'Twas the Night Before a New Math Curriculum Review

'Twas the night before reviewing a new math curriculum and many thoughts are running through my head. I'm worried. I'm worried a lot because the decision that's made will have a significant impact on student learning.

Many math textbooks are improving and are more aligned to standards, which is a good thing. But, in my opinion, they are not there yet. I've found that most teachers follow the textbook page by page, problem by problem, so what you see in the textbook is what you'll probably get. (Unless the kids don't get it. Then we have a bigger problem because the march through the textbook waits for no one.)

So what does there  look like?

• Less scaffolding - Let kids grapple with the math. They are way smarter than we often give them credit for, even (especially) our kindergarteners. Plus, math is so much more than procedures and steps to follow. It's a beautiful thing to listen to how kids think about math!
• Connections to Prior Knowledge - And by this I don't mean, "Yesterday, we did ..." Begin lessons with a problem that uncovers something important about the day's new learning. Think of it as an entry point into the new learning, something students can use to solve the day's task. 
• Open Ended Tasks - Again math is so much more than a set of steps to follow. Open Ended Tasks connect what students already know with new learning. They show the connectedness of math ideas, and they encourage discussions about student's thinking. The new learning is highlighted either by students or by the teacher, and then students can solve a couple of similar problems with a partner to practice the new learning.
• More problems with context - And the flip side, less naked problems. For example, what might addition and subtraction look like in real life? Have you ever looked at a kindergarten textbook? How many times do we count the bugs on leaves in real life?
• Suggestions for In-the-Moment Scaffolding - For example, if I have students in 2nd grade who are still struggling with adding single digit numbers, I would like ideas to support them within the context of the lesson on adding three-digit numbers. Maybe not every school or classroom needs this, but for the teachers that do, it would be a lifesaver.

There's so much more that I would like to add, like math work stations, review and enrichment worksheets, and even technology, but I'll save those thoughts for another day.

When is Too Soon, Too Soon?

The Importance of Teaching Why Before How

Have you ever given up on learning something because you thought, "I don't really need to know this?" You might know a short cut or learned something different that makes the learning obsolete, at least to you eyes.


When I was in college, I majored in airport management for a semester. As part of the course work, I had to pass ground school for a pilot's license. Did you know pilots have to know exactly how the engine of the airplane works? I could tell you what the carburetor does, how the spark plugs work, and how the fuel makes the engine run. I learned it and understood it because I needed to know it. This learning was a prerequisite for future learning.

In contrast, when preparing to get my driver's license, I learned some of this information in my driver's ed. class, but none of it stuck with me. The reality was that I had a short cut if my car wasn't running right. It was 1-800-CALLDAD. He always took my car to get fixed. I never had to think about it.

Learning math can be a lot like this, if we let it be. If we teach students short-cuts or tricks before they've developed conceptual understanding, they will not see a reason to learn the reasoning behind the shortcut. In reality, we're robbing our students of important learning opportunities.

For example, if we teach the standard algorithm for addition and subtraction before students have a full understanding of place value, many students will not see the purpose in learning or practicing with place value. You may have heard students say, "I already know this. Why do I have to do it this way?" (i.e., with place value drawings) Students cannot see the bigger picture of mathematics. They cannot see that place value will be important for rounding, estimating, and even multiplication and division. Without place value understanding of whole numbers, how will students ever make sense of decimals?

Learning the standard algorithm too early and without understanding, can also lead to misconceptions that interfere with future learning. Earlier this year, I completed subtraction running records with some of our fourth graders. While I knew that students may have unfinished learning, I found that several of the students I tested, had major misconceptions based on unfinished learning with place value. When asked to solve the problem, 14 - 9, students wrote the problem in vertical form in the air or with their fingers on the desk. They then proceeded to solve 4 - 9. When they realized that they couldn't subtract 4 - 9, they "went next door", crossed out the one, and then wrote the one next to the four to end up with the same problem 14 - 9. Only then were they able to solve it. (This is how they explained their thinking to me.) Some students even did this when they were asked to solve 12 - 11!

It was clear that the standard algorithm was taught too early. Students learned it rotely and followed it rotely. Place value understanding was missing, and, as a result, number sense was also not evident.

What if we put the textbook aside for a moment and taught our students to add and subtract using place value concepts. What if we followed a developmental progression, as suggested by Michael Battista, and provided the time and space for students to make the connection between manipulatives and drawings to the standard algorithm? As students build understanding first with connecting cubes, then with base ten blocks, and finally, with pictures, they uncover how the standard algorithm works and are able to explain it. They have the understanding they need to round numbers, estimate based on place value, and eventually see that place value can also include numbers less than one.

And maybe, they won't be the adults who hates math because it makes little sense.

Have you taught addition and subtraction this way? I'd love to hear your stories.

Golf - What's Math Got to do With It?

You may be wondering what Jordan Spieth has to do with math? The USGA has devoted a portion of
their site to information about STEM in golf. For example, did you know a golf ball can't go any further than 317 yards when hit at 120 miles per hour by the USGA's test robot? (www.USGA.org)

But that's not the connection that I'm going for today. I want to make a connection between Jordan's Spieth's last round during this year's British Open because this is the type of attitude that I want to cultivate in my students. As required by Math Practice 1, I want my students to persevere in solving problems.

Imagine you're going into the last round of the British Open leading by 3. The day doesn't start out that well for you though. You make some mistakes during the first three holes and end up tied for the lead. The "bad luck" continues and you eventually end up one behind your opponent. Just when you think it can't get any worse. You end up making a terrible shot to end up way out of position.

What would you do?  Would you complain about Murphy's Law and give up? Would you throw up you're hands in despair saying, "I don't get it"? Would your inner voice start telling you that you might as well give up because you'll never get it?

If this were a student in your class, what would he or she do?

I think we've all felt that way at some point, but, hopefully, as adults, we've learned to manage the self-doubt, quiet the negative inner voice, and dig our heels in to complete the task. This is what we need to model and discuss with our students. What does it mean to persevere when things get difficult? What are strategies that students can use when the going gets tough?

This is the conversation I want to have with Jordan Spieth. I want to know what he was thinking as he stood by the ball after his bad shot on the 13th hole and as he struggled to find a way to make the shot. Did he ever think, "I should just give up"? Or was he so focused on succeeding that he never even considered failing? I want to know how he considered his options and how he decided on the best path moving forward. And I want to know this so I can share it with my students. I want to be able to tell them that a 23 year old golfer persevered where others may have given up, and that they can do it to on a smaller scale every single day.

Here's the rest of Jordan Spieth's story. Instead of wallowing in self-doubt and cursing the rules of golf, Jordan made a nearly perfect shot to put him back into contention. For the remaining holes, he was on fire! Not only had he overcome any negative self-talk, he had sent it running with it's tail between its legs. He won the British Open that day. Skill clearly played a part, but his attitude, perseverance, and grit are what brought him back from the edge.

I plan on sharing this with my students. I want them to feel the energy that comes from refusing to fail in a small way, every day.



Disclaimer: I know there's probably a lot being written about Spieth's prior collapses in tournaments and even about his behavior as he looked for a solution on the 13th hole, but I watched the tournament through the eyes of an educator rather than those of an avid golf fan.

Putting an End to Math Confusion

Confusion leads to frustration which leads to despair and ultimately to the belief that failure is inevitable. Confusion is often the foundation upon which math anxiety grows.

A teacher came to me with a problem similar to this in the first grade textbook. She felt frustrated because it barely made sense to her, and she knew that she wasn't explaining it clearly to her students.

So what's going on here?

Students are solving a three number addition problem by decomposing a number to make an easier problem to solve. In this case, students are making a ten. This is something that people often do mentally because it is easier to add tens. But, as the strategy is presented for first graders, it is more than likely too difficult for them to understand. The textbook is forcing a strategy, and students are expected to use this strategy to solve the lesson's problems.

What's a better way to introduce this strategy to students?

Allow students to model the situation with different color counters and ten frames. Start with only two numbers, 7 and 5. Ask, "How might we solve this problem?" Students should be encouraged to explain different strategies that can be used to solve the problem. Highlight the make a ten strategy by moving three of the red counters onto the ten frame to make a ten, and have students explain that 7 + 5 = 10 + 2. Next show 7 + 5 + 6, as shown above. Repeat the same process to uncover the students' understanding of adding three numbers. The make a ten strategy can be encouraged and discussed. Don't be surprised if some students use 3 from the group of 5 to make 10, while other students use 3 from the group of 6 to make 10. Just be sure to ask them to explain their thinking!

Strategies should be introduced, discussed, revisited, and encouraged. They should not be forced upon students. Math is built upon a foundation of understanding. Teachers provide experiences for students to stretch their thinking and build new understandings. For many students, forced procedures and strategies interfere with meaning making and create confusion, frustration, and even math anxiety.

Creating Misconceptions in Math by Teaching Concepts Too Soon

Teaching math concepts too early can create mathematical misconceptions that

are difficult for students to overcome. That's because often times, when concepts are taught too early, they are taught rotely, where students are shown a set of steps to follow to solve a problem. When a teacher attempts to correct the misconception by building understanding of why the rote way works, students often respond that they already know how to do it. With this kind of mindset, new learning is difficult if not impossible.

I see this very often with the standard algorithm for addition and subtraction. Students are taught early on to carry a one for addition For subtraction they're taught to cross out the number in the place to the left, put a one next to the number to the right and over the crossed out number write the number that's one less. Where's the math? Where's the understanding of place value that makes these steps work? Yet this is what many textbooks have our first graders doing.

Math is a sense making activity, just as reading is. We need to make a commitment to our students that we will help them understand math, not just be able to do math.

Frustration as a Positive Force - My Commitment

How might you translate the energy of frustration into a positive force for change? 

Dan Rockwell wrote this in his Leadershipfreak blog this morning, and even though I'm not in a traditional leadership role, it rang true to me. 

I am frustrated about mathematics instruction and the reliance on textbooks that encourage conformity rather than development of mathematical understanding. I'm frustrated that it's taking so long for change to happen, and mostly, I'm frustrated that many students are still learning to dislike mathematics at a young age.

Rockwell also wrote,
Determine if they want to do something with their frustration or just complain. Don’t follow complainers.

This year, I commit to stretching myself and doing more with my frustration to impact positive change. I will be the squeaky wheel. Our kids deserve to understand and like math, and I will be part of this change.

Fractions Are Not Shapes

Fractions are numbers, and we need to make sure that we are not creating misconceptions in the primary grades that will interfere with this understanding.

On the math word wall in many second grade classrooms that I visit, I see similar definitions for fractions. Numerator: The number on top, the number of boxes shaded. Denominator: The number on the bottom. The total number of boxes.

How does that connect to a fraction as a number? It doesn't connect. The emphasis as students initially learn about fractions needs to be on equal shares. This understanding in second grade builds to the understanding in third grade that a fraction represents equal shares on the number line. This progression is important for students to build understanding. Fractions are not pizzas or rectangles divided into squares. Fractions are numbers that can be used to represent equal shares of shapes.