Putting a Value on Place Value

One of the most fundamental concepts taught in math are the numbers themselves, and that they always come in a specific and consistent order. Just as crucial to the development of number theory, however, is place value. Understanding the relationship between ones, tens, and hundreds (and later tenths and hundredths) is crucial to expressing different quantities themselves and articulating the relationships between them.

Many children find difficulty with this skill initially, however. Without knowledge of place value, students may see 13 and 31 as the same number–both containing the same two digits. Place value puts an emphasis on digit order, and gives reason for why 31 is larger than 13. An reference article found on education.com says, “Learners can correct [any] misunderstandings by solving real-world problems with hands-on materials and learning aids such as counters, base ten manipulatives, and place value charts.”

1) How to use base-ten blocks:

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Particularly catering to the visual and kinesthetic learners, base ten blocks allow students to manually display and manipulate each place value of a number. Yet, because each unit is indented into each piece (10 units = 1 long, yet 10 units can still be seen and counted), they reaffirm each student’s work.

This blogger cut up pool noodles (and an accompanying book–yay interdisciplinary!) to use instead as base-ten blocks. The same idea, but the larger pieces may appeal to younger students and add a stronger kinesthetic element.

pool noodle place value Collage

There are so many activities that teachers can do with these manipulatives (tackled later in this post), but the biggest goal is to have students practice modeling 1, 2, and 3 digit numbers with whatever tools you decide to use! Click here to find the worksheet below–a perfect way to guide students in their independent exploration:

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2) How to use place-value charts:

While this tactic is much more visual, it can still be a helpful organization tool for students to see the relationship between the digits of a number, especially larger expressions. Here is an example below:

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An activity like this contains a few different approaches. Students can practice reading numbers that are already on the chart (coming up with the expression “three thousand, six hundred, eighty four”), or they could write the numbers on the chart based on verbal dictation of a four, five, or six digit number. These charts put an emphasis on the labeling of place values, and not as much on the composition of numbers (like the base-ten blocks do).

3) Fun with Place Value! 

While the manipulatives listed above are crucial learning tools, I say that there can always be fun found in math! The following are some very fun ideas that I’ve stumbled across. Click the link to see the original source and to learn more about the game or craft.

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I hope these ideas spark some inspiration for place value emphasis in your classroom!

Differentiation–A Daily Challenge

In my about me section, I mentioned that I work as a 1:1 paraprofessional. Specifically, I am an intervener for a middle school student who is deafblind. While my student does have residual hearing and vision, we do things a lot differently in the classroom than her peers do. I, therefore, am always on the lookout for how teachers are able to differentiate their lessons–particularly more visual lessons, like those found in math. My student needs a mix of all three learning styles, I’ve found, in order to be the most successful. A quick review of the three primary learning styles for those who are a little rusty:

Learning-Styles

And this need for differentiation doesn’t exclusively relate to the needs of my student, but it extends to the challenge all teachers face of differentiating for all students’ learning styles. A student doesn’t need to have low-vision to prefer auditory lessons, etc. We as human beings prefer to learn information in different ways. And in a classroom of 30 students, one will usually find 30 different learning styles.

So how can teachers tackle this challenge? How can teachers appeal to visual, auditory, and kinesthetic learners all in one lesson? An article produced by Glencoe explains that while many strategies can be used, the primary adjustment should be a shift away from lecture. Instead, have visual worksheets, outlines, and/or models to appeal for visual students. Auditory students’ needs can be met through discussion and group work, and kinesthetic learners will find success with manipulatives and hands-on activities and projects. Having a combination of all three, however, is ideal.

Why is important to cater to different learning styles? Shouldn’t students learn how to “deal with it”, much as they would have to in the adult world? Perhaps to an extent. However, in an era when math anxiety and damaging self-fulfilling prophecies are a legitimate concern in the classroom, it is our job as teachers to do everything we can to make students feel not only at ease, but set them up for success and confidence as mathematicians. Differentiating for their preferred learning style not only appeals to the student, demonstrates respect and empathy, and engages them during the lesson, but it also ideally sets the student up for longterm understanding and acquisition of the topic. Differentiation is key!

Marvelous Multiplication!

Ah, multiplication. I’ve worked in classrooms ranging from early elementary to middle school and troubles with multiplication are, unfortunately, not a rarity. Here are some strategies that classroom teachers can use to explain the concept of multiplication and to answer what that “x” symbol really means!

1) Equal Group Beads

This set up provides students with tangible units that can be used for counting and stimulating the repeated-addition model. For example, the drawing below shows 2 ways that students could compute 3 X 4:

Equal Group Beads--strands of 3 and 4
Equal Group Beads–strands of 3 and 4

Each line represents a pipe cleaner, and each colored dot represents a bead. This is an easy manipulative to let students experiment with visually and kinesthetically to compute multiplication problems.

2) Arrays

While these rectangular shapes can be made with a marker and graph paper, they can also be found in multiple everyday situations. Take this picture below:

A very delicious array!
A very delicious array!

Let’s say a friend brought in donuts for her birthday. How many did she bring? Having the conceptual knowledge of arrays prevents you from counting every single donut individually. Instead, you can count the bottom row (4) and the side column (4) and multiply these numbers together (16). When students are first learning multiplication, these are a great tool to use, as they can offer evidence to the correct answer (students could count the total to confirm their answer).

3) Timelines

I recently acquired this strategy through a class that I took (Math for Elementary Teachers I). The use of timelines can also serve as a model for showing how multiplication relates to repeated addition. Let’s look at this example:

A timeline demonstrating  3 x 3 = 9
A timeline demonstrating 3 x 3 = 9

By showing how 3 units are counted 3 times (indicated by the vertical lines at the end of each curve), students can see that 3 x 3 really means 3 + 3 + 3, both of which equal 9. This strategy also reinforces number line skills and aids in number theory.

4) Review of Multiplication Strategies

When students get the general concept of multiplication down, it may be helpful to remind them of the different patterns that emerge when multiplying by certain numbers. For example:

  • A number multiplied by 0 is always 0
  • A number multiplied by 1 is always the other factor
  • A number multiplied by 2 is doubled
  • A number multiplied by 5 is equal to that number multiplied by 10, then cut in half

Reinforcing that multiplication is a systematic operation may put students at ease. Multiplication isn’t a crazy monster, it’s just patterns!

Favorite Educational Blogs

Just like I hope to share what I’ve learned and found useful through my posts, here are some folks who are doing the same thing!

1.These ladies have some really great examples of hands-on formats for various topics.

2. This blog shows a variety of manipulatives that can be helpful to explain different concepts. Plus, she has a section of math literature titles for interdisciplinary lessons!

3. This blog, while catering to upper elementary, shows how critical thinking projects can yield fun and achievement during math class.

4. Another example of how fun really can be found in the math classroom–especially through the use of anchor charts. Click here to see a fun, lively source of inspiration for all things elementary!

5.  This blog serves as a fun and animated example of what can happen when passion and collaboration are combined. He has a lot of great ideas on here!

I hope these help you as much as they’ve helped me! 🙂

Unequal Education in our Own Backyards

It is no surprise that the US public education system is facing criticism in light of several years of evidence that show inadequate student progress in an international context. 2012 PISA (Program for International Student Achievement) Test results reveal that the Untied States ranks 30th (out of 65 participating nations) in mathematics , a shocking ranking considering the amount of resources available to our country. Yet the United States also lacks from a state-to-state, as well. According to a report from the US Department of Education regarding the 2012 PISA, three separate states participated as their own comparative samples: Massachusetts, Connecticut, and Florida. Of these students, 19% of the Massachusetts sample showed mathematical proficiency, 16% of the Connecticut sample, and only 6% in Florida.

While several different factors play into the creation, execution, and maintenance of a productive education system, one aspect seems to be a crucial determinant of success—attitude.

Deputy Director for Education and Skills of the OECD says one factor that he sees present in successful education systems, but lacking in less productive ones, is a higher value being placed on education, both in regards to societal influences and those of national governments. For example, he says Germany, in response to dangerous racial achievement gaps, began increasing investment in immigrant student education. Since this change, student achievement has made remarkable progress.

The heartbreaking truth of our current education system.
The heartbreaking truth of our current education system.

But the more prevalent area for change is attitude of the students themselves. Understanding that “effort is going to make a difference, not talent”, which is the opposite of how many American students take on academics, is crucial to being successful. Discussions on gender and mathematics give a more concrete example of the power that thought really can have on performance. Lisa Wade explains that, “Girls and boys internalize the idea that they will be bad and good at math respectively because of [toys] like the ‘Math class is tough!’ Barbie”. If we are telling half of society that they are inadequate and don’t have the skills to do something, yet are expecting mastery at the highest level, we won’t get the results that we want.

“Research published in Child Development found that hard work and good study habits were the most important factor in improving math ability over time”. So the challenge remains—how do we execute preventative measures? How do we change these patterns and trends of negativity at the tender age of 5? Will multi-phased government guided plans and trainings be the answer, or is something simpler than that? I believe it is an affective factor that must stay consistent from coast-to-coast, student-to-student, that everyone can succeed with effort, time, and perseverance, regardless of race, gender, socioeconomic status, and ability.