U.S. patent application number 14/329811 was filed with the patent office on 2014-10-30 for apparatus and method for tools for mathematics instruction.
The applicant listed for this patent is CONCEPTUA MATH, LLC. Invention is credited to Arjan KHALSA, Edward Murphy.
Application Number | 20140322681 14/329811 |
Document ID | / |
Family ID | 46162584 |
Filed Date | 2014-10-30 |
United States Patent
Application |
20140322681 |
Kind Code |
A1 |
KHALSA; Arjan ; et
al. |
October 30, 2014 |
APPARATUS AND METHOD FOR TOOLS FOR MATHEMATICS INSTRUCTION
Abstract
In a computer-implemented educational environment, four
particular features are provided, which may be used separately or
in combination, to drive the instruction of a particular
curriculum, e.g. the instruction of fractions. In another
implementation, educators and authors may use permutations and
combinations of such features to construct in a graphical user
interface a progression of interactive elements which help guide a
student from visual concepts to procedures, such as equations and
formulas. In a particular implementation, by using these features
carefully, a curriculum author may design instructional sequences
that provide careful, step-by-step sequences that help students
progress from understanding the visual concepts, e.g. of fractions,
to mastering procedures, e.g. fractions equations.
Inventors: |
KHALSA; Arjan; (San Rafael,
CA) ; Murphy; Edward; (Novato, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
CONCEPTUA MATH, LLC |
Petaluma |
CA |
US |
|
|
Family ID: |
46162584 |
Appl. No.: |
14/329811 |
Filed: |
July 11, 2014 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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12960337 |
Dec 3, 2010 |
8790119 |
|
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14329811 |
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Current U.S.
Class: |
434/201 |
Current CPC
Class: |
G09B 19/02 20130101;
G09B 19/025 20130101; G09B 3/02 20130101 |
Class at
Publication: |
434/201 |
International
Class: |
G09B 3/02 20060101
G09B003/02 |
Claims
1. A method for facilitating a user to progress from visual
concepts to procedures, said system comprising providing at least
one processor programmed for: presenting one or more models in
which at least one of a denominator, a numerator, and a whole
number, where present, is unlocked, wherein the value of the at
least one of a denominator, a numerator, and a whole number can be
directly changed by the user, and in which the user changes one or
more unlocked values in a presented model to create a correct
answer; presenting one or more models in which at least one of a
denominator, a numerator, and a whole number, where present, is
unlocked and that includes one or more introduced numbers in which
at least one of a denominator, a numerator, and a whole number,
where present, is locked, wherein the value cannot be directly
changed by the user, and in which at least one of a denominator, a
numerator, and a whole number, where present, is linked to
corresponding values in the one or more models, wherein a change in
the one or more models results in a change in a corresponding value
in the one or more numbers, and in which the user changes one or
more unlocked values in the one or more models to create a correct
answer; presenting one or more models in which at least one of a
denominator, a numerator, and a whole number, where present, are
locked, and that includes one or more introduced numbers in which
at least one of a denominator, a numerator, and a whole number,
where present, is unlocked, in which a presented model is not
linked to corresponding values in the one or more introduced
numbers, wherein a change in the one or more introduced numbers has
no effect on the corresponding values in the one or more models,
and in which the user changes one or more unlocked values in the
one or more introduced numbers to create a correct answer by making
one or more unlocked values in the one or more introduced numbers
match corresponding values in the one or more models; presenting
one or more models in which at least one of a denominator, a
numerator, and a whole number, where present, are locked, and that
includes one or more introduced numbers in which at least one of a
denominator, a numerator, and a whole number, where present, is
unlocked and is linked to corresponding values in the one or more
presented models, wherein a change in the one or more introduced
numbers results in a change in a corresponding value in the one or
more models, and in which the user changes one or more unlocked
values in the one or more introduced numbers to create a correct
answer; and presenting one or more numbers in which at least one of
a denominator, a numerator, and a whole number, where present, is
unlocked, and in which the user changes one or more unlocked values
in a presented number to create a correct answer.
2. The method of claim 1, wherein a correct answer includes any of:
with regard to all presented models, the numerator, the
denominator, and whole number, if present; and with regard to
introduced numbers, a valid value that is neither empty,
uninitialized, nor otherwise not a number.
3. The method of claim 1, wherein a correct answer includes: an
aggregate value of a resulting model that is equivalent to an
aggregate value of a corresponding resulting number, when both
models and introduced numbers are presented.
4. The method of claim 1, wherein a correct answer includes: an
aggregate value of a resulting model that is exactly equal to an
aggregate value of a corresponding resulting number. when both
models and introduced numbers are presented.
5. The method of claim 1, wherein a correct answer includes: models
in which all of the numerator, the denominator, and the whole
number of a resulting model is exactly equal to the corresponding
value of a corresponding resulting number, when both models and
introduced numbers are presented.
6. The method of claim 1, wherein a correct answer includes: all of
the numerator, the denominator, and the whole number, if present,
in all presented models, and said introduced numbers, are
mathematically correct based on a context comprising any of
addition, subtraction, multiplication, and division.
7. The method of claim 1, further comprising: displaying on a
display any of four layers, said four layers comprising: a model
layer; an equivalence layer; a fractions layer; and a sentence
layer; and any of: hiding and showing any layer of said four
layers; linking and unlinking any two layers of said four layers;
and locking and unlocking any portion of any of said four
layers.
8. The method of claim 7, wherein said linking and unlinking any
two layers of said four layers further comprises linking and
unlinking any of: said model layer and said fractions layer; said
model layer and said equivalence layer; and said equivalence layer
and said fractions layer.
9. The method of claim 1, wherein said type of model comprises any
of: a fractions circle; a vertical fraction bar; a horizontal
fraction bar; a dot model; a discrete model; a number line; a
numeric representation; a decimal; a decimal grid; sets; rods; two
dimensional arrays; ten frames; place value blocks; number line;
and hundreds chart.
10. The method of claim 1, further comprising: determining whether
a user input is correct and, when it is determined that said user
input is not correct, providing feedback on a display for said
user.
11. The method of claim 10, further comprising: determining whether
a user input is correct based on a particular order of
correction.
12. The method of claim 10, further comprising: providing
corrective feedback incrementally as a user corrects errors on the
display.
13. The method of claim 10, wherein said feedback excludes
providing answers to problems.
14. The method of claim 10, further comprising: generating feedback
based on student inputs, which layers are visible, which components
are linked, which components are not linked, which components are
locked, which components are unlocked, and which model is
selected.
15. The method of claim 1, further comprising: scaffolding
instruction to drive student instruction by providing a particular
sequence of steps that are taken by said student, wherein said
particular sequence of steps help said student learn a topic, and
wherein said particular sequence of steps comprises a particular
problem built from a prior problem.
16. The method of claim 7, further comprising: linking and
unlinking any two layers of said four layers using an observer
pattern wherein linked components emit events when any of said
components change, wherein corresponding components listen for said
emitted events and synchronize themselves to a corresponding
changed value, and wherein when any of said components are
unlinked, such components stop listening for emitted events or such
components ignore said emitted events.
17. The method of claim 7, wherein said locking and unlocking any
portion of any of said four layers, further comprises any of:
causing a locked component to stop listening to user input from a
mouse or keyboard; causing a locked component to ignore user input
from a mouse or a keyboard; and hiding and removing controls which
change the value of a locked component.
18. The method of claim 7, wherein said locking and unlocking any
portion of any of said four layers prepares fractions for
manipulation on a display and wherein said locking and unlocking
any portion of any of said four layers has a utilitarian function
that allows a user to start interacting with the fractions.
19. The method of claim 1, further comprising: providing a multiple
models feature allowing a user to interchange models, to select a
specific model, and to select a set of models for a given
problem.
20. The method of claim 19, further comprising: said multiple
models feature using an object oriented approach, said object
oriented approach creating a template class that declares an
application programming interface for the methods and properties
common to corresponding fraction models.
21. The apparatus of claim 20, wherein said object oriented
approach further comprises a model container that holds shadow
copies of corresponding model values and that synchronizes new
models with said shadow copies of corresponding model values when
any of said new models are selected.
22. A computer implemented for facilitating a user to progress from
visual concepts to procedures, comprising: providing at least one
processor programmed to perform the steps of: providing a plurality
of visual models; and allowing a user to change the type of the
model presented and interact with a selected model, wherein said
model type comprises any of: a model in which the denominator of a
resulting model of the presented models is locked; a model allowing
a user to create an entire fraction in the resulting model; a model
for introducing numbers and requiring the user to provide a
numerator in the resulting model; a model for allowing the user to
strategically change the model to promote flexibility; a model
requiring the user provide the numerator and the denominator; a
model presenting procedures only and requiring the user to
calculate the numerator; and a model requiring the user to
calculate the numerator and input the denominator.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application is a continuation of U.S. patent
application Ser. No. 12/960,337, filed on Dec. 3, 2010, which is
currently allowed, and which is incorporated herein in its entirety
by this reference thereto.
BACKGROUND OF THE INVENTION
[0002] 1. Technical Field
[0003] This invention relates generally to the field of
computer-implemented educational tools in mathematics. More
specifically, this invention relates to computer-implemented
educational tools for facilitating the teaching of mathematics by
employing inventive approaches and techniques that support the
teacher-student collaboration desired for a student mastering an
area of mathematics.
[0004] 2. Description of the Related Art
[0005] Mathematics may be difficult to each and may be difficult
for students to learn. Fractions, for example, have been found to
be both difficult to teach and difficult for students to learn. At
the same time, fractions, as well as other areas of mathematics,
are a pivotal topic in mathematics education. Strategic use of
technology can help support the teacher-student collaboration
required to master this wide-ranging subject area.
[0006] An example: The challenges with fractions.
[0007] "Learning about fractions in the upper elementary grades is
hard. Really hard! Fractions are hard not only for children to
learn but for teachers to teach." This is how Marilyn Burns, the
highly esteemed author on elementary mathematics education, begins
her first of three extensive books on teaching fractions (Burns,
2001). The National Math Advisory Panel identified fractions as an
area that requires special attention: "Difficulty with fractions
(including decimals and percents) is pervasive and is a major
obstacle to further progress in mathematics, including algebra"
(National Math Advisory Panel, 2008, p. xix). This challenge is
understandable. Fractions present major conceptual leaps for
students.
[0008] Consider these factors: [0009] Fractions can describe many
different things. When Sarah drinks half of the water in the
bottle, she is consuming part of a whole (1/2). When Jeremy eats
three of nine carrot sticks, he is consuming part of a set ( 3/9 or
1/3). When Justin reads for 15 minutes, that represents 1/4 of a
common unit of time. When Hannah swims across the lake her feat is
a measure of length (e.g., 11/3 miles). And on it goes, from one
quarter of a dollar, to 1/3 cup of flour, to 1/2 of an acre of
land. Fractions represent so many different things! [0010]
Sophisticated reasoning is required to evaluate any fraction. Upon
entering the topic of fractions, students must analyze the
relationship between two numbers in order to understand a single
value. For example, 1/8 is smaller than 1/4, and 3/8 is larger than
1/4. [0011] The real value of fractions is dependent upon the unit,
or whole, of which they are a part. 3/4 is not always greater than
1/4! 3/4 of a county is a smaller region than 1/4 of a continent.
[0012] Fractions present a plethora of new terms for students to
master: numerator, denominator, equivalent, common, uncommon,
proper, improper, and more. [0013] Students must first learn what
fractions mean, and then they must perform operations on these
fractions. Some of these operations, like addition and subtraction
with uncommon denominators, require multiple steps. Other
operations, like multiplying and dividing fractions, seem very
abstract to many people, and disconnected from anything in real
life. (Can you find a real life example of 1/8/1/3? It is possible,
but certainly not trivial!)
[0014] It would be advantageous to provide computer-implemented
educational tools that address and target the particular challenges
for both the student and teacher about the teaching of and the
learning of particular areas of mathematics, e.g. fractions, as
described hereinabove.
SUMMARY OF THE INVENTION
[0015] In a computer-implemented educational environment, four
particular features are provided, which may be used separately or
in combination, to drive the instruction of a particular
curriculum, e.g. the instruction of fractions. In another
implementation, educators and authors may use permutations and
combinations of such features to construct in a graphical user
interface a progression of interactive elements which help guide a
student from visual concepts to procedures, such as equations and
formulas. In a particular implementation, by using these features
carefully, a curriculum author may design instructional sequences
that provide careful, step-by-step sequences that help students
progress from understanding the visual concepts, e.g. of fractions,
to mastering procedures, e.g. fractions equations.
BRIEF DESCRIPTION OF THE DRAWINGS
[0016] FIG. 1 is an example screen shot of models, equivalence, and
fraction layers in tool mode, according to an embodiment;
[0017] FIG. 2 is an example screen shot for adding fractions,
according to an embodiment;
[0018] FIG. 3 is an example screen shot showing a linked model
layer and fractions layer, according to an embodiment;
[0019] FIG. 4 is an example screen showing an model layer,
equivalence layer, and feedback in the fractions layer, according
to an embodiment;
[0020] FIG. 5 is an example screen shot showing a model layer,
equivalence layer, and a sentence layer, according to an
embodiment;
[0021] FIG. 6 is an example screen shot showing icons for the hide
or show models layer, hide or show equivalence layer, and hide or
show fractions layer, according to an embodiment;
[0022] FIG. 7 shows three example screen shots showing models only,
models and fractions only, and fractions only, respectively,
according to an embodiment;
[0023] FIG. 8 is an example screen shot for a model layer and
fraction layer which are unlinked, according to an embodiment;
[0024] FIG. 9 shows a model in its unpopulated state (before the
students starts her work), its partially populated state (the
student is partially done), and its completed state, according to
an embodiment;
[0025] FIG. 10 shows example screen shots of a problem where a
student is adding fractions with common denominators and the linked
option has been employed, according to an embodiment;
[0026] FIG. 11 shows example screen shots showing at left, the
model layer unlinked to the fraction layer, and at right, the model
layer unlinked and one model changed by the student to match the
first addend, according to an embodiment;
[0027] FIG. 12 shows link icons between two layers in three
particular placements, according to an embodiment;
[0028] FIG. 13 is an example screen shot showing equal parts in a
model that are locked, according to an embodiment;
[0029] FIG. 14 shows two screen shots for adding fractions
including changing the denominator in the model layer, according to
an embodiment;
[0030] FIG. 15 shows a lock icon, according to an embodiment;
[0031] FIG. 16 shows particular types of models, according to an
embodiment;
[0032] FIG. 17 shows a set of icons each of which corresponds to a
particular model, according to an embodiment;
[0033] FIG. 18 is an example screen shot showing corrective
feedback targeted to specific errors, according to an
embodiment;
[0034] FIG. 19 is an example screen shot showing that the
denominator of the model in the sum is locked and that the fraction
layer is hidden, according to an embodiment;
[0035] FIG. 20 is an example screen shot for subtracting the
fractions using models, according to an embodiment;
[0036] FIG. 21 is an example screen shot for subtracting the
fractions using models where the denominator and numerator of the
difference are both unlocked, according to an embodiment;
[0037] FIG. 22 is an example screen shot showing subtracting the
fractions where the models and the fraction layer are linked, and
in the fraction layer, the numerator in the difference is unlocked
and the denominator is locked, according to an embodiment;
[0038] FIG. 23 is an example screen shot showing the discrete
model, also called the dot model, according to an embodiment;
[0039] FIG. 24 is an example screen shot showing unlinked models
and unlocked numerator and denominator in the difference fraction,
according to an embodiment;
[0040] FIG. 25 is an example screen shot where the model layer is
hidden and in the difference fraction, the numerator is unlocked
and the denominator is locked, according to an embodiment;
[0041] FIG. 26 is an example screen shot showing only the fractions
layer where both the numerator and the denominator in the
difference are unlocked, a final step in a sequence of visual
concepts to procedures, according to an embodiment;
[0042] FIGS. 27A-27D are a part of a sample worksheet of a sample
spreadsheet, according to an embodiment;
[0043] FIGS. 28A and 28B are a table containing screens of an
example where each screen represents like problems on each floor of
the scaffolding, according to an embodiment;
[0044] FIGS. 29A and 29B are a table containing screens, each of
which help demonstrate the results of using the spreadsheet to
invoke Hide and Show, according to an embodiment;
[0045] FIG. 30 is a block schematic diagram of a system in the
exemplary form of a computer system, according to an embodiment;
and
[0046] FIG. 31 is an example screen showing WYSIWIG authoring,
according to an embodiment.
DETAILED DESCRIPTION OF THE INVENTION
[0047] In a computer-implemented educational environment, four
particular features are provided, which may be used separately or
in combination, to drive the instruction of a particular
curriculum, e.g. the instruction of fractions. In another
implementation, educators and authors may use permutations and
combinations of such features to construct in a graphical user
interface a progression of interactive elements which help guide a
student from visual concepts to procedures, such as equations and
formulas. In a particular implementation, by using these features
carefully, a curriculum author may design instructional sequences
that provide careful, step-by-step sequences that help students
progress from understanding the visual concepts, e.g. of fractions,
to mastering procedures, e.g. fractions equations.
[0048] While much of the discussion herein is about a particular
area of mathematics, namely, fractions, it should be appreciated
that using fractions as a particular subject area of mathematics is
to aid in the understanding of the invention only and is not meant
to be limiting. Put another way, one skilled in the art would
readily recognize that the particular details about fractions could
be replaced by details of other areas of mathematics and not stray
from the scope and spirit of the invention.
Introduction--Features, Tools, and Modes
[0049] Four (4) Features. An embodiment of a computer-implemented
fractions methodology ("the computer-implemented methodology" or
"system") is provided.
[0050] Such methodology provides four primary feature sets that
power conceptual fractions tools. The four features provided by the
methodology is as follows: [0051] Hide and Show [0052] Link and
Unlink [0053] Lock and Unlock [0054] Multiple Models
[0055] These four features can be used separately or in combination
to drive the instruction provided by the computer-implemented
methodology. By using these features carefully, a curriculum author
can design instructional sequences that provide careful,
step-by-step sequences that help students progress from
understanding the visual concepts of fractions to mastering
fractions equations. With the computer-implemented methodology,
educators and authors use these permutations and combinations to
construct a progression from visual concepts to procedures
(equations, formulas). This is an educational core of the
computer-implemented methodology. These four features are discussed
in more details hereinbelow in this document.
[0056] Fifteen (15) Tools. In an embodiment, technical,
instructional capabilities of the computer-implemented methodology
are built into fifteen tools. Each tool is an application designed
to teach a specific fractions topic. The tools are in two
categories: Concepts to Procedures and Higher Order Thinking. The
Concepts to Procedures tools make extensive use of the four primary
feature sets. The Higher Order Thinking tools each employ at least
one of the feature sets, but do not involve the same level of
feature integration as they do not emphasize the progress from
concepts to procedures.
Concepts to Procedures Tools
[0057] Identifying Fractions [0058] Comparing Fractions [0059]
Adding Fractions with Common Denominators [0060] Subtracting
Fractions with Common Denominators [0061] Equivalent Fractions
[0062] Equivalent Fractions with Word Sentences [0063] Finding
Common Denominators [0064] Adding Fractions with Uncommon
Denominators [0065] Subtracting Fractions with Uncommon
Denominators [0066] (Multiplying Fractions--not yet developed)
[0067] (Dividing Fractions--not yet developed)
Higher Order Thinking Tools
[0067] [0068] Order Fractions on a Number Line [0069] Unitizing
with Pattern Blocks [0070] Unitizing Fractions on a Grid [0071]
Estimating with benchmark Fractions
[0072] It should be appreciated that the procedure tool, Comparing
Fractions, is a tool in this set that does not employ Hide and
Show, Link and Unlink, or Lock and Unlock. However, it does employ
a special case version of lock and unlock. For example, in the
Comparing Fractions tool, the user is given the opportunity to
place their own fractions onto the screen, and compare them to
determine which is smaller and which is larger. The user may
compare either two or three fractions. The user enters a numerator
and denominator in each of two or three blank fractions. When the
user is satisfied with the fractions they have entered, they lock
these fractions, e.g. by using the icon displayed in FIG. 15. Once
this locking action has been accomplished, the user may display the
fractions in the form of various models and may place the fractions
in the "bins" that are used for comparison purposes. In this
instance, the user does not have an "unlock" option. The fractions
start in the unlocked state. Once the user locks them, the user may
employ them and compare them, but the user cannot unlock them and
re-state their value. In order to compare a new set of fractions,
the user resets the tool, effectively completing their session with
one set of fractions, and is then given two or three unlocked
fractions. These unlocked fractions are ready to be populated,
locked, and compared.
[0073] Three (3) Modes. In an embodiment, the four features are
employed in three different modes of interactions: Tool Mode,
Activity Mode, and Authoring Mode. Each of these modes employs the
tools referenced above.
Tool Mode
[0074] Tool mode involves using the computer-implemented
methodology in a most native mode. For example, in an embodiment,
such tools may include the features that are free of actual numeric
content. Tool Mode is intended for use by a teacher in whole class
instruction. The teacher may select his or her own problem to
investigate with the students, enter the problem into the tool, and
then lead a discussion that involves the progression from visual
models to procedures. Tool Mode is also for students who are
investigating a problem, or type of problem, and benefit from an
open-ended environment with a choice of fractions models and visual
representations. FIG. 1 shows an example of an embodiment of the
Tool Mode for an equivalent fractions investigation. FIGS. 2-4 were
all created in Tool Mode, and numbers were entered to populate each
screen. That is, FIG. 2 is an example screen shot for adding
fractions, according to an embodiment. FIG. 3 is an example screen
shot showing a linked model layer and fractions layer, according to
an embodiment. And, FIG. 4 is an example screen showing a model
layer, equivalence layer, and feedback in the fractions layer,
according to an embodiment.
Activity Mode
[0075] In an embodiment, an Activity mode is provided that may
involve using student activities, or units, that are pre-authored.
For example, authored activities may be provided and sold as a
premium subscription. Activities are comprised of a set of one or
more problems arranged in a sequence. Each problem within each
activity set uses a distinct tool, such as Identifying Fractions or
Adding Fractions with Common Denominators.
[0076] In an embodiment, activities created for this mode are
authored using a spreadsheet. An authoring spreadsheet is the
document that contains the information, or data, that defines one
or more Activities, and the Problems within each activity.
Activities consist of a set of problems, usually between 20 and 30,
that a child is expected to explore and answer in a given session,
e.g. of about five to 15 minutes. Each activity culminates in a
six-problem Skills Check, during which data is collected.
[0077] Each worksheet within a spreadsheet defines an activity or
set of problems. An embodiment of such spreadsheet using such type
of worksheet can be understood with reference to FIGS. 27A-27D.
FIGS. 27A-27D are a part of a sample worksheet of a sample
spreadsheet, according to an embodiment, each of which show parts
of a worksheet according to an embodiment, as follows. FIG. 27A
shows rows 1-21 and columns A-J of this particular spreadsheet.
FIG. 27B shows rows 22-51 as well as columns A-J; FIG. 27C shows
rows 1-21 and columns K-Q; and FIG. 27D shows rows 22-51 and
columns K-Q. Such rows 1-51 and columns A-Q are for illustrative
purposes only and are not limiting. The rest of the rows and the
rest of the columns of the particular spreadsheet are not
shown.
[0078] The columns of the worksheet defined each individual
problem. There is one column for each problem. The rows of the
spreadsheet define the variables for each problem. Some of these
variables are used in a number of tools and some variables are
specific to one kind of tool. For instance, variables used in a
number of tools include: the name of the problem (a unique
identifier), the tool upon which the problem is based, the text
presented at the top of the screen (instructions for the student),
whether the problem is a skills check problem, the elements that
are visible on the screen (models, fractions, anything that will
appear on the screen), the values of every fraction in the problem
(whether represented as a model or number), and which values are
locked or unlocked, which values are linked or unlinked, the
specific visual models (if any) that are employed in the screen,
and the answer field(s) that the student must populate in order to
complete the problem. An example of variables that are specific to
a single tool are the variables that define the tessellation
pattern in the Unitizing with Pattern Blocks tool. In an
embodiment, there are approximately 100 variables per problem, with
the amount dependent upon the tool that is being employed.
[0079] In an embodiment, the spreadsheet may contain information on
the Activity level as well, as opposed to the problem level. This
information may include the name of the activity and such
information identifies the activities to which the student will
progress in the case when the student passes or fails the Skills
Check.
[0080] Thus, as can be gleaned from the above, in an embodiment,
the spreadsheet may identify technical variables for each problem,
including the numbers (fractions) employed, models used, and the
use of hide and show, link and unlink, lock and unlock, and other
specifications for a given problem. These variables set the
presentation of the tool for a specific problem, so that the tool
does not appear agnostic and content-free, as it does in tool mode.
Rather, it appears populated and dedicated to a very specific task,
and that task is denoted as a problem for a student to solve. In
this regard, problem authoring is accomplished using a spreadsheet
to assign the values of the four features to a given tool. Problems
are aggregated together into activities or activity sets.
Activities are presented in Activity Mode.
[0081] In another embodiment, problems created for Activity mode
are authored using a GUI interface that implements a
What-You-See-Is-What-You-Get (WYSIWYG) editing environment where
the author can specify values for the technical variables for each
problem, including the numbers (fractions) employed, models used,
and the use of hide and show, link and unlink, lock and unlock, and
other specifications for a given problem, by setting the state of
GUI controls, such as buttons, sliders, text editing boxes,
sliders, scrollbars and the like. The state of the GUI controls are
reflected in the technical variables that comprise the problem
data. For instance, by selecting the icon located next to the
numerator editing box and making it appear locked, the author can
specify that the numerator cannot be changed in activity mode. FIG.
31 shows an example screen shot of WYSIWYG authoring, according to
an embodiment.
Authoring Mode
[0082] An embodiment provides an authoring mode that may employ the
fifteen tools described above, with authoring, saving, and sharing
capabilities built-in. With the embodiment, a teacher may be able
to accomplish some authoring tasks for Activity Mode without using
a spreadsheet. In authoring mode, the user is presented with
on-screen icons to set variables for each object that can be
controlled. For the purposes of the discussion herein, an icon
means any graphical user interface (GUI) control that the user
interacts with to set, clear or change a value in the software.
Some examples of GUI controls are a graphic image, a button, a
radio button, a check box, a list control, a scrollbar, a slider or
the like. This list is not exhaustive and is not meant to be
limiting and is meant to illustrate that there are a wide range of
possible GUI controls. An embodiment provides, but is not limited
to, the following capabilities: [0083] Icons that are present in
Tool mode may be included. These include icons for multiple models,
link and unlink, and hide and show. [0084] Additional icons may be
provided for lock and unlock. In an embodiment, there may be one
icon for each editable element. This means that there may be a lock
and unlock option for the numerator and denominator of any fraction
or model displayed on the screen. [0085] There are icons that allow
a user to mask the presence of any given manipulative model, so
that students cannot see or employ the model in the given screen.
[0086] There is an ability to SAVE the authored page. The saved
page may include the settings of each model, such as
hide/show/link/unlink, and lock/unlock decision made by the author.
The save operation may also include any numbers that have been
placed on the screen, whether locked or unlocked. [0087] There is
the ability to OPEN a SAVED document. This means that after an
author has created a saved problem to his/her specifications, that
author, or another user, may open the item to experience the
benefits of, e.g. to use, the authored problem.
[0088] An embodiment may provide a What-You-See-Is-What-You-Get
(WYSIWIG) editor for problem variables, such as that illustrated in
FIG. 31.
[0089] Thus, for example, in an embodiment, a teacher is able to
create and save problems that employ hide and show, link and
unlink, lock and unlock, and multiple models. These functionalities
allow a teacher to use the Tools to create his or her own problems,
save those problems, assign them to students, and share them with
colleagues.
Benefits of the Features, Tools, and Modes
[0090] The ability to scaffold instruction. An embodiment provides
the ability to scaffold instruction. It should be appreciated that
in an embodiment, many of the provided tools offer up to 13,000
combinations of the variables dictated by the feature sets.
[0091] For example, the tool called Adding Fractions with Common
Denominators may be analyzed in order to elucidate these
combinations. The tool may include the following elements on the
screen: six manipulative models and three fractions, each with a
separate field for the numerator and denominator. An author may
choose to employ these elements in a variety of ways in order to
accomplish specific educational objectives. Hereinbelow in no
particular order are some examples of how a problem, such as
2/3+3/8, may be presented: [0092] a) Show the problem with locked
models in the addends, models only, no numbers using one specific
model, not allowing the student to choose their own, different
model. In the answer field, divide the sum into the appropriate
number of equal parts (8), and ask the student to shade the
requisite number of equal parts in order to represent the answer
(5), (6 possible renderings, given the different models) [0093] b)
Show the problem exactly as described in (a), except do not include
the pre-determined division into 8 equal parts. Instead, ask the
student to make their own partitions and to shade the appropriate
number (5/8). (6 possible renderings, given the different models)
[0094] c) Show the problem exactly as described in (a) above,
except give the student the option to display specified alternative
models. The author has one model set as the default, and the
student may choose either one other model, two other models, or
whatever set of the five other models the author has selected. (60
possible renderings) [0095] d) Show the problem exactly as
described in (a) above, except unlock the numerators of the
addends, forcing the student to shade their own addends in each
model. Require the student to represent the problem, 2/8+3/8. (6
possible renderings, given the different models) [0096] e) Show the
problem exactly as described in (d) above, except unlock the
numerator of one of the addends, forcing the student to shade the
addends in one model. Require the student to represent the problem,
2/8+3/8. (12 possible renderings, given the different models, and
the choice to have the student shade one or the other
numerator)
[0097] The permutations and combinations above describe 90
different possible representations of the same problem. Many more
representations are possible as the screen is evolved to display
models and numbers together, and numbers only. When a given problem
includes more than one layer (the layers in this instance being
models and numbers), then there exists the possibility to Link or
Unlink these layers. Link/Unlink may increase the number of
variables substantially.
[0098] Each combination potentially offers a unique level of
scaffolding to drive student instruction. For the purposes of the
discussion herein, scaffolding instruction is a term that describes
the careful sequence of steps that may be taken to help a student
learn a topic. For example, the problems may be designed to be
instructional. Thus, the purpose for a problem set is to provide
the student with a sequence of interactions that leads the student
down a particular pedagogical path. One problem builds from the
prior problem to help the student gain greater insight into the
fractions topic. Much like a scaffold on the side of a building
helps painters work at higher and higher floors, carefully
scaffolded instruction helps students understand sequentially more
advanced mathematical topics. Thus, as shown hereinabove and
according to an embodiment, authors may use the spreadsheets and
teachers may use the Tools to build such scaffolds and lead
students to higher understanding.
[0099] In an embodiment, a broadly accepted tenant in scaffolded
instruction is progressing from concrete-representational (or
pictorial) to abstract (numbers). Much research has been conducted
that points to the value of this progression. It has been found
that students learn better when they learn mathematics first with
hands-on manipulatives (such as blocks), then pictures of those
manipulatives (such as the models in the system discussed herein),
and finally numbers. This sequence may be referred to as concrete,
to concrete representational, to abstract. [0100] Scaffolding,
according to an embodiment, can be understood with reference to
FIGS. 28A and 28B. FIGS. 28A and 28B are a table containing screens
of an example where each screen represents like problems on each
floor of the scaffolding, according to an embodiment. The example
in FIGS. 28A and 28B demonstrates scaffolding in one problem set
that is included in an embodiment. In the example, the topic is
Adding Unit Fractions. A "unit" fraction has the number "1" in the
numerator. The screens displayed are extracted from a set of 30
problems. Each screen represents six like problems. In other words,
there are six problems on each "floor" of the "scaffolding."
[0101] Thus, using the Adding Fractions with Common Denominators
Tool, a teacher may render each of the problems in real time. This
scaffolding may be presented live (real-time) in tool mode,
designed into the problems included in activity mode, or saved by
teachers in authoring mode.
[0102] Progressing from Visual Concepts to Procedures. The four
features, added together, provide a unique environment to present
lessons that are organized to guide students from understanding
visual models to understanding numeric procedures.
[0103] Correlating with different curricula. Given the great
flexibility built into the computer-implemented methodology, it is
possible to adapt particular representations of the
computer-implemented methodology to match published mathematics
curricula.
[0104] Useful in many different classroom settings. Flexibility
makes the computer-implemented methodology applicable and useful in
many modes of instruction; teacher-led whole class instruction,
computer labs, one on one instruction, independent student practice
(including at home) and intervention environments.
[0105] Answer checking and corrective feedback. An embodiment
provides a unique capability of checking the accuracy of student
responses and providing corrective feedback. It should be
appreciated that an embodiment provides such capability for
problems rendered in tool mode and activity mode. Because the
computer-implemented methodology is aware of the complex
relationships between the many features on the screen, it provides
responses and feedback that are customized around the student's
individual work.
The Four Features
Feature #1: Hide and Show
[0106] Purpose: In order to accomplish the goal of revealing one,
two, three or four of the layers discussed above, the
computer-implemented methodology provides a Hide and Show feature.
In a particular educational approach to teaching fractions, it is
desirable for teachers and students to appreciate the relationships
between pictorial models and numeric equations, i.e. or procedures.
An embodiment provides either two or three elements to hide or
show, depending upon the fractions topic. Such elements are
referred to herein as layers. Earlier topics in the learning
process may contain two layers, the models layer and the fractions
layer as shown for example in FIG. 2. More advanced topics may
include a layer that is devoted to equivalence; this equivalence
layer offers a third option for these topics, as shown for example
in FIG. 4. One specific topic, equivalent fractions with sentences,
includes a sentence layer. In this instance, the emphasis of the
progress is from visual models to sentences, instead of visual
models to procedures. FIG. 5 depicts an example screen shot showing
a sentence layer. An embodiment contemplates similar tools for
multiplying and dividing fractions, which are not shown. An
embodiment contemplates employing the sentence layer for such tools
for multiplying and dividing fractions.
[0107] Procedure: In an embodiment, in Tool Mode, Hide and Show is
implemented with a simple click of an icon that appears on the
right side of the screen, for example. An embodiment provides an
icon for each of the model, equivalence, and fraction layers, such
as shown in FIG. 6. When a layer is hidden, its icon is grayed out.
In Activity Mode, Hide and Show is implemented in the authoring
spreadsheet. There are up to five hide and show variables in tools
that have three layers: show layer 1, show layer 3, show layers 1
and 2, show layers 2 and 3, show layers 1 and 3. In Authoring Mode,
Hide and Show may be implemented with the same icons as in Tool
Mode.
[0108] An embodiment using the spreadsheet to invoke Hide and Show
can be understood with reference to FIGS. 29A and 29B. FIG. 29A and
FIG. 29B, are a table containing screens, each of which help
demonstrate the results of using the spreadsheet to invoke Hide and
Show. The first four images are taken from an example Big Idea 5,
which covers the topic of equivalent fractions. Such images employ
the Equivalent Fractions Tool. The author used different settings
within the spreadsheet to Hide or Show different layers on the
screen. This was done to help students progress from
concrete-representational to abstract, numbers-only
representations.
[0109] Another embodiment can be understood with reference to FIG.
6, which illustrates icons for the hide or show models layer, hide
or show equivalence layer, and hide or show fractions layer,
according to an embodiment and FIG. 5, which illustrates an example
placement of the Hide and Show icons in the upper right corner of
the screen.
[0110] Result: By using the Hide and Show feature a teacher can
present the students a sequence of educational visual
representations. The first step in the sequence may be a
presentation that includes only pictorial models. The second step
may be a combination of pictorial models and numeric
representations, using the fractions themselves. The third step may
include the fractions, omitting the pictorial representations.
Refer to FIG. 7 for example screen shots of the above discussed
three-step sequence.
Feature #2: Link and Unlink
[0111] Purpose: In an embodiment, Link and Unlink determines the
causal relationship between editing numbers and pictures, or
editing numbers and words. At times this relationship may be a
one-to-one correspondence, where there is a close association
between numbers and models. For example, when the user types a
fraction (or enters numbers into a sentence), a pictorial model may
be rendered that corresponds to the user's typed numbers. As
another example, when the user populates a visual model, the
corresponding fractions (or numbers in sentences) may become
populated. At other times it is desirable to disassociate these
fields and erase the one-to-one correspondence. In the
disassociated state, numbers may be entered, sentences completed,
and models populated without any causal effect in the companion
representation. Thus, Link and Unlink enables such states of
association and disassociation.
[0112] In an embodiment, Link and Unlink may be implemented using
an observer pattern where the linked components emit events when
they change. In this pattern, the corresponding components listen
for these events and synchronize themselves to the changed value.
For instance, in the identifying fractions interaction, if the
fraction model component and the fraction number component are
visible and linked and the user types into the numerator box, then
the fraction number component calculates its new value and if this
new value is different than its previous value, then it emits a
NUMERATOR_CHANGED event. When the fraction model component receives
this event, it reads the numerator and denominator from the
fraction number component and uses these values to set its own
numerator and denominator and then redraws itself using these new
values. When components are unlinked, such components simply stop
listening for these synchronization events.
[0113] Consider the use of Unlink in an early topic of learning
about Identifying Fractions. In Tool Mode, a teacher may present a
visual model of a fraction and ask the students to identify the
name of the fraction. This can be accomplished by unlinking the
model and the fraction. When the teacher creates the model, the
fraction is not populated. Instead, it is left blank. This feature
gives the student the opportunity to fill in the blank. An example
is illustrated in FIG. 8.
[0114] Consider a linked presentation of Identifying Fractions. As
an example, suppose the students are given the following
instruction: "Come to the whiteboard in the front of the room. I
will indicate a fraction and you can make that fraction either with
the model or the numbers." A student comes to the front of the room
and the teacher states, "Please show the class three fifths." The
student chooses to populate the model. FIG. 9 shows the model in
its unpopulated state (before the students starts her work), its
partially populated state (the student is partially done), and its
completed state. Upon accurately completing the model, the fraction
is rendered because the model and the fraction are linked.
[0115] The Linked and Unlinked feature according to an embodiment
may play a pivotal role in Activity Mode. FIG. 10 shows an example
of a problem where a student is adding fractions with common
denominators and the linked option has been employed within the
spreadsheet created by the author. The screen on the left displays
the problem as it is first encountered by the student. As is the
case with Activity Mode, the problem includes a written
instruction. In this case, the instruction reads, "Add the
fractions. The models will help you." The student has only one task
to accomplish: writing in the numerator of the sum. Because Linked
is built into this problem, the model becomes populated when the
student types the number. It is through this linkage, i.e. this
association between the number and the model, that the student
receives the visual confirmation in the model that their selection
of "3" makes sense and is the correct answer.
[0116] It should be appreciated that in an embodiment, the
spreadsheet author invokes this linkage by setting a variable for
this specific problem. The spreadsheet contains a row, e.g. row 26,
which is labeled as linked/numberandmanipulative. That label is
entered on the first column, to the left side of the spreadsheet.
The columns to the right set the state of each variable for each
problem. In this column, the text in row 26 is "true." This sets
the variable for "Linked". If the text read "false", then the
manipulative and numerator in this problem would be unlinked.
[0117] As the student moves forward in the activity, a subsequent
problem uses the unlinked state. The student is asked to add two
fractions and the instructions state: "Use the models if you wish."
Here, as depicted in FIG. 11, the models are presented as a
voluntary support. If the student knows the answer and has
internalized the logic of adding fractions to the extent that they
no longer require the support of models, they can simply add the
fraction. On the left, the student has added the fraction without
ever touching the models. Because this mode is in the unlinked
state, the models remain neutral or unpopulated. On the right, a
different student has populated the first model only and then
realized that he/she no longer required visual support to complete
the problem. It is with this type of flexibility that an embodiment
of the computer-implemented methodology provides the
carefully-stages scaffolding that allows students to internally
construct mental representations of fractions.
[0118] Procedure: In an embodiment, in Tool Mode, Link and Unlink
is implemented with a simple click of an icon that appears on the
right side of the screen. There is one icon for tools that have two
layers, and two icons for tools that have three layers. When an
icon is set to unlink, its icon is grayed out. Refer to FIG. 12 for
example icons according to an embodiment. In Activity Mode, Link
and unlink is implemented in the authoring spreadsheet. There are
up to three variables for link and unlink in the spreadsheet
authoring system. The tools that have three variables for Unlink
and Unlink include the tools that have three-layers. These include,
but are not limited to: [0119] Equivalent Fractions [0120]
Equivalent Fractions with Word Sentences [0121] Finding Common
Denominators [0122] Adding Fractions with Uncommon Denominators
[0123] Subtracting Fractions with Uncommon Denominators
[0124] One of these tools, Adding Fractions with Uncommon
Denominators, is depicted in FIG. 4 and in FIG. 18. The three
layers are clearly shown. The top layer displays the manipulative
models. The middle layer shows the fractions, some of which are
multiplied by a "one fraction," or x/x. In FIG. 4, there are two
"one fractions," 2/2 and 1/1. The bottom layer, or third layer,
contains the fractions that result from the multiplication
performed in the middle layer. In an embodiment, these three layers
are called: the manipulative layer, the equivalence layer, and the
number layer.
[0125] According to an embodiment, In the spreadsheet and in
authoring mode, the three link and unlink variables that may be
controlled are 1) linking and unlinking the number and the
manipulative, 2) linking and unlinking the equivalence and the
manipulative, and 3) linking and unlinking the equivalence and the
number. For example, these three variables may be named
numberAndManipulative, manipulativeAndEquivalence, and
numberAndEquivalence. In the spreadsheet, these variables are
controlled by indicating whether any of these three conditions is
"true" or "false." In Authoring Mode, Link and Unlink may be
implemented with the same icons as in Tool Mode.
[0126] Result: The Link and Unlink feature provides flexibility in
activity development. FIGS. 8, 9, and 10 provide examples of the
progression from concepts to procedures that is facilitated through
the use of the feature. It is with this type of flexibility that
the computer-implemented methodology provides the carefully-stages
scaffolding that allows students to internally construct mental
representations of fractions.
Feature #3: Lock and Unlock
[0127] Purpose: In an embodiment, Lock and Unlock is an authoring
feature that limits the responses students are required to make in
a given problem. It is with Lock and Unlock that an author may
require a student to enter, for example, numerators only,
denominators only, or otherwise focus student responses on a
specific element. Lock and Unlock may be applied to any portion of
the screen that can be altered by a student. Locking an element
omits the ability to change that value. Unlocking the element and
allows the student to edit that value. Commonly, unlocked fields
are left blank.
[0128] In an embodiment, Lock and Unlock pertain to authored
activities and they are designed to provide focus to a student in a
constrained environment when there is no guarantee of a teacher
being present. Consider the example in FIG. 13: Locked equal parts.
The student is asked to shade the model and make the fraction "one
third." The activity is constrained so that the student is not
required to create the equal parts on their own. In this example,
the denominator of the fraction (as represented by the model) has
been set as three (thirds) and locked. The student cannot change
its value, thus, as shown in the figure, there is no slider
available for changing the number of equal parts.
[0129] Lock and unlock determine the editability of components. A
locked component cannot be directly edited by the student, an
unlocked component can be directly edited by the student.
Editability is distinguished from mutabililty in that an locked
component's value, even though it cannot be directly edited by the
user, can still be changed if it is linked to another component and
that component's value changes. In an embodiment, locking may be
accomplished in any number of ways. One way may be to make the
locked component stop listening to user input from the mouse or
keyboard. Another way may be to make the locked component ignore
user input from the mouse or keyboard. For instance, the fraction
number component sets a locked property to true. Once this property
is set to true, any user input received from the keyboard is
ignored. When the component is unlocked, the locked property is set
to false and a subsequent keyboard input is processed.
[0130] According to an embodiment, another way to implement locking
is for GUI controls that change the value of the component to be
removed or hidden. For instance, if the denominator (the number of
equal parts) of the pie model is locked, then the slider control
that changes the number of equal parts is hidden. This prevents the
student from directly changing the model's denominator.
Importantly, this implementation of locking is distinguished from
making the model's denominator immutable. If the denominator were
immutable, then linking another component to the denominator would
have no effect. This is undesirable; it is important that a
component that is locked can also be linked to another component
and if that other component's value changes, then the locked
component's value should change. This may be understood with
reference to FIG. 9, which shows an embodiment that uses the
identify fractions tool, with a locked fraction pie component and
an unlocked fraction number component that are linked. In this
case, the student cannot directly change the value of the locked
fraction pie by using the slider, but the student can change the
fraction number component by typing numbers into it. Because the
fraction number is linked to the fraction pie, it in turn changes
the value of the fraction pie. This is an important interaction,
because the student may construct a visual model by inputting
numbers only. This example illustrates that the locked fraction
pie's value remains mutable in even in the locked state.
[0131] The example of a locked numerator and unlocked denominator
may be understood with reference to FIG. 14: Locked numerator in a
model. FIG. 14 demonstrates a pedagogical design that is unique to
the computer-implemented methodology or system. FIG. 14 depicts an
addition with uncommon denominators problem. The student is asked
to solve the problem using visual models only. Initially, the
models represent 1/2+1/4. The sum has already been rendered in a
model that has a locked denominator (fourths) and an unlocked
numerator which is blank. The final answer requires the student to
populate three of the four equal shares in the model for the sum.
As part of answering the question, however, the student is required
to "change the denominator on the first fraction.", in other words,
they are required to create common denominators. The shaded area
has been pre-determined and locked. One half of the entire circle
has been shaded. This addend model is the opposite of the sum
model. The addend model has a locked numerator (one half) and an
unlocked denominator (number of equal parts). On the right side of
the figure, the student has changed the number of equal parts and
effectively re-stated the fraction as 2/4. Now, the student can
visually see that 2/4 can be added to 1/4 in order to make 3/4.
[0132] A common use of Lock and Unlock is in constructing a problem
where entire portions are required to be fixed and unaltered. In
both FIG. 10: Activity Mode--Linked and FIG. 11: The Flexibility of
using Unlinked, the addends of the equations are completely
locked.
[0133] In an embodiment, three tools employ a special case version
of Lock, cited hereinbelow. In these tools, the lock feature is
employed as a method for preparing fractions for manipulation on
the screen. These tools include Comparing Fractions, Order
Fractions on a Number Line, and Unitizing with Pattern Blocks. In
all three cases, lock has a utilitarian function and must be
employed in order for a user to start interacting with the
fractions. The user is given the opportunity to place their own
fractions onto the screen. The user enters a numerator and
denominator in each of one, two, or three blank fraction(s). When
the user is satisfied with the fraction(s) they have entered, they
lock these fractions, e.g. by using the icon displayed in FIG. 15.
Once this locking action has been accomplished, the user may
display the fraction(s) in the form of various models, and may act
on the fractions according to the function of the specific tool.
One example, involving the Compare Tool, is described in this
document. In Comparing Fractions and Ordering Fractions on a Number
Line, the user does not have an "unlock" option. The fractions
start in the unlocked state. Once the use locks them, the user may
employ them, but the user cannot unlock them and re-state their
value. In order to act on a new set of fractions, the user resets
the tool, effectively completing their session with one set of
fractions, and is then given two or three unlocked fractions. These
unlocked fractions are ready to be populated, locked, and acted
upon.
[0134] Procedure: In an embodiment, a version of lock may use the
icon such as displayed in FIG. 15: The Lock Icon. In an embodiment
in Activity Mode, Lock and Unlock is implemented in the authoring
spreadsheet. There may be many locations on a given page where lock
and unlock may be employed. In an embodiment, there are up to
fourteen variables for lock and unlock in the spreadsheet authoring
system. For the purposes of the discussion herein, the following
examples, each related to a Figure, elucidate the various variables
that may be controlled by Lock and Unlock: [0135] In FIG. 19, the
two fractions that constitute the minuend and subtrahend (3/4 and
1/4) are locked in both the numerator and denominator. As a result,
the student cannot change the problem. In the difference, or
answer, the denominator is locked (fourths), but the numerator is
unlocked. The student must create the numerator. [0136] In FIG. 21,
the two fractions that constitute the minuend and subtrahend (3/5
and ) are locked in both the numerator and denominator. As a
result, the student cannot change the problem. In the difference,
or answer, both the denominator and the numerator are unlocked. The
student must create the denominator and the numerator. [0137] In
FIG. 23, the two fractions that constitute the minuend and
subtrahend ( 7/9 and 3/9) are locked in both the numerator and
denominator. This is true for both models and numbers. Thus, there
are eight different locked elements in these two fractions. As a
result, the student cannot change the problem, nor are they asked
to enter their own numbers or change the pictures in the minuend or
subtrahend. In the difference, or answer, the numerators of both
the model and number are unlocked. The student must create both
numerators in order to answer the problem.
[0138] In an embodiment in Authoring Mode, Lock and Unlock may be
implemented using the same icons as in Tool Mode, but the icon may
be attributable to specific text fields and portions of models. It
is possible to Lock or Unlock the numerator and denominator of each
fraction model, e.g. the equal parts, the shaded parts, and each
fraction, e.g. a text field for each numerator and denominator.
[0139] Result: According to an embodiment, some important uses of
Lock and Unlock are those that involve strategically focusing the
student on specific responses. Student activities as provided in an
embodiment may place emphasis on the sequence of problems with
which students are presented. Lock and Unlock allow an activity
author to focus a student's attention on one specific answer field,
or one specific portion of a model, as illustrated in FIG. 10. It
is with this type of authoring that the computer-implemented
methodology provides the focus that allows students to clearly
understand expectations, and provides the scaffolding to build
cognitive models.
Feature #4: Multiple Models
[0140] Purpose: When teaching fractions and other mathematical
subjects, research has shown that students should be subject to the
use of multiple visual models. Thus, an embodiment provides a
multiple models implementation that allows a student or a teacher
to easily switch the visual representation, i.e. the visual model.
The research also shows that different models represent different
ways of thinking about fractions. For instance, an
area-of-a-rectangle model portrays the fraction as a portion of a
whole. In this model, 2/3 and 4/6 occupy the identical area. A
model that employs dots is often described as a discrete model.
When there are three dots and two of those are shaded, the fraction
being represented is two thirds. There is no "whole", other than
the collection of three dots. It should be appreciated that an
embodiment provides configuring the system such that a student may
fill in any of the circles in the model where the filled-in circles
are not required to be contiguous. For example, the foregoing may
mean that in a collection of eight dots, the student may shade two
dots that are not next to one another in order to portray 2/8. A
number line may be used to portray a distance model, where 2/3 is
the distance traveled between the zero marker on the left and the
number one on the right. A fraction pie or fraction circle that is
segmented into discrete sections depicts a fraction in a manner
that combines the concept of area (the portion of the whole circle
that is shaded) with the discrete model (one of three equal shares
is shaded). An embodiment can be understood with reference to FIG.
16: The Visual Models employed in the Identifying Fractions tool.
Such tool shows the manner in which the fraction 2/3 is displayed
by the six models that are incorporated in the Indentifying
Fractions tools.
[0141] The Multiple Models feature as provided in an embodiment
makes it possible to interchange models, to select a specific
model, or to select a set of models for a given problem. In an
embodiment, the tools use interchangeable models with the exception
of Unitizing with Pattern Blocks and Unitizing on a Grid. Some
models may be employed in any of the fractions computations that
have been discussed. The dot model or discrete model may not be
useful in interactions that involve equivalent fractions or the
addition and subtraction of fractions with uncommon denominators.
This is because it may not be helpful to attempt to show that
one-out-of-three dots are equivalent to two-out-of-six dots. In an
embodiment, the tools which are used either on their own or as a
basis for authoring student activities employ six interchangeable
models. By employing this capability, the system offers the benefit
of flexible thinking about fractions, as indicated in the research,
and offers the related benefit of having students understand
fractions to have more than one meaning: parts of a whole, part of
a set, a distance along a line, a numeric value, and an area. An
embodiment includes, but is not limited to, the following models:
[0142] fractions Circle [0143] vertical fraction bar [0144]
horizontal fraction bar [0145] dot model, or discrete model [0146]
number line [0147] numeric representation [0148] decimal [0149]
decimal grid
[0150] Procedure: In an embodiment, in Tool Mode, Multiple Models
is implemented with a simple click of a set of icons that appears
on the left side of the screen. There is one icon for each
available model. FIG. 17 illustrates a particular implementation of
a set of icons. As well, in an embodiment, in Activity Mode, the
set of available models may be customized in the authoring
spreadsheet. As a result, students do not always receive a choice
of models, as there are some problems where the author has
determined that the interaction should be constrained to just one
model. In an embodiment, there are up to 63 combinations of visual
model choices in most of the tools, as 63 is the total number of
combinations of any six items. It is contemplated that in Authoring
Mode, the ability to use or make available the visual models as is
in the spreadsheet activity mode may be available.
[0151] Multiple models may be implemented on a computing system in
a number of ways. In one embodiment, an object oriented approach is
taken. A template class is created that declares an application
programming interface for the methods and properties common to all
fraction models. A default implementation, e.g. of such methods and
properties, is created. Each individual fraction model class is
derived from this template class. Each individual model class
implements, e.g. on a computer, the logic to layout the model, draw
the model, handle hit-testing on the model and other functions
common to all models; however, a custom implementation may be
created for each given model. For instance, the hit testing method
of the template class declares a function called pointToNumerator,
that takes a horizontal pixel value and a vertical pixel value that
represent the local coordinates of the mouse pointer within the
model. The computer-implemented method returns the index of the
equal share that the point is over in the model or -1 if it is not
over an equal share. To implement its version of this common
method, the computer-implemented fraction pie model first
determines if the given point is within the fraction pie's
rectangular bounds. If it is not, -1 is returned. If it is, then
the fraction pie computer-implemented logic uses the given pixel
coordinates and its center pixel coordinate to compute the angle in
degrees of the line created by the center point and the given
point. The computer-implemented logic then computes the arc length,
in degrees, represented by one equal share by dividing 360 by the
denominator. It then divides the angle in degrees by the arc in
degrees to compute the index of the equal share. The same
computer-implemented method is implemented differently for the
number line model. This model does a similar bounds check,
returning -1 if the given pixel value is not within its bounds. If
it is within its bounds, it computes this segment's horizontal
length by subtracting the horizontal pixel value of the zero point
on the number line from the given horizontal point value. It then
computes the horizontal length of an equal share by computing the
horizontal pixel length of the number line and dividing it by the
total number of equal shares. It then divides the segment length by
the length of one equal share to compute the index of the equal
share. In addition to implementations of common methods, some
models implement model specific methods or properties. For
instance, the number line model implements a maximum value property
that determines the value on the right end of the number line.
[0152] Importantly, when the student selects a model from the
available models, the link/unlink feature and other aspects of the
system that change the values in the currently selected model must
be reflected in the newly selected model. This can be done in a
number of ways. In an embodiment, the system logic may synchronize
all changes to all available models, so that when a new model is
selected, it already has the necessary, e.g. updated, values. In
one embodiment, a model container holds shadow copies of the
necessary model values and the embodiment synchronizes new models
with these values as such new models are selected. In addition, the
embodiment may delay allocation of a new model until the first time
such new model is selected.
[0153] Result: The Multiple Models feature leads to several unique
aspects. One aspect is that it enables the computer-implemented
methodology to correlate closely with most any published
mathematics curricula for elementary and middle school. A key
purpose for the computer-implemented methodology is to supplement
the basal curriculum employed in the schools. Different math
curricula emphasize different visual models at particular points in
the fractions curriculum. It should be appreciated that most of the
models employed in such curricula are embodied in embodiments of
the computer-implemented methodology. For example, by employing the
Tool Mode or the Authoring Mode, teachers may correlate daily
lessons directly to the visual models employed in the basal
curricula.
[0154] A second aspect is that the computer-implemented methodology
adapts easily to different learning and teaching styles. Fractions
are a difficult topic to teach and learn. When a teacher finds it
challenging to convey a given topic using one model, he/she can
instantly adapt their lesson and try an alternative model.
Similarly, when a student finds one visualization to be
confounding, he/she can switch the model and provide an alternative
model for an alternative learning path for themselves.
[0155] A third aspect in the use of Multiple Models involves
flexibility of thinking. It has been found that the research is
clear. When a student understands fractions interactions as
represented in a variety of visual, representational formats, he or
she has a heightened comprehension of fractions content. Various
research projects have pointed to the importance of using fraction
circles, the area of a rectangle, dots (a discrete model), and the
number line. It has been found that the researchers of such
research appear to agree that it is desirable for students to be
comfortable with multiple renderings. The field is converging on
the belief that ultimately, students should recognize the value of
fractions on a number line as a measure of their absolute value as
a numerical entity. Thus, the computer-implemented methodology,
through at least the Multiple Models feature, makes it possible to
educate to these lofty goals.
Correction and Feedback
The Salient Feature: Error Detection and Corrective Feedback
[0156] One salient feature embodied throughout the
computer-implemented methodology is persistent awareness of the
correct relationships between models and numbers, addends and sums,
and the other relative components of the mathematical equations.
This is true for Tool Mode, Activity Mode, and Authoring Mode. For
instance, when the problem 7/8- 2/8 is input into the system, the
system knows implicitly that the answer is 5/8. Similarly, the
system knows that a model that accompanies 7/8 must accurately
represent 7/8. Such awareness by the system of such correct
relationships enables a corrective device in an embodiment. In the
linked state, where the models are rendered automatically and in
direct concert with the numerical representations, this corrective
device may be redundant and unnecessary. However, in the unlinked
state, such relationships may become very important and serves as a
corrective device for analyzing the accuracy of student work.
Similarly, when the student adds two fractions, such as 1/6+1/6,
and performs such addition with representational models, the system
knows that the sum will be a model comprised of 2/6, whether or not
the system displays such sum in the model. As well, the system also
knows that the model one third is equivalent to 2/6. Authoring
capabilities built into the spreadsheet for Activity Mode allow an
author to determine whether an equivalent fraction will be regarded
as an acceptable answer. It should be appreciated, however, that an
author or tools user need not consider how to account for correct
or incorrect answers, as the system is configured to be aware of,
i.e. contain in persistent or non-persistent memory, the correct
answer. It should be appreciated that the knowledge of the
relationships by the system as described above may be very powerful
and their uses by educators may be endless in the pursuit of
teaching students.
The Corrective Feedback Subsystem
[0157] In an embodiment, the system is aware of the status of the
four features and the mathematical relationships between the items
on the screen, as described above. Therefore, when an error does
occur, the system knows where on the screen, e.g. which item, such
error or errors is located. The system provides feedback that
focuses the student's attention on any area of the problem that
needs correction. An example can be understood with reference to
FIG. 18. In FIG. 18, a student attempted to add fractions which
have different denominators. The system provided corrective
feedback in this example by circling each of the denominators and
by provided a message in a message box saying, "All of the
fractions in the equation must use a common denominator."
[0158] An embodiment provides algorithms that determine an order of
correction. For instance, any mismatch between models and numbers
gets the first attention. Errors in equal shares or denominators
receive attention before errors in shaded parts or numerators. This
way, the system provides corrective feedback as a student repairs
the errors that are on the screen. It should be appreciated that
implementing a particular order of correction may require the
student to fix and recheck his or her work several times.
[0159] In an embodiment, the system never provides the student with
the answer. Rather, the system guides the student in solving the
problems. The system does not provide specific inputs or the
answer.
[0160] In an embodiment, such unique feedback subsystem is made
possible because of the thorough amount of data that passes between
all aspects of the system. The various combinations of the four
features informs the feedback system;
[0161] feedback is generated based on the students inputs, which
layers are visible, which components are linked, which components
are not linked, which components are locked, which components are
unlocked and which model is selected. For instance, if a component
is not visible, then it is not checked for correctness and plays no
part in the feedback. If a component is visible and it is linked to
another component, then feedback is generated to take into account
that the component and its linked component represent a single
value, rather than two separate values. This reduces the verbosity
of the feedback and reinforces the conceptual correspondence
between the linked values. In another example, if the values are
not linked, then individual feedback is generated for each
component, so the student knows that each component must be
addressed to create a fully correct answer. In another example, if
a visible component is locked, then the feedback system still
considers its value when determining correctness and generating
feedback, however, since the component cannot be directly changed
by the user, the feedback only refers to its value and does not
indicate to the student that they may change the value to create a
correct answer. In another example, if a visible component is
unlocked, then feedback may be generated that indicates to the
student that the value can be changed to create a correct
answer.
Putting the Features into Action: Progressing from Visual Concepts
to Procedures
[0162] FIGS. 19-26 depict the steps taken, in Activity Mode of the
system, to help students progress from Concepts to Procedures in
studying a particular topic of Subtracting Fractions with Common
Denominators, in accordance with an embodiment. It should be
appreciated that such screens and comments paint a picture of how
the four features of an embodiment are used in combination to
facilitate the cognitive progression.
[0163] FIG. 19 shows Step 1, models only, where the denominator is
determined by being provided in the difference. That is, the
denominator in the difference model is locked. The fractions layer
is hidden.
[0164] FIG. 20 shows Step 2, for a different problem, a different
model (pie circle) is selected. The pie circle icon is highlighted
because it is selected.
[0165] FIG. 21 shows Step 3, the student creates the entire
fraction in the difference. FIG. 21 shows a problem using the
vertical bar model. In the example, both the numerator and the
denominator of the difference model are unlocked. Thus, the student
must build the answer in the difference model from scratch.
[0166] FIG. 22 shows Step 4, numbers are introduced and the student
provides the numerator of the difference model. FIG. 22 shows
linked rectangle models and fractions. However, the numerator in
the difference fraction is unlocked while its denominator is
locked. Thus, the student may either fill in the model or type in
the correct number in the fraction to solve the problem.
[0167] FIG. 23 shows Step 5, strategically changing the model to
promote flexibility. FIG. 23 shows the discrete model icon is
selected. Thus, discrete models are shown and the fractions layer
is also shown.
[0168] FIG. 24 shows Step 6, the student provides the numerator and
the denominator. FIG. 24 shows that the vertical bar icon is
selected. The vertical bar models are shown and the fractions layer
is shown. In this particular example, the models are unlinked to
the fractions. As well, both the numerator and the denominator in
the difference model are unlocked. Such example shows the
increasing sophistication and learning that has taken place by the
student, because the student presumably wants to solve the problem
using fractions only and not the models, which may be presented for
support.
[0169] FIG. 25 shows Step 7, procedures only, the student
calculates the numerator. FIG. 25 shows the models layer is hidden.
In this embodiment, the set of models icons is also hidden. In the
difference model, the numerator is unlocked and the denominator is
locked.
[0170] FIG. 26 shows Step 8, the student is independently
completing the equation. FIG. 26 shows the models layer is hidden.
In this embodiment, the set of models icons is also hidden. In the
difference model, both the numerator and the denominator are
unlocked. This is the final step in the sequence of visual concepts
to procedures.
Alternative Implementations
[0171] It should be appreciated that, among other areas of
mathematics, alternative implementations are provided. Exemplary
implementations for whole numbers and for decimals and percents are
provided hereinbelow.
Whole Numbers
[0172] Provided below are particular models and tools used for
teaching and learning whole numbers, according to an
embodiment.
Models
[0173] Sets [0174] Rods [0175] Two dimensional arrays [0176] ten
frames [0177] Place value blocks [0178] number line [0179] hundreds
chart
Tools
[0179] [0180] Number identification [0181] Counting [0182] Compose
(Combine) and Decompose (separate) [0183] Counting on/Counting
back/Skip Counting [0184] Comparison/Ordering [0185] Number line
[0186] Automaticity [0187] Word problems in 4 operations [0188]
Place value and Rounding [0189] Addition to 20 [0190] Addition Fact
Families (5+7-12,7+5=12,12-5=7,12-7=5) [0191] Place value addition
(expanded notation) [0192] Subtraction under 20 [0193] Place value
subtraction (expanded notation) [0194] Primes, Factors, Multiples,
Least Common Multiple [0195] Multiplication within 100 (number line
& arrays) [0196] Multiplication Fact families
(5*7=35,7*5=35,35/5=7,35/7=5) [0197] Place value multiplication
[0198] Factors and multiples, Even/Odd, Prime and composite numbers
[0199] Division within 100 [0200] Place value Division (long
division)
Decimals and Percents
[0201] Provided below are particular models and tools used for
teaching and learning decimals and percents, according to an
embodiment.
Models
[0202] decimal chart [0203] decimal blocks [0204] numberline [0205]
all models in the Fractions product
Tools
[0205] [0206] Identifying, writing decimals [0207] Composition,
Decomposition [0208] Comparison, Sorting [0209] Number line [0210]
Correspondence to fractions, fractions as division [0211] Rounding
[0212] Addition [0213] Subtraction
An Example Machine Overview
[0214] FIG. 30 is a block schematic diagram of a system in the
exemplary form of a computer system 1600 within which a set of
instructions for causing the system to perform any one of the
foregoing methodologies may be executed. In alternative
embodiments, the system may comprise a network router, a network
switch, a network bridge, personal digital assistant (PDA), a
cellular telephone, a Web appliance or any system capable of
executing a sequence of instructions that specify actions to be
taken by that system.
[0215] The computer system 1600 includes a processor 1602, a main
memory 1604 and a static memory 1606, which communicate with each
other via a bus 1608. The computer system 1600 may further include
a display unit 1610, for example, a liquid crystal display (LCD) or
a cathode ray tube (CRT). The computer system 1600 also includes an
alphanumeric input device 1612, for example, a keyboard; a cursor
control device 1614, for example, a mouse; a disk drive unit 1616,
a signal generation device 1618, for example, a speaker, and a
network interface device 1628.
[0216] The disk drive unit 1616 includes a machine-readable medium
1624 on which is stored a set of executable instructions, i.e.
software, 1626 embodying any one, or all, of the methodologies
described herein below. The software 1626 is also shown to reside,
completely or at least partially, within the main memory 1604
and/or within the processor 1602. The software 1626 may further be
transmitted or received over a network, 1630 by means of a network
interface device 1628.
[0217] In contrast to the system 1600 discussed above, a different
embodiment uses logic circuitry instead of computer-executed
instructions to implement processing entities. Depending upon the
particular requirements of the application in the areas of speed,
expense, tooling costs, and the like, this logic may be implemented
by constructing an application-specific integrated circuit (ASIC)
having thousands of tiny integrated transistors. Such an ASIC may
be implemented with CMOS (complimentary metal oxide semiconductor),
TTL (transistor-transistor logic), VLSI (very large systems
integration), or another suitable construction. Other alternatives
include a digital signal processing chip (DSP), discrete circuitry
(such as resistors, capacitors, diodes, inductors, and
transistors), field programmable gate array (FPGA), programmable
logic array (PLA), programmable logic device (PLD), and the
like.
[0218] It is to be understood that embodiments may be used as or to
support software programs or software modules executed upon some
form of processing core (such as the CPU of a computer) or
otherwise implemented or realized upon or within a system or
computer readable medium. A machine-readable medium includes any
mechanism for storing or transmitting information in a form
readable by a machine, e.g. a computer. For example, a machine
readable medium includes read-only memory (ROM); random access
memory (RAM); magnetic disk storage media; optical storage media;
flash memory devices; electrical, optical, acoustical or other form
of propagated signals, for example, carrier waves, infrared
signals, digital signals, etc.; or any other type of media suitable
for storing or transmitting information.
[0219] Although the invention is described herein with reference to
the preferred embodiment, one skilled in the art will readily
appreciate that other applications may be substituted for those set
forth herein without departing from the spirit and scope of the
present invention. Accordingly, the invention should only be
limited
* * * * *