Thursday, September 8, 2016

Elementary Content Specialists: Is it Time?

Over the last 10+ years, I've had the privilege of working with hundreds of K-5 teachers in math professional development. The longer I work, the more frequently a question comes to mind:

Is it time that elementary teachers specialize in subject content?


Sheer Amount of Content
New standards change and increase the amount of math and language arts standards that each grade level teacher must know. With new standards comes new curricula, hopefully of high quality (check yours on EdReports) and hopefully with plenty of supportive PD.

For some, new curricula (or good curricula!) is not purchased and teachers piece together their own. (For a recent statement on this trend, read a reflection by NCTM President, Matt Larson.) If one is to learn even one new set of content area standards--and a new, related curricula--it takes time to develop proficiency. Add in another major content area (or 2, 3, 4...) and you have a recipe for overwhelmed teachers.

Depth of Content
Over the last 10+ years, I watched many standards appear to come down a grade level (example: what was in 4th is now in 3rd.) Then it seemed to happen again. Much of what we learned when we were kids is now taught 1-2 grade levels earlier. I routinely work with 4th and 5th grade teachers to learn how to teach math content that they first encountered when they were in middle school. 

Please Note: I fully believe teachers capable of learning the mathematics. (See Jo Boaler's work.) But they don't always have the TIME to learn it.

If K-5 elementary teachers specialized, they could focus, teaching one subject (or a major content subject and related subjects) twice a day. For example, a language arts specialist might team up with a math content specialist, sharing a class that rotates between two locations. Each could add related subjects like social studies or science, or another specialist could take a third area. Teachers could focus professional development time, standards, and curricula on a single subject.

Students would benefit from teachers who not only understand the content, but know how to teach it well. Teachers would be trained in developmental stages in a given content area, allowing them to deeply reflect on current student understanding and what individuals need in order to advance.

Completely self-contained classrooms and all the related advantages would disappear. However, if paired well, teachers could work together to establish common expectations and similar classroom climates. Students would still only see two teachers per day. And with today's emphasis on constant change of focus (thinking of video games and t.v....always quickly changing), perhaps it would even help to keep student attention?

What do you think? Are you or your teachers overwhelmed? How can we help teachers to learn all they must know to be successful?

Friday, July 22, 2016

Tears to Cheers! (Perseverance!)

If you were a fly on the wall in my house this morning, you would have heard this coming from my 10yo during math...

[sniff, sniff]

[sniff, sniff]

[sniff, sniff]

Background: At the beginning of the summer, I noticed that my son struggled to solve multi-digit subtraction problems. So, for the last several weeks, we've been adding strategies to his toolbox. He's caught on fast. Yesterday, as we discussed one strategy he grinned and said that this made sense!

Fast forward to today.

He's working on a problem. I ask him to call me over when he's ready to talk about it. When he does, my heart sinks. (More on that later!) I immediately see that he attempted to apply the strategy from yesterday. But it didn't work. Basically, he tried to make the strategy into a series of rules. He tried to follow "the rules." He forgot "the rules." And when it didn't work, he just accepted whatever answer came from the procedure. He forgot that math MAKES SENSE.

My gut reaction? I wanted to immediately jump in and show him where he went wrong. I sorta did. That was sorta terrible on my part. Luckily, it was short-lived and he went back to working on the problem on his own. He struggled. And struggled. And struggled.

[sniff, sniff]

Then he got it. Sorta.

At this point, you might say his knowledge was shaky. He did figure out the problem. He did use his strategy. But it was very unclear as to whether he really "got it."

So he continued to the next problem. Wherein my heart sank again. (Yeah, yeah. More on that later...) He used the same strategy in the same, wrong way. This time, I asked him to talk me through what he'd done on his visual model. His words made sense, but he couldn't show it on the visual model. I repeated what he'd said, but explained that I couldn't see that on his model. He looked at the model, obviously perplexed. He KNEW that it didn't match what he was saying. It was dawning on him that this DIDN'T MAKE SENSE! I asked if he could revise his model to represent what he said and told him to call me back when he was ready.

[sniff, sniff]

[sniff, sniff]

Now here, folks, is where I'd usually jump in with two never-let-my-child-suffer feet. I mean the kid is CRYING! (Or at least sniffling!) Who wants to see their kid in agony?

I wanted to intervene. I wanted to get-him-back-on-the-happy-track. Maybe ask a good (pointed!) question. Or suggest an avenue that might lead him to discover where he'd gone wrong. I wanted bunnies. Pink ponies. And rainbows.

But today, with every-ounce-of-my-being, I kept my mouth shut.

Guys, it was SO HARD!

Time passed. It felt like FOREVER. I think it must have been at least, what, TEN MINUTES! An eternity!

[sniff, sniff]

[sniff, sniff]  

But then, tentatively, he calls me over. He talks me through his visual. He explains why it makes sense. And it DOES! It REALLY DOES!

So here's what I said to him: "You know what was cool today? You persevered. Do you know what persevere means? [no] It means you kept going. You stuck with it. And look what you did! You kept going until it MADE SENSE." I cheered!

Herein, the child grins. Tears falling, while grinning.

Guys, he GOT IT! And I ALMOST, with my Momma-doesn't-want-to-see-you-suffer-mentality, TOOK THAT AWAY FROM HIM!

So back to my heart sinking...

When I see a child make a mistake, my natural reaction is to cringe. It's in me. I admit it. I want to FIX IT! But we know that mistakes grow the brain. We have to let kids make mistakes. Toss, turn, and roll in their mistakes. Until they discover that MATH MAKES SENSE.

You know what would have happened if I'd intervened today: a few less tears and a whole lot less learning.

Obviously, I don't want math to hurt. Shoot, no one wants to see kids in pain. But sometimes there is a little pain in perseverance. And if we never let them persevere? Well you know what would happen then... (Eeesh.)

Our job? We must also letting them struggle. We must allow them to bask in moments of disequilibrium. For it's in those moments, those oh-how-I-want-to-save-my-child-moments that real learning happens.


Update p.s.
The first week of school this child came home and related an assessment he'd just taken in math. "I used that strategy, Mom! It worked!"

Wednesday, June 29, 2016

Chalk, Animals, & Tile: Summertime Fun with Area & Perimeter

On my summer reading list: Jo Boaler's Mathematical Mindsets: Unleashing Students' Potential through Creative Math, Inspiring Messages and Innovative Teaching.

Yesterday, I read her challenge to teach math content using a question rather than following a procedure. (p. 78) She offers this suggestion:
"Instead of asking students to find the area of a 12 by 4 rectangle, ask them how many rectangles they can find with an area of 24."
And voila...we have a summer afternoon activity for a slightly bored child whose siblings are all at camp!

I posed the situation this way...

A farmer has pens that each contain an area of 24 square units. The farmer wants to know how many different rectangular pens he can make.

Then we got out the chalk, tile, and animals.  The first pen he built, 4 x 6, was for the cows:

I asked what other pens fit the criteria of 24 square units. He thought for a bit. "6 x 4?" We agreed that since this had the same dimensions, we wouldn't built it. He soon thought of another: 3 x 8.

He asked if this farm could have penguins. Sure, why not? The next pen took a bit more thought, but after some think time he built a 2 x 12 for the pigs.

I asked if this included all the possibilities for pens with an area of 24. He wasn't sure, so we made a list. (Ideally, he would have played around with 24 tile, exploring how many different rectangles could be made, but the farmer was getting tired.)

Although he didn't want to make the 1 x 24, he talked about how LONGGGG that one would be.

Then I asked another question:

The farmer needs to buy fencing for each of the pens. One section of fence covers one side of a tile. Which pen has the cheapest fencing? Which has the most expensive?

At first he predicted that the "biggest" pen would have the most fence. (At this point, in his mind, the 4x6 pen was "biggest." After all, it did contain the cows! Later on, I asked about pen size and he was able to say that they are all the same.)

His findings:
4x6 area = 20 sections of fence
3x8 area = 22 sections of fence
2x12 area = 28 sections of fence
1x24 area = 50 sections of fence

His eyes got really big when he heard it would take 50 sections of fence. He remarked that the chunkier pens have less fence because more of the edges are in the middle. I asked if he knew another name for the "fence" or the distance around. He named it perimeter.

Thanks, Jo, for a great summertime exploration!

p.s. Try making your own farms with pens of 36, 100, or other areas!

Tuesday, June 28, 2016

Student Notebook Strategy Posters

This quick, easy idea is one that works well for student notebooks. When I work with students on strategies, I often create a classroom anchor chart for the wall. I like to record the name of the student who used the strategy, along with a title that clearly describes the strategy. Kids love to see their own names in print and when they're asked to name what happens in the strategy, they often delve into rich mathematical thinking and discussion to define exactly what it is that they've done.

To give students greater ownership in the process, I invite students to make their own posters to go in their math notebooks or journals. I give each child an 11" x 17" paper, folded near (but not on) the halfway mark and 3-hole punched on the left. This way, the poster can be folded and added to their math notebooks as permanent reference.

Today, we made posters for Addition Strategies. If you click on the photos, you can see that we depict and name a variety of strategies. You'll also notice that this exercise is appealing to the artists in the crowd.

Monday, June 27, 2016

Give (and Take!) Me a Great Addition Strategy

Exciting work continues in our summer math sessions!

Last week, my 10yo son and I started exploring strategies to help with multi-digit addition fluency. The "Give and Take"* strategy has given us inspiration and taken away some of our math anxiety. Here's how it works...

Let's say you're asked to add two, somewhat unfriendly, numbers.

97 + 78

Yuck. Not a great combination.

But what if you could do a little give-and-take to make it easier?

97 + 78 = 97 + (3 + 75) = (97 + 3) + 75 = 100 + 75 = 175

Which would you rather solve?

97 + 78


100 + 75

The consensus was pretty clear around here!

How about:

443 + 289

What if we "take" 11 from 443 (443 - 11 = 432) and "give" it to 289 (289 + 11 = 300)? Is it easier to now add 432 + 300?

 My 10yo explains the strategy in his math journal, in the photos you see here.

So "witch" would you rather add? :)

After journaling, to solidify the concept, he made up his own problem:

270 + 665

He took/gave 30:

300 + 635 = 935

And today, he applied it to a story problem where he had to add 275 + 168. He took/gave 25 to end up with 300 + 143. He bubbled with excitement ("MOM!!!!!"), telling me how great the give/take strategy works!

I hope this gives you a little inspiration to take back to class!

P.S. This also works well with decimals!

*The Bridges Curriculum calls this the "Give and Take" strategy.

Wednesday, June 22, 2016

Multiplication Strategies: x2, x4, x8

Over the past week, my son and I have made tremendous progress with multiplication. (Intro post here.) Each day, we add to his fluency toolbox by looking at specific strategies. The (related) strategies for 2s, 4s, and 8s, have been especially fruitful. Let's look at why...

2s...Dare to DOUBLE!
Twos are easy-peasy. Just a matter of doubling. We can see an example in this array.

If you multiply something by 2, you only need to double. Instead of 1 group of 6, you have 2 groups of 6; you just double 6.

In Bridges in Mathematics, the strategy for multiplying by 4s is called Double-Double. It's easy to see why.

We already doubled when we multiplied by 2. To go from 2x a number to 4x a number, we double. So we double, then double again.

6 is doubled to 12 (x2)
12 is doubled to 24 (x4)

I bet you can guess what's coming next!

We call the strategy for 8s Double-Double-Double.
For 8x, we double 3 times:

6 is doubled to 12 (x2)
12 is doubled to 24 (x4)
24 is doubled to 48 (x8)

Can you see it in the model?

This is not a multiplication "trick" but rather a strategy with meaning behind it. Children need to see the visual model and understand what "Double-Double-Double" means. Once they understand the concept, they can apply it in wonderful ways.

I asked my son (just developing fluency with single digit multiplication) to consider these problems.

8 x 15 = ?

He doubled 15 and got 30. He doubled 30 (60). And doubled once more to get 120. So 8 x 15 = 120.

4 x 13 = ?

Double 13 to get 26. Double 26 to get 52. (Of course he then wanted to keep going and figure out 8 x 13. Double 52 and get 104!)

25 x 8 = ?

50, 100, 200, done! This problem was also a great opportunity to talk about another strategy. Do you know what it is? Leave your ideas in the comments below to start a RICH exchange.

Hope you're having a double dose of summer fun!

Little Girl Graphic from:
Number Frames (free app) from:

Friday, June 17, 2016

Multiplication Fluency: Summer Practice

Ready for a start-to-summer quiz?
Question: What happens when the math coach's child begins the summer by taking a multiplication fluency assessment in which he answers 20 problems in 4.5 minutes when the fluency guideline is 20 problems in 1 minute?

Answer: Summer math!  (Don't you wish you lived at my house?)
In case anyone else is in a similar predicament, here are a few resources to get you started...

First, "fluency" does not equate memorization. If you're interested in the difference between "by memory" and "memorization," check out this article. Fluency means accurate, efficient, and flexible mathematical thinking. Think about reading fluency. A fluent reader is not just fast. 120 words-per-mind counts for nothing without comprehension. Fluent readers AND mathematicians are  accurate, efficient, and flexible.

Although every child needs to master all three areas, he may demonstrate challenges in one area over the others. In our case, flexibility is an issue. Although my child knows some strategies for working with multiplication, it doesn't appear to be something that's been emphasized in his education. To that end, we are working to increase his strategy toolbox.

I pulled the Multiplication & Division Discussion Cards from Opening Eyes to Mathematics. (Cards are located on pp. 32-35 in this pdf, free from The Math Learning Center.) We flip through several cards a day and talk about what strategies could be used to solve a problem. For example:

What multiplication expression is represented here? (8 x 8)

How could we look at pieces of this array to help us solve the problem? Maybe we could see it as two parts: 8 x 5 and 8 x 3.

So 8 x 8 = (8 x 5) + (8 x 3) = 40 + 24 = 64

Or maybe you see it as two groups of 4 x 8:

What about this one? Could you use what you already know about 10 x 7 to help you figure out
9 x 7?

Strategies become critical when you get to larger multiplication problems, so this is definitely something we want to work on now.

Although fluency doesn't equate speed, it is generally expected that students be able to complete 20 problems in 1 minute to meet fluency standards. With the strategies in hand, I plan to assess his progress using the free online program, Xtra Math. I would not use this in isolation as I don't want to overemphasize speed, but when combined with visual models and strategies, it's a reasonable way for  both of us to track his progress.

Tuesday, March 15, 2016

Standard Algorithm For Subtraction: Sometimes Yes, Sometimes NO!!!!

Introductory note: For the past year, I've been working as a K-5 Math Coach. Not surprisingly, I have learned a lot. I hope the following blog post expresses just a bit of the wonder of the past year...

I developed a new appreciation for the power of strategies taught in Bridges when a third grader approached me for help on a worksheet he received in his (non-Bridges) classroom. The “Zero-Concept” worksheet included 36 problems with multi-digit subtraction, intended for practice with borrowing across zeros, solely using the standard algorithm.

Although this was the intent (and yes, the way many of us were taught!), it quickly became obvious that several other strategies might produce more efficient results. The third grade standard 3.NBT.2 specifically calls for this:

“Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.”

One of the key words, “algorithms” is plural for a reason. We want students to develop fluency defined by accuracy, efficiency, and flexibility. In this standard, students demonstrate fluency using multiple, flexible strategies--strategies selected because of their strength with a particular set of numbers.

The very first problem, 420-115, seemed a good candidate for Partial Place Value Splitting, one of several strategies explored in Bridges.

The student could mentally solve the problem using this strategy; he was surprised by how easy it was to break the subtrahend into manageable pieces and then subtract.

Another problem, 200-189, seemed ideal for Finding the Difference.

Again, once he understood the strategy, the problem was easy to solve mentally. In comparison, the standard algorithm was very complex and inefficient, leaving a lot of room for error.

The Removal Strategy (using a Number Line) worked for 500-333. He also noted that this could be done mentally using Partial Place Value Splitting, taking away 300, then 30, then 3.

Once again, borrowing across multiple zeros seemed unnecessarily complex with a high possibility of error.

A problem like 703-187 became a prime candidate for Constant Difference. Here it's illustrated on a number line:

He agreed that it was far easier to solve 716-200 than 703-187. And it's so simple to get there. Just add 13 to both the minuend and the subtrahend.

Looking over the worksheet we noted that while the standard algorithm might be an efficient method for a handful of problems, for the majority it was not. But perhaps the most surprising to my student: the number of problems that could be completely solved with mental math, using one of the above strategies.

If we think of fluency in terms of accurate, efficient, and flexible thinking, students are best served when they have a variety of strategies from which to choose. By the time we were done, my young friend heartily agreed!

Saturday, August 29, 2015

Fly on the Math Teacher's Wall: Personalized Math Notebook Covers

Here's an idea for getting your math year off to a great start...

When I posted about Math Journals & Notebooks, I mentioned that I loved the idea of having children make covers relating math to themselves as Courtney shares at A Middle School Survival Guide. Ideally, I'd begin the lesson by reading aloud a book that relates math to everyday life. (I mention several suggestions here.)

When I first considered what might go on a cover, I didn't have a lot of ideas. I just thought about # of siblings or children, year of birth or age, height or weight, etc. But the more I considered, the more ideas multiplied! I'd definitely want to do this as a brainstorming activity with students rather than giving them a list. See what your collective brain energy can come up with! How is math related to our daily life? Here are some of the things we thought of:
  • time you wake up/go to sleep
  • # of favorite ____________ (sports, colors, hobbies)
  • # of years _____________ (teaching, being a student, playing an instrument or sport)
  • time each day that you ___________ (exercise, go to school, watch tv, read, play video games)
  • # of _____________ that you own (pets, video games, books)
  • # of years until you (finish school, turn 21, want to get married or have kids)
  • cost of your favorite (restaurant meal, soda, candy bar)
  • amount you spent per week on (lunch, snacks, coffee)
The possibilities are endless!

These covers then become a fabulous jumping-off point for PROBLEM SOLVING.

After students finish their covers, have them generate several problems on 3x5" notecards that use the information they created. For example, on my cover, I posted the following:

I went ahead and wrote my problem on the cover itself, but would have students write on cards. My question, "How many hours do I sleep each night? Each week?" could then be posed to other students. In the classroom, I could put my cover under the document camera and ask students to answer the question posed on my card(s). They could then share a variety of strategies for solving the problem. In a homeschool setting, children could write problems for siblings or parents to solve. Problems could be written at a wide variety of levels, making them grade and age appropriate.

At the Northwest Math Conference I went to a workshop entitled, "Taking the Numb Out of Numbers" by Don Fraser (Ontario, Canada). He began by telling the group of 30 of us, "Did you know that in a group of 23 or 24 there is a 50% chance that at least two people in the group will have the same birthday?" He then gave us a graph showing us the probability of sharing the same birthday in groups of varying sizes. In a group our size--30 people--the likelihood was 70%. We graphed the days/months for birthdays in the room. Interestingly enough, none of us shared the same birthday...we were in the 30%. After looking at the data, Don asked us to come up with problem solving questions--real life questions--based on the information we'd collected. It was amazing to see how many questions we could generate, at all different levels of mathematical knowledge and proficiency.

Don encouraged us to begin each day by reading a "story" and having kids make up a question/word problem. Going back to the math notebook covers, imagine the possibilities if you put ONE child's notebook cover up each day and asked kids to generate questions from the "stories" found there. The problem solving possibilities are endless!

Do your students make personalized math notebook covers? What interesting stats have they included? Comment below with your stories and then visit Mrs. Balius and read what she has to say about setting up daily math routines!!! :)

Saturday, February 21, 2015

Fly on the Math Teacher's Wall: Squashing Fraction Misconceptions

The irony is not lost on me. Fractions, the math concept I most struggled with in elementary school, is now one of my favorites to teach. In this Blog Hop, my math blogging friends and I will be exploring fraction misconceptions. Here we go...

After years of operating with whole numbers, it's new territory to see fractions and understand the what numerator and denominator mean. What do each of those numbers really mean?

In this example, we'll look at an egg carton. First, we'll consider what the whole this case the whole is the entire egg carton.

Look at the following examples and ask yourself:

1. What does the string show?
2. What do the tile show?


The string shows how many parts our whole is divided into (our denominator) and the tile show how many of those parts have been filled (our numerator.)

A common misconception results when students look at the pieces in the model without taking the meaning of numerator/denominator into consideration. For example, in the first photo above, a student might say that they have 6 pieces, so it's 6/2. Most students, however, can readily tell you that the top example shows one-half, so a bit of probing (Where do you see the 1 in 1/2? Where do you see the 2 in 1/2?) helps to reestablish context.

In a similar example, I've heard students struggle with the question, "What fraction of a dollar is a nickel?"

Many students will answer "one-fifth" because they are thinking of 5 cents; if it has a 5 in it, it must be 1/5.

I like to pull out Money Value Pieces and again revisit the concept of numerator and denominator. We first talk about what our "whole" is: 100 cents.

I might ask, "What does 1/5 look like on our model?" Since we've explored numerator and denominator, they know that the whole would be broken into five portions:

It doesn't take long for someone to say, "One-fifth of a dollar is 20 cents!" (They can check this using the dime piece, a ten strip.) Then, using the nickel model, they explore how many pieces it would take to cover the dollar. "Twenty! So a nickel is 1/20 of a dollar!"

Students need many opportunities to explore the concepts of numerator and denominator using a variety of manipulatives and visual models. (More love2learn2day examples here.) I ask them to record their thinking in a variety of venues: math notebooks, class anchor charts, and video productions. In this ScreenChomp example, you'll hear a pair of students explain the meaning of numerator and denominator; notice that they use more than one visual to explain their thinking.

To continue on the Fraction Misconceptions Blog Hop, please visit my friend Jamie at Miss Math Dork!

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