Do Multiple Ways Matter: Using Tech to Make the Connection

Let's be clear counting in small quantities is a skill most kids do naturally without much prompting or coaching needed, in fact the region of the brain we use for counting includes the same portion of the brain that controls our fingers. Research suggests this may be attributed to the fact that our ancestors first experience with numbers involved the fingers (Devlin, 2000).  However counting and calculations that go beyond our friendly finger tips are likely to result in errors. 

When it comes to calculations students need strategies and not just one.  Research suggests students will most likely use a particular strategy that they find to be a more efficient solution for a particular type of problem (Sieger & Jenkinds, 1989). Take for example the problem 6 x 42 . Students who have proficiency with breaking apart numbers can determine they will need to calculate 6 X 40 (240) and 6 X 2 (12) and mentally calculate the total of 252, but when these numbers become much larger 656 X 3245 the standard algorithm may be more efficient. 

Common core mathematics shifts the focus from learning one-way and one-algorithm to understanding the underlying principles of a concept and applying multiple algorithms.   This approach certainly lends itself to going deeper with math through multiple representations and ways of showing what you know.   

The idea that students don't begin with the end in mind but begin with understanding and developing concepts is at the heart of the common core. 

This shift in standards does not guarantee a shift in learning, this will only happen when teachers change the way they teach and curriculum evolves from focusing on some learners, to all learners in the classroom.  From gifted and talented to students with special needs and English language learners, our approach and modes of instruction need to be flexible and supportive of our classroom population.  Students need a variety of pedagogical approaches from number talks that support discussions of strategies and mental calculations, to manipulatives that allow students to construct models and make meaning of concepts.  


So when parents ask, "Why can't they just memorize their multiplication facts"  we can assure them that memorization does not promote understanding and automaticity will develop with practice.  Elementary teachers should begin introducing concepts by building on what children already know and albeit this may be intuitive, it can lead to a deeper understanding of the concepts.  

Rooting math in the lives of the students we teach can support building conceptual understanding as well as transfer the learning of math  (number words, symbols and quantities) into their informal learning experiences such as the park, playing games and with friends.  Take for example the idea of using arrays to introduce the concept of multiplication.  Arrays are all around students but this knowledge needs to be brought forward during instruction and through practice.  It's not enough just to talk about where you might see arrays kids also need to  construct , discover, apply and identify. We should understand that what works for one kind of learner might not work for another.  Woodward and Baxter found that students with disabilites in math tend to make significantly less growth in discussion-oriented classes (1997) than traditional ones.  

Technology can be a great mediator to support, and challenge students with open ended tasks and flexibility.  It can also be useful to move from the abstract to the real-life connection.  

Available as a Google Slide here

Repeated addition is the knowledge students can start with to build an understanding of multiplication, but some students in your class might not have developed automaticity of their addition facts.  Working with arrays that are highly contextualized and not just on a piece of graph paper can provide practice in addition while also introducing the concept of multiplication.  In this video a second-grader works on a Google Slide presentation his teacher made to drag and drop cars into a parking lot.  

 

If our ancestors first counting tool was their fingers then digital devices might be consider the cultural tools for our students today.  

 Using this digital lesson teachers can scaffold instruction and allow students to work at their own pace.  While some students might work with benchmark numbers 2's, 5's, and 10's to construct an array, students who understand the concept of multiplication and have strong number sense in this area can move to more complex equations.  

Are you using technology to support students conceptual understanding in your math class? Share your ideas here and join the conversation on our Facebook Group! 

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3 Reasons to Use Virtual Manipulatives in 3 Easy Steps

Manipulatives come in all shapes and sizes, from base ten blocks that support number sense concepts to geometric shapes that dispel the myth of a solid being flat.  We love manipulatives because it provides students with the tools to make sense of math concepts and model with mathematics in an intuitive way. What we don't love about manipulatives is they get lost, need to be organized, we may not have enough, and perhaps they are being used for something other than what we intended.  

Virtual Manipulatives can become somewhat of a panacea to the woes of tactile manipulatives.  There is plenty for everyone, no need to organize and you won't see one being thrown across the room or lost in a student's backpack.  Virtual manipulatives also may include a mat to organize students placement and self-checking features to give students instant feedback.  

Oh yes~whether their real or virtual, manipulatives should and must be a part of instruction and this belief is echoed in the Standards for Mathematics Practice which informs teachers on how to teach math, and explicitly states students should "Model with Mathematics" 

Beyond the drag and drop attraction of virtual manipulatives (VM) students really get into the flow of learning with virtual manipulatives. Perhaps for the same reasons as to why kids love to play video games, virtual manipulatives have some of the same appeal: competence, autonomy, and relatedness.  

Competence: When students use VM tools they get instant feedback and have the capacity to keep trying until they achieve mastery.  

Autonomy: When students work with virtual manipulatives they have the control over what they use and how they use them.  They don't need to share, and they are free to make mistakes without someone telling them it' wrong and why.  They may get feedback but it's not a real person!   

Relatedness:  Virtual manipulatives have some of the same features as videogames with choice in tools and how they use them.  They sometimes get points and rewards just like videogames do too! 

Teaching with VM is very similar to how you would use real-life manipulatives.  Here are my quick three steps: 

1. First, plan how you will have your students use the VM and what you will have them do with them.  Some tools provide problems for students to solve others will not.  If problems are not provided create a worksheet that students can record their response.  If problems are provided have students transfer the skills and ideas they are learning about with VM into the procedures and processes they are using without VM.  This might include either recording the problem and solution or writing out the steps to solve with VM.  If students receive a score at the end of play have them record so they can track their progress over time (growth mindset).

2.  Second, you model with the students the way the tools can be used and what each of the tools represent. Not all VM tools are created equally and you will need to explain and demonstrate how to use the tools. Solve a few problems and demonstrate with your students before they work independently,

3.  Check in on students and determine misconceptions and areas for support.  After you set them free walk around the room and help troubleshoot then bring everyone back together to address misconceptions and share out how students are using VM.  

Okay so now you want to know what VM I recommend.  Here we go: 
    National Library of Virtual Manipulatives: create a worksheet of  problems to solve. Go deeper with concepts in visual form; I love the equivalent fractions tools.  
    National Council of Teaching Mathematics Illuminations has a variety of tools across k-12 span.  The algebra tiles tool is my favorite to build equations and solve.  
    Glenco Math Tools: these tools are fun and provide much space for students to explore, create and craft situations around math in the real work.

Got a Virtual Manipulative tool to share or way to use it that has captured your students attention please LEAVE A COMMENT! 

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Getting Start with An Hour of Code

Ask the average 8 year old boy what he wants to be when he grows up and you will probably hear “make video games”.  Boys love video games because it gives them an opportunity to use their logic and reasoning skills in the context of a virtually crafted world that includes tools to build and create.

The idea of having “video game” time in any classroom would probably raise some eyebrows by parents and teachers alike.  But we know that creating an engaging learning environment means tapping into students interests so why not use the concept of video games to get students to read, write, problem solve and activate higher order thinking skills.

So that is exactly what I did, I finally took the plunge this week and let my son engage in an “Hour of Code”

As an elementary teacher I could easily see the connection to mathematics, reading and 21 Century Skills.  My son had to read the directions to figure out what to do, he had to watch the video to see a visual demonstration of the process (multiple representations), and then he had to try his code out to see if the program worked.  When the line of code he wrote was not successful at achieving the task he went back and tried again.  The Mathematics Practice Standard “Make Sense of Problems and Persevere in Solving Them” was so evident in his thinking process.

The deeper he went in the coding process the more evidence for higher level mathematics.  He could use the “repeat loop” to write a line of code where a process is completed at a faster rate just like multiplication.  Then during more complex tasks he was introduced to the “If___Then____” statement which are conditional statements in math.  I was so jazzed to see the higher mathematics application being introduced to kids at a much younger age and they are motivated to do it because it is in the context of video games

The first few tasks my kindergarten completed as it required him to count the number of blocks and write a line of code that would get Steve a character in Mindcraft to shear a sheep oh yes the coding was about video games and they also have a Star Wars and Frozen version.  My eight year old son was so in the flow he promised me that he would write a blog post about his experience coding if I let him finish his task.

As a teacher I am all about creating an interdisciplinary lesson that includes not just math and reading but also writing so feel free to check out my son’s blog about what he learned from an hour of code.

Braeden’s Blog: The Code of Mindcraft

Please share your experiences on how you can or have used coding in the classroom?

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