# Problems with Problem Solving? Focus on the structure

Word problems are at the heart of mathematics; and yet, mathematical problem solving is one of the most challenging topics for students and teachers alike. What does research say about the best ways to teach word problems?

One strategy shown to improve students’ problem solving is focusing on the underlying structure of word problems (Fuchs et al., 2004; Jitendra et al., 2013; Gersten et al., 2009).

Schema-based instruction (SBI) teaches students to identify the schema of a problem, or the organizing structure, and use appropriate visual models (e.g. diagrams, graphic organizers, equations) to help solve the problem. The first step, however, is being able to determine what type of problem students are solving and then being able to figure out what part of the situation is missing.

Within addition/subtraction word problems, there are 4 main types of schema:

Take from

Put together/Take apart

Compare

Each of these schema represent a different type of scenario or action that is occurring. We can help students identify and distinguish between these schema by using different visual representations or graphic organizers that model each situation.

Let’s look more closely at the Add to schema. Within this type of problem, there are 3 different possible unknowns: start, change or result unknown. Here are examples of each of these:

Getting students to understand what is happening in a word problem and be able to represent the problem using a visual model, like the open number lines above, is key to improving their mathematical problem solving.

# Rekenrek – a great K-2 manipulative

A rekenrek is one of my favorite early elementary math manipulatives. First designed by Adrien Treffers from the Freudenthal Institute in Holland, a rekenrek is a set of 20 beads organized into two rows of 10, with 5 red and 5 white beads on each row.

You can easily create and recognize each number from 1 – 20 using the groups of 5 beads. For example, if I push the whole first row to the left and 2 beads on the bottom row, I can quickly recognize the number 12.
The ability to “see” numbers rather than counting them is called subitizing and is an important early indicator of number sense. Normally we are only able to subitize small numbers (1-4), but the rekenrek helps children quickly see larger numbers as combinations of sets. For example, I can quickly see that there are 8 beads if I see 5 red beads and 3 white. I could also show this with linker cubes, but it takes a lot longer to count out and construct stacks of linker cubes.

The rekenrek is also a quick way to demonstrate and practice addition strategies. You can demonstrate counting on with the problem 6 + 3 by first sliding over 6 beads as a group, saying “6,” and then sliding over three more beads individually, “7, 8, 9.” Remember to say “7, 8, 9” rather than “1, 2, 3” as you are sliding over the beads.

You can practice near doubles (or doubles plus one) by first creating a doubles problem such as 4 + 4, by sliding over 4 beads in each row. Then slide over one more bead on the top row to create 5 + 4 or one more bead on the bottom to create 4 + 5.
Demonstrating near tens is easy on the rekenrek. Let’s say you’re trying to solve the problem 8 + 4, first slide over 8 on the top row. Then slide over 2 on the top to complete the 10 and 2 more on the bottom to make 12.

Here are a few extra resources: rekenrek activites and instructions for making a class set*.

*I made my class set of rekenreks with pipe cleaners instead of string because that way my kids could help. Watch the video below for tips on making a class set.

# Can counting with fingers improve math scores?

Ever feel your fingers move unconsciously when you’re figuring out simple math problems? It may be you’re sensing the strong neural connection between our fingers and numbers. Scientists became interested in this topic when noticing that the same areas of the brain that control the fingers light up when solving math problems even though people were not actually moving their fingers.

Researchers have found a strong connection between math skills and finger gnosia, or knowledge of your fingers (e.g. Penner-Wilger & Anderson, 2008). They test young kids by asking them to put their hands in a box with an open side. The researcher then touches one or more of the kid’s fingers, removes the box, and asks which finger was just touched. Children’s ability to recognize which finger has been touched predicts how well that child will perform on a variety of math tests over years.

Yet anytime we talk about how one skill predicts another, as researchers we need to figure out if there is a causal relationship between the skills. Is it that kids who are already predisposed to be good at math have great finger gnosia when they are young? Or can we teach finger gnosia to young children and see effects later in math performance?

There is some early evidence that direct finger awareness training can affect students’ later math performance (Gracia-Bafalluy & Noel, 2008; Jay & Betenson, 2017). For instance, in a recent study of 137 six and seven year olds, researchers worked with students for 4 weeks either playing finger gnosis activities or having them play number games or both, to see the effects on students’ quantitative skills. They found that while finger gnosis skills and number games alone did not improve quantitative skills, when combined together, they did significantly improve students’ quantitative skills. Here are a few of the games, they played with students:

-Counting by 1s, 2s, 5s, and 10s on fingers

-Teacher asks students to “show me” a specific number on fingers

-Holding up fingers to represent operations (Show 3+ 4 on your fingers)

See more of the activities and number games here.

So, Kindergarten and First Grade teachers, don’t discourage finger counting! And remember, you can also promote more advanced counting strategies with fingers (e.g. counting on, counting by, etc.)…

 Kara

# Exploring Number Lines

One Saturday morning, I decided to try out an estimation task with my five year old daughter while we were eating pancakes. I gave her a piece of paper and drew a blank number line on it and labeled the ends with a 0 and 10.

I started by asking her to place where the 5 should go. I wanted to give her a landmark to see how that would shape her placement of the other numbers. Here’s what she did:

I was curious whether the number line would prompt her to connect that 5 is half of 10 and should therefore go in the center. It did not.

Then I asked her to fill in the 9. I was trying to be strategic about first asking her to place 5 and then 9, so she would be more inclined to space the numbers throughout the number line. From there, I asked her to fill in the rest of the numbers. Here’s what she did:

She quickly realized that her number line didn’t look right, and she wanted to fix it. So, she did another number line below and ran into trouble between 9 and 10.

I asked her to do a third number line, and this is what she did next:

You can see from this number line that she placed 5 at the midway point. She also corrected herself more than once to make the number line more accurate and linear.

Having kids experiment with number lines is an easy and fun way for them to think more deeply about number relations and estimation. There are some fascinating studies about how kids start with a logarithmic pattern of estimating (e.g. my daughter’s first number line, where the majority of the numbers lie towards the left of the number line) and with practice and age get more linear (evenly spaced).

Here’s a great research article by Dr. Robert Siegler (Carnegie Mellon Univ.) and Dr. Julie Booth (Temple Univ.) on children’s numerical estimation:  http://www.cs.cmu.edu/~jlbooth/sieglerbooth-cd04.pdf

Want to try this type of activity with your students? Here are some blank number lines you can cut out and give to your students.

# Is four times always the same as two times doubled?

One of the trickiest part of teaching multiplication deals with the distributive property. Because of this, we made the distributive property the focus of Mt. Multiplis, our multiplication app.

The Common Core State Standards expect third graders to use the distributive property when multiplying (which you and I learned in middle school): “Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56” (CCSM.3.OA.B.5). That’s one complicated equation, and it’s hard to imagine that most 9-year-olds, who are just learning how to multiply, are able to make sense of that notation. So, we set out to see if we could not only make this complex content more approachable, but also fun.

At its core, the distributive property is about groups. It connects multiplication and addition by letting you break up a multiplication problem into smaller chunks. For example, when solving 7 x 8, you can think of it as 7 groups of 8. But, how does that help you multiply? Let’s say you forgot what 7 x 8 is, but you know 5 x 8 is 40. You can then think of 7 x 8 as 5 groups of 8 plus 2 more groups of 8, that’s 40 + 16.

In Mt. Multiplis students explore this property by using groups of wooden planks to build bridges. To fill a 7 by 8 area, students drag over 5 planks of 8 and then drag over another 2 planks of 8.

The best part of playtesting has been watching kids discover the applications of the distributive property on their own. While playing Level 5, which features a lot of 11 x __ problems, one fourth grader literally said, “Aha!” as he realized for the first time, why the 11s rule works. He explained to me that the 11s facts are easy (i.e. 5 x 11 = 55, etc), but that he didn’t know why the shortcut works, until playing Mt. Multiplis. Solving 11 x 7, he dragged over 10 groups of 7 to make 70 and then dragged over 1 more 7.

Another third grader was working through several 4 x __ problems. She would consistently drag out 2 planks and then use the Double-It card to make another set of 2 planks. She first saw the problem 4 x 6, dragging out 2 six-planks to make 12 then used the Double-It card to make another group of 12. She did not fully trust that the answer was actually going to be 24, but she entered it in anyway and was surprised when it was correct. Next, she solved 4 x 3 and then 4 x 8 in the same way. After solving 4 x 8, thinking about 16 doubled, she asked me, “Does that always work…with the 4 times problems…you can just double the 2 times?” Yep, it always works. And that’s some pretty sophisticated algebraic thinking. Another way to phrase her question: Does 4y = 2y + 2y? Yes, because of the distributive property.

Kara

# What Does Mickey Mouse Have to Do with Counting?

One night before bed, in the spirit of Piaget, I decided to videotape my 2-year old daughter counting. As most parents know, counting is a fundamental part of children’s everyday mathematics experiences.

Here’s an example of my daughter counting. She’s starting to show understanding of certain aspects of counting and patterns. Pay close attention to the end of the video:

Before kids slide down a slide or begin a race, they chant, “1,2,3, go!” Kids learning to count often spend what feels like hours counting everyday objects, like fingers and toes, cheerios for breakfast or stairs they are climbing.  Children learn counting as if it were a song, a rhythmic chant.  Eventually, they begin to see patterns in the song that they can extend.

Young kids are often able to count higher than you might expect if you help them with a few key numbers, particularly the decades.  For example, a 2.5-year-old might count up until 19 and then get stuck on what comes next, but after you tell her it is 20, she can keep counting until 29.  If you tell her 30 comes next, she will keep going until 39, etc. In a past blog post, I wrote about how language makes our counting words particularly tricky for English-speaking kids, especially compared with their Asian peers. See that post here: http://bit.ly/1dnOJ4c

But knowing “the counting song” is just the tip of the iceberg in learning how to enumerate, or count objects.  Researcher Rochel Gelman from Rutgers University outlines 5 key principles that underlie the ability to enumerate.

1. The stable ordering principle: This is what most of us think of when we say, “My child knows how to count.” It refers to knowing the counting sequence (or the “counting song”). But knowing the count sequence doesn’t necessarily mean you can count objects or use the sequence in any meaningful way.
2.  The one-to-one principle: Each object to be counted gets one and only one counting word.  In other words, no double counting jelly beans and no skipping objects when you count.
3. The cardinal principle: Knowing that the last number you say when counting objects refers to the entire set of objects, not just the last object. This is one of the hardest principles for little kids to grasp and has been the subject of much research.  The classic Piagetian task testing this principle is to ask a 3-year-old to count a set of blocks.  The child may be very good at carefully counting, “1,2,3,4” as he points to each of the 4 objects; however, if you then immediately cover the objects with your hands and ask how many objects he just counted, the child doesn’t know how to answer.  He needs to count them all over again, starting from 1 because he doesn’t realize that the “4” he just said refers to the set. One way to help reinforce this idea with kids is to have them always state the cardinal value after they count, i.e. “1,2,3,4. 4 blocks.”
4. The abstraction principle: You can count just about anything (blocks, candies, ideas or all three mixed together)
5. The order-irrelevance principle:  Items can be counted in any order.  This idea can be tricky for young kids, too.  If you line up blocks in a row, many kids will think you have to start from one specific end and not from the other end or even from the middle of the line.

Here’s an example of my daughter demonstrating this principle:

Dana

# Teachley Research at a Glance

At Teachley, we take research seriously (it’s in our tagline, after all).  Our products are developed with grant funding from the U.S. Department of Education, Institute of Education Sciences (IES) and the National Science Foundation (NSF). As part of these grants, we conduct ongoing research to evaluate the effectiveness of our apps in improving students’ mathematical abilities.

Design research. When developing our apps, we utilize an iterative design research process during which we build initial prototypes of the app and its features, such as levels and scaffolds, then give them to kids to play. We observe as they interact with the app, ask questions, and gather valuable feedback to inform the refinement of the app.

Efficacy research. To evaluate whether our apps impact learning, we conduct research studies during which we explore changes in students’ abilities before and after they play. To help synthesize some of our research efforts, we’ve put together two short briefs and link to them below.

Teachley Operations. Students who played our operations apps improved their fluency more than students who played traditional fluency games. Further, students using our Mt. Multiplis app were significantly more likely to use the distributive property when explaining how they solved problems. These results were also found when looking specifically at children who struggle in mathematics. Read the full Teachley Operations brief here.

Teachley Fractions. Initial research on a prototype of our first fractions app, Fractions Boost found significant effects on students’ ability to estimate fractions on a number line. Read the full Teachley Fractions brief here.

# Tiggly joins the Teachley platform

## Tiggly games now sync with Teachley

Tiggly makes interactive, educational iPad apps that your students will love! We are thrilled to announce that 4 of Tiggly’s apps now sync with Teachley Connect*. When you download any of the 4 free Tiggly apps below on an iPad that also has the Teachley Connect* app, the games will automatically sync with your Teachley class list for personalized gameplay. Download the apps and start playing today!

For grades PreK-K, join Tiggly to play a number line adventure! This app helps your students get familiar with the number line, learn about number relations, and count in more than 10 languages. Learn about the number line, number relations, and counting in different languages including Spanish and Mandarin. Download here.

For grades PreK-1, this math game helps your students learn addition and represent addition problems with drawings, verbal explanation, and math equations, all while helping Chef prepare his signature Spicy Hula Monkey Cake and over 40 other outrageous dishes. Learn about: early addition, representing addition problems with drawings, verbal explanations, and math equations, composing numbers less than or equal to 10 in more than one way, and understanding the meaning of plus (+) and equal signs (=). Created by Tiggly, a Teachley partner. Download here.

### Tiggly Chef Subtraction

For grades PreK-1, learn subtraction math concepts while experimenting in Chef’s super-duper secret kitchen laboratory to create the most silly and flavorful creations the world has ever witnessed! Learn about: mental subtraction, conceptually understanding subtraction as “taking away,” decomposing numbers less than or equal to 20 in more than one way, understanding the meaning of minus symbol (-) and equal sign (=), and representing subtraction problems with drawings, verbal explanation, and math equations. Created by Tiggly, a Teachley partner. Download here.

### Tiggly Cardtoons

For grades PreK-K, count, drag, match, and enjoy as the seemingly simple buttons you create come alive becoming part of a wildly imaginative cornucopia of storytelling. Tiggly Cardtoons will help your students learn basic math ideas such as one-to-one matching, counting, and equal sets while stretching their imagination and sense of wonderment. Features 25 imaginative stories focused on numbers, unique illustrations with textures taken from the real world, and guided counting gestures important for developing counting skills. Created by Tiggly, a Teachley partner. Download here.

Please note: Tiggly also makes physical manipulatives that seamlessly interact with the apps and are available for purchase separately. These manipulatives are not required to play the apps. The apps work by finger touch as well.

Teachley customers are eligible for a 10% special promotion on Tiggly connected manipulatives with the code TigglyTeachley. To claim your discount, email

# Teacher Appreciation Sweepstakes Winners

Congratulations to the winners of our Teacher Appreciation Sweepstakes!

Basic Account winners:

Erin Berthold

Cara Bartlett

Tammy Evarts

Julie Bormett

Amber Hoerner

Pilot School winners:

Cathie Herbers

Maurren Driscoll

Merle Goess

Jami Zimmerman

Mrs. Owens

Crystal Malloy

Pam Gray

# App Settings

At Teachley, we’re making using iPads in the classroom even easier and more efficient for you and your students. Teachley’s App Settings enables teachers to customize our apps right from the Teachley dashboard to meet each student’s needs.

Here are just some examples of ways you can customize the apps:

Turn off the Speed Round in Addimals EDU Subtractimals EDU

Select specific multiplication factors in Fact Flyer EDU

Assign a pop-up quiz in Fractions Boost EDU

To see how App Settings work, watch this short video: bit.ly/2nDMrb5

App Settings work with the EDU versions of Teachley’s math apps (and select third party apps) and is available now for premium Teachley subscribers. Simply log into your teacher dashboard at www.teachley.com or in the Teachley Connect app. Not a Teachley subscriber? Sign up for a free pilot here.