Musings and Strategies From the Teachers Next Door

Posts tagged ‘math’

The Struggle is Good

If you haven’t yet discovered the Open Middle site, may I introduce you.  Welcome to the world of challenging, thought provoking problems.  Dan Meyer introduced this concept, creating problems that have a definite beginning and end, but have an “open middle” that creates the space for students to solve problems multiple ways.  The open middle is what I love about math, why math is so much more than learning a set strategy.  It allows for understanding based on how you think and how you understand numbers.  I have seen kids come up with strategies that I would have never thought of…oh, how I love those conversations.  For that reason, this site is a gift.

This past week, I used a fraction problem with my 3rd graders.  Here is it: (source:  Open Middle)

fraction_number_line from openmiddle

Students were asked to place 3/4 on this number line.

While I anticipated my students to be a little stumped by the fact that 3/3 would not be at the end of the line, I had no idea the level of difficulty this would pose.  I wish I had recorded all the great discussion that ensued.

They justified their position, even tried to form a coup and overthrow me, when I challenged them placing 3/3 at the end of the line.  Guiding students through questioning is an art that I am practicing daily and there were moments that I wanted to just break down and “teach.”  But then it happened….my favorite sound in the world…”Ohhh, I see it.”  First there was one, one brave soul that walked over to the dark side and placed 3/3 on the number line correctly.  She even grabbed a ruler, measured the distance from 0 to 1/3, and proceed to place 2/3 and 3/3 exactly in the right place.  As soon as she began to measure, the kids got so excited as they SAW it.

At this point, I thought it was smooth sailing.  Next came the true task of placing 3/4, so the first thing they wanted to do was place 1/2.  Perfect.  I loved how they were using benchmark fractions, and I was sure that their misconceptions were suddenly gone.  Well….here is how is actually went down.1:2

The black line is where one student placed 1/2 and the red is where another tried to emphasize and defend that 1/2 would indeed be placed there.  I used all the skills I learned in my acting class and asked them to talk about what they knew was true.  The told me that 3/3 was the same as one whole.  We discussed and discussed.  Finally, they did come to realize where 1/2 should be and eventually 3/4, but not without a clear struggle.  These are not kids I teach everyday, but they are bright students with a wonderful teacher.  I don’t know if it was because they have only looked at fraction number lines with ending points of 0 and 1, but this experience reiterated the importance of exposing students to problems in multiple ways.  If they cannot transfer the information they learned from one type of problem to the next, then their understanding is basic and needs to be developed so much further.  I’m reading everything I can get my hands on…I want to be the best teacher I can be…I want to challenge, push, equip and help students make  a rich web of connections.

Feel free to pass on any wisdom you have.  It takes a village…

It’s GAME time!

What better way to celebrate a weekend and family than to engage in a friendly battle of Wits and Wagers. Have you played it? It is a completely awesome game that challenges players to estimate answers to different questions. Then, after everyone has made an estimate, you have a chance to place your bets on the estimate that seems closest to the answer. The thing that I love about this game (aside from the absolute awesomeness of everyone putting into practice their powers of number sense) is that everyone is on equal ground. Talk about a low entry point. That makes it perfect for the whole family! Even if the youngest family member is 5. Or 6 in my case. And as in my case, those young ones can win! Big time! I love the togetherness that games within families create and I love them even more when we sharpen our math skills while we have fun.

So that night of family fun and this post by GFletchy got me really thinking about the mathematical value of games. I love that Graham went into a classroom and literally turned the teaching upside down! I don’t teach lower grades, so I don’t have one of those 0-99 charts, but if I did, you can bet that I would turn it upside down come Monday. Milton Bradley, I bet, was onto this technique. Take a look at the game boards:chutes-and-ladders-1 orenstein_Candyland-1980s

Even Candy Land, though it has no numbers, kids start counting from the bottom. They win when they reach the top. They make gains as they move towards the top. Maybe it’s time to break these goodies out in class!

Kudos to Graham for giving this a try! I see the look and use of the 0-99 chart doing a 180!

But what are the KEYWORDS?




My team collab teacher was absent the other day. In her place was an older man whom I had never met. He was quite friendly and immediately introduced himself and was very quick to ensure that I understood that he was a retired teacher. Had been a teacher for 30 years and that he was there to help. Awesome! I was truly excited to have such a seasoned educator come in and contribute to the class.

At the time of the initial meeting, the students were in rotation (their specials classes). As they returned and entered the classroom, they fell right into their bell work. On this particular day, there was a story problem dealing with decimals that they needed to work through. They have been instructed on the procedure for these questions, so all of them are very familiar with the overly repeated chant of, “We make sense of problems and persevere in solving them!”

After some time of the students having been busily engaged the entire time, knowing that they are never finished and there is always another way to solve a problem, we began to discuss our solutions. This is the part of class that I love because we listen to one another, we are able to ask questions of one another or comment on work that we really like. It is our practice time of SMP #3 (construct viable arguments and critique the reasoning of others).

As we are discussing, the well seasoned substitute cuts in and asks, “But what are the KEYWORDS that told you to solve the problem that way?”


All eyes were on me. Pleading with me to bail them out. Questioning why we were looking for keywords. Asking why we don’t have those words posted? SCREAMING to me…KEYWORDS?! You never taught us KEYWORDS!!!

I just smiled. And restated his question in more familiar terms, which of course redirected them to explain how they made sense of the problem.

I am certain the substitute was aggravated that I wouldn’t just make it simple for them and teach them the keywords to look for. Well….I will never again teach keywords. I did in the past, and it didn’t take long to realize that keywords get confusing.  Keywords, as well as “rules”, often have expiration dates.

Here is a great article that lists 13 of these rules.  Interestingly enough,  keywords made the list…  number 2, actually.

What rules should be added to the list?

It’s True! Math Makes Sense!

I had been losing sleep. I had been wracking my brain. I kept coming up with NOTHING.

Even the shower, which is normally where I do my very best brainstorming, was not helping me with this ever perplexing problem.

How in the WORLD am I going to help my students understand division??

Oh, I had tried all the strategies I could think of. I had tried manipulatives, acting it out, actually having the students divide anything I could think of to try to help them “get it”. And they did. For about a minute or until they spotted a double digit divisor and then it was as if the concept of dividing had morphed into a calculus problem. I had a problem. And it was pretty huge. We work a lot with division in fifth grade.

And then….a life line. I hadn’t asked. It just appeared as if out of nowhere and at EXACTLY the right moment. Graham Fletcher came through with this post. What I love about this post is that now I have been introduced to Joe Schwartz and Nicora Placa. Two more awesome educators who share some really awesome ideas, strategies, and experiences aimed at helping elementary students grow into great thinkers, problem solvers, DIVIDERS. I’m pretty sure that isn’t a word, but who cares?? Do you know what they shared? This treasure!!!!

My small groups this week underwent a division intervention. As we went through the lesson, and we’re folding strips of paper, making quotients, and subtracting, inevitably, in EVERY group I heard a collective, “AAAAHHHH!” It was a gasp. The room got a little brighter. They UNDERSTOOD!!! One of my students said, “Oh. So THAT is why you subtract!”

One of my students who has struggled to understand math probably most of her elementary years, was in one of my groups. She diligently went to work. She was very quiet the first day, not sharing much, but always busy. She worked hard. The next day, as we went to groups, I met with those students again. This day was quite different from the previous day. The girl who had struggled, was now the teacher. She took over the group and taught division as if she had been born dividing.

Now THAT is an awesome week! I love it when my kids take over and teach one another! I love it when educators come together and share! So a huge shout of thanks goes to these awesome collaborators. Math really DOES make sense!


TED Talk Tuesday “Math Makeover”

Math Minded

Do you have a go to place where you look for inspiration and innovation?  For us, it’s TED Talks so from now on, Tuesdays are TED Talk Tuesday. Here is our pick of the week.

In this talk, Dan Meyer clearly illustrates the difference between poor math instruction and math instruction that engages students in deep thinking while sharing how he develops real world questions and problem posing.

He states the obvious, that “Math reasoning is hard to teach.  The way we teach it in the U.S. all but ensures they won’t get it.”  We know that, we see it in classrooms often.  Are you wondering how successful you are?

Five Signs You’re Doing It Wrong

1.  Lack of Initiative – Your students are not self starters.

2. Lack of Perseverance – Your students give up, they lack grit.  If the answer is not immediately clear, they expect you to hold their hand and walk them through it.

3. Lack of Retention– You find yourself teaching, and reteaching, and reteaching.  It just doesn’t hold.

4. Aversion to Word Problems-They hate word problems.

5. Eagerness for a Formula…to solve said hated word problems.

We’re guessing you are reading this because you value education and you want to get it right.  You desire your students to excel and your instruction to create depth and understanding in those whose minds you’re shaping.  So why do educators find it necessary to go back to the old multiple choice stand by?  We’ve witnessed students answering multiple choice questions like nobody’s business, but change one nuance and they are lost.

Meyer makes a great point.   Have you ever faced a problem in which you were given all the information necessary to solve it… in advance? Or better yet been given several options to choose from…wisely? We haven’t either.

We were so excited when we saw this talk.  He breaks the development of good questioning down, starting with a picture and getting kids talking, bringing in math, as the kids see the need for finding ways to talk about what they see, and then having the kids pose questions that need answering.

The beauty of starting with a picture, or something that all kids know about, is that it levels the playing field.  One of his examples deals with filling a water tank.  He shows it and then asks kids how long it would take to fill it.  Kids see the picture and watch a video of it being filled.  Now, when they talk about filling the tank, every kid is on equal footing, even those kids that have decided they don’t need to talk about math, because someone else has the answer – every kid has filled something, so they have experience upon which they can draw.  He then takes guesses and writes them down.  He found that in the course of one semester, he could put a new problem on the board, something they have never see,n and the kids would have conversations about it.

The takeaways:

1.  Use multimedia in your class  -as often as you can, the best quality you can.

2.  Encourage student intuition – to level the playing field for all students.

3.  Ask the shortest question you can – and then let the other, more specific questions develop through conversation.

4.  Let students build the problem.

5. Be less helpful – knowing it will develop patient problem solvers.

“It is an amazing time to be a math teacher.”  We have the tools at our fingertips.

For more insight, follow his blog.


April and Diane




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