In these initial phases, we are teaching kids how to reflect. To look over their week and explain certain images where they learned, wondered, or were confused about. Many have never been asked to “reflect”, especially in math. As we begin this year, we are trying to introduce this and make it a habit. The hope is that the mathematical language becomes stronger and the thinking becomes deeper and meaningful to understanding concepts better.
Precision teaching is the idea that relates to the book, Visible Learning by John Hattie. The premise behind Hattie’s research is the correlation between surface, deep, and transfer learning.
The findings go on to address the influence that different aspects of the education system have on our students and the math classroom. Basically, the idea of getting the “most bang for your buck”.
Every school district, school, and classroom teacher wants a magic formula for success. With limited time and funding, how do we ensure our academic goals are being met in the math classroom?
As math teachers, how do we know if our students are learning?
Setting routines are one of the cornerstones of a
successful classroom. It is important for teachers to understand the difference
in a compliant classroom and an environment conducive to learning.
When beginning the school year it is crucial to
ask ourselves a few questions when considering routines and procedures.
Do your students know what they are supposed to be learning? And more importantly why?
Are your students benefitting from your expertise?
Are your students capable of connecting their learning to what they’ve already learned and are they able to see where this learning goal will lead?
Are your students able to manage their own learning?
At the end of the lesson, are your students able to hold themselves accountable for what they have learned? (leading into self-assessment)
Self-assessment has one of the largest effect sizes for student learning (Hattie, 2009). The process of self-assessment requires several different elements implemented in the classroom.
One example of self-assessments in the math classroom is the use of Math Journals. Beyond a journal prompt containing the “content goal” teachers have an amazing opportunity to go beyond a right/wrong answer by providing students with a voice for self-reflection.
Below is an example of a pre, mid, and post year self-assessment used in a fourth-grade classroom to gauge confidence and attitude toward mathematics.
This same questioning is also used in this classroom in the aforementioned Math Journal activity. The example seen below addressed the math topic of Place Value. The students were asked to describe the value of the 8 in the number 88,888.
They could use pictures and words. The prompt was actually shared at the beginning of class as an informal discussion. Students were asked to read the prompt and think about what they already knew about place value, what they remember from previous lessons that would help them with today’s goal, if they felt like the topic was going to be difficult and if so, why?
They were given time to think about their responses and write them in their journal, then after the lesson, they returned to their journal to work on the prompt and reflect on their thoughts from the beginning of class. Once journals were complete, they were allowed to share the journals with their group.
The outcome of this part of the activity is to foster a safe space for math discourse, more importantly, mathematical talk is proven to lead to the ultimate level of transfer of learning and metacognitive awareness.
If self-assessment is the goal and math journaling is one tool, what is the payoff? How do these techniques help spark success in our math students? Metacognition is the ability to think about our thinking (almost sounds too simple).
While it may sound simple it is the cornerstone of all actual and true learning. Metacognition happens when students practice self-reflection on their level of thinking. In addition, when students can relate this to a target it becomes powerful enough to increase understanding and motivation.
This knowledge promoted in the classroom is invaluable. As teachers, we all dream of that day when our students are intrinsically motivated. A moment in the classroom when the value of the content rises above a state standard or an arbitrary notion but becomes a self-fulfilling desire to learn and grow. But students need guidance and tools to develop their metacognitive awareness and become confident in the ability to self-question.
Formative evaluation is the process of gathering evidence to inform instruction. In simpler terms, it is the process for teachers and students to communicate about the learning that is taking place and the direction that instruction should go. Formative evaluation should drive instruction decisions, gathering real-time data is crucial in guiding how the teacher will proceed with the delivery of the lesson.
In the book, Visual Learning, Hattie speaks of several internal questions
that drive learners:
Where am I going? What are my goals?
How am I going there? What progress is being made towards my goal?
Where to next? What activities need to be undertaken next to make better progress?
Consequently, these are the same three questions that teachers must ask about instruction as they make adjustments based on the data they gather from students. Some examples shown below, include response cards (whiteboards), and exit tickets, journal entries may also serve as a way to inform instruction.
Follow our Journey. Lesson Study with a Focus on Visualization.
Jugyou kenkyuu, a Japanese phrase gives us the term “Lesson Study”. Introduced in the U. S. in the late 1990s, interest in Japanese lesson study remains strong in the education world throughout the United States. Our Lesson Study this year will focus on visualization and metacognition.
Lesson Study & Mathematics
Lesson study works well across education and in particular, in improving mathematics education. We will wrap up professional summer reading on visualization in September with a look into the routines we create in classrooms that promote visualization. During “Introduction to Lesson Study” in October, we will explore what lesson study is, how it works, how to use it, and best practices with a focus on creating metacognition in students.
Pre-Lesson Study Questions
We engaged our focus group from St. Edward School in Vero Beach by asking the following questions:
What attracted you to this Lesson Study?
Overall the participants felt this lesson study would improve their ability to use visualization strategies in their own classrooms. They felt the experience would allow them to “dig deeper” into learning the best way to improve their teaching skills to build visualization.
What do you hope to learn from this Lesson Study
Participants generally responded similarly, wanting a deeper understanding of the science behind visualization, learning how to integrate visualization into their daily teaching, and using visualization to help students see concepts in a different way.
What is visualization to you?
It is creating a picture in your mind, being able to ‘see’ what you are hearing or reading to help you better understand the lesson, and it brings life to situations, assisting a student in understanding the concepts being taught.
What do you feel you already know about visualization? (before reading)
The response to this question was consistent with all participants. All felt that visualization was a way of seeing something in your mind to better or fully understand it and using it in math as well would bring life to situations and assist students in better understanding the concepts being taught.
Ideas on how to get kids to visualize math?
Using various concrete and pictorial models
Incorporating color in our board witing to connect ideas
Relating ideas especially in the operations
Have children create a short movie in their minds with each math concept so they can ‘see’ the process and verbalize it before computing
What questions do you have before we start the lesson study?
Can all students visualize?
How are other teachers using visualization?
Does the brain have any physical limitations with visualizing?
How do we teach visualization to students so they use it seamlessly when seeing a math problem?
What forces the brain to want/have to visualize?
We will be holding a private Lesson Study at St. Edward’s School, Vero Beach, FL in September.
Follow us through this Lesson Study.
We’ll be at Oak Hill High School in Nashville, TN, October 2, 2019 – October 4, 2019. Seats still open!
Click here to register for this event and for details on this Lesson Study.
As schools begin to embrace technology in the classroom, one question that Sarah often receives is, “What apps do you use in your workshops and classrooms?”
With a multitude of apps available, it can be challenging to
find the right app that can help teachers teach better and engage students more
Sarah shares her list of frequently-used apps below.
Drawing and Writing Tools
Algebra Balance (iOS $1.99)
Bamboo Paper (free for iOS, Android and Windows)
Turn your screen into a paper with Bamboo Paper! Draw and write effortlessly with the various pens and brushes available. The zoom function in this app allows you to write in small spaces and fit more notes onto a page. This app also allows you to add images to your notes and write on them. Notebooks created within the app can also be shared and exported to other mobile platforms and cloud services.
Noteshelf (iOS $9.99)
Personalize your notes with Noteshelf! Individual notebooks can be easily customized in Noteshelf using the extensive library of cover designs and page templates in the app. Writing and drawing tools are available to help you write notes and draw precise geometric shapes with ease. There are also several fonts to choose from within the app if you prefer to type.
In Noteshelf, notes can be written across pages, just like a physical notebook, making it easy to create sections. Noteshelf also allows you to insert photos and write on them. Another excellent feature is its audio recording function, which means you never have to worry about missing out on the nitty-gritty details. Making annotations and signing PDFs is also a useful feature in Noteshelf. Other features include scanning of documents, password protection of notes, and syncing with Evernote and cloud services.
Pencil Box (iOS $0.99)
Pencil Box is a drawing tool that will help anyone who wants to draw with precision. To start, pick a drawing tool and begin drawing or import an image to draw on. Besides the many brushes and shapes available, Pencil Box also allows you to add and edit layers of your drawing, making it easy to add or delete parts of a drawing without having to start over.
Geometry Pad+ (iOS $6.99, Android $5.99)
Geometry Pad+ is a dynamic app that allows you to construct geometric shapes, such as circles, squares, triangles, rhombuses, and parallelograms, with precision and ease. Beyond creating shapes, this app also allows you to mark out angles within shapes and draw arcs using the in-app tools. The app also comes with a measuring tool that can measure lines and other properties. Other shapes properties, such as perimeter and area, can be calculated using the in-app tools as well. All documents can be saved and exported, making it convenient for future use and reference.
GeoBoard (free for iOS, Windows, and web browsers)
Learn about shapes in a fun and interactive manner! In Geoboard, bands can be stretched around pegs on a virtual board to create line segments and different types of polygons. Available on both mobile and web-based platforms, GeoBoard can also be used to illustrate area, perimeter, fractions, angles, and many other concepts. With three board types to choose from and multiple band colors to work with, this app is bound to bring a lot of fun to the classroom.
Number Lines (free for iOS and web browsers)
Available as a mobile and web app, Number Lines helps you to customize number lines easily for your classroom. This app allows you to create number lines with whole numbers, fractions, decimals, and even negative numbers. Students can visualize number sequences and demonstrate strategies related to counting, comparing, adding, subtracting and much more. Elements of the number line can also be hidden to encourage creative thinking.
Pieces Basic (free for iOS and web browsers)
This interactive app helps students learn about place value and develop their computation skills with multi-digit numbers. Number Pieces Basic is a great visual aid for teachers to use in classrooms to engage students as pieces can be dragged around to illustrate addition and subtraction problems. Equations and expressions can also be written using the text tool. The app can be used on-the-go on a mobile device or from a web browser on a computer.
is a great visual aid for teachers to use in classrooms to engage students as pieces can be dragged around to illustrate addition and subtraction problems. Equations and expressions can also be written using the text tool. The app can be used on-the-go on a mobile device or from a web browser on a computer.
Manipulative of the Week (free for iOS)
This app bundle provides a free manipulative every week and contains the 14 most popular manipulatives used by students and teachers. Some of the apps that Sarah uses frequently are listed below.
Algebra Tiles (iOS $1.99)
Algebra Tiles provides an engaging experience for students to explore algebraic concepts. Using tiles to represent algebraic expressions, students can learn to add and subtract integers as well as solve algebraic equations.
Base Ten Blocks (iOS $1.99)
Base Ten Blocks are virtual blocks to help students learn about place value, addition, subtraction, regrouping and more. Students can work with blocks representing ones, tens, hundreds and thousands to explore addition, subtraction, multiplication and division strategies. The place value chart is especially useful when working with decimals as well.
Color Tiles (iOS $1.99)
Use Color Tiles to build number frames, make shapes, find fractions and more. The user-friendly interface gives users the freedom to create and customize manipulatives to suit their learning needs. Color Tiles helps students to develop their computing skills while enabling them to learn how to sort and classify objects.
to build number frames, make shapes, find fractions and more. The user-friendly interface gives users the freedom to create and customize manipulatives to suit their learning needs. Color Tiles helps students to develop their computing skills while enabling them to learn how to sort and classify objects.
Cuisenaire® Rods (Number Rods) (iOS $1.99)
Number rods can be placed along a number line to compare numbers and fractions. By adding and removing rods, students can visualize the addition and subtraction of integers and fractions. This app also enables students to view ratios and proportions at a glance.
Fraction Circles (iOS $1.99)
In Fraction Circles, fraction pieces can be moved, rotated, overlapped and put together. It is a versatile app as fractions, decimals and percentages can be used to label each fraction piece. Each fraction piece is color-coded, making it easy to identify and work with as well.
Fraction Manipulatives(iOS free)
Hundred Board (iOS $1.99)
Linking Cubes (iOS $1.99)
Linking Cubes are virtual multi-colored cubes that help students to visualize numbers in a pictorial manner. Students can learn how to add, subtract, multiply and divide using different colored cubes. Place value can also be represented using the place value background available in the app. The built-in graph background makes it easy to create graphs as well.
Little Bit Studio Bugs & Buttons (iOS $2.99)
Number Frames (iOS free)
Osmo Tangram (iOS free)
Pattern Blocks (iOS $1.99)
Pattern Blocks is the closest thing you can get to physical pattern blocks. With a wide selection of two-dimensional shapes, including hexagons and trapezoids, students can learn about geometry, patterns, fractions, and decimals in a fuss-free manner.
Place Value Disks (iOS $1.99)
Software Smoothie Felt Board App (iOS $2.99)
Two-Color Counters (iOS $1.99)
Two-Color Counters is a minimalistic app for those looking for a simple yet effective way to understand numbers, integers, and fractions. Only two colors are used in this app as its focus is on illustrating operations concepts. Teaching ideas aligned to Common Core are also featured in this app.
Are you inspired? Try out these apps in your own teaching or learning journey and let us know the results!
Math facts are simply the basics: addition, subtraction; multiplication; division. They are basic number combinations and calculations we do every day. So why is it that learning math facts creates such huge problems for teachers and students in our classrooms?
Common subtraction mistake when children are learning math facts.
Is it that some of us are just naturally better at math than others? Perhaps. The good news is you as a teacher can help anyone improve their math skills. You might be wondering how. Well, rote memorization alone won’t get us there.
Often we teach the meaning of equality as the equal sign indicating “put the answer.” To prepare students for future mathematics, we need to rethink the way we communicate this representation to students. The equal sign is too important to attach such a limited meaning, especially when students are moving to abstraction. When moving to abstraction, it is important to use this symbol as a “reader” versus a “speller.” What do I mean by that? Let me explain. First, a speller sounds foreign because the student is trying to make sense of abstract symbols that they don’t fully understand. A reader connects meaning that they can comprehend while connecting the story to the symbol.
For example, when teachers record the following equation,
Using a simple addition problem as a reader versus a speller.
A speller would say, “Four plus five equals nine”.
A reader would say, “Four apples and five apples is the same as saying ‘I have nine apples’” .
In the journal prompt below, we want students to see the statement as 5 and 4 is the same as 2 less than 11. If we focus too much as a speller, students see symbols as performing actions rather than relating ideas, which leads to misconceptions of the equal sign.
Less is more. Hesitate in thinking math becomes harder by creating larger numbers, rather provide depth early on using children’s intuitive understanding of basic number operations. A simple journal entry can help assess what students understand about operations and the meaning of equality. Once you find that a student understands the meaning of addition and subtraction, it is time to create tasks that dive deeper in understanding equality. These task can be as simple as asking questions such as, 4 + 5 = ____ + 6. Additional equality tasks can be found in the Algebraic Thinking and Reasoning with Numbers books (Groundwork Series) by Greenes, Carole, et al., (https://www.mheducation.com/prek-12/program/MKTSP-O8302M06.html).
In the book, “Thinking Mathematically,” Carpenter, Thomas P., et al. suggest some benchmarks to keep in mind while moving towards the conceptual understanding of equality.
Establish a starting point. What are students’ initial ideas regarding the equal sign?
Mix it up. Don’t always write equations in the form a + b = c, rather c = b + a.
See the equal sign as a relational symbol. Emphasize students proving that each side is the same as the other.
Compare sides without calculations. Encourage students to look for relationships without performing calculations.
Listen to students and be aware of where they are in the process. Encourage them to question if it is true. If so, why is it true? By listening and looking we may be pleasantly surprised at what we find out.
Below is student work demonstrating various levels of understanding within the concept of equality.
Spiky says that 5 + 4 = 11 – 2 is false.
Curly says that it is true.
Who do you agree with and why?
This student understands how to simplify expressions but lacks applying the meaning of equality.
This student is confused in understanding what it means to have expressions on each side of the equal sign.
This student understands the meaning of equality. What can we ask next?
This student demonstrates a shallow understanding of equality. What can we do next?
This student can solve for an answer but needs practicing in mixing up the form of an equation.
How can we help student refocus on understanding versus just an answer?
Too often when we pose a problem and students shout out an answer. We need to ask ourselves do students understand the concept or are they obtaining the correct answer by fitting the symbols and numbers into a structure they know?
In this example, the anchor task was to subtract 34 from 87. The teacher wanted to screen the children first wondering, Do students know what it means to subtract one number from another? To find out more the teacher posted the following problem, removing the numbers.
Do students have the conceptual understanding or just fit the numbers into a given structure?
Students were asked to set up the expression that could represent the situation. The student A on the far right was the only student in the class that seemed to understand. When asked to explain his thinking, many observing teachers felt he understood the concept and that he gave us a platform to generate a discussion.
Do students understand the part-whole relationship here?
Following this analysis, the numbers were inserted into the problem. Subtract 34 from 87. It was interesting to see student A’s work. Much to our surprise, Student A who seemed to understand the concept had a hard time transferring that knowledge to another situation. Notice his equation is not correct but he gets the final answer.
Can student A transfer knowledge from one setting to the next? Look and listen to help guide students to conceptual understanding. Looking at answers does not tell us the whole story.
Less is more. Spend more time on conceptual understanding and listening and watching students versus solving more problems. The answer does not help assess student reasoning or how we can extend or guide the learning process.