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Where & Why Do Fact Strategies Fit In (Adapted from Baroody, Teaching Children Mathematic)
The Texas State Standards states that students are to become fluent in mathematics, developing a robust sense of number. This is backed by research and defines fluency as “skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.” This is directly applicable to problem solving that may take time, effort and perseverance.
Counting Strategies, Reasoning Strategies and Mastery are the three phases through which children typically progress in mastering the basic number computations. Fact strategies are considered a crucial second phase in a three-phase program for teaching students basic math facts. This is a significant contribution to developing our pedagogical awareness by observation and listening for teaching. (Dr. Robert Wright, Developing Number Knowledge, 2012)
The first phase is concept learning. Here, the goal is for students to understand the meanings of addition and subtraction. In this phase, students focus on actions (i.e. “joining/combining and separating) that relate to addition and subtraction concepts.
An important instructional bridge that is often neglected between concept learning and memorization is the Reasoning Strategy phase, which are fact strategies. There are two goals in this phase. First, students need to recognize there are clusters of addition and subtraction facts that relate in certain ways. Second, students need to understand those relationships. These lessons are designed to assist with the second phase of this process. If you have students that are not ready, you will need to address the first phase of concept learning.
The third phase Mastery is memorization of the basic facts. Here the goal is for students to master sums and differences so they can recall them carrying out procedures flexibly, accurately, efficiently, and appropriately
It is all about our targeted instruction
The Principles and Standards for School (NCTM) states, “Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships” (p. 152).
Teaching Student-Centered Mathematics
When students count on their fingers or make marks in the margins they have not mastered their facts because they have not developed efficient methods of producing a fact answer based on number relationships and reasoning. Drilling inefficient methods does not produce mastery! Over many years, research supports the notion that basic fact mastery is dependent on the development of reasoning strategies. These reasoning strategies are essential to fact development.
Guided intervention is an effective research-informed method for fact mastery. Teachers should design sequenced tasks and problems that will promote students’ invention of effective strategies. Then, students need to clearly articulate these strategies and share them with peers. This sharing is often best carried out in think-alouds, in which students talk through the decisions they made and share counterexamples. SCUC has developed the Making the Most of Small Group Instruction website with best practices, strategies and tools for implementing this
Effective Drill and Practice
1. Avoid inefficient practice. Practice will strengthen strategies and make them increasingly automatic. Do not subject any student to fact drills unless the student has developed efficient strategies for the facts being practiced.
2. Individualize practice. Different students will bring different number tools to the task and will develop strategies at different rates. This means there are few drills that are likely to be efficient for a full class at any given time. That is why we need to create a large number of practice activities promoting different strategies and addressing different collections of facts.
3. Practice strategy retrieval. When students are involved in a drill exercise that is designed to practice a particular strategy, it is likely they will use that strategy. Organize the students’ practice problems according to a selected strategy.
Teaching Student-Centered Mathematics: Volume 2, Van de Walle, p. 94 – 95
Avoid All Tricks
Tricks are not reasoning strategies. If the mathematical concept cannot be explained through math concepts and understanding it is a trick and some point down the road the student will not be able to use the trick. Hitting a brick wall stopping them in their progression of complex concepts and understandings.
Three Steps on the Road to Fluency with Basic Facts
Make sure that there is understanding and strategies before moving to extensive practice. Without the understanding and strategies the practice will result in a lack of flexibility in problem solving.
Teach for understanding with models.
Teach the computational strategies.
Practice, Practice, Practice!
Limitations and Risks of Timed Mathematics Tests
Timed tests offer little insight about how flexible students are in their use of strategies or even which strategies a student selects. Evidence suggests that efficiency and accuracy may actually be negatively influenced by timed testing. A study of nearly 300 first graders found a negative correlation between timed testing and fact retrieval and number sense (Henry and Brown 2008). Children who were frequently exposed to timed testing demonstrated lower progress toward knowing facts from memory than their counterparts who had not experienced as many timed tests. In fact, growing evidence suggests that timed testing has a negative impact on students (Boaler 2012, Henry and Brown 2008, Ramirez et al. 2013). (from Teaching Children Mathematics, April 2014, pp 488 – 497)
Defining Fluency
Jennifer Bay-Williams & Gina Kling (from Teaching Children Mathematics, April 2014) A variety of interpretations exist for what procedural fluency (in general) and basic fact fluency (specifically) mean. Fortunately, recent standards, research, and reports provide a unified vision of what these terms signify. Part 3 of the Introduction to the Texas Essential Knowledge and Skills for Mathematics “For students to become fluent in mathematics, students must develop a robust sense of number. The National Research Council's report, "Adding It Up," defines procedural fluency as "skill in carrying out procedures flexibly, accurately, efficiently, and appropriately." As students develop procedural fluency, they must also realize that true problem solving may take time, effort, and perseverance.” Likewise, Baroody (2006) describes basic fact fluency as “the efficient, appropriate, and flexible application . . . an essential aspect of mathematical proficiency” (p.22). These definitions reflect what has been described for years in research and standards documents (e.g., NCTM 2000, 2006; NRC 2001).
Mathematization means bringing a more mathematical approach to some activity. For example when a student pushes some counters aside and solves an addison task without the, we say they are mathematizing, since it is mathematically important to reason about relations independent of concrete materials. (Dr. Robert Wright, Developing Number Knowledge 2012) To assess basic fact fluency, all four tenets of fluency (flexibly, accurately, efficiently, and appropriately (strategically)) must be addressed. There is an Early Learning Progression of Mathematics that can assist in planning appropriately for student success. The Numeracy Continuum Chart outlines a progression of learning that can be used when observing students working on problems in Mathematics. The aspects described should not be regarded as distinct from one another, not developing in a fixed order. Rather, the aspects are overlapping and interrelated. Additionally, assessments must provide data on which facts students know from memory. Timed tests are commonly used to supply such data—but how effective are they in doing so?
Fluency: So much more than Fast and Accurate!
Linda Gojak (NCTM Past-President) – from NCTM Summing It Up, Nov. 1, 2012 Our students enter school with the misconception that the goal in math is to do it fast and get it right. Do we promote that thinking in our teaching without realizing it? Do we praise students who get the right answer quickly? Do we become impatient with students who need a little more time to think? As we strive for a balance between conceptual understanding and procedural skill with mathematical practices, we must remember that there is a very strong link between the two. Our planning, our instruction, and our assessments must build on and value that connection. Fluency entails so much more than being fast and accurate!
Balance is Needed
Fortunately, children can learn facts effectively without the use of timed testing. In a longitudinal study of twenty second graders, Kling found that without any timed testing or other rote fact activities, by the end of the year, the children demonstrated automaticity with addition facts (solved within 3 seconds) 95 percent of the time. Interestingly, the children performed strategies (e.g., making ten or near doubles) so quickly that it was impossible to distinguish between highly efficient strategy application and “knowing from memory.” Since the beginning of first grade, fact practice for these children had involved (a) activities within textbook lessons (b) weekly fact games and (c) activities such as Quick Images with ten frames that were used to foster discussion around strategies. This research suggests that timed assessments and drill may not be necessary for children to achieve fact mastery. (Gina Kling and Jennifer M. Bay-Williams, Basic Fact Fluency April 2014)
Using formative assessment strategies
Think about how you assess reading fluency. Does your assessment plan involve listening and observing as children read as well as asking reading comprehension questions? Now imagine what you might learn about students’ reading fluency if you used only timed quizzes. How would your confidence in your assessment change? Formative assessments—including observations, interviews, performance tasks, and journaling—have become common practice in many classrooms, with a recognition that by using different ways to assess children, we gain a more comprehensive, accurate picture of what they know, what they do not know, and their misconceptions. These data are then used to design instruction accordingly (Wiliam 2011). Yet, in spite of this trend in other areas of education, timed, skill-based assessments continue to be the prevalent measure of basic mathematics facts achievement. As a result, many rich opportunities for assessing basic fact fluency are lost. There are a variety of ways to formatively assess basic fact fluency. With an eye on the aspects of fluency (flexibly, accurately, efficiently, and appropriately (strategy)), use various assessment strategies to see what students know (and do not know) and determine what next instructional steps might be. All approaches have been used with children in grades 1–4, and when used in combination with one another, these methods provide a comprehensive picture of mastery.
How do you organize your planning for each student?
Review this chart if we strategically plan for instructional practices that support student knowledge of plus 1, doubles, five plus, ten plus and partitions of 10 covers the majority of summation using 1-10. Noted in white are those summations requiring relational thinking. Six summations whose totals are less than 10 that require relational thinking and 36 of the total 100 summation. This specifically shows us that deliberate instruction in the areas of these strategies can result in an understanding of ⅔ of the summation up to 20.
Multiplication Strategies and Basic Facts Charts.
Multiplication Strategies and Basic Facts Charts.
Assessment should be for guiding Instruction
Leverage personal conversations: Conversing with students remains one of the most powerful and meaningful ways to check for understanding. In our distance learning environment, we run the risk of being further isolated. By scheduled individual sessions with students, we can assess their learning and provide feedback with a real human connection. In fact, we should be focusing more on these types of assessments in the distance learning environment, Video tools can bring a human element to the assessment process.
Make it useful: Data is useless unless it is used. When we collect and examine formative assessments, we need to use what we learn from them to inform instruction.
How Do Math Fluency Workstations Work?
After completion of the Universal Screener and possibly AVMR assessment interview, , the teacher interprets the findings and identifies each student's stage of learning for each area assessed. Based on the class data the teacher then selects a beginning focus area from the strands assessed for Number in K-2 (Counting, Numeration, Addition and Subtraction) or the Grades 3-5 strands (Addition and Subtraction, Multiplication, Division and Decimals). Students are placed in partnerships according to the stage they are at on the continuum for the selected strand and center activities are chosen accordingly.
Students work on math fluency Workstations in partnerships for 12-15 minutes 2-3 days a week, often repeating the same center for 5-6 sessions. This repetition of the same task plays a crucial role in the success of these Workstations. On the first day a center is introduced students are often focused on the rules, new materials, or ‘how to do’ the task. Repeating the center several times allows students to engage with the math skills and concepts at a deeper level. As students work in their center they are expected to play the game, explain their strategy to their partner and record their thinking in a journal. This notebook provides a record of the child's progress throughout the year. Strategy Cards can be a support to remind students of different mental strategies.
Resources
Resources
Addition and Subtraction Strategies
Addition and Subtraction Strategies
Multiplication and Division Strategies
Multiplication and Division Strategies
Resources for developing workstations:
SCUC implements the best practices developed through US Math Recovery and in particular Add+Vantage Math Recovery training and materials. We have SCUC Math Foundation Courses under the umbrella of NUMBERS. Specifically we have broken the course down into two extensive book studies and developed websites with supporting information and resources:
Contact Math Coordinator for enrolling in one of these Foundational Courses.
As students work in the math fluency Workstations the teacher observes individual students and takes conference notes when in small group instruction on students progress. Particular notes are taken as to the student's progress with the 'Next Steps' for the particular stage/level/construct they are working on. If the teacher feels that a student has mastered the focus skill a new fluency center is introduced to provide an opportunity to work on another skill within that stage on the continuum. Once the student demonstrates mastery of all skills listed in a stage on the continuum a new center aligned with the next stage on the continuum is introduced. Below are printable workstations that accompany the Numbers materials.
K-2 FLUENCY WORKSTATIONS
K-2 FLUENCY WORKSTATIONS
GRADES 3-5 FLUENCY WORKSTATIONS
GRADES 3-5 FLUENCY WORKSTATIONS
When is the Best Time to do Math Fluency Workstations?
In order for fluency Workstations to be effective it is important to set aside a 12-15 minute block of time 2-3 times per week. This could be the first thing in the morning when students enter the classroom, at the beginning of a math workshop, or straight after lunch or built into workstations. Develop a schedule that works best for your students and stick with it. This way Fluency Workstations become part of the classroom routine and you are less likely to skip them when things get busy.
How Should I Introduce Fluency Workstations?
When introducing Fluency Workstations for the first time begin slowly. You may want to give the whole class the same workstation for the first day or two so that you can focus on setting up the expectations for sharing materials, taking turns, packing up materials etc. Model how to play new games and role play how to solve possible problems that may arise without teacher assistance. After a few days introduce a second center and run two Workstations simultaneously for a day or two. Gradually build up the number of different Fluency Workstations that are running until all students are working in a group that is appropriate to their stage on the continuum. These mini groups can exist within your workstation groups.
As with any new venture the first few weeks of Fluency Workstations are the most difficult. By preparing materials in advance and taking the time to make students aware of expectations you will soon begin to see an improvement in the number sense of all students in your classroom.
How Can I Organize Fluency Workstations to Run Smoothly?
How Can I Organize Fluency Workstations to Run Smoothly?
As fluency Workstations are only scheduled for 12-15 minutes per session (ideally3 minutes set up, 10 minutes playing the game, 2 minutes pack up) it is crucial to have good structures in place to make the best use of this short period of time. Many teachers like to display a digital timer on an interactive whiteboard so that students can keep track of the time and begin to pack up at the 18 minute mark without any teacher prompting.
Having all the materials needed for a specific center activity in one heavy duty plastic bag or plastic envelope makes for a smoother start to fluency center sessions and makes pack-up time easier to manage. Bags can be stored on a stand or in baskets to ensure that they are easily accessible by students. Designating one student per group to collect and return materials can also help things to run smoothly.
Printing Fluency Workstations on cardstock, or printing on paper and laminating, will extend the life of the Workstations and allow you to reuse them from one year to the next. Reusable dry erase pockets are another option. These are useful for protecting and extending the life of center activities that involve the use of a game board, but are particularly useful for Workstations in which students need to record on a game board or sheet of paper as they eliminate the need to continually photocopy recording sheets. Simply insert a game board and have students use a dry erase marker to write on the pocket. When the center is completed, the marker is wiped off the pocket and is ready to be used again. Task display stands can also be used to keep task instructions clean instead of laminating task cards. Simply slip the task into the clear stand and place in a central spot so that all group members can read the instructions.
Why do Some of the Workstations Have Sentence Frames?
In order to develop students’ mathematical language sentence frames have been developed for use with many of the center activities. These frames provide language support for ELL students and help to develop the math language of all students in a meaningful context.
What is a Goals Chart?
An ancient Chinese proverb notes that no wind is favorable if one does not know to which port one is sailing. Setting goals can have a positive impact on student self-efficacy and achievement. Goals give students a clear picture of what the expectations are as well as something to strive for. This is important because it helps to motivate students and also provides a sense of accomplishment when goals are reached. The photographs show examples of class and individual fluency goal charts from K-2 classrooms. On the class chart on the left students wrote their name on a post-it and placed it alongside their current goal. On the class chart in the center each students' name was written on a clothes peg and moved as goals were met and new ones set.
It is all about the TARGETED Instruction you provide
It is all about the TARGETED Instruction you provide
There is a great deal of research and information gathered to assist with Small Group Instruction on the SCUC Making the most of Small Group Instruction. There you will find organization and planning templates that will help you in providing Targeted Instruction to advance students' mathematical thinking and success.
There is a great deal of research and information gathered to assist with Small Group Instruction on the SCUC Making the most of Small Group Instruction. There you will find organization and planning templates that will help you in providing Targeted Instruction to advance students' mathematical thinking and success.
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