be a very good day. It was.
     I don't have my own classroom yet, so until I do I am going to continue enjoying every day I have the opportunity to positively impact the lives of children. Teachers can change the world, one student at a time.

Look at This!

     Yesterday I experienced another example of students taking pride in their products when the audience is greatly expanded by publishing their work on the web. (See The Purpose and the Audience.) I was volunteering at the after-school homework program when a student opened the school's website on his computer. I heard him exclaim, "Look at this!" He turned his screen toward me and, pointing to a video, he excitedly explained that it was a math project his classmates made. He hadn't even worked on the published project, yet he was thrilled that it was on the web for all to see.