Below is a review of two applets and one app that I found beneficial for students in grades K-2, specifically focusing on number system and rational numbers (counting, cardinality, number and operation in Base 10):
Applet #1: Grouping and Grazing
Applet #1: Grouping and Grazing
Source:
-From
Illuminations: Resources for Teaching Math
Summary:
This applet is intended for grades
Pre-K-2 and the math content is number and operations specifically focusing on
counting, adding, and subtracting. Students learn about grouping, tallying, and
place value. This applet has an arrow in which you can select a category. There
are four categories on it to choose from: 1. Count by 5s, 2. Count by 10s, 3.
Add, or 4. Subtract. When a student clicks each category, the directions are
given and the applet orally reads the directions as well. In the count by 5s
and count by 10s categories, the screen displays green grass that is divided
into two parts (a top half and a bottom half). For the count by 5s category,
there are a certain number of cows displayed on the bottom half of the grass.
Students have to hit the “select” button and count 5 cows and then hit the
“group” button. Once the student presses the “group” button, a space ship will
then automatically lift the 5 cows and group them together on the top half of
the grass. The students continue this selection and grouping process until they
cannot make any more groups of 5s. Any cows that are leftover, the student must
drag to the top half of the grass. As the students make groups of 5’s and as
they drag the leftover cows to the top, a tally chart is shown keeping track of
how many cows there are. Once all the cows are dragged to the top half of the
grass, the student presses the “check” button. Then a box comes up that says,
“How many cows are in the field?” and the student can either press, “count with
me” or, “count alone.” When they press, “count with me,” the applet counts the
total number of cows aloud with the student as the student clicks the groups of
cows and leftover cows. When they press, “count alone,” a calculator comes up
and students click the number of cows they think there are and then they press
the check button. The applet verifies this amount. If the amount is too much,
the applet will tell you it’s too much. Likewise, if the amount is not enough,
the applet will tell you it’s not enough. Once the student correctly answers the
problem, they are done with the problem. They then click “new set,” and another
problem is displayed with a different number of cows. For the count by 10s
category, the same process occurs except students have to group the cows by
groups of 10 rather than 5s. The addition and subtraction categories are
slightly different. In the addition category, there are two halves, but the
bottom half is split in half again. An addition problem is displayed on the
bottom half. For example, the problem 16 +6 may be displayed. The first part of
the addition problem is on one bottom half and the second part of the addition
problem is on the other bottom half. Taking the 16+6 example, 16 of the cows
would be displayed on one bottom half and 6 of the cows would be displayed on
the other bottom half. The student has to hit the “select” button and bring all
of the cows to the top half where it says “sum.” The student also has the
option of breaking groups and re-grouping. For example, if there is a group of
10, the student can break up the group by pressing the “break group” button so
there are 10 individual cows. They may then, for instance, choose to regroup
cows by groups of 5 by clicking the “group” button. A space ship comes in and
automatically groups or breaks apart the cows. Once all cows are dragged into
the sum section, the student presses the “check” button and a box appears that
says, “How many cows are in the field?” A calculator is shown and the student
clicks the number of cows they think there are and presses the check button. The
applet verifies this amount. If the amount is too much, the applet will tell
you it’s too much. Likewise, if the amount is not enough, the applet will tell
you it’s not enough. Once the student
correctly answers the problem, they are done with the problem and the correct
equation is displayed and orally stated. They then click the arrow button for a
new addition problem or if they click the wheel button, they can create their
own addition problem. In the subtraction category, there are also two halves
but the top half is split in half again. A subtraction problem is displayed on
the bottom half and the word “difference” is displayed on one of the top
halves. For example, the problem 14-9 may be displayed on the bottom half. The
number of cows displayed is based on the first number (the highest number) of
the subtraction problem. For instance, in the 14-9 example, 14 cows would be
displayed on the bottom half. Students first have to select the number of cows
that are being subtracted. For the 14-9 example, the student would click on 9
cows. They would then drag those cows into the difference section. Students may
find that they need to use the “break group” or “group” buttons. For example,
if there are 14 cows and there is a group of 10 and 4 individual cows, the
student has to break apart the group of 10 in order to drag 9 cows into the
difference section. After the student drags the cows into the difference
section, they select those cows and press the “subtract” button. A space ship
comes in and automatically moves those cows to the other top half (so the
difference section is now empty). The student then selects the remaining cows
that are in the bottom half and drags them into the difference section. In the
14-9 example, the student would move the five cows that are shown on the bottom
half into the difference section. Then, the student presses the “check” button
and a box appears that says, “How many cows are in the field?” A calculator is
shown and the student clicks the number of cows they think there are and presses
the check button. The applet verifies this amount. If the amount is too much,
the applet will tell you it’s too much. Likewise, if the amount is not enough,
the applet will tell you it’s not enough.
Once the student correctly answers the problem, they are done with the
problem and the correct equation is displayed and orally stated. They then click
the arrow button for a new subtraction problem or if they click the wheel
button, they can create their own subtraction problem.
With teacher demonstrations, I
think this applet is easy to use and engaging for students. Once students
practice with it, they will likely understand what to do in no time! Also, the
oral directions and the clear buttons with corresponding pictures (for example,
the “select” button says “select” and has a picture showing an arrow) are
helpful to make this applet easier for students. The visual presentation of
this applet is appealing for students. The bright colored green grass
background and brown spotted cows make this applet exciting for students and can
contribute to a more positive math learning experience. With a bright colored
background and cows, students may be more visually stimulated and drawn to the
game (which may help them stay engaged for a longer period of time). While I
think this applet is great for young elementary students, as described below, I
think the main challenge for students is that there are so many buttons that it
can be overwhelming or confusing. If teachers do not model how the applet works
and clearly explain what each button is used for, students will likely be lost,
irritated, overwhelmed, and lose focus more quickly (however, once teachers do
explain it, I think students can successfully work through this applet independently).
Critique:
I think
this applet is a wonderful tool to aid students in their number and operation
knowledge. This applet does a great job at helping the students practice and
master their counting, adding, and subtracting skills. It is truly interactive
by allowing students to select, group, and/or break apart the cows. This applet
isn’t a matching game nor is it a simple “circle the correct answer” game.
Rather, it requires that students think about how they can group numbers and
different ways they can group numbers when adding or subtracting. Plus, this
applet is highly engaging because most young students like physically “doing”
something rather than watching a video for example. Also, most young students
are simply engaged by the cute cows and cool space ship! In addition to the
fact that this applet is interactive, engaging, and visually appealing, there
are many other strengths of this applet. First, I like how that once the
student checks their answer, there is immediate feedback given (it tells the
student if their answer is not enough or too much). I especially like how the
applet doesn’t tell the student the correct answer if they are incorrect. This
forces the student to rethink the problem. I also like how there is no timer so
students can take all the time they need to practice the problems they are
given. Additionally, there are instructions for teachers to read, which are
beneficial for them to understand the applet as well as helpful for them when
they explain how to use the applet to their students. Moreover, one aspect of
this applet I find particularly interesting is that in the addition and
subtraction categories, the student (or teacher) could plug in their own
addition or subtraction problem. This would be a great assessment tool for
teachers! For instance, a teacher can create a worksheet that has a set of
addition problems and subtraction problems. Students will have to plug these
problems in and solve them. This would allow the teacher to see if students are
understanding the content or if they are struggling. Moreover, I like how this
applet gives the instructions visually and orally so students can see and hear
them (which helps students who are visual and auditory learners). In addition
to the fact that this applet has many buttons and may be confusing for children
to follow, one other weakness is that when teachers or students create their
own addition or subtraction problem, the highest number they can type in is 20.
For students who have mastered the skill and are ready to be more challenged,
this can be a downfall (because they are continuing to practice what they
already know). Despite these weaknesses,
I think this applet is effective overall for both students and teachers. It can
help students develop their basic counting, addition, and subtraction skills
and help teachers gain a better understanding as to whether the students are
grasping the content. I think this applet is ideal for students who are
beginning to learn about addition and subtraction (and grouping and regrouping).
As they learn about this concept from their teacher through discussions,
worksheets, and activities, this concept can be reinforced through them
engaging with this applet.
Applet #2: Base
Blocks
Source:
-From the
National Library of Virtual Manipulatives
Summary:
This applet is intended for grades
Pre-K-2 and the math content is number and operations specifically focusing on
illustrating using base ten blocks to represent a number. Students are working
on representing a number, counting, grouping, and place value. The applet is a
grid where the student (or teacher) can adjust the number of decimal places
(either 0, 1, 2, 3 decimal places; for example, 9, 0.9, 0.09, or 0,009), the
bases (base 2, 3, 4, 5, or 10), and the number of columns (2, 3, or 4 columns;
for example a rod and a unit is two columns, a flat, rod, and a unit is three
columns, and a cube, flat, rod, and a unit is four columns). Once a student
adjusts their settings, they click “show a problem” and the directions appear
saying “Use blocks to show: _____” (in the blank, a number appears). Students
then have to use the base blocks to appropriately represent the number. There
is a “clear” button, which erases their work completely (this may be useful if
a student realizes they made many errors), and there is a trash can where
students can drag a block they don’t need (for example, if they accidently have
5 units instead of 4, they can drag one unit to the trash can). When students
finish a problem, they can click the button that says “next problem.” As a
student drags the base blocks, numbers appear on the right side showing how
many they have so far. For example, if the problem is “77” as students drag 1
rod out it shows “10,” as students drag another rod out it shows “20,” etc. In
short, it shows students how many their base blocks equal as they drag them. Therefore,
students know when their answer is correct (immediate feedback) when their
total equals the problem total. Also, a student can drag between columns to
break down the cube, flat, or rod. For example, if the problem is “39” and the
student chooses to drag a rod into the unit column, the rod will break up into
individual units (the student will then have to put one unit in the trash for
the problem to represent 39).
This applet is very easy to use.
There are no instructions for students or teachers. However, teachers can look
at it and easily understand what to do, and students can figure it out by
simply exploring for themselves (without any teacher instruction). There are
only a few different buttons, and they are easy to use. The visual presentation
of this applet is unattractive. The background is all gray and the base blocks
are blue. However, it is nice that the blue pops out and that the base blocks
are clear and easy-to-see. There are no other colors or stimulating visual
features. One main challenge this applet has for students are the various
decimal places, bases, and columns to choose from. Students may be overwhelmed
or confused when they see decimal places, bases other than base 10, or multiple
columns (that show a cube or flat, for example).
Critique:
This “base block” applet is a great
instrument for students to use as they explore numbers, counting, grouping, and
place value. It is great because it is interactive, allowing the students to
drag the base blocks to represent the number given. Students are able to really
analyze a number. Rather than just being given the number “32” and circling 32
ducks in a pond or making 32 dots for example, students are thinking about what
32 means. It means there are 3 tens and 2 ones, and they can represent it by
using 3 rods and 2 units. In addition to the fact that this applet is
interactive and allows students to critically think about numbers, there are
other strengths as well. As mentioned above, I like how the applet counts with
them as they drag the base blocks out. I also like how the applet can be easily
adapted for students. For example, a student who is struggling or lower
achieving may work with base 10 blocks containing just rods and units and no
decimals. A student who is excelling and/or is high achieving or gifted may
work with base 10 blocks containing flats, rods, and units (and maybe even
cubes) and/or decimals. While I enjoy the various adaptations this applet has,
I also see these adjustments as a weakness. Students may be overwhelmed or
confused when they see decimal places, bases other than base 10, or multiple
columns (that show a cube or flat, for example). Specifically, at this age, I
believe students should only be working with base 10 blocks so I do not like
that other bases are even an option. I think it can confuse and even hinder
student learning. Another weakness of this applet is that it does not allow the
student (or teacher) to create a problem. Problems are automatically given.
This may be nice if the teacher likes the problems this applet presents, but
otherwise, it limits the students to only the applet problems. This limits
students’ exploration because they cannot create their own problems to further
analyze or discover the content. Also, teachers are limited because they cannot
create problems to assess student learning. Assessments may be challenging,
because all students will have different numbers and the teacher cannot choose
which numbers students will get. I think teachers could assess students by having
them record the problem they are given and draw how they represented the
problem using base blocks. Or, the teacher can give students problems, have
them practice using the applet, and then have them draw their representation
using the base blocks. Another weakness of this applet is that there are no
written or verbal directions. Teachers can easily figure out how to work
through the applet, but students may struggle (so teachers should explain how
this applet works before students explore independently). Also, some of the
words are hard to read (such as “columns”), and since there aren’t verbal
directions, teachers must make sure they review this vocabulary with the
students prior to them using the applet. Although I mentioned many weaknesses,
I do think this applet can be valuable for students’ to explore numbers,
counting, grouping, and place value. Students can delve deeper as to what makes
up a number and learn how to represent that number with base blocks. As
students progress in math (for example, as they work with addition and
subtraction), their knowledge of grouping and place value that they learned
working with this applet (along with class activities), will come in handy. I
think this applet is ideal for students who have already been introduced to
base 10. I don’t think it would be appropriate to have students use this applet
if they haven’t been introduced or practiced with base 10 before. Once students
have worked with base 10 (and have worked with physical manipulatives), then
they can further practice their number knowledge through this applet.
App #1: Teaching
Number Lines
Source:
-From the iTunes store-free app
Summary:
“Teaching Number Lines” by Aleesha
Kondys is a free app that can be accessed from the iTunes store and downloaded
to iPhones or iPads. This app is targeted for young students (I would say
ideally kindergarten and first graders). Students work with number lines, which
is a visual model, to help with basic math skills such as counting, ordering,
addition, and subtraction. There are various math activities to choose from in
this app: Jump forwards, jump backwards, sequencing skip counting (practice),
skip counting (test), and teaching tool. When students click the “jump forward”
lily pad, directions appear on a chalkboard saying, “Jump forward to show
__________” (in the blank is an addition problem; i.e. 3+9). A number line (0-15) is displayed and the
frog is placed on the first number. The student has to move the frog to the
total (ex. if the problem was 3+9, the frog would already be placed on the 3
and the student would have to add 9 and place the frog on 12). When the student
is correct, the number turns green, a dinging noise is heard, and the app will
show something such as “Bravo!” or “Nice work” (immediate feedback). When
students click the “jump backward” lily pad, directions appear on a chalkboard
saying, “Jump backwards to show: ________” (in the blank is a subtraction
problem; i.e. 13-3). The rest of this activity is very similar to the “jump
forward” activity. A number line (0-15) is displayed and the frog is placed on
the highest number. The student has to move the frog to the difference (ex. if
the problem was 15-3, the frog would already be placed on the 15 and the
student would have to take away 3 and place the frog on 12). When the student
is correct, the number turns green, a dinging noise is heard, and the app will
show something such as “Bravo!” or “Nice work” (immediate feedback). When
students click the “sequencing” lily pad, directions appear on a chalkboard
saying, “Complete the number line.” A number line (0-15) is displayed with
missing numbers. The missing numbers are shown on frogs above the number line.
Students have to drag the frog with the number in the correct location on the
number line. For instance, the number line may be missing numbers 4, 7, and 2.
These numbers would appear on the frogs. The student would have to drag the 4,
7, and 2 in their appropriate spot on the number line. These sequence problems
may have 2, 3, or 4 numbers missing (depending on the problem given). When the student is correct, the number turns
green, a dinging noise is heard, and the app will show something such as
“Bravo!” or “Nice work” (immediate feedback). When students click the “skip
counting (practice)” lily pad, directions appear on a chalkboard saying, “Jump
by _____” (either 2, 3, 4, or 5). A number line (0-15) is displayed and a frog
is displayed at the zero of the number line. If the chalkboard says “Jump by
3’s” the number line will already have the numbers highlighted by 3’s and the
student has to practice moving the frog by 3’s. For example, 3, 6, 9, 12, and
15 would be highlighted in green. The student would then practice moving the
frog from 0 to 3, 3 to 6, 6 to 9, 9 to 12, and 12 to 15. On the chalkboard, the
student can click the 2, 4, or 5 to practice counting by those groups. This
activity is simply practice. When students click the “skip counting (test)”
lily pad, directions appear on a chalkboard saying, “Jump by ______” (either
2’s, 3’s, 4’s, or 5’s). A number line (0-15) is displayed and a frog is
displayed at a certain number (this number changes). For instance, the frog may
be placed on the number 1 or it may be placed on the number 4. If the frog is
placed on the number 1 and the chalkboard says “Jump by 2’s,” the student has
to move the frog by counting by 2’s (so the student would move from 1 to 3, 3
to 5, 5 to 7, 7 to 9, 9 to 11, 11 to 13, and 13 to 15). As the student moves
the frog to the correct number, that number gets highlighted in a green color
(so as the student moves the frog from 1 to 3, the 3 turns green). When the
student is correct, a dinging noise is heard, and the app will show something
such as “Bravo!” or “Nice work” (immediate feedback). The last lily pad is
known as the “teaching tool” lily pad. This activity has frogs with a blue dot,
yellow dot, and red dot (and as little or as many of these frogs can be used).
There is a blue and a red marker (which can actually be clicked on and drawn
with) and an eraser. Also, there is a number line (0-15) displayed. The student
or teacher can work on any of the math skills shown in the other activities
(ex. addition, subtraction, sequencing, and/or skip counting).
This number lines applet is easy to
use. Students can figure it for themselves by clicking the different lily pads
and exploring the various activities (without any teacher instruction). Teachers
can investigate the app and easily understand what to do in each of the
activities (they can also read the directions on the chalkboard). Plus,
teachers can go to the iTunes store or Google and research this game to find
out more detailed information about the activities. Moreover, there are only
six lily pads students can click on, so there are not too many buttons students
need to worry about using. The visual presentation of this applet is highly
attractive. The home page is bright blue with 6 lime green lily pads. Within
each activity (except the teacher tool lily pad), there is a chalkboard that
displays the directions and a clear number line below. The frog (s) in each activity
are a dark green and cream color, which also add visual interest. With these
bright colors and fun features (such as the chalkboard and frogs), this app can
help students be more engaged and interested in the math content. One main
challenge this app has for students is that there are no oral directions. This
is a big problem since most of these students are still learning how to read.
Plus, most of the words displayed are not appropriate grade-level words (they
are too challenging). Therefore, a teacher or aid must recite the directions
aloud, which is an extra hassle and takes away some of the students’ freedom to
independently explore and have fun with the app.
Critique:
This app is awesome for students to
use as they work on counting, ordering, addition, and subtraction. Not only is
it interactive, engaging, and highly kid-friendly as well as visually
appealing, but it also allows students to have a visual representation through
a number line. With a wonderful visual model such as a number line, students
are likely to visually see the problem, the physical position of a number, and
recognize patterns and relationships. Plus, once students really start to work
and practice with the number line, often it will likely be a mental image they
will remember in their heads and can use almost as second nature. In addition
to the fact that this applet is interactive, engaging, visually appealing (with
a great visual model-the number line), there are other strengths to be noted. The
“skip counting (test)” lily pad is a great way to assess students’ counting
skills. A teacher can assess students counting skills by watching them do this
test activity or by having them write what the directions in the chalkboard say
(ex. “Jump by” 3’s”) and have them draw on a pre-made number line the frog
moving by 3’s. Teachers can assess student learning of the other activities in
this app the same why (either by observing them on the app or by recording the
directions and filling out a pre-made number line). Moreover, one of my
favorite activities on this app is the teaching tool. Teachers can give
students a counting, addition, or subtraction problem and watch as the students
work through this problem on the number line. Or, teachers can work one-on-one
with a student or in small groups and practice/review the concepts in the
activities. This may be especially helpful for lower achieving and/or
struggling students. While this app is appealing in many ways, there are some
weaknesses to mention. As previously stated, one huge factor to be aware of is
that there are no oral directions to go along with the words. Since these
students are still so young (K-2), many of them are still learning and
progressing in their reading. Therefore, a teacher must recite the directions
aloud. Another weakness I noticed in the “jump forwards” and “jump backwards”
is that the students may just drag the frog where it goes but not actually
understand the content. For example, in the “jump forward” lily pad, students may
just drag the frog until they hover over the number that turns green (green
means they got the correct answer). If students were to do this, they are not
gaining any mathematical knowledge. Hopefully, students will actually think
through the problem and use the number line appropriately. Also, teachers can
monitor as well as assess students to make sure they are engaged and
understanding the content. One other weakness of this app is that the numbers
on the number line do not go very high (the number line only goes to 15). This
is great as students are beginning to practice counting, ordering, addition,
and subtraction. However, once students start mastering these skills, more
challenging problems that are beyond the scope of this app will need to be
provided. Even though I stated a few main weaknesses, this app can be
advantageous for students as they are working on basic counting,
ordering/sequencing, addition, and subtraction. The number line is one
important strategy/model students can use to help them understand numbers and
operations. If students continually use models, such as number lines, to help
them with numbers and operations, they will probably have a better
understanding for the problems and begin to do these problems mentally. I think
this app is ideal for students who are just being introduced the concept of
counting, ordering, addition, and subtraction. As students learn about this
concept from their teacher through discussions, worksheets, and activities,
this concept can be reinforced through them engaging with this app.
Final Thoughts:
While each of these applets and app
have their strengths and weaknesses, I could envision myself using all of these
in my future elementary classroom to aid, positively contribute, and reinforce
math concepts to my students. I want my students to realize that learning math
does not need to be boring or a complex process. Rather, exploring math
concepts online through applets and apps will show students that math can be
fun and they will have a richer, more interactive, and more meaningful math
learning experience. Plus, nowadays students thoroughly enjoy when they get to
use technology, so by having students use these math applets and apps, they
will be excited about their learning. I hope to use these applets and app in my
future classroom amongst many others that I learn or find out about along the
way in my teaching journey.
Very thorough and nice job, Hallie:) Thank you! Also, these would be helpful for students to become more fluent in their understanding of these facts:)
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