We spent some time in Toni Laughrey’s classroom in December. She had so many creative ideas in play in her classroom, we thought we just had to share some here. Here are a few of our favorites.

Support collaborative work and communication with a chalkboard table. Toni turned an old table into a chalkboard table with some chalkboard paint. This provided a great space where we saw teachers and students working collaboratively. Chalkboard paint is available at many department and home improvements stores, and prices range from $6-$15. Here’s one site we found to create both a table and then tips on painting it.

Greet your students with fun, individualized notes they can’t miss. Toni used a dry-erase marker to write encouraging notes to her students directly on their desks. What a great way to boost morale! Dry erase markers are widely available, in a local department store or office supply store. Here’s an example of another teacher sharing motivational notes with her students as well!

Who says you can’t write on your desk? Toni turned her desk into a whiteboard desk! She used it to lay out the plan for the day for the substitute teacher, but this can be a great way to leave notes for yourself, too! Did you know there’s dry erase paint? Check out how this couple transformed an old desk into one with a dry erase surface. Dry erase paint is available from several manufacturers, so check out your local home improvement store – there are also rolls of dry erase material you could apply to your desk as well!

What creative ideas have your students enjoyed? Do you have any tips for our T.I.P.S. (Teaching is Problem Solving) readers? Share in the comments, on our Facebook page, or on Twitter!

Robert C. Schoen, Ph.D., is the Associate Director of LSI’s Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR–STEM), as well as the founder and director of Teaching is Problem Solving. His research involves mathematical cognition, the mathematical education of teachers, the development and validation of educational and psychological measurement instruments, and evaluation of the effectiveness of educational interventions.

Classroom observations were once a part of my Cognitively Guided Instruction (CGI) Teacher Professional Development (PD) sessions. I loved bringing teachers to an expert CGI teacher’s classroom so that they could watch a math lesson. The teachers told me that they loved seeing these lessons. Although everyone loved these visits, I started to wonder what teachers were actually learning from these observations.

Teaching is a complex problem-solving endeavor. We know that watching other people solve problems isn’t the best way to become a better problem solver and we structure our math classes to ensure that students are solving problems and explaining their thinking when they share their strategies with each other. When teachers observe another teacher teach a lesson, they aren’t engaged in solving problems and don’t have access to the thinking behind the teacher’s actions. I knew that having students watch someone else solve math problems wasn’t a good way to help them become better mathematical problem solvers. Why did I think that teachers watching someone else teach was a good way for them to become better teachers?

About 13 years ago, my CGI colleagues at Teachers Development Group and I developed the Classroom Embedded Protocol as a way to engage teachers in solving the problems of teaching math in the context of a real classroom. Although this protocol includes teachers observing another teacher, we spend the majority of the day actively engaged in solving the problems of teaching in the context of a real classroom.

Each Classroom Embedded Day is designed to support teachers’ understanding of children’s thinking around a particular mathematical concept. Teachers learn CGI research frameworks about this concept in seminar-style PD before the Classroom Embedded Day. These frameworks include problem types to engage students with a specific concept and the trajectory of strategies that students use to solve these problems. Our Classroom Embedded Days typically work through these seven steps:

1. Assessment. We begin by interviewing all of students in the host classroom to assess each student’s level of understanding of the concept. Teachers analyze the data from the interviews to develop a classroom profile that describes where each student falls along the CGI learning trajectory for this particular concept.

2. Set learning goals. Teachers use the classroom profile, their understanding of CGI learning trajectories, and State Math Standards to set learning goals for these students. Students will engage with these learning goals for several weeks or months; they are not goals to be mastered in one lesson. There are usually different learning goals for different groups of students.

3. Design a problem. Teachers use the classroom profile along with their understanding of a CGI problem types to design a problem for the lesson. We support teachers to choose specific numbers for the problem that will engage students with their learning goals. Sometimes different students need different numbers in their problems.

4. Plan the sharing of strategies. Teachers anticipate how these students will solve the problem and choose which strategies will be shared with the whole class based on which strategies are most likely to support groups of students to engage with their learning goals.

5. Lesson Observation. Teachers return to the host classroom and observe the classroom teacher or the CGI PD leader teach the lesson.

6. Debrief.When debriefing the lesson, we focus on how students engaged with their learning goals. Teachers also reflect on teacher moves that worked well and teacher moves that didn’t work so well. Because teachers have been engaged in the problem solving around the lesson, they can better understand the teacher moves.

7. Plan a lesson for their students. Teachers have already posed a problem or two from the interview to their own students. They work with teachers at their grade level to plan a lesson for their own students using a protocol similar to the one the used for the host classroom.

This process takes about 5-6 hours. We know that teachers will never have this much time to spend on a single lesson in the real world of teaching. We slow down the problem solving process to increase teachers’ understanding and problem solving capacities much like an athlete or musician might slow down a process to increase their skill.

Teaching is a complex problem-solving endeavor. We should adhere to the same principles when teaching teachers as we use for teaching mathematical problem solving. Assessing students’ thinking, setting learning goals, choosing problems, anticipating students’ strategies, and planning the sharing of strategies are complex problem solving endeavors that US teachers almost always do on their own. Our Classroom Embedded Days are designed so that teachers can work together to develop their knowledge, strategies and skills for solving these complex problems.

Teachers typically report that they learn more from a Classroom Embedded Day than they learn from solely observing another teacher teach. Are you someone who has participated in a CGI Classroom Embedded Day? If so, please comment below or tag us on Twitter – @LLeviCGIMath and @teachingsolves with the hashtag #tips_blog on what you learned from this process.

Linda Levi has been a member of the CGI research and development team since 1989. She is the Director of Cognitively Guided Instruction (CGI) Initiatives for Teachers Development Group where she designs and supports the instruction of CGI professional development for elementary school teachers and teacher leaders nationwide. Dr. Levi is a co-author of Children’s Mathematics: Cognitively Guided Instruction, Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School and Extending Children’s Mathematics: Fractions and Decimals. She loves talking about math with children as well as discussing children’s mathematical thinking with teachers.

Pop quiz: True or false? 1=1. You’re probably thinking, “Of course it’s true! Obviously 1 equals 1!”

Is it really obvious, though? I’d like to think that this statement is true as well. I would also like to think that my 7-year-old self would have said the same thing. However, that may not be the case for little-me or other first-grade students.

Let’s try another one: Read this equation and decide whether it is true or false.

8 = 5 + 3. It’s a no-brainer, right? The quantity ‘8’ is equivalent to itself, which in this case looks like ‘5 + 3’, so this, again, is true. Now imagine you’re back in second grade. Your teacher is writing an equation on the board (or over-head projector, remember those?) like this one. Is the problem written as or?

There’s not much difference between the two equations, but most elementary students see equations like 5 + 3 = 8 much more often than ones like 8 = 5 + 3. If you saw the first orientation of that equation a lot in elementary school, raise your hand. Now that most of you are raising your hand, let’s take a moment to think. If this orientation was the only way you ever saw equations throughout elementary school, how would you react to the previous question?

What seems true to us now may not have seemed to be “true” when we were young. The rules didn’t change, our understanding of the rules did! That includes how we reason through math questions.

In 2014 and 2015, a team of researchers conducted more than 1,500 student interviews to see how students reason through math questions, including true/false questions like the ones above. I was one of the assistant researchers who coded videos of the student interviews to document how kids were thinking about the math problems. I later transcribed student responses to several true/false questions about equations. In this blog post, I’ll discuss three true/false questions in particular that I thought had intriguing answers; you may find the result and response to be surprising. I sure did!

A similar true/false question to 1=1 was asked in the interviews with first- and second- graders. Approximately 68% of the students from the transcribed interviews said the statement was “true”. There were variations in reasoning, but the most common response was that they are the same number, as in the quantity ‘1’ is equivalent to itself. Yay! This is good, two thumbs up!

But let’s not get ahead of ourselves yet – more than 30% of the students said the statement was false. Why? Because there was no ‘plus or minus’ sign. These students noted that the equation didn’t include an addition or subtraction sign, so there have been something wrong with the equation. Little-me (and maybe little-you!) would have probably noticed the same thing.

So how about the 8 = 5 + 3 problem? A similar true/false question was asked in the interviews and, once again, many of the students said “true,” but this time only 53% of them said it was true. The reasoning was that the quantity ‘8’ is equivalent to ‘5’ added to the quantity ‘3’ (or the reverse). Students would use their fingers or manipulatives to confirm this statement. Other students said it was true, but that the equation was “backwards” or “switched around” –interesting!

On the flip side for students who said it was false, their common reasoning was that ‘8’ does not equal ‘5’. Another explanation was that the equals sign and the addition sign were in the wrong place. Kind of similar to students who noticed that the equation was “backwards,” but these students said that it made the statement false.

Pop quiz again: does 5 + 4 = 3 + 6? I’ll admit; it’s a bit of a head-scratcher if you’re not paying attention, but the sum of ‘5 + 4’ is equivalent to the sum of ‘3 + 6’. So this statement is true.

You might have guessed by now: not many students (only 31%) got the question like this one correct. The students who said the statement was true mostly said that each side is ‘9’, so overall, the equation is true. However, a lot of students who stated it was false said that ‘5 + 4’ does not equal ‘3’. Oddly enough, even when they calculated both sides of the equation (with their fingers or blocks) to be ‘9’, many students stuck with their original answer.

Are you surprised by the results? I was and wasn’t at the same time. This study found that some elementary mathematics curricula do not expose students to “nonstandard equation types.” Exposure to these types of equations can make a difference in how students answer and reason these true/false questions.

Let me know what you think! Were you surprised with some of the ways students are explaining these true/false questions? If you can recall, how did you reason with these true/false questions? It’s okay if little-you would not have given the correct response. I know little-me wouldn’t have, and that’s okay!

Lastly, please use the comments section (or send us a tweet and tag #tips_blog) to share some teaching strategies that can help students think beyond these types of reasoning. Every little bit of effective advice can go a long way!

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Shelby McCrackin has been working with the Learning Systems Institute at FSU since the summer of 2015. During that time, she has coded cognitive interviews with students, entered data for student mathematics tests, analyzed textbooks, provided feedback on assessment drafts, and served on adjudication committees for constructed-response items on student assessments. In addition, she is conducting research on undergraduate minority STEM students, their interactions with their STEM professors, and, of course, how elementary students see the equals sign. Shelby is double-majoring in Mathematics and Mathematics Secondary Education and will complete the requirements for her bachelor’s degree in spring 2018.

The equals sign has a deep, almost sacred, meaning in mathematics. Despite the importance of = as a relational operator in mathematics that is used at all levels K–20 (and beyond), many elementary, secondary, and even post-secondary mathematics students have a limited understanding of the intended meaning of that symbol in mathematics.

For seven years and counting, the mathematics curriculum standards in Florida (and almost every state in the United States) have set explicit learning goals for student understanding of the equals sign. In most states, the following two content standards are in place for first grade:

The writers of the standards, in my view, did students a great service by translating research on student thinking related to the equals sign into policy.

Many teachers look at these standards and ignore them or think they’re trivial. Or, maybe they interpret the intent of the standard to be focused on students knowing basic facts rather than the meaning of the equals sign as a relational symbol. As many other teachers know, those teachers are overlooking an important component of student understanding.

Several years ago, I found myself in a fortunate position to lead an amazing team of teacher-researchers working to develop tasks and rubrics for (commonly called MFAS). Our goal was to develop at least four good mathematics problems, or tasks, aligned with each of the content standards at each grade level and designed to provide teachers with insight into student understanding that could inform their instructional decisions. Each task had a corresponding, task-specific rubric with four levels. Over several years, we conducted thousands of student interviews in the process of piloting and refining the tasks and rubrics. We had a group of partner teachers in four school districts in Florida who kindly allowed us to interview their students. In exchange, we would tell the teachers what we learned about their students.

In the development of tasks and rubrics for the first-grade equals-sign standards, we interviewed dozens of first-graders in four school districts. We had created a set of mathematical statements in the form of simple equations (some true, some not true) and asked first-grade students to read each equation and tell us whether or not the equation was true. While almost all of the students performed the operations on the numbers accurately and used vocabulary words such as equation correctly, none of the students answered the true-false questions about these equations correctly.

The first-grade students we interviewed were very consistent in their explanations for why they decided the statement 7 = 7 was false, or not correct. They explained that the equation did not have a plus or minus sign, so it wasn’t correct. They told us that 8 = 4 + 4 is not correct, because the plus sign was on the wrong side of the equals sign. They knew vocabulary terms such as equation, subtraction, plus, add, etc. They also knew how to evaluate expressions such as 4 + 4. Across dozens of first-grade students, they were almost 100% consistent in saying 7 = 7 was not correct, because there was no addition or subtraction symbol and that 8 = 4 + 4 was not correct, because the plus sign was on the wrong side of the equals sign.

We are certain that teachers never told their students that 6 = 6 is a false statement! They probably never even said that it was an incorrect way to write an equation. It seems most plausible that the first-graders had simply never seen equations written in that form. Rather, they probably had only seen them in the form a + b = c or a – b = c. Because they had never been exposed to equations in other forms, they assumed those other forms to be incorrect.

This was a problem for us, because we weren’t getting examples of what students would say if they were in the higher levels of the rubrics, and we wanted to have real examples of what students at each of the four levels of the rubric would do.

Fortunately for us, the partner teachers were serious about teaching mathematics! When we reported what we learned about student understanding of the equals sign to their teachers, the teachers were often surprised. As soon as they learned about this issue, they went right to work on trying to improve and expand their students’ understanding of the intended meaning of =. Many teachers asked for recommendations for how to address this pervasive problem.

Here are three recommendations that came from those conversation with teachers. While we think these recommendations apply to first-grade students, we also think they apply to kindergarten students, second-grade students, and even students at the secondary and post-secondary level.

Discuss the mathematical meaning of the equals sign with students. Don’t just use the symbol and assume students will grasp the intended meaning of = in mathematics. Talk explicitly about its intended meaning with your students, and reinforce it many times throughout the year.

Expose students to non-standard forms of equations (e.g., 8 = 4 + 4) frequently, and state explicitly when those expressions of mathematical ideas can be perfectly fine and correct to use in mathematics.

Introduce = as a relational symbol rather than as a command. In other words, try to create a closer association between =, <, and > in your students’ minds than there is between = and other operators such as + or –.

After incorporating these recommendations, our partner teachers reported that some students’ understanding of the equals sign changed quickly (and for the better). Other students’ ideas were more resistant to change. Like many other topics in mathematics (See Place Value), students can appear to fully understand the intended meaning of the equals sign one day and then appear to not have that same knowledge later.

***

We’ll have more blog posts in this series about the equals sign in the coming weeks. Each writer will share their unique perspective on the topic of the equals sign. I hope you will join the conversation to share your experiences and perspective on this topic. If you want, you can even write a guest post (email me for details). We invite you to share your own unique experiences in the comments so that we can work together to address this interesting problem of mathematics teaching.

Robert C. Schoen, Ph.D., is the Associate Director of LSI’s Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR–STEM), as well as the founder and director of Teaching is Problem Solving. His research involves mathematical cognition, the mathematical education of teachers, the development and validation of educational and psychological measurement instruments, and evaluation of the effectiveness of educational interventions.

[1]This task and the associated rubrics, including videos of student responses, can be found here. Several hundred other tasks and task-specific rubrics were created through the MFAS project and available free of charge through CPALMS.

One of the exciting features of Teaching Is Problem Solving is the collection of stories our team is writing about some classroom experiences studying mathematics teaching and learning. The What’s Next? stories document professional development experiences and may be useful for individuals or groups of teachers who want to study real stories of teachers engaging in formative assessment and lesson study based around the frameworks for student thinking in the Cognitively Guided Instruction and Extending Children’s Mathematics programs. Read more about the What’s Next stories here.

In the story titled Bailey and Emma Use Direct Modeling to Solve an Addition Word Problem, the teachers discuss student thinking, discuss learning goals for students on that day, plan a lesson designed to advance those students’ understanding, carry out the lesson, and reflect on the experience.

This lesson is an example of why we say teaching is problem solving! The teachers identify a problem to solve (i.e., how to help these kids learn), make a plan, carry out the plan, assess whether they have solved the problem, and think about how to adjust.

Have you done something similar? Let us know how it went!

Figure 5. The teacher’s record of students’ strategies for 6 + 5.

Figure 4. The teachers’ record of strategies shared for 4 + 4, 5 + 5, and 6 + 4.

Figure 3. Sequence of addition expressions for students to evaluate during the classroom lesson.

Figure 2. Classification of the students by the most prevalent strategies observed in the addition fact interview.

Figure 1. Set of addition expressions used in one-on-one interviews (posed in sequence down the left column and then down the right).

Figure 5. Issabella's strategy is shown in red with orange parentheses

Figure 3. The sequence of items teachers decided to use during the planned lesson with the goal of encouraging students to begin using fact-based strategies to evaluate addition expressions

Figure 3. The sequence of items teachers decided to use during the planned lesson with the goal of encouraging students to begin using fact-based strategies to evaluate addition expressions

Figure 2. Classification of the students on the basis of the strategies they most often used

Figure 1. Items used in the fact-fluency assessment

Figure 3. Gianluca's invented algorithm strategy for the Pete's-rocks problem.

Figure 2. Timothy's direct modeling with tens strategy for the Pete's-rocks problem.

Figure 1. Classification of the interviewed students based on strategies used.

Figure 3. The teacher’s notation of three fact recall—derived fact strategies for 8 + 8.

Figure 2. Classification of the students on the basis of the strategies they used most often.

Figure 1. Set of items used in the fact-fluency interview.

Figure 5. Jack's work after the teacher added the notation.

Figure 4. Recording notation to match a student's strategy for the follow up problem.

Figure 3. Recording notation to match a student's strategy.

Figure 2. The sorting of the students by strategy used on the sandwich problem.

Figure 1. A direct modeling solution to the sandwich problem.

Figure 4. Students' names, sorted according to the type of strategy they used to solve Problem A.

Figure 3. An example of a student’s additive coordination—sharing groups of items strategy.

Figure 2. An example of a student's additive coordination–sharing one item at a time strategy.

Figure 4. Students' names, sorted according to the type of strategy they used to solve Problem A.

Figure 6. Brandon's notation of his counting by tens strategy for the Kevin's-candies problem.

Figure 5. Erick's work depicting a counting by tens strategy for the Kevin's-candies problem.

Figure 4. Adrian's work depicting a counting by ones strategy for the Kevin's-candies problem.

Figure 3. Strategies teachers anticipated students would use to solve the Kevin's-candies problem and ideas for the lesson implementation.

Figure 2. Summary of strategies used by the second-grade students on the interview tasks.

Figure 1. Children's strategies for solving grouping-by-tens problems.

Figure 2. Details of two students' strategies for Problem D.

Figure 1. Strategies used by students in a first-grade class to solve problem D.

Figure 7. The teacher's notation of Carter's invented algorithm strategy next to Kayla's base-ten block strategy.

Figure 6. Kayla touching each small cube embedded in the base-ten rods as she uses counting by ones to verify the quantity.

Figure 5. Trinity's cubes organized to make counting easier.

Figure 4. Trinity's direct modeling with ones representation of 20 and 24 rocks.

Figure 3. The working plan developed by teachers for the class discussion of the Pete's rocks problem.

Figure 2. Differentiated learning goals for subgroups of students.

Figure 1. Multidigit computation strategies used by students in a first-grade class to solve the Pete's rocks problem.

Figure 5. The teacher writes numbers corresponding to the students’ verbal counting.

Figure 3. The teacher shows Denise’s explanation on the board.

Figure 3. Denise's model of both 42 and 83.

Figure 2. Denise's initial model for the Harmony's Rocks problem

Figure 1. Teachers' classification of students' strategies for problems A and B.

Figure 4. A snapshot of the strategies used by the students in the class.

Figure 3. The strategy of a student who shared groups of items to distribute 11 brownies among four people

Figure 2. The strategy of a student who shared one item at a time distribute 11 brownies equally among four people.

Figure 1. An example of a non-anticipatory sharing strategy

Figure 13. Kevin's student work for the initial Skittles problem.

Figure 12. David's student work for the initial Skittles problem.

Figure 11. Breanna's student work for the initial Skittles problem.

Figure 10: Alexis' student work for the initial Skittles problem.

Figure 9. Jason's student work for the initial Skittles problem.

Figure 8. Gracie's student work for the initial Skittles problem.

Figure 7. Teachers' Classification of Students' Strategies for Problem C.

Figure 6. Teachers' Classification of Students' Strategies for Problem B.

Figure 5. Lane's work for Problem B.

Figure 4. Aubrey's work for Problem B.

Figure 3. Teacher's Classification of Students' Strategies for Problem A.

Figure 2. John's student work for Problem A.

Figure 1. Tyler’s student work for Problem A.

Figure 6. A chart of what the teachers felt worked well during the classroom embedded lesson.

Figure 5. Student strategies for each of the three interview problems.

Figure 4. Counting by tens student strategy for problem B.

Figure 3: Counting by ones student strategy for problem B.

Figure 2: Direct modeling with adding on in chunks for problem A.

Figure 1: Direct modeling strategy for problem A.

Figure 23. Brandon’s work on the new problem.

Figure 22. Emma’s work on the new problem.

Figure 21. Hayden’s work on the new problem.

Figure 20. The teacher demonstrates a way to notate Austin’s strategy.

Figure 19. The teacher demonstrates how to notate Austin’s strategy.

Figure 18. Austin’s strategy for solving Problem D.

Figure 17. Brandon’s strategy for solving Problem D.

Figure 16. Emma draws a circle around ten “ones”.

Figure 15. Emma models both numbers in the problem (27 and 35).

Figure 14. Mya’s second step in modeling her strategy.

Figure 13. Mya’s second step in modeling her strategy.

Figure 12. Hayden’s vertical representation of the problem after recognizing that 12 includes one ten.

Figure 11. Hayden’s vertical representation of the problem.

Figure 10. Hayden’s model for solving Problem D.

Figure 9. Hayden writes t's and o's above the digits that represent the tens and ones in the word problem.

Figure 8. Classification chart with students selected to share Problem D strategy indicated.

Figure 7. Teacher’s demonstration (to other teachers) of Emma’s strategy

Figure 7. Teacher’s demonstration (to other teachers) of Emma’s strategy

Figure 7. Teacher’s demonstration (to other teachers) of Emma’s strategy

Figure 7. Teacher's demonstration (to other teachers) of Emma's strategy. (2/5)

Figure 7. Teacher’s demonstration (to other teachers) of Emma’s strategy. (1/5)

Figure 6. Teacher's analysis and explanation, using formal notation, of the properties of operations that justify various steps in Kara's strategy.

Figure 5. Kara Ann's student work.

Figure 4. Interviewer's notation of Jace's strategy.

Figure 3. Interviewer's notation of Mya's strategy.

Figure 2. Interviewer's notation of Ethan's strategy.

Figure 1. Classification of students' strategies.

Figure 3. A valid strategy with nonanticipatory thinking.

Figure 2. A nonvalid, no coordination between sharers and shares strategy without equal shares.

Figure 1. A nonvalid, no coordination between sharers and shares strategy with not all the clay being shared.

Figure 9. Direct modeling and counting by tens strategy for the boxes-of-markers problem.

Figure 8: Direct modeling and counting by ones strategy for the boxes-of-markers problem.

Figure 7. Students organized in two main target groups. 2/2

Figure 7. Students organized in two main target groups. (1/2)

Figure 6. Student strategies 67 divided by ten

Figure 5. Student strategies eight groups of ten

Figure 4. Student strategies for five groups of six

Figure 3. Student's work and explanation written down by interviewer.

Figure 2. Student's work from Problem A.

Figure 1. Teacher’s anticipation of student strategies for interview problems. 3/3

Figure 1. Teacher’s anticipation of student strategies for interview problems. (2/3)

Figure 1. Teacher's anticipation of student strategies for interview problems (1 of 3)

Figure 5. Maxwell's counting by tens strategy for the pizza-party problem.

Figure 4. Gina's counting by tens strategy for the pizza-party problem.

Figure 3. Paul’s counting by tens strategy for the pizza-party problem.

Figure 2. Strategies used by these second-grade students.

Figure 1. Three levels of sophistication in the strategies students typically use to successfully solve grouping-by-tens problems.

Figure 9. A student’s solution to the Abby’s Valentine Cards problem

Figure 8. Student who added the tens and ones separately and recorded his thinking with an equation.

Figure 7. Student sharing plan for strategies discussion on the Tylesha’s books problem

Figure 6. The final sorting of students after some discussion.

Figure 5. A chart showing names of students organized according to the strategies they used to solve the Tylesha’s-books problem.

Figure 4. Symbolic representation written by a student who used a combining the same units strategy.

Figure 3. A representation of a direct modeling with tens strategy for the Tylesha’s-books problem.

Figure 2. A representation of a direct modeling with ones strategy for the Tylesha’s-books problem. This is a photograph of the teacher’s notes, which were a replication of the child’s drawing along with numerals and words to record what the child said.

Figure 1. A representation of a direct modeling with ones strategy for the Tylesha’s books problem.

Figure 1: Teachers’ classification of students’ strategies for Problem B.

Figure 7: Jeovani's, Desean's, and Cesar's strategies recreated on chart paper and presented side by side to facilitate comparing and contrasting the strategies.

Figure 6: Cesar's strategy.

Figure 5: Desean's strategy

Figure 4: Jeovani's strategy

Figure 3: The sorting of the students by strategy used.

Figure 2: An example of a student's written representation of an incrementing strategy.

Figure 1 - A student's drawing of a direct modeling with tens strategy for the Jamie's-stickers problem.

Figure 3. Emma's work showing a direct modeling with tens strategy.

Figure 2: Avery's work showing a direct modeling with tens strategy.

Figure 1. The classification of student strategies on the cubes problem.

Figure 4. Two students’ strategies for solving 14-8

Figure 3. Jordan's and Max's strategy for solving 13-6

Figure 2. Lily’s strategy for solving 12-4

Figure 1. Classification of the students on the basis of the strategies they most often.