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.